BOUGHT WITH THE INCO 
 FROM THE 
 
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 THE GIFT OF 
 
 Henrs W. Sage 
 
 ^ 1 1S91 
 
 ME 
 
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 ,«,„ MUSIC 
 
borneii University Library 
 ML 3805.H31 1887 
 
 Handbook of acoustics lor the use of mus 
 
 3 1924 022 201 697 
 
Cornell University 
 Library 
 
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 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924022201697 
 
HANDBOOK 
 
 ACOUSTICS 
 
 FOE THE USE OE MUSICAL STUDENTS. 
 
 T. F. HAEEIS, B.Sc.,F.c.s. 
 
 Lecturer on Acoustics at the Tonic Sol-fa College. 
 
 THIRD EDITION. 
 
 J. CfUEWEN & SONS, 8 & 9 WARWICK: LANE, E.G. 
 BEDUCEL) PKIOE 3/^. 
 
LONDON : 
 J. CUEWEN & SONS, MITSIC PBINTEE8, 
 PLAISTOW, E. 
 
PREFACE. 
 
 AtTHOTTGH many works on the subject of Acoustics have been 
 written for the use of musical students, the author of this book 
 has not met with one which gives, in an elementary form, more 
 than a partial view of the science. Thus, there are several 
 admirable treatises on the purely physical and experimental 
 part, but most if not all of them stop short just when the 
 subject begins to be of especial interest to the student of music. 
 On the other hand, there are many excellent works, which treat. 
 of the bearings of purely acoustical phenomena on the science 
 and art of music, but which presuppose a knowledge of such 
 phenomena and their causes on the part of the reader. Thus 
 the ordinary musical student, who can probably give but a 
 limited amount of time to this part of his studies, is at the 
 disadvantage of having to master several works, each probably 
 written in a totally different style, and possibly not all agreeing 
 perfectly with one another as to details. This disadvantage has 
 been felt by the author, in his classes for some years past, and 
 the present work has been written with the object of furnishing 
 to the student, as far as is possible in an elementary work, a 
 complete view of Acoustical science and its bearings on the art 
 of music. 
 
PREFACE. 
 
 In the arrangement of the subject, the reader should observe 
 that up to and including the 7th Chapter, the sounds treated of 
 are supposed to be simple; the next four chapters treat of 
 sounds — both simple and compound — singly, that is to say, only 
 one tone is supposed to be produced at a time ; the phenomena 
 accompanying the simultaneous production of two or more 
 sounds are reserved for the remaining chapters. 
 
 The movable Sol-fa names for the notes of the scale have been 
 used throughout, as they are so much better adapted to scientific 
 treatment than the fixed Staff Notation symbols. It may be 
 useful to readers not acquainted with the Tonic Sol-fa Notation 
 to mention that in this system, the symbol d is taken to 
 represent a sound of any assumed pitch, and the letters, r, m, f, 
 S, 1, t, represent the other tones of the diatonic scale in 
 ascending order. The sharp of any one of these tones is denoted 
 by placing the letter e after its symbol : thus, the sharp of s is 
 se ; of r, re ; and so on. The flat of any tone is denoted 
 by placing the letter a after its symbol : thus the flat of t is ta; 
 of m, ma ; and so on. The upper or lower octaves of these 
 notes are expressed by marks above or below their symbols : 
 thus Sl is one octave, d'' two octaves above d ; Si is one octave, 
 Sa two octaves below s- Absolute pitch has been denoted 
 throughout by the ordinary symbols, C representing the 
 note on the ledger Une below the treble stafi. Its successive 
 higher octaves are denoted by placing the figures 1, 2, 3, &c., 
 above it, and its lower octaves by writing the same figures 
 beneath it; thus, C, C», C«, &c.; Cj, Cj, Cj, &c. 
 
 It is perhaps as weU to observe, that although Helmholtz's 
 theory as to the origin of Combination Tones given in Chap. 
 XII is at present the received one, it is possible that in the 
 future it may require modification, in view of the recent 
 researches of Preyer, Koenig, and Bonsanquet. 
 
PREFACE. 
 
 Although this book has not been written expressly for the use 
 of students preparing for any particular examination, it will be 
 found that a mastery of its contents will enable a candidate to 
 successfully work any papers set in Acoustics at the ordinary 
 musical examinations, including those of the Tonic Sol-fa 
 College, Trinity College, and the examinations for the degree of 
 Bachelor of Music at Cambridge and London. The papers set 
 at these examinations during the last two years, together with 
 the answers to the questions, will be found at the end of the 
 book. 
 
 The text will be found to be fuUy illustrated by figures, of 
 which Nos. 1, 2, 20, 21, 23, 36, 37, 50, 51, 64 are taken from 
 Deschanel's Treatise on Natural Philosophy, and JSTos. 4, 5, 58, 
 72, 78 from Lees' Acoustics, Light, and Heat, by permission of 
 Messrs. Blackie & Sons and Collins & Son respectively. All the 
 other figures have been cat expressly for this work. 
 
 School of Science, 
 
 Bbomley, Kent. 
 
CONTENTS. 
 
 -♦- 
 
 CHAPTER I. PAGES 
 
 Introductory— The Origin of a Musical Sound .... 1-7 
 
 CHAPTER II. 
 The TiansmiBsion of Sound 8-22 
 
 CHAPTER m. 
 On the Ear 23-28 
 
 CHAPTER IV. 
 On the Pitch of Musical Sounds 29-43 
 
 CHAPTER V. 
 The Melodic Relations of the sounds of the Common Scale . . 44^51 
 
 CHAPTER VI. 
 On the Intensity or Loudness of Musical Sounds .... 52-56 
 
 CHAPTER VH. 
 Resonance, Co-vibration, or Sympathy of Tones .... 57-68 
 
 CHAPTER Vm. 
 On the Quality of Musical Sounds 69-85 
 
 CHAPTER IX. 
 On the Vibrations of Strings 86-97 
 
 CHAPTER X. 
 Flue-pipes and Heeds 98-117 
 
 CHAPTER XI. 
 On the vibration of Rods, Plates, &c 118-127 
 
 CHAPTER Xn. 
 Combination Tones 128-135 
 
CONTENTS. 
 
 CHAPTER Xni. PAOE8 
 
 On Interference 136-153 
 
 CHAPTER XIV 
 On DisBonance 154-172 
 
 CHAPTER XV. 
 The Definition of the Consonant Intervals 173-185 
 
 CHAPTER XVI. 
 On the Relative HarmoniousneBs of the Consonant Intervals . 186-203 
 
 CHAPTER XVH. 
 Chords 204-225 
 
 CHAPTER XVni. 
 Temperament 226-247 
 
 QUESTIONS 248-263 
 
 EXAMINATION PAPERS. 
 
 Trinity CoUege, London 264-267 
 
 Cambridge 268-271 
 
 London University 272-283 
 
HMD-BOOK OF ACOUSTICS. 
 
 CHAPTER I. 
 
 Inteodttctoby : The Origin of a Mitsioai, Sotjmd. 
 
 It must be evident to every one, that the cause of the sensation -we 
 term " sound," is something external to us. It is almost ecjually 
 obvious that this external cause is motion. To be convinced of this 
 
 Fia. 1. 
 
 fact, it is only necessary to trace any sound to its origin ; the sound 
 from a piano, for example, to its vibrating string, or that from a 
 harmonium to its oscillating tongue. If the glass bell (fig. 1) be 
 
 B 
 
HAND-BOOK OF ACOUSTICS. 
 
 bowed, it will emit a sound, and the little suspended weight wiU be 
 violently dashed away ; the rattle of the moying glass against the 
 projecting point will also be plainly heard. Even where the move- 
 ment cannot be seen, as in most wind instruments, it may easily be 
 felt. 
 
 Although all sounds are thus produced by motion, movements 
 do not always give rise to the sensation, sound. We have therefore 
 to ascertain, what particular kind of motion is capable of producing 
 the sensation, and the conditions necessary for its production. 
 Sounds may be roughly classified as musical or unmusical. As we 
 are only concerned here with the former, it will be as well first to 
 distinguish as far as possible between the two classes. For acoustical 
 purposes, we may define a. musical sound to be that, which, 
 whether it lasts for a long or short period of time, does not vary in 
 pitch. In other words, a musical sound is a steady sound. In an 
 ordinary way, we say that a sound is musical or unmusical, accord- 
 ing as it is pleasant or otherwise, and on examination, this wiU be 
 found to agree fairly well with the more rigid definition above, 
 especially if we bear in mind the fact, that most sounds consist of 
 musical and unmusical elements, and that the resulting sound is 
 agreeable or the reverse, according as the former or the latter pre- 
 dominate. For example, the sound produced by an organ pipe 
 consists of the steady sound proper to the pipe, and of the unsteady 
 fluttering or hissing sound, caused by the current of air striking the 
 thin edge of the embouchure ; but, as the former predominates 
 greatly over the latter, the resulting sound is termed musical. 
 Again, in the roar of a waterfall we have the same two elements, 
 but in this case, the unsteady predominates over the steady, and an 
 unmusical sound, or noise, is the result. 
 
 We have just seen that the external cause of a musical sound 
 is motion ; we shaU further find on examination, that this motion is 
 a periodic one. A periodic motion is one that repeats itself at equal 
 intervals of time ; as, for example, the motion of a common 
 pendulum. In order to satisfy ourselves that a musical sound is 
 caused by a periodic motion, we will examine into the origin of the 
 sounds produced by strings, reeds, and flue pipes. 
 
 A very simple experiment will suffice in the case of the first 
 named. Stretch a yard of common elastic somewhat loosely 
 between two pegs. On plucking it in the middle, it begins 
 vibrating, and although its motion is somewhat rapid, yet 
 we have no difficulty in counting the vibrations ; or at any 
 
THE OEiaiN OF A MUSICAL SOUND. 
 
 rate, we can see that they follow one another regularly at 
 equal intervals of time. Further, we may notice, that this is the 
 case, whether we pli^ok the string gently or violently, that is, 
 whether the vibrations are of large or small extent. The 
 motion of the string is therefore periodic, — its vibrations are all 
 executed in equal times. If now the elastic be stretched a 
 little more, the vibrations become too rapid for the eye to foUow. 
 We see only a hazy spindle, yet we cannot doubt but that the kind 
 of motion is the same as before. Stretch the string still more, 
 and now a musical sound is heard, which is thus caused by the 
 rapid periodic motion of the string. 
 
 A similar experiment proves the same fact with regard to reed 
 instruments. Fasten one end of a long thin strip of metal in a 
 vice (fig. 2). Displace the other end (d) of the strip, and let it go. 
 
 
 r 
 
 Fig. 2. 
 
 The strip vibrates slowly enough for us to count its vibrations, and 
 these we find to recur regularly ; that is, the motion is periodic. 
 Gradually shorten the strip, and the vibrations will follow one 
 
HAND-BOOK OF ACOUSTICS. 
 
 another faster and faster, till at length a musical sound is heard. 
 Although we cannot now follow the rapid motion of the strip, yet, 
 as in the case of the string above, we may fairly conclude that its 
 character remains unaltered ; that is, the motion is stiU periodic. 
 
 In such an instrument as a flue pipe, the vibrating body is 
 the air. Although this itself is invisible, it is not difficult to render 
 its motion visible. Fixed vertically in the stand (fig. 3) is a glass 
 
 Fig. 3. 
 
 tube A B, about 2ft. long and an inch in diameter. Passing into the 
 lower end of the tube is a pin-hole gas jet, (/), joined to the ordinary 
 gassupplyby india-rubber tubing. Before the jet is introduced 
 into the tube, it is ignited, and the gas turned down, until the flame 
 is about an inch or less in height. On inserting this into the glass 
 tube, after a little adjustment, a musical sound is heard coming 
 from the tube. It is, in fact, the well-known singing flame. The 
 particles of air in the tube are in rapid vibration, moving towards 
 
THE ORIGIN OS A MUSICAL SOUND. 6 
 
 tlie centre and from it, alternately. The air particles at that part 
 of the tube, where the flame is situated, will therefore be alternately, 
 crowded together and scattered wider apart ; that is, the pressure of 
 the air upon the flame will be alternately greater and less than the 
 ordinary atmospheric pressure. The effect of the greater pressure 
 upon the flame wUl be to force it down, or even extinguish it 
 altogether ; the effect of the lesser pressure wiU be to enlarge it. 
 Thus the flame will rise and fall at every vibration of the air in the 
 tube. These movements of the flame are too rapid, however, to be 
 followed by the eye, and the flame itself will still appear to be at 
 rest. In order to observe them, recourse must be had to a common 
 optical device. First, reduce the tube to silence, by lowering the 
 position of the jet. Having then darkened the room, rotate a miiTor 
 (M) on a vertical axis behind the flame. The latter now appears in 
 the mirror as a continuous yellow band of light, for precisely the 
 same reason, that a lighted stick, on being whirled round, presents 
 the appearance of a luminous circle. Now restore the jet to its 
 former position in the tube. The latter begins to sing, and on 
 rotating the mirror we no longer see a continuous band of light> 
 but a series of distinct flames (o p) joined together below by a very 
 thin band of light. This clearly shows, that the flame is alternately 
 large and very small ; that is, alternately rising and falling, as 
 described above. Now while the mirror is being rotated at an even 
 rate, notice that the intervals between the flames are all equal, and 
 also that the flames themselves are all of the same size. Prom what 
 is stated above, it will be seen that this proves our point, namely, 
 that the sound in this case is produced by the periodic or vibratory 
 motion of the particles of air. 
 
 By examining in this way into the origin of other sounds, it will 
 be found that all musical tones are caused by the periodic motion 
 of some body. Further, a periodic or vibratory motion will always 
 produce a musical sound, provided, (1), that the vibrations recvu: 
 with sufficient rapidity; (2), that they do not recur too rapidly; (3), 
 that they are sufficiently extensive, and the moving body large 
 enough. The following experiments wiU illustrate this. 
 
 Pig. 4 represents an ordinary cogwheel (B), having some 80 or 90 
 teeth, which can be rapidly rotated by means of the multiplying 
 wheel (A). Holding a card (E) so as just to touch the cogs, we 
 slowly turn the handle of the multiplying wheel. The card is 
 
HAND-BOOK OF ACOUSTICS. 
 
 Fio. 4. 
 
 lifted slightly by each cog as it passes, but is almost immediately 
 released, and falls back against the succeeding one ; that is, it 
 ■vibrates once for eyery cog that passes it. As long as the wheel 
 is revolTing slowly, the card may thus be heard striking against 
 each cog separately. If, however, the speed be increased, the taps 
 will succeed one another so rapidly as to coalesce, and then a 
 continuous sound will be heard. 
 
 The well-known Trevelyan's rocker is intended to illustrate the 
 same thing. It consists of a rectangular-shaped piece of copper 
 about 6 inches long, 2 J inches broad, and 1 inch thick. The lower 
 side is bevelled, and has a longitudinal groove running down the 
 middle, as shown in fig. 5. Attached to one end is a somewhat 
 
 Fio. 5. 
 
 slender steel rod, terminating in a brass ball. If we place the 
 rocker, with its bevelled face resting against a block of lead, and 
 with the ball at the other end resting on the smooth surface of a 
 table, it will rock from side to side on being slightly displaced, but 
 not quickly enough to produce a musical sound. If, however, we 
 hold the rocker in the flame of a Bunsen burner, or heat it over a 
 fire, for a few minutes, and then place it as before against the 
 leaden block, we shall find it giving forth a clear and continuous 
 sound. This phenomenon may be explained thus : — ^When any 
 
THE OBiaiN OF A MUSICAL SOUND. 7 
 
 point of the heated rocker on one side of the groove touches the 
 lead, it imparts its heat to the latter at that spot. This causes the 
 lead at that particular point to expand. A little pimple, as it were, 
 darts iip<and pushes the rocker back (fig. 6), so that it falls over on 
 
 FiQ. 6. 
 
 to the other side of the groove. A second heat pimple is raised 
 here, and the rocker is tilted in the contrary direction, and thus 
 the motion is continued, as long as the rocker is sufficiently hot. 
 
 Stimmart. 
 
 The cause of the sensation sound, is motion. 
 
 A musical sound is one which remains steadily at a definite pitch. 
 
 A musical sound originates in the periodic motion of some body. 
 
 A periodic or regular vibratory motion will always give rise to 
 a musical sound, if the vibrations recur with sufficient, but not too 
 great rapidity, and provided they are extensive enough. 
 
CHAPTER II. 
 
 The Teansmissiokt of Sotjitd. 
 
 We have seen in the preceding chapter, that all sounds originate in 
 the vibratory movements of bodies. This vibratory motion is 
 capable of being communicated to, and transmitted by, almost all 
 substances, to a greater or less extent. Wood, glass, water, brass, 
 iron, and metals in general may be taken as examples of good 
 conductors of sound. But the substance, ■which in the vast ma- 
 jority of oases transmits to our ears the vibratory motion, which 
 gives rise to the sensation, sound, is the air. 
 
 A very little reflection is sufficient to show us that some medium 
 is absolutely necessary for the transmission of sound; for inasmuch 
 as soimd is caused by vibratory motion, it is plain that this motion 
 cannot pass through a vacuum. This fact can, however, be easily 
 proved experimentally. Under the receiver of an air-pump a beU 
 is suspended. We shake the pump, and the bell may be heard ring- 
 ing ; for its vibrations are communicated by the air to the glass of 
 the receiver, and by the latter to the outer air, and so to ova: ears. 
 We now pump as much air as possible out of the receiver, and again 
 ring the bell. It can still be heard faintly, for we cannot remove 
 all the air. We now allow dry hydrogen to pass into the receiver, 
 and on again ringing the bell, there is very little increase of 
 sound, this gas being so very light — only about -Jj- as heavy as air. 
 If we now exhaust the receiver, we shall be able still further to 
 attenuate the atmosphere within it, and then, although we may 
 violently shake the apparatus, no sound will be heard. 
 
 As the air is generally the medium, by means of which the 
 vibratory motion reaches our ears, we shall now have to carefully 
 study the manner in which this transmission takes place. When 
 we stand on the sea shore, or better still, on a clifi near it, and 
 watch the waves rolling in from afar, our first idea is, that the 
 
THE TRANSMISSION OF SOUND. 
 
 water ia actually moving towards us, and it is difiScult at the time 
 to get rid of this notion. If, however, we examine a little more 
 closely, we see that the boats and other objects floating about, do 
 not travel forward with the wave, but simply rise and fall as it 
 passes them. Hence we conclude that the water of which the 
 wave is composed, is not moving towards us. What is it, then, 
 that is being transmitted ? What is it that is moving forwards ? 
 The up and down movement — the wave motion. Now the vibratory 
 movements which give rise to sound are, as we shall presently 
 see, transmitted through the air much in the same way; and 
 therefore, although a sound wave is not exactly analogous to a 
 water wave, yet a brief study of the latter will help us more easily 
 to understand the former. 
 
 Let ADEOB (fig. 7) represent the section of a water wave, and 
 the dotted line AEB the surface of stUl water. That part of the 
 wave ADE above the dotted line, is termed the crest, and the part 
 EOB below, the trough. Through D and 0, the highest and 
 lowest points, draw DH and EO, parallel to AB, and from the 
 same points draw DP and OH perpendicular to it. Then the 
 distance AB is termed the length of the wave, DE or CH is its 
 amplitude, and the outline ADECB is its form. 
 
 
 D JT 
 
 
 
 
 
 \^ 
 
 ^/ 
 
 ^ 
 
 f c 
 
 
 Fio. 7. 
 
 These three elements completely determine a wave, in the same 
 way as the length, breadth and thickness of a rectangtilar block of 
 wood, determine its size; and further, as any one of these three 
 dimensions may vary independently of the other two, so any one 
 of the three elements — length, amplitude, and form — may vary, the 
 other two remaining constant. Thus in fig. 8 (1), we have three 
 waves of the same amplitude and form, but varying in length ; in 
 fig. 8 (2), we have three waves of same length and form, but of 
 
JO 
 
 HAND-BOOK OF ACOUSTICS. 
 
 FiQ. 8 (1). 
 
 different amplitudes ; while in fig. 8 (3), we have three waves of 
 same length and amplitude, but of varying forms. 
 
 Fio. 8 (2). 
 
 FiQ. 8 (3). 
 
 We now have to inquire into the nature of the movements of the 
 particles of water which form the wave. The following apparatus 
 will assist the student in this inquiry. Fig. 9 represents a box 
 
 Fio. 9. 
 
 about four feet long, four inches broad, and six inches deep, with 
 either one or both the long sides of glass, and caulked with marine 
 
THE TRANSMISSION OF SOUND. 11 
 
 glue, so as to be •water-tigM. Tliis is more than half filled witli 
 ■water, in ■which are immersed at different depths, balls of ■wax 
 mixed ■with just so much iron filings, as ■will make them of the 
 same specific gravity as ■water. No'w alternately raise and depress 
 one end of the box, so as to give rise to ■wa^yes. The balls ■will 
 describe closed curves in a vertical plane, the horizontal diameter 
 of -which -will much exceed the vertical. If a deeper trough be 
 taken, the difference bet^ween the horizontal and vertical diameters 
 of these curves ■will become less ; aud in fact, in very deep ■water 
 the t-wo diameters become of the same length ; that is, the closed 
 curves become circles. It ■wiU. be noticed also, that each particle of 
 ■water describes its curve and returns to its original position, in the 
 same time as the vrave takes to move through its own length. 
 Bearing in mind these facts, ■we may plot out a ■water ■wave in the 
 manner sho-wn in fig. 10. In the top ro^w are 17 dots, representing 
 
12 
 
 HAND-BOOK OF ACOUSTICS. 
 
 (T) Qffi<D®CDCp(I>Q 
 
 (D (D O d) (D Ct> CD 
 OiO(DCD(t>CD ■ 
 
 0) 
 
 
 
 ® 
 
 (pq>Q 
 
 O® (1)0 
 
 0<i)®®®® 
 
 ct)Od)®0®® 
 
 Fio. 10. 
 
THE TRANSMISSION OF SOUND. 13 
 
 17 equidistant particles of water at rest, and tlie circles in wMoh 
 they are about to move, in the same direction as the hands of a 
 clock. In (0), the position of each of these particles is given at 
 the moment when they form parts of the two complete waves A and 
 B, which are supposed to be passing from left to right ; the 8th 
 particle from the left has just passed through Jth of its journey, 
 that is ^th of the whole circumference ; the 7th particle has passed 
 through I = i; the 6th through f ; the 5th has just performed ^ 
 its journey; the 4th, %; the 3rd, |; the 2nd, f ; and the first has 
 just completed its course and regained its original position. In (1) 
 each particle has moved through ^ more of its curve, and the wave 
 has passed through a space equal to i of its length. In (2) each 
 particle has moved through ^ more, and the wave has again 
 advanced as before; and so on, in (3), (4), (5), (6), (7), and (8). 
 Each particle in (8) has the same position as in. (0), having made 
 one complete j oumey ; and the wave has advanced one wave length. 
 Thus we see, as in the experiment with the trough, that each 
 particle makes one complete journey in the same time as the wave 
 takes to travel its own length. 
 
 Another variety of wave motion may be studied thus : FiU an 
 india-rubber tube, about 12 feet long and J inch in diameter, with 
 sand, and fasten one end to the ceiling of a room. Hold the other 
 end in -the hand, and jerk it sharply on one side. Notice the wave 
 motion thus communicated to the tube, the wave consisting of two 
 protuberances, one on each side of the position of the tube at rest ; 
 the one corresponding to the crest of the water wave, and the other 
 to its trough. Let the uppermost row of dots in fig. 11, represent 
 
14 
 
 HAND-BOOK OF ACOUSTICS. 
 
 r^ 
 
 Fia. 11. 
 
THE TRANSMISSION OF SOUND. 15 
 
 17 equidistant particles of the tube at rest ; (0) gives tke positions 
 of the particles when they form parts of two complete waves A and 
 B; and (1) to (8) show their positions at successive intervals of time, 
 each equal to J- of the time that it takes a particle to perform one 
 complete vibration. By a careful inspection of this diagram, it will 
 be seen that while each particle performs one complete vibration, 
 the wave travels through a space equal to its own length. Taking 
 the 1st particle in A, for example : — in (1), it has risen half way to 
 its highest point ; in (2), it has reached its highest point ; in (3), it 
 is descending again ; m (4), it is passing downwards through its 
 original position; in (5), it is stUl slaking; in (6), it has reached its 
 lowest position ; in (7), it is ascending again ; in (8), it has reached 
 its original position, having made one complete vibration. Now it 
 is obvious from the figure, that dtiring this time the wave A has 
 passed through a space equal to its own length. 
 
 Let the curve in fig. 12 represent the outline section of the wave 
 A in fig. 11. Through the deepest and highest points {d) and (c) of 
 the protuberances, draw the dotted straight lines parallel to AB, 
 the position of the tube when at rest. Through [d) draw {df) at 
 right angles to AB ; then as before {df) in the amplitude, of the 
 wave. Now on comparing this with fig. 11 it will be seen that this 
 amplitude is the same thing as extent of vibration of each particle. 
 
 Fig. 12. 
 
 If we suppose, that a particle takes a certain definite time, say 
 one second, to perform its vibration, it is evident that the number 
 of different modes in which it may get over its ground in this 
 time is infinite. Thus it may move slowly at first, then quickly 
 and again slowly; or it may start quickly, then slacken, again 
 quicken, slacken again, and finish up quickly; and so on. In fig. 13, 
 two waves of equal length and amplitude are represented after the 
 manner of fig. 12. The extent and time of particle vibration being 
 the same, they only differ in the mode of vibration. In (6) the 
 particle at first moves more quickly than in (a) ; it then moves 
 more slowly ; and so on. Thus for example : supposing the time of 
 a complete vibration to be one second, the particle in (a) reaches its 
 highest position in one quarter, and its lowest, in three quarters of 
 
16 
 
 HAND-BOOK OF AC0V8TIC8. 
 
 a second after the start ; wliile in (J) the particle reaches the same 
 points in J, and -^ of a second, respectively. Now it is plain from 
 the figure, that this difference in mode of particle vibration is 
 accompanied by a difference in the form of the wave, and a little 
 reflection wiU show that this must always be the case. We have 
 thus seen how the length, amplitude, and form of a wave are 
 respectively connected with the time, extent, and mode of particle 
 vibration. 
 
 Fio. 13. 
 
 Having now made some acquaintance with wave motion in 
 general, we may pass on to consider the particular case of the 
 sound wave. The first point to which we must turn our attention, 
 is the nature of the medium. Air, lite every other gas, is 
 supposed to be made up of almost infinitely small particles. These 
 particles cannot be very closely packed together; in fact, they 
 must be at considerable distances apart in proportion to their size, 
 for several hundred volumes of air can be compressed into one 
 volume. When air is thus compressed, the particles of which it 
 consists, press against one another, and against the sides of the 
 containing vessel in all directions ; and that with a force which is 
 the greater, the closer the particles are pressed together ; in other 
 words, the air is elastic. An ordinary popgun will illustrate this ; 
 a certain volume of air having been confined between the cork and 
 
THE TRANSMISSION OF SOUND. 17 
 
 the rammer of the gun, the latter is gradually pushed in ; the air 
 particles being thus crowded closer and closer together, exert an 
 ever increasing pressure, untU at length the weakest point, the 
 cork, gives way. 
 
 m 
 
 A B 
 
 (2) 
 
 (3) 
 
 A B 
 
 (4) 
 
 A B 
 
 — b - 
 
 (5) 
 Fig. 14. 
 
18 HAND-BOOK OF ACOUSTICS. 
 
 Let A, fig. 14 (1), represent a tuning-fork, and BO a long tube 
 open at both ends. Suppose now, that the fork begins vibrating, 
 its prongs first performing an outward journey. The air in BO 
 will be condensed, but in consequence of the swiftness of the fork's 
 motion, and the great elasticity of air, this condensation will be 
 confined during the outward journey of the prongs, to a compara- 
 tively small portion of the tube BO : say to the portion {ah) 
 fig. 14 (2). The air in {ah), being now denser than that in advance 
 of it, will expand, acting on the air in (60) as the tuning-fork 
 acted upon it, thus causing a condensation in {he) fig. 14 (3), 
 while the air in {ah) itseK, overshooting the mark, as it were, 
 becomes rarefied, an effect which is increased by the simultaneous 
 retreat of the prong A. Again, the air in {he) fig. 14 (3), being 
 denser than that before and behind, expands both ways, forming 
 condensations {cd) and (ah) fig. 14 (4), in both directions ; while in 
 its place is formed the rarefaction (ic), the formation of the con- 
 densation {ah) being assisted by the outward journey of A. Next 
 the air in {ah) and {cd) fig. 14 (4), being denser than that on either 
 side, expands in both directions, forming the condensations (he) 
 and {de) fig. 14 (5) ; rarefactions being formed in (aJ) and {cd), 
 the former assisted by the retreat of the prong A. By further 
 repetitions of this process the sound waves {ac) and (ce) will be 
 propagated along the tube. 
 
 For the sake of simplicity, the motion has been supposed to take 
 place in a tube. This restriction may now be removed. The 
 movement of the air passing outward in every direction from the 
 sounding body, the successive condensations and rarefactions form 
 spherical concentric shells round it. 
 
 In fig. 14 (5), {ac) and (ce) form two complete sound waves, {he) 
 and (de) being the condensed, and {ah) and {cd) the rarefied parts. 
 In studying the motion of the particles of air forming these sound 
 waves, it will be simplest to consider those adjoining the prong A, 
 for their motion will necessarily be similar to that of the prong 
 itself. The first point to notice is, that the direction of particle 
 vibration in this case is the same as that of the wave motion, and 
 not transversely to it as in the case of the sand tube. In the next 
 place, it will be observed, that the tuning-fork makes one complete 
 vibration while the wave passes through a space equal to its own 
 length ; that is, each particle executes one complete vibration in 
 exactly the same time as the wave takes to pass through a distance 
 equal to its own length. Again, the amplitude of the wave is 
 
TEE TRANSMISSION OF SOUND. 
 
 1? 
 
 evidently the distance through which the individual particles of air 
 vibrate ; that is, the extent of particle vibration. Further, it is 
 plain that, the greater the swing of the prong A, the greater will be 
 the degree of condensation and rarefaction in (aJ) ; that is, the 
 greater the extent of particle vibration, the greater will be the 
 degree of rarefaction and condensation. Lastly, just as the particles 
 in the case of the sand tube could perform their vibrations in an 
 infinite number of modes, so the fork A, and therefore the air 
 particles, may perform their vibrations in an endless variety of 
 modes, giving rise to as many wave forms. 
 
 It is often convenient to represent a sound wave after the same 
 manner as a water wave. Let fig. 15 represent the section of a 
 
 water wave, the dotted line being the surface of the water at rest. 
 This may also represent a sound wave the length of which is {ai) 
 distance above the dotted line representing extent of forward, and 
 distance below extent of backward longitudinal vibration. For 
 example, the air particles which are at the points I and / when at 
 rest, would be at d and g, {Id^ce and hk=g/) in the state of things 
 represented in the figure. The curve in the figure, related in this 
 way to the wave of condensation and rarefaction, is termed its 
 associated wave. 
 
 The velocity with which these sound waves travel is very great, 
 compared with that of water waves. The method of dete rminin g 
 it is very simple in principle. Two stations are chosen within sight 
 of each other, the distance between them being accurately known. 
 A gun is fired at one station, and an observer at the other counts 
 the number of seconds that elapse between seeing the flash and 
 hearing the sound. As light passes almost instantaneously over 
 any terrestrial distance, the time that sound takes to travel over a 
 
20 HAND-BOOK OF ACOUSTICS. 
 
 given distance is thus known. Dividing this distance by the number 
 of seconds, we determine the space through which sound travels per 
 second. This is found to vary considerably with the temperature. 
 At 0° (Centigrade) or 32° (Fahrenheit) the velocity of sound is 
 1,090 feet per second. M. Wertheim gives the following results at 
 other temperatures : — 
 
 Tempebatitre. Velocity 
 
 Centigrade. Fahrenheit. or Soitni). 
 
 _-5° 31-1° 1,089 
 
 a-l" 35-8° 1,091 
 
 8'6° 47-3° 1,109 
 
 12° 53-6° 1,113 
 
 26-6° 79-9° 1,140 
 
 The velocity at any temperature may be approximately found by 
 adding on 2 feet for a rise of one degree Centigrade, and 1 foot for 
 a degree Fahrenheit. 
 
 The velocity of sotind in any medium varies directly as the square 
 root of the elasticity, and inversely as the square root of the density 
 of that medium. Therefore, as all gases have the same elasticity, 
 the velocity of sound in gases varies inversely as the square root of 
 their densities. Thus oxygen is 16 times as heavy as hydrogen, 
 and the velocity of sound in the latter gas is 4 times the velocity 
 in the former. 
 
 Sound travels more quietly through water than through gases. 
 CoUadon and Sturm proved that the velocity of sound in water is 
 4,708 feet per second, at 8° Centigrade. Their experiments were 
 conducted on the Lake of Geneva, on the opposite sides of which 
 the observers were stationed in two boats. The sound was emitted 
 under water by striking a beU with a hammer, and after travelling 
 through a known distance was received by a long speaking tube, 
 the larger orifice of which, covered with a vibrating membrane, was 
 sunk beneath the surface. The same movement which gave rise to 
 the sound, also at the same instant ignited some gunpowder, and 
 the number of seconds elapsing between the flash and the sound 
 was determined by a chronometer. 
 
 . In solids, the velocity of sound is usually greater than in liquids. 
 It may be calculated from the formula 
 
 when the elasticity E and density d of the solid are known. 
 
THE TRANSMISSION OF SOUND. 
 
 21 
 
 The various experimental methods that have been employed for 
 determining it, will be given later on. The following are some of 
 the principal determinations made by Wertheim. 
 
 Lead 4,030 feet per second at 20" 0. 
 
 Gold 5,717 „ 
 
 Silver - 8,553 „ 
 
 Copper 11,666 ,, 
 
 Platinum 8,815 ,, 
 
 Iron - - 16,822 „ 
 
 Steel - - 16,357 „ 
 Wood (along fibre) from 10,000 to 16,000 
 
 „ (across „ ) „ 3,000 to 5,000 
 
 The superior conducting power of elastic solids may be illustrated 
 by a variety of experiments. Thus, strike a tuning-fork and place 
 the stem against the end of a rod of wood 12 or 15 feet long. So 
 perfect is the transmission, that if a person appKes his ear to the 
 other end, the sound wiU appear to come from that part of the rod. 
 A still better result is obtained by placing one end of the rod 
 against a door panel, and applying the vibrating tuning fork to the 
 other. Again, place a watch well between the teeth, without 
 however touching them, and note the loudness of its tick. Now 
 gently bite the watch, and observe how much more plainly it can 
 be heard. In the first case, the vibrations pass through the air to 
 the ears, in the second ease, through the solid bones of the skuU. 
 This is the principle of the audiphone. 
 
 Stjmmakt. 
 
 Sound cannot be transmitted through a vacuum. 
 
 The transmission of sound is a particular case of wave motion, of 
 which, water waves and rope waves are other examples. The 
 peculiar characteristic of a, wave motion is, that the material 
 particles through which the wave is passing, do not move onwards 
 with the wave, but simply oscillate about their position of rest. 
 In the rope wave, for example, the particles of the rope oscillate at 
 right angles to the direction in which the wave is advancing ; 
 while, on the other hand, in the sound wave, the air particles 
 oscillate in the same direction as the wave is moving. 
 
 Just as a water or rope wave consists of two parts, a crest and a 
 trough, so a sound wave is made up of two portions, viz., a 
 condensation and a rarefaction. 
 
22 HAND-BOOK OF ACOUSTICS. 
 
 A sound wave, like a water or rope wave, is determined by three 
 elements, viz., its length, aTmpUtude, and form. The length of a wave 
 is the distance from any point in one wave to the corresponding 
 point in the succeeding one ; the amplitude is measured by the 
 extent of vibration of its air particles; while the form is determined 
 by the varjdng velocities of these particles as they perform their 
 excursions. 
 
 The greater the amplitude (that is, the extent of particle vibra- 
 tion), the greater will be the degree of condensation and 
 rarefaction. 
 
 In an associated sound wave, the direction of particle vibration is 
 represented, for the sake of convenience, as being at right angles 
 to its true direction. 
 
 The velocity of sound at the freezing point is 1,090 feet per second, 
 and increases as the temperature rises. 
 
 The velocity of sound in water is 4,708 feet per second at 8° 
 Centigrade. 
 
 The velocity of sound in elastic solids, such as iron and wood, is 
 much greater than the above. 
 
23 
 
 CHAPTER 
 
 On the Ear. 
 
 The Human Ear is separable into three distinct parts — the External, 
 the Middle, and the Internal. 
 
 The External ear consists of the cartilaginoiis lobe or Auricle, and 
 the External Meatus (E. M. fig. 16). The latter is a tube about an 
 inch and a quarter long, directed inwards and slightly forwards, 
 and closed at its inner extremity by the Tympanum. (Ty., fig. 16) 
 or di'um of the ear, which is stretched obliquely across it. 
 
 Fig 16. Diageammatio Section of the Human Eab. 
 
 The Middle ear, which is separated from the External by the 
 Tympanum, is a cavity in the bony wall of the skull, called the 
 Tympanic Cavity (Ty. C, fig. 16). Erom this, a tube, about an 
 inch and a half long, termed the Eustachian Tube (Eu., fig. 16) leads 
 to the upper part of the throat, and thus places the air in the 
 Tympanic Cavity in communication with the external air. On the 
 side opposite to the tympanum, there are two small apertures in 
 the bony wall of the cavity, both of which, however, are closed with 
 
24 HAND-BOOK OF ACOUSTICS. 
 
 membrane. Tlie lower of these two apertures, wliich is about tbe 
 size of a large piu's head, is called from its shape the Fenestra 
 Rotunda, or round window (P.R., fig. 16); the upper and rather 
 larger one, the Fenestra OvaHs, or oval window (F. 0., fig. 16). 
 
 A chain of three small bones, stretches between the Tympanum 
 and the Fenestra Ovalis. One of these, the Malleus or Hammer 
 Bone (Mall., fig. 16) is firmly fastened by one of its processes or 
 arms to the Tympanum, while the other process projects upwards 
 into the cavity, and is articulated to the second bone — the Incus or 
 Anvil (Inc., fig. 16). The lower part of this last again, is articulated 
 to the third bone— the Stapes or Stirrup Bone (St., fig. 16), which 
 in its turn ia firmly fastened by the flat end of the Stirrup to the 
 membrane which closes the Fenestra Ovalis. These three bones 
 are suspended in the tympanic cavity in such a way, that they are 
 capable of turning as a whole upon a horizontal axis, which is 
 formed by processes of the Incus and Malleus, which processes fit 
 into depressions in the side walls of the cavity. This axis in fig. 16 
 is perpendicular to the plane of the paper, and would pass through 
 the head or upper part of the Malleus. 
 
 There are also two small muscles in the t3rmpanio cavity, by the 
 contractions and relaxations of which, the membrane of the Fenestra 
 Ovalis and the tympanic membrane can be rendered more or less 
 tense. One, called the Stapedius, passes from the floor of the 
 tympanic cavity to the Stapes, and the other — the Tensor Tympani — 
 from the wall of the Eustachian tube to the tympanum. 
 
 Before going further, let us consider the functions of these various 
 parts. The sound waves which enter the External Meatus, pass 
 down it, assisted in their passage by its configuration, and strike 
 against the Tympanum. Now, as the air in the cavity of the 
 tympanum is in communication with the outer air, as long as the 
 latter is at rest, they will be of the same density, and hence the 
 Tympanum will sustain an exactly equal pressure from bolii sides. 
 When, however, the condensed part of a sound wave comes into 
 contact with it, this equilibriiim will no longer exist ; the air on the 
 outer side will exert a greater pressure than that on the inner, and 
 the Tympanum will bulge inwards in consequence. The condensed 
 part of the wave will be immediately followed by the rarefied part, 
 and now the state of things is reversed ; there will be a greater 
 pressure from the inside, and consequently the Tympanum will 
 move outward, a movement assisted by its own elasticity. We see 
 therefore, that the Tympanum will execute one complete vibration 
 
ON THE EAR. 25 
 
 for every -wave that reaches it ; that is, as -we know from the pre- 
 ceding chapter, it -will perform one vibration for eveiy vibration of 
 the sound-producing body. 
 
 Further, 'it follows from the arrangement of the Malleus, Incus, 
 and Stapes, and the attachment of the first and last to the Tympanum 
 and the Fenestra OvaUs respectively, that every movement of the 
 Tympanum must cause a similar movement of the Fenestra Ovalis. 
 We see, therefore, that this latter membrane faithfully repeats 
 every vibration of the Tympanum, that is, every movement of the 
 sound-originating body. 
 
 The Internal ear is of a much more complicated nature than 
 the parts already described. It consists essentially of a closed 
 membranous bag of a very irregular and intricate shape, which 
 contains a liquid termed Endolymph, and which floats' in another 
 liqtiid called Perilymph. Both the Endolymph and the Perilymph 
 are little else than water. This membranous bag may be divided, 
 for the purpose of explanation, into two parts — the Membranous 
 Labyrinth (M. L., fig. 16) and the Scala Media of the Cochlea 
 (Sea. M., fig. 16). 
 
 The Membranous Labyrinth comprises the Utriculus (Ut., fig 17), 
 the Sacculus, and the three Semi-circular Canals. All these parts 
 
 Axf.rrs.c. 
 
 ^^ J PoslVSP. 
 
 HorSr. 
 
 Fig. 17. 
 
 communicate with one another, forming one vessel ; the two first 
 lying one behind the other, and the three canals springing from the 
 Utriculus. One end of each of the Semi-circular Canals is dilated 
 at the point where it joins the Utriculus into a swelling called an 
 Ampulla (A, fig. 17). Of the three canals two are placed vertically 
 and the other horizontally ; hence their names — the Anterior 
 Vertical Semi-circular Canal (Ant. V. S. 0., fig. 17), the Posterior 
 Vertical (Post. V. S. C, fig. 17) and the Horizontal (Hor. S. C, fig. 
 17). All these parts are contained in a bony casing, which follows 
 
26 HAND-BOOK OF ACOUSTICS. 
 
 their outline pretty closely, so that there is a bony lab3Tinth, bony 
 semi-circular canals, and so on. There is however a space between 
 the bony casing and the membranous parts, which is filled by the 
 Perilymph referred to above. 
 
 Every part of the Membranous Labyrinth is lined by an exceed- 
 ingly delicate coat — the Epithelium — the cells of which, at certain 
 parts, are prolonged into very minute hairs, which thus project into 
 the Endolymph. In close communication with these cells are the 
 ultimate fibres of one branch of the Auditory Nerve, which, 
 ramifying in the waU of the Membranous Labyrinth, pierces its 
 bony casing and proceeds to the brain. Floating in the Endolymph, 
 there are also nunute hard solid particles called Otoliths. 
 
 The remaining part of the Internal ear is more difficult to describe. 
 It consists essentially of a long tube of bone closed at one end, of 
 which fig. 18 is a diagrammatic section. A thin bony partition — 
 
 .LS. 
 
 Fio. 18. 
 
 the Lamina Spiralis, L. S. — projecting more than half way into the 
 interior, runs along the tube from the bottom, nearly but not quite 
 to the top, so that the two chambers — Sc. V. and So. T — communi- 
 cate at the closed end of the tube. These two chambers — the Scala 
 Vestibuli and the Scala Tympani — are filled with Perilymph. 
 Diverging from the interior edge of the Lamina Spiralis and ter- 
 minating in the bony wall of the tube, are two membranous 
 partitions, the Membrane of Eeissner — m.r. — and the Basilar 
 Membrane — 6. m. The chamber between these two membranes 
 which is termed the Scala Media — So. M. — is filled with Endolymph. 
 So far the tube has been described for simplicity as if it were 
 straight. Now, we must imagine it forming a dose coil of two and 
 a half turns round a central axis of bone — the Modiolus; the Scala 
 Media being outwards and the Lamina Spiralis springing from the 
 axis. The whole arrangement is not unlike a small snail's shell, 
 
ON THE EAR. 27 
 
 and is termed the Cochlea. It is so placed with reference to the 
 other parts of the ear, that the Scala Tympani is closed at the lower 
 end by the Fenestra Ectunda (F. E., fig. 16) while the Perilymph 
 of the Scala VestibuH is continuous with that of the Labyrinth. 
 On the other hand, the Endolymph which fills the Scala Media is 
 in communication with that of the Sacculus. 
 
 Besting on the upper side of the elastic Basilar membrane are the 
 arches or rods of Corti. Each of these rods consists of two filaments, 
 joined at an angle like the rafters of a house. Altogether there 
 are some three or four thousand of them, lying side by side, 
 stretching along the whole length of the Scala Media Hke the keys 
 of a pianoforte. A branch of the auditory nerve, entering the 
 Modiolus, gives off fibres which pass through the Lamina Spiralis, 
 their ultimate ramifications probably coming into close connection 
 with Corti's rods. 
 
 Not much is known for certain of the functions of the various 
 parts of the Internal ear. We have seen how the vibrations of the 
 sounding body are transmitted to, and imitated by, the membrane 
 of the Fenestra Ovalis. These vibrations are necessarily taken up 
 by the Perilymph, which bathes its inner surface, and hence com- 
 municated to the Endolymph of the Membranous Labyrinth and 
 the Scala Media. It was formerly thought, that both of these 
 organs were necessary to the perception of sound, but the re- 
 searches of Goltz have shown that the special function of the 
 Labyrinth, is to enable us to perceive the turning of the head. 
 Thus, the only part of the Internal Ear which is engaged in 
 transmitting sound vibrations to the auditory nerves, is the 
 Cochlea. 
 
 The Basilar Membrane of the Cochlea consists of a series of 
 radial fibres, lying side by side, united by a delicate membrane, 
 and it is believed by Helmholtz, that the faculty of discriminating 
 between sounds of different pitch is due to these radial fibres. The 
 Basilar Membrane gradually increases in width, that is, its radial 
 fibres gfradually increase in length, as we pass from the Fenestra 
 Ovalis to the apex of the Cochlea, being ten times as long at the 
 latter, as at the former. Helmholtz likens them to a series of 
 stretched strings of gradually increasing lengths, the membranous 
 connection between them simply serving to give a fulcrum to the 
 pressure of the fluid against the strings. He further assumes 
 that each of these fibres is timed to a note of definite pitch, and 
 capable of taking up its vibratory motion. He considers that the 
 
28 HAND-BOOK OF AG0U8TIC8. 
 
 arches of Oorti whioli rest on these fibres, serve the purpose of 
 transmitting this vibratory motion to the terroinal appendages of 
 the nerve, each arch being connected with its own nerve ending. 
 
 As already mentioned, there are about 3,000 of these arches in 
 the human ear, which would give about 400 to the octave. When 
 a simple tone is sounded in the neighbourhood of the ear, the 
 radial fibre of the Basilar Membrane in ujiison with it, is supposed 
 to take up its vibrations, which are then transmitted by the arch 
 of Corti in connection with it, to the particular nerve termination 
 with which it is in communication. 
 
 For a complete discussion of this theory, the reader is referred to 
 Helmholtz's " Sensations of Tone," Part I, Chap. vi. 
 
 Summary. 
 
 The Human Ear may be divided, for descriptive purposes, into 
 three parts : the JExternal, Middle, and Internal ears. 
 
 The External ear consists of the Lohe, and the tube or Meatus 
 leading inwards, which is closed by the Tympanum. 
 
 The Middle ear contains a chain of three small bones, the Malleus, 
 Incus, and Stapes, which serves to connect the Tympanum with the 
 Fenestra Ovalis. The air in the cavity of the Middle ear is in 
 coinmunication with the external air, by means of the Eustachian 
 Tube. 
 
 The Internal ear consists essentially of ti membranous bag of 
 exceedingly complicated form, filled with a liquid — Endolympk. 
 This bag floats in another liquid — Perilymph — contained in a bony 
 cavity, which is separated from the cavity of the Middle ear by the 
 membranes of the Fenestra Ovalis and Fenestra Botunda. 
 
 The nerves of hearing ramify on the walls of this membranous 
 bag, and their ultimate fibres project into the Endolymph therein 
 contained. 
 
 The alternate condensations and rarefactions of the sound waves 
 which enter the External ear, strike against the Tympanum and 
 set it vibrating. These vibrations are then transmitted by means 
 of the small bones of the Middle ear to the Fenestra Ovalis, and 
 from this again through the Peiilymph and Endolymph to the 
 minute terminations of the auditory nerve, which lie in the latter 
 liquid. 
 
 One particular part of the membrane of the Internal ear, in 
 which lie the Fibres of Oorti, is specially modified, and is supposed 
 to be the region of the ear which serves to discriminate the pitch 
 and quality of musical sounds. 
 
29 
 
 CHAPTER IV. 
 
 On the Pitch of Mtjsicai Sounds. 
 
 In order to fully describe a Musical Sound, it is necessary, besides 
 specifying its duration, to particularize three things about it, viz., 
 its Pitch, its Intensity, and its Quality. By pitch is meant its 
 height or depth in the musical scale, by intensity its degree of 
 loudness, and by its quality, that character which distinguishes it 
 from another sound of the same pitch and intensity. 
 
 In the present chapter ■we have only to do with Pitch. Inasmuch 
 as we have seen that the Sensation of Sound is due to the vibratorj' 
 motion of softie body, the first question that arises is, — What change 
 takes place in the vibratory movement, when the pitch of the sound 
 changes ? 
 
 If we stretch a violin string loosely, and pluck it, we get a low 
 sou:nd ; stretch it more tightly, and again pluck it, a higher sound 
 is obtained, and at the same time, we can see that it is vibrating 
 more rafidly. Again, take in the same way a short and a long 
 tongue of metal, and the former will be found to give a higher 
 sound and to vibrate more quickly than the latter. 
 
 But the instrument that best shows the fact, that the more rapid 
 the rate of vibration, the higher is the pitch of the sound produced, 
 is perhaps the Wheel Syren (fig. 19). This consists of a disc of 
 zinc or other metal, about 18 inches or more in diameter, mounted 
 on a horizontal axis, and capable of being rapidly revolved by means 
 of a multiplying wheel. The disc is perforated by a number of 
 holes arranged in concentric circles, all the holes in each circle being 
 the same distance apart. In order to work the instrument, a current 
 of air is blown through an India-rubber tube and jet, against one 
 of the circles of holes while the disc is slowly revolving. Whenever 
 a perforation comes in front of the jet, a pufi of air passes through. 
 
30 
 
 HAND-BOOK OF ACOUSTICS. 
 
 causing a condensation on the 
 other side. The current of air is 
 immediately cutofi by the revolu- 
 tion of the disc, and consequently 
 a rarefaction follows the previous 
 condensation — in short, a complete 
 sound wave is formed. Another 
 hole appears opposite the jet, and 
 another wave follows the first, 
 and so on, one wave for each hole. 
 While the disc is being slowly 
 revolved, the waves do not follow 
 one another quickly enough to 
 give rise to a musical sound, — 
 each puS is heard separately; but 
 on gradually increasing the speed, 
 they succeed each other more and 
 more quickly, till at last a low 
 musical sound is heard. If we 
 now still further increase the 
 speed, the pitch of the sound will 
 gradually rise in proportion. 
 
 Savart's toothed wheel (fig. 4, 
 p. 6 ) is another instrument which 
 proves the same fact. While the 
 wheel is being slowly turned, each 
 tap on the card is heard separately, but on increasing the speed, 
 we get a musical sound, which rises in pitch as the rate of 
 revolution increases, that is, as the rate of vibration of the card 
 increases. 
 
 We see, therefore, that the pitch of a sound depends upon the 
 vibration rate of the body, which gives rise to it. By the vibration 
 rate, is meant the number of complete vibrations — journeys to and 
 fro — which it makes in a given time. In expressing the vibration 
 rate, it is most convenient to take a second as the unit of time. 
 The pitch of any sound may therefore be defined, by stating the 
 number of vibrations per second required to produce it; and to 
 avoid circumlocution, this number may be termed the *' vibration 
 number" of that sound. We shall now proceed to describe the 
 principal methods which have been devised for ascertaining the 
 vibration number of any given sound. 
 
 Fio. 19. 
 
ON THE PITCH OF MUSICAL SOUNDS. 
 
 31 
 
 One of the earliest instruments constructed for this purpose was 
 Savart's toothed -wheel (fig. 4) which has just been referred to. A 
 registering apparatus, H, which can be instantly connected or dis- 
 connected with the toothed wheel by touching a spring, records the 
 number of revolutions made by it. In order to discover the 
 vibration number of any given sound, the velocity of the wheel 
 must be gradually increased until the sound it produces is in imison 
 with the given sound. The registering apparatus is now thrown 
 into action, while the same speed of rotation is maintained for a 
 certain time, say a minute. The number of revolutions the wheel 
 makes during this time, multiplied by the number of teeth on the 
 wheel, gives the number of vibrations performed in one minute. 
 This number divided by 60 gives the number of vibrations per 
 second required to produce the given sound, that is, its vibration 
 number. 
 
 Cagniard de Latour's Syren is an instrument constructed on the 
 same principle as the Wheel Syren, but different in detail. Two 
 
 IT 
 
 [^ *iriisSM^ 
 
 *§ ^- J. 
 
 1 1. 
 
 • ii I ' 
 
 I 
 
 FiQ. 20. Fio. 21. 
 
 brass discs, rather more than an inch in diameter, are each simi- 
 larly pierced with the same number of holes arranged in a circle 
 (figs. 20 & 21). The upper disc revolves with the least possible 
 
32 HAND-BOOK OF ACOUSTICS. 
 
 amount of friction on the lower one, wliicli is stationary, and forms 
 the upper side of a -wind chest. When air is forced into this chest 
 by means of the tube below, and the disc is made to revolve, a pufE 
 will pass through each hole, every time the holes coincide. Thus, 
 supposing there are 15 holes, they will coincide 15 times for every 
 revolution, and therefore one revolution will give rise to 15 
 sound waves. By an ingenious device, which, however, in the 
 long run is a disadvantage, the current of air itself moves the 
 upper disc. This is brought about by boring the holes in the upper 
 disc in a slanting direction, while those in the lower one are bored 
 in the opposite direction, as shown in fig, 21. On the upper part 
 of the axis of the revolving disc, a screw is cut, in the threads of 
 which work the teeth of the wheel shown on the left in fig. 21. 
 Further, the right hand wheel in fig. 21 is so connected with the 
 left hand one, that while the latter makes one revolution the former 
 moves only one tooth. Each of these wheels has 100 teeth. On 
 the front of the instrument (fig. 20) are two dials, each divided into 
 100 parts corresponding to the 100 teeth on each of the wheels, the 
 axles of which passing through the centre of the dials, carry the 
 hands seen in the figure. Thus the right hand dial records the 
 number of single revolutions up to 100, and the left hand one the 
 number of hundreds. Finally, this registering apparatus can be 
 instantly thrown in and out of gear, by pushing the nuts seen on 
 both sides. 
 
 Suppose now we wish to obtain, by means of this instrument, the 
 vibration number of a certain sound on a harmonium or organ. 
 We first see that the registering apparatus is out of gear, and then 
 placing the syren in connection with an acoustical bellows, gradually 
 blow air into the instrument. This causes the disc to revolve, and 
 at the same time a sound, at first low, but gradually rising in pitch, 
 is heard. As we continue to increase the wind pressure, the pitch 
 gradually rises until at last it is in unison with the sound we are 
 testing. Having now got the two into exact unison, we throw 
 the registering apparatus into gear, and maintain the same rate of 
 rotation, that is, keep the two sounds at the same pitch for a 
 measured interval of time — say one minute. At the expiration of 
 this time the recording apparatus is thrown out of gear again. We 
 now read ofi the number of revolutions: suppose the dial on the 
 left stands at 21, and that on the right 36, then we know the disc 
 has made 2,136 revolutions. Mtdtiply this by 16 (for we have 
 seen that each revolution gives rise to 16 waves), and we get 32,040 
 as the number of waves given ofl in one minute. Divide this by 
 
ON THE PITCH OF MUSICAL SOUNDS. 33 
 
 60, and the quotient, 534. is the vibration number of the given 
 sound. 
 
 Flo. 22. 
 
34 
 
 HAND-BOOK OF ACOUSTICS. 
 
 Although, this instmmeiit does very well for purposes of lecture 
 illustration, it has several practical defects. When, for instance, the 
 registering apparatus is thrown into gear, the increased work which 
 the wind has to perform in turning the cogfwheels, slackens the 
 speed, and consequently lowers the pitch. Then, again, it is very 
 difficult to keep the blast at a constant pressure — that is, to keep the 
 sound steady. Further, there is an opening for error in noting the 
 exact time of opening and closing the recording apparatus. From 
 these causes, this form of Syren cannot be depended upon, according 
 to Mr. Ellis, within ten vibrations per second. 
 
 Helmholtz has devised a form of Syren, in which these sources of 
 error are avoided. It consists reaUy of two Syrens, A and B, fig. 
 22, facing one another, the discs of which are both mounted on 
 the same axis, and driven with uniform velocity, not by the 
 pressure of the wind, but by an electro-motor. Each disc has 4 
 circles of equidistant holes, the number of holes in the circles 
 being, in the lower disc 8, 10, 12, 18, and in the upper 9, 12, 15, 16 
 respectively. By means of the handles 1, 2, 3, &o., projecting 
 from the wind chests, any or all of these circles may be closed or 
 opened, so that two or any number of sounds up to eight, may be 
 heard simultaneously. 
 
 Knowing the length of a stretched string, the stretching weight, 
 and the weight of the string itself, it is possible to ascertain the 
 vibration number of the sound it gives by calculation. The 
 instrument used in applying this method, is termed a Sonometer 
 
 or Monoohord. It consists (fig, 9.Z) of a sound box of wood, 
 about 5 feet long, 7 inches broad, and 5 inches deep. A steel 
 wire is fastened to a peg at one end, passes over bridges as shown 
 
ON THE PITCH OP MUSICAL SOUNDS. 35 
 
 in the figure, and 18 stretched by a weight at the other. The sound 
 is evolved by drawing a violin bow over the wire. In order to find 
 the vibrational number of any given sound, the wire is first tuned 
 in unison with it, by varying the length of the wire (by means of 
 the movable bridge shown in fig. 23) or the stretching weight. 
 This weight is then noted, and the vibrating part of the wire 
 measured and weighed. If the number of grains in the stretch- 
 iug weight (including the weight of the adjacent non- vibrating 
 part of the wire) be denoted by "W; the number of inches in the 
 vibrating wire by L ; and the number of grains in the same by w ; 
 then the vibrational number is 
 
 ^ / WXP 
 2 'V wX-^ 
 P being the length of the seconds pendulum at the place of obser- 
 vation, (= 39'14 at Greenwich), and tt the constant 3'14159. 
 Although theoretically perfect, the practical difficulties of determin- 
 ing the unison, measuring the length, ascertaining the weights, 
 obtaining perfect uniformity in the wire, keeping the temperature 
 constant, together with those arising from the thickness of the wire, 
 axe so great, as to render this method of no avail where accuracy is 
 required. 
 
 A far more accurate method of counting vibrations is that known 
 as the Graphic method, which is, however, only applicable in general 
 to tuning-forks. The principle of this method will be readily 
 understood on reference to fig. 24. A light style is attached to one 
 of the prongs of a tuning fork. A piece of paper or glass, which 
 
 Fig. 24. 
 
 has been coated with lamp-black by holding it in the smoke of 
 burning camphor or turpentine, is placed below the tuning fork so 
 that the style just touches it. If now, while the glass remains at 
 rest, the fork be set in vibration, the style will trace a straight line 
 on the glass or paper, by removing the lamp black in its path as it 
 moves to and fro. But if the glass be moved rapidly and steadily 
 
36 
 
 HAND-BOOK OF ACOUSTICS. 
 
 a 
 
 in the direction of the fork's length, a complete record of the motion 
 of the fork ■will be left on the lamp-blacked surface in the form of 
 a wavy line, each double sinuosity in which will correspond to a 
 
ON THE PITCH OF MUSICAL SOUNDS. 
 
 37 
 
 complete vibration of the fork. A more convenient arrangement 
 is to fasten the lamp-blacked paper round a rotating cylinder, -which 
 is also made to travel slowly from right to left by means of a screw 
 on the axis, so as to prevent the tracing from overlapping itself. If 
 this cylinder be kept rotating for a certain number of seconds, the 
 number of sinuosities traced in that time can be counted; we have 
 then only to divide this number by the ntimber of seconds, to obtain 
 the vibration number of the fork. 
 
 This method has been brought to a very high degree of perfection 
 by Professor Mayer, of New Jersey. He uses lamp-blacked paper, 
 wrapped round a rotating metallic cylinder or drum, as above 
 described. The wave curve is traced on this by an aluminium, style, 
 attached to the end of one of the prongs of the tuning fork under 
 examination. A pendulum {a, fig. 25), beating seconds, has a 
 platinum wedge fastened to it below, which, at every swing, makes 
 contact with a small basin containing mercury. This basin is in 
 communication with one pole of a battery (J), a wire from the other 
 pole being attached to the primary coU (p) of an inductorium, from 
 which again a wire proceeds to the top of the pendulum. The wire 
 from one end of the secondary coil (s) is attached to the stem of 
 the tuning fork (t), and that from the other end to the revolving 
 cylinder (d). When the apparatus is at work, it is obvious that 
 a spark vrill pierce the smoked b 
 
 paper at every contact of the pen- 
 dulum with the mercury, leaving 
 a minute perforation. The number "^ 
 of complete sinuosities between 
 two consecutive perforations, will 
 consequently be the vibration 
 number of the fork. 
 
 A new instrument for counting 
 rapid vibrations, called by its in- 
 ventors, the Oydoscope, was de- 
 vised a few years ago by McLeod 
 and Clarke. A diagrammatic re- 
 presentation of the essential parts 
 of the apparatus is shown in fig. 
 26. A is a drum, rotating on a 
 horizontal axis, and capable of 
 being revolved at any definite rate. 
 
 Fio. 26. 
 
38 EANB-BOOE OF AOOUSTIOS. 
 
 Eound tMs is fastened a strip of dark paper B, ruled witli equidistant 
 ■wMte lines. is a 2" objective, giving an image of the wHte lines, 
 which is viewed through the microscope, D. The tuning-fork (t) to 
 be tested is fastened vertically iu a vice, so that one of the prongs 
 is situated in the common focus in such a way as to obscure about 
 one-fourth of the field of view. Thus, on looking through the 
 microscope when everything is at rest, an image of the white lines 
 is seen, and the part of the prong (a, fig. 27 A), of the tuning-fork 
 
 Fio. 27 (A). FiQ. 27 (B). 
 
 partially obsouring them; (5) is a scale fixed in the eye-piece. 
 "When the fork is set vibrating and the drum is rapidly rotated, 
 the lines can no longer be separately distinguished; but, just 
 as, in the Graphic method, we found a wave to result from two 
 movements at right angles, so by the composition of the fork's 
 motion with that of the white lines, a wave makes its appearance, 
 (fig. 27, B). If the white lines pass through a space equal to the 
 distance between two of them, while the fork makes one vibration, 
 then the length of the wave wiU be the distance between two white 
 lines as seen through the microscope ; furthermore, if this is the 
 case exactly, the waves wiU. appear stationary. If, however, the 
 drum rotates faster or slower than this, the wave will have a slow 
 apparent motion, either upwards or downwards. As the velocity 
 of rotation of the drum, is under the control of the observer, it 
 is easy to keep it at such a speed, that the wave appears 
 stationary, that is, at such a speed that the white lines pass 
 through a space equal to the distance between two of them while 
 the fork makes one vibration. It follows, therefore, that the 
 number of white lines that pass over the field of view during the 
 time of the experiment, is equal to the number of vibrations 
 executed by the fork in that time. An electric pendulum, beating 
 seconds, gives the time, while an electric counter records the num.- 
 ber of revolutions made by the drum. A fine pointed tube, filled 
 with magenta, automatically marks the paper on the drum at the 
 
ON THE PITCH OF MUSICAL SOUNDS. 
 
 39 
 
 be giTiTiin g and end of the experiment, so as to give the fractional 
 part of a revolution. The number of white Hues that pass the 
 field of -vie-w is thus easily obtained, by multiplying the number of 
 ■white lines round the drum by the number of revolutions and 
 fractional part of a revolution made by it. This, divided by the 
 number of seconds the experiment lasted, gives the vibration 
 number of the fork. In the hands of the inventors, this.uistrum.ent 
 has given extremely accurate results. 
 
 Another very accurate counting instrument is the Tonometer, an 
 account of -which is deferred, until the principles on -which it is 
 constructed have been explained (see Chap. xiii). 
 
 The exactness -with which pitch can now be determined, is shown 
 by the folio-wing abridged table, taken from Mr. EUis's "History of 
 Musical Pitch," p. 402. In the first column are the names of five 
 particular forks, the -vibrational numbers of which are given in the 
 second, third, and fourth columns, as determined independently by 
 McLeod -with the Cycloscope, Ellis with the Tonometer, and Mayer 
 with his modification of the Graphic method, respectively. 
 
 Name of fork. 
 
 Me Leod. 
 
 Ellis. 
 
 Mayer. 
 
 Conservatoire 
 
 Tuileries 
 
 Feydeau 
 
 Versailles 
 
 Marloye 
 
 489-55 
 434-33 
 423-02 
 895-83 
 255-98 
 
 439-54 
 434-25 
 423-01 
 395-79 
 255-96 
 
 439-51 
 434-33 
 422-98 
 395-78 
 256-02 
 
 Eno-wing the velocity of sound in air, and having ascertained the 
 vibrational rate of any sounding body by one of the preceding 
 methods, it is easy to deduce the length of the sotmd waves emitted 
 by it. For, taking the velocity of sound as 1,100 feet per second, 
 suppose a tuning-fork, the -vibration number of which is say 100, 
 to vibrate for exactly one second. During this tune it -will have 
 given rise to exactly 100 waves, the first of which at the end of the 
 second -will be 1,100 feet distant from the fork ; the second one will 
 be immediately behind the first, and the third behind the second, and 
 so on : the last one emitted being close to the fork. Thus the com- 
 bined lengths of the 100 waves -will evidently be 1,100 feet, and as 
 they axe all equal in length, the length of one wave -will be 
 1100 
 
 Too = " ''■ 
 
40 HAND-BOOK OF ACOUSTICS. 
 
 To find the waye length of any sound, therefore, it is only necessary 
 to divide the velocity of sound by the vibration number of the 
 soimd in question. It will be noticed that as the velocity of sound 
 varies with the temperature of the air, so the wave length of any 
 particular sound must vary with the temperature. 
 
 We have seen that the pitch of any sound depends intimately 
 upon the rapidity with which the sound waves strike the tympanum 
 of the ear, and usually this corresponds with the rate of vibration 
 of the sounding body. If, however, from any cause, more sound 
 waves from the sounding body, strike the ear, in a given time, than 
 are emitted from it during that time, the apparent pitch of the note 
 will be correspondingly raised, and vice versa. Such u, case occurs 
 when the sounding body is moving towards or from us, or when we 
 are advancing or receding from the sounding body. When, for 
 example, a, locomotive, with the whistle sounding, is advancing 
 rapidly towards the observer, the pitch will appear perceptibly 
 sharper than after it has passed. This fact may be easily illustrated 
 by fastening a whistle in one end of a piece of india-rubber tubing 
 4 or 5 feet long, and blowing through the other ; at the same time 
 whirhng the tube in a horizontal circle above the head. A person 
 at a distance will perceive a rise in pitch as the whistle is advancing 
 towards bi-m and a fall as it recedes. 
 
 The lowest sound used in music, is found in the lowest note of 
 the largest modern organs, and is produced by 16J vibrations per 
 second ; but so little of musical character does it possess, that it is 
 never used except with its higher octaves. The musical oharactei 
 continues to be very imperfect for some distance above this limit ; 
 in fact, until we get to above twice this number of vibrations per 
 second. The highest limit of musical pitch at the present time is 
 about 0*, a sound corresponding to about 4,000 vibrations per 
 second. Very much higher sounds than this can be heard, but 
 they are too shrill to be of any use in music. Fig. 28, which 
 explains itself, shows the limits of pitch of the chief musical 
 instruments. 
 
 The note C in the treble stafi is the sound that musicians usually 
 take as a basis of pitch ; this, once fixed, all the other sounds of 
 the musical scale are readily determined. But, imfortunately, at 
 the present day, there are a very large number of vibration 
 numbers, corresponding to this note ; in other words, there is no 
 universally recognised standard. It appears from Mr. Ellis's paper 
 on "The History of Musical Ktch" that the vibration numbei 
 
ON THE PITCH OF MUSICAL SOUNDS. 
 
 41 
 
 
 C4I6 
 
 C332 
 
 C264 
 
 Cil28 
 
 256 
 
 01512 
 
 02 
 
 1024 
 
 0-' 
 
 2048 
 
 0* 
 
 4096 
 
 -m- 
 
 
 
 
 
 
 
 
 
 
 
 
 ^— 
 
 Ss= 
 
 Sg=r 
 
 ^^e: 
 
 ^^^ 
 
 ^5—~ 
 
 ^r= 
 
 ^^= 
 
 4= 
 
 Huxuan Voice. 
 
 = 
 
 -»■ 
 
 ' 
 
 
 tJ -m- 
 
 Z7 
 
 tJ 
 
 
 ^ 
 
 
 
 
 
 ■ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 » 
 
 — 
 
 ~ 
 
 
 
 
 
 
 
 Bow Instrnments. 
 
 Violin 
 
 
 
 
 
 Viola 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 , , 
 
 ^^ 
 
 
 
 ^^ 
 
 
 
 
 
 Pizzicato Instru- 
 ments. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 " 
 
 Harp 
 
 
 
 
 
 
 
 _ 
 
 Keed Instruments. 
 
 Oboe 
 
 
 
 
 _^ 
 
 — 
 
 — 
 
 - 
 
 - 
 
 ■RngliHli TTn-m 
 
 
 
 
 
 
 — 
 
 
 
 
 
 
 
 Tube Instruments. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Wind Instruments 
 of Brass. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 _ 
 
 
 Key-Board Instru- 
 ments. 
 
 
 
 
 
 
 "■^^1 
 
 
 
 
 
 
 
 
 1 1 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 Fig. 28. 
 
42 
 
 HAND-BOOK OF ACOUSTICS. 
 
 corresponding to C lias been gradually rising for many years. The 
 following are a few of the cHef standards as determined by him, 
 and given in the paper just referred to. !Por fuller details the 
 paper itself should be consulted. 
 
 Date. 
 
 Standard. 
 
 A 
 
 C 
 
 1842 
 
 HuUah's C Fork 
 
 
 524-8 
 
 1860 
 
 Society of Ai-ts C fork (528) 
 
 
 
 534-46 
 
 1877 
 
 Tonic Sol-fa C Fork 
 
 — 
 
 513 
 
 After 1877 
 
 tT )l J> 
 
 — 
 
 507 
 
 1751 
 
 Handel's A Fork 
 
 422-5 
 
 507 
 
 1877 
 
 Westminster Abbey Organ . . 
 
 438 
 
 520-9 
 
 1877 
 
 St. Paul's Organ 
 
 — 
 
 528-7 
 
 1877 
 
 Albert Hall Organ 
 
 — 
 
 641-2 
 
 1877 
 
 Crystal Palace Organ 
 
 — 
 
 540 
 
 1874 
 
 Broadwood's Piano's (highest) 
 
 454-7 
 
 540-8 
 
 1877 
 
 Collard's Pianos 
 
 — 
 
 535 
 
 1879 
 
 Erard's Pianos 
 
 455-3 
 
 541-5 
 
 1859 
 
 Diapason Normal (French Std.) 
 
 435 
 
 522 
 
 In the above table, where there is an entry in column headed A, 
 the note A was the one actually tested, the corresponding calculated 
 value of C being placed in the next column. 
 
 SlTMMAET. 
 
 A musical sound has three elements : Pitch, Intensity, and Quality. 
 The Pitch of a musical sound depends solely upon the vibration 
 rate of the body that gives rise to it. 
 
 The Vibration Number of a given musical sound is the number 
 of -vibrations per second necessary to produce a sound of that par- 
 ticular pitch. 
 
 The principal instruments which have been used from time to 
 time in determining the vibration numbers of musical sounds are : 
 Savart's Toothed Wheel. 
 The Syren. 
 
 The Sonometer or Monochord. 
 Mayer's Graphic Method. 
 The Oycloscope. 
 The Tonometer. 
 The last three are by far the most accurate. 
 
ON THE PITCH OF MUSICAL SOUNDS. 43 
 
 To ascertain tlie ■wave length of any given sound — Divide the 
 velocity of sound ty its vibration number. 
 
 The wave lengtb of any given sound, increases with tbe tem- 
 perature. 
 
 The temperature remaining constant, the length of the sound 
 wave determines the pitch of the sound produced. 
 
 The range of musical pitch is from about 40 to 4,000 vibrations 
 per second. 
 
 The only authentic Standard of Pitch is the French Diapason 
 
 Normal, viz : 
 
 A = 435 
 corresponding in true intonation to 
 
 CI = 622 
 
44 
 
 CHAPTER V. 
 
 The Melodic Relations of the Sotjitds of the Common Scale. 
 
 In describing tlie form, of Syrea deyised by HelmTioltz, it was 
 mentioned, tbat the lower revolTing plate was pierced with four 
 circles of 8, 10, 12, and 18 holes, and the upper with four circles of 
 9, 12, 15, and 16. If only the "8-hole circle" on the lower and 
 the " 16-hole " circle on the upper be opened, while the Syren is 
 working, two sounds are produced, the interval between which, the 
 musician at once recognises as the Octave. When the speed of rotation 
 is inoreased,both soxinds rise ia pitch, but they always remain an 
 Octave apart. The same interval is heard, if the circles of 9 and 18 
 holes be opened together. It follows from these experiments, that 
 when two sounds are at the interval of an Octave, the vibrational 
 number of the higher one is exactly twice that of the lower. An 
 Octave, therefore, may be acoustically defined as the interval 
 between two sounds, the vibration number of the higher of which 
 is twice that of the lower. Musically, it may be distinguished from 
 all other intervals by the fact, that, if any particular sound be 
 taken, another sound an octave above this, another an octave above 
 this last, and so on, and all these be simultaneously produced, there 
 is nothing in the resulting sound unpleasant to the eai\ 
 
 Since the ratio of the vibration numbers of two sounds at the 
 interval of an octave is as 2 : 1, it is easy to divide the whole range 
 of musical sound into octaves. Taking the lowest sound to be 
 produced by 16 vibrations per second, we have 
 
 1st Octave, from 16 to 32 vibrations per second. 
 2nd „ 32 to 64 
 
 to 128 
 
 to 266 
 
 to 512 
 
 to 1,024 
 
 to 2,048 
 
 to 4,096 
 
 Thus all the sounds used in music are comprised within the compass 
 of about eight octaves. 
 
 Srd 
 
 64 
 
 to 128 
 
 4th 
 
 128 
 
 to 256 
 
 5th 
 
 256 
 
 to 512 
 
 6th 
 
 512 
 
 to 1,024 
 
 7th 
 
 „ 1,024 
 
 to 2,048 
 
 8th 
 
 2,048 
 
 to 4,096 
 
MELODIC BELATIONS OF SOUNDS OF TEE SCALE. 45 
 
 Eeturning to the Syren: if the 8 and 12 "hole circles" be opened 
 together, vre hear two sounds at an interval of a Fifth, and as in 
 the case of the octave, this is the fact, whatever the velocity of 
 rotation. The same result is obtained on opening the 10 and 15, 
 or the 12 and 18 circles. When, therefore, two sounds are at an 
 interval of a Fifth, for every 8 vibrations of the lower sound, there 
 are 12 of the upper, or for every 10 of the lower there are 15 of the 
 upper, or for every 12 of the lower there are 18 of the upper. But 
 
 8 
 
 12 : 
 
 : 2 
 
 3 
 
 10 
 
 15 : 
 
 : 2 
 
 3 
 
 12 
 
 18 : 
 
 : 2 
 
 3 
 
 Therefore two sounds are at the interval of a Fifth when their 
 vibration numbers are as 2 to 3; that is when 2 vibrations of the 
 one are performed in exactly the same time as 3 vibrations of 
 the other. This may be conveniently expressed by saying that the 
 vibration ratio or vihration fraction of a Fifth is 3 : 2 or |^. 
 Similarly the vibration ratio of an Octave is 2 : 1 or ^. 
 
 Again, on opening the circles of 8 and 10 holes, two sounds are 
 heard at the interval of a Major Third. The same interval is 
 obtained with the 12 and 15 circles. Now 8 : 10 : : 4 : 5 and 
 12 : 15 : : 4 : 5. Therefore two sounds are at the interval of a 
 Major Third, when their vibration numbers are as 4 : 5 ; or more 
 concisely, the vibration ratio of a Major Third is |. 
 
 With the results, thus experimentally obtained, it is easy to 
 calculate the vibration numbers of aU the other sounds of the 
 musical scale, when the vibration number of one is given. For 
 example, let the vibration number of d be 288, or shortly, let 
 d = 288 ; then the higher Octave d' = 288 X 2 = 576. Also the 
 vibration ratio of a Fifth =: ^ ; therefore the vibration number of 
 8 is to that of d, as 3 : 2; that is, s = | X 288 = 432. Similarly 
 the interval { ^ is a Major Third ; but the vibration ratio of a 
 Major Third we have found to be, |; therefore n : d : : 5:4, that 
 is n = f X 268 = 360. Again, {g is a Major Third; therefore 
 t = f X 432 = 540. Further, j^' is a Fifth; therefore 
 ri = -^ X 432 = 648, and its lower octave r = ^-^ = 324. It only 
 remains to obtain the vibration numbers of f and 1. Now j f is 
 a Fifth, thus the vibrational number of f is to that of d' as 2 : 3 ; 
 therefore f = f X 576 = 384 ; and j J is a Major Third, con- 
 sequently 1 = 384 X -S- = 480. Tabtilatuig these results we have 
 
46 HAND-BOOK OF ACOUSTICS. 
 
 di 
 
 = 
 
 576. 
 
 t 
 
 = 
 
 540. 
 
 1 
 
 = 
 
 480. 
 
 s 
 
 = 
 
 432. 
 
 f 
 
 =z 
 
 384. 
 
 n 
 
 = 
 
 360. 
 
 r 
 
 = 
 
 324. 
 
 d 
 
 ^ 
 
 288. 
 
 The vibration numbers of tbe upper or lower octaves of these notes, 
 are of course at once obtained by doubling or halving them. 
 
 It win be noticed that a scale may be constructed on any vibration 
 number as a foundation. The only reason for selecting 288 was, to 
 avoid fractions of a vibration and so simplify the calculations. As 
 another example let us take d = 200. Proceeding in the same way 
 as before, but tabulating at once, for the sake of brevity, we get 
 
 d' = 200 X 
 
 2 = 400. 
 
 (2). 
 
 t = aio X 
 
 f = 375. 
 
 (5). 
 
 1 = 266f X 
 
 f = 3331. 
 
 (8). 
 
 S = 2|i X 
 
 1 = 300. 
 
 (3). 
 
 f = ittH X 
 
 1 = 266|. 
 
 (7)- 
 
 n = 2fa X 
 
 1 = 250. 
 
 (4). 
 
 r = 3|jix|: 
 
 X 1= 225. 
 
 (6). 
 
 d = 
 
 = 200. 
 
 (1). 
 
 We may now adopt the reverse process, that is, from the vibration 
 numbers, obtain the vibration ratios. For example, using the first 
 scale, we find that the vibration number of t is to that of Pi as 
 540 : 360, that is (dividing each by 180, for the purpose of simpli- 
 fying) as 3 : 2 ; or more concisely 
 
 I t __ 640 __ 3 
 
 ( m 360 2 
 
 The interval j ^ is therefore a perfect Fifth. Again, 
 11 _ 4^ 
 f r 324 
 
 Now the vibration fraction of a perfect Fifth = f ^ |fS, therefore 
 i ], is not a perfect Fifth. "We shall return to this matter further 
 on, at present it will be sufficient to notice the fact. The student 
 
MELODIC RELATIONS OF SOUNDS OF THE SCALE. 47 
 
 must take particular care not to subtract or add vibration numbers, 
 in order to find tbe interval between tbem; tlius the diHerenoe 
 between tbe vibration numbers of t and n in tbe second scale ia 
 375 — 250 = 125, but tbis does not express the interval between 
 them, viz., a Fifth, but merely the difference between the vibration 
 numbers of these particular sounds. To make this clearer, take 
 the diSerence between the vibration numbers of d and s ia the 
 second table = 300 — 200 = 100, and between d and s ia the 
 first = 432 — 288 = 144. Here we have different results, although 
 the interval is the same. Take the ratio, however, and we shaU get 
 the same in each case for 
 
 aoo _ S- and ±|| = 3., 
 
 200 2 288 2 
 
 We shall now proceed to ascertain the vibration ratios of the 
 mtervals between the successive sounds of the scale, using the first 
 of the two scales given on the preceding page : — 
 
 d' 
 t 
 
 = 
 
 B76 
 540' 
 
 = 
 
 96 
 90 
 
 = 
 
 16 
 15 
 
 t 
 1 
 
 = 
 
 640 
 480 
 
 = 
 
 54 
 48 
 
 = 
 
 9 
 8 
 
 1 
 
 
 480 
 
 
 120 
 
 
 10 
 
 s 
 
 
 432 
 
 
 108 
 
 
 9 
 
 s 
 f 
 
 = 
 
 432 
 384 
 
 = 
 
 54 
 48 
 
 = 
 
 9 
 
 8 
 
 f 
 
 = 
 
 384 
 360 
 
 = 
 
 96 
 
 90 
 
 = 
 
 16 
 15 
 
 m 
 
 _ 
 
 360 
 
 _ 
 
 90 
 
 _ 
 
 10 
 
 r 
 
 
 824 
 
 
 81 
 
 
 9 
 
 r 
 d 
 
 = 
 
 324 
 
 288 
 
 = 
 
 81 
 
 72 
 
 = 
 
 9 
 8 
 
 There are, therefore, three kinds of intervals between ' the 
 consecutive sounds of the scale, the vibration ratios of which are 
 A, J-fi, and J-^. The first of these intervals, which has been termed 
 the Greater Step or Major Tone, occurs three times ia the diatonic 
 scale, viz., 
 
 |{ 1? \l 
 
 The next is the Smaller Step or Miaor Tone, and is found twice, 
 viz.. 
 
48 HAND-BOOK OF ACOUSTICS. 
 
 The last is tlie Diatonic Semitone, and also occurs twice, viz., 
 
 !t In 
 We may now calculate the vibration ratios of the remaining 
 intervals of the scale. I ^ may be selected as the type of the 
 Fourth. Taking again the vibration numbers of the first scale, the 
 vibration ratio of this interval is 
 
 384 96 8 4 
 
 288 — 72 — 6 — 3 
 
 This result may be verified on the Syren by opening the 12 and 9 or 
 16 and 12 circles. 
 Taking j ^ as an example of a Minor Third, its vibration ratio is 
 
 432 1? _ ? 
 
 360 — 40 — 6' 
 
 This can also be verified by the S3Ten with the 12 and 10 circles. 
 Again, the vibration ratio of I ^ a Minor Sixth, is 
 
 676 72 8 
 
 360 45 5" 
 
 and this, too, may be confirmed on the Syren, with the 16 and 10 
 circle. 
 The vibration ratio of j i, a Major Sixth, is 
 
 480 60 6 
 
 288 — 36 3' 
 
 which may be confirmed with the 15 and 9 circles. 
 The vibration ratio of the Major Seventh j ^ is 
 
 540 135 15 _ 
 
 iS — W 8 ' 
 
 and this can be verified with the 15 and 8 circles. 
 The vibration ratio of the Minor Seventh I * is 
 
 384 96 16 
 
 216 64 9 ' 
 
 capable of verification with the 16 and 9 circles. 
 The vibration fraction of the Diminished Fifth \tis 
 
 384 64 
 
 270 ~~ 45 ' 
 
 and that of the Tritone, or Pluperfect Fourth j | is 
 
 640 90 45 
 
 3S4 ^^ 64 ^ 82' 
 
 In order to find the vibration ratio of the sum of two intervals, 
 the vibration ratios of which are given, it is only necessary to 
 
MELODIC RELATIONS OF SOUNDS OF THE SCALE. 49 
 
 multiply tliem together as if they were vulgar fractions, thus, 
 gi^en {^ = f, and {^ = f ; to find Jil- 
 ls _66_6_ 3 
 
 ■which we already know to be the case. The reason of the process 
 may be seen from the following considerations. From 1 ^ = 5 > 
 and f ^ = f we know that, 
 
 for every 6 vibrations of s, there are 5 of pi ; 
 
 and ,, ,, 5 „ „ n, „ „ 4 „ d; 
 
 Therefore ,, ,, 6 ,, ,, s, ,, ,, 4 ,, d; 
 
 that is ,, ,, 3 ,, ,, s, „ ,, 2 ,, d. 
 
 Again, in order to find the vibration ratio of the difference of two 
 intervals, the vibration ratios of which are given, the greater of 
 these must be divided by the less, just as if they were vulgar 
 fractions. For example, given i ^' =: f , and ) ™ =: f , to find 
 
 Im 
 
 \^' = - — - = - X - = ? 
 In 1 ■ 4 1 5 6' 
 
 The reason for the rule will be seen from the following considera- 
 tions. From the given vibration ratios we know that, 
 for every 2 vibrations of d', there is 1 of d ; 
 that is „ ,, 8 „ ,, d', „ are 4 ,, d; 
 
 and ,, ,, 4 ,, ,, d, ,, ,, 5 ,, n; 
 
 therefore „ „ 8 „ „ d', „ „ 5 „ n. 
 
 We shall apply this rule, to find the vibration ratios of a few 
 other intervals. The Greater Chromatic Semitone is the difference 
 between the Greater Step and the Diatonic Semitone, j ^ is an 
 example of the Greater Chromatic Semitone, being the difference 
 between { S a Greater Step, and j ^ a Diatonic Semitone. Now 
 j I := f , and j f^ = i-f (for it is the same interval as j ^') ; therefore 
 
 f f 8 ■ 15 8 16 128' 
 
 The Lesser Chromatic Semitone is the difference between the 
 Smaller Step and the Diatonic Semitone ; j Y, for example, which 
 is the difference between j^ and |Jg. Now jl = ^ and 
 {^ = jl' = If; therefore 
 
 10 . 16 10 ,, 16 25 
 
 24* 
 
 ise ^_:.i5 — y - 
 
 ? S 9 -16 9 '^ 16 
 
50 HAND-BOOK OF ACOUSTICS. 
 
 This is also the difEerence between a Major and a Minor Third, for 
 
 6.6 5 B 25 
 
 4"="B 4 ^ 6 24 
 
 The interval between the Greater and Lesser Chromatic Semitones 
 will be 
 
 185 . 25 135 24 81 
 
 i28"^24 128 ^ 25~~80' 
 
 which is usually termed the Comma or Komma. 
 
 Eeferring to the first table of vibration numbers on page 46, we 
 have 1 = 480, and r := 324 ; therefore 
 
 1 1 480 40_ 
 
 / r 324 27 ' 
 
 and thus, as noticed above, it is not a Perfect Fifth. To form a 
 Perfect Fifth with 1, a note r' would be required, such that 
 
 ?r' 2 
 It is easy to find the vibration number of this note if that of 1 be 
 given, thus : — 
 
 1 3 
 
 
 
 that is, ^= ?; 
 ' r' 2 
 
 
 
 leref ore !_=:-, 
 
 480 3' 
 
 r' = i 
 
 X 
 
 480 
 
 1 
 
 ^ = 320. 
 1 
 
 This note has been termed rah or grave r, and may be conveniently 
 
 written, r\ Similarly j J is not a true Minor Third, for its vibration 
 
 ratio is 
 
 asA == 95 = i2 • 
 
 324 81 27 ' 
 
 but j ^» is a true Minor Third, for its vibration ratio is 
 
 384 _. i_a = fi 
 320 40 5' 
 
 The interval between r and r^ is the comma, its vibration ratio 
 being evidently 
 
 324 _- £1 
 320 80" 
 
 SUMMABT. 
 
 The sounds used in Music lie within the compass of about eight 
 Octaves. 
 
 The vibration ratio or vibration fraction of an interval, is the 
 ratio of the vibration numbers of the two sounds forming that 
 interval. 
 
 The vibration ratios. of the principal musical intervals have been 
 exactly verified by Helmholtz's modification of the Double Syren. 
 
MELODIC RELATIONS OF SOUNDS OF THE SCALE. 51 
 
 It may be sho-wn, by means of this instrument, that the vibration 
 numbers of the three tones of a Major Triad, in its normal position 
 
 — I E, or j m, for example, — are as 
 4:5:6. 
 Starting from this experimental foundation, the vibration numbers 
 of all the tones of the modern scale can readily be calculated on any 
 basis ; and from these results, the vibration ratio of any interval 
 used in modern music may be obtained. 
 
 Vibration ratios must never be added or suhfracted. 
 
 To find the vibration ratio of the sum of two or more intervals, 
 multiply their vibration ratios together. 
 
 To find the vibration ratio of the difference of two intervals, divide 
 the vibration ratio of the greater interval by that of the smaller. 
 
 The vibration ratios of the principal intervals of the modem 
 musical scale are as follows : — 
 
 Komma - - - fj 
 
 Lesser Chromatic Semitone - 2? 
 
 Greater ,, ,, - ^|| 
 
 Diatonic Semitone " " T5 
 
 Smaller Step or Minor Tone -1^ 
 
 Greater Step or Major Tone ^ 
 
 Minor Third - f 
 
 Major Third - - f. 
 
 Fourth - ... * 
 
 Tritone ^| 
 
 Diminished Fifth f f 
 
 3 
 
 Fifth 
 
 Minor Sixth |- 
 
 Major Sixth - f 
 
 Minor Seventh - - ~r 
 
 Major Seventh - - i/ 
 
 Octave X 
 
 To find the vibration ratio of any of the above intervals increased 
 by an Octave, multiply by -f ; thus the vibration ratio of a Major 
 
 Tenth is 
 
 5. V 2 — ] — 5 
 
52 
 
 CHAPTER VI. 
 
 On- THE Intensity or Lotjuness of Mtjsicai, Sotinbs. 
 
 We have seen that the pitch of a sound depends solely upon the 
 rapidity ■with which the vibrations succeed one another. "We have 
 next to study the question: "Upon what does the Loudness or 
 Intensity of a sound depend ? " 
 
 Gently pluck a violin string. Notice the intensity of the result- 
 ing sound, and also ohserve the extent or amplitude of the string's 
 vibration. Pluck it harder; a louder sound is heard, and the string 
 is seen to vibrate through a greater space. Pluck it harder still ; 
 a yet louder sound is produced, and the amplitude of the vibrations 
 is still greater. We may conclude, from this experiment, that as 
 long as we keep to the same sounding body, the intensity of the 
 sound it produces, depends upon the amplitude of its vibrations ; 
 the greater the amplitude, the louder the sound. This fact may be 
 strikingly illustrated by the following experiment. Fasten a style 
 of paper, or better still, parchment, to one prong of a large 
 tuning-fork. Coat a slip of glass on one side with lamp- 
 black, and lay it, with the coated side upwards, on a smooth board, 
 
 Fig. 29. 
 
 having previously nailed on the latter a straight strip of wood, to 
 serve as a guide in subsequently moving the glass slip. Now 
 strike the fork sharply, and immediately hold it parallel to the 
 glass, in such away, that the vibrating style just touches the lamp- 
 
INTENSITY OF MUSICAL SOUNDS. 53 
 
 black. Move the glass slip slowly along under the tuning-fork. 
 The latter, as it vibrates, will remove the lamp-black, and leave a 
 clean wedge-shaped trace on the glass, as seen in fig. 29. As the 
 width of the trace at any point is evidently the amplitude of the 
 vibration of the fork, at the time that point was below it, we see 
 that the amplitude of the vibrations of the fork gradually decreases 
 till the fork comes to rest; and as the sound decreases gradually tUl 
 the fork becomes sUent, we see that the intensity of its sound 
 depends upon the amplitude of its vibrations. 
 
 It is obvious, that the greater the amplitude of the vibrations of 
 a sounding body, the greater will be the amplitude of the vibrations 
 of the air particles in its neighbourhood ; thus we may conclude, 
 that the intensity of a sound depends upon the amplitude of vibra- 
 tion of the air particles in the sound wave. But it is a matter of 
 common experience, that a sound becomes fainter and fainter, the 
 farther we depart from its origin ; therefore, we must limit the 
 above statement thus : the intensity of a given sound, as perceived 
 by our ears, depends upon the amplitude of those air particles of 
 its sound wave, which are in the immediate neighbourhood of our 
 ears. This leads us to the question : " At what rate does the in- 
 tensity of a sound diminish, as we recede from its origin ? " We 
 may ascertain the answer to this question, by proceeding as in the 
 analogous case of heat or light. Thus, let A, fig. 30 be the origm 
 of a given sound. At centre A, and with radii of say 1 yd., 2 yds., 
 3 yds., describe three imaginary spheres, B, 0, D. Now, looking 
 on sound, for the moment, as a quantity, it is evident that the quan- 
 tity of sound which passes through the surface of the sphere B is 
 identical with the quantity that passes through the surface of the 
 spheres and D. But the surfaces of spheres vary as the squares 
 of their radii ; therefore, as the radii of the spheres B, 0, and D 
 are 1, 2, and 3 yds. respectively, their surfaces are as 1= : 2' : 3^; 
 that is, the spherical surface C is four times as great, and D 9 times 
 as great, as the spherical surface B. We see, therefore, that the 
 quantity of sound, which passes through the surface of B, is, as it 
 were, spread out fourfold as it passes through 0, and ninefold as it 
 passes through D. It foUows, therefore, that one square inch of 
 will only receive ^ as much sound as a square inch of B, and one 
 square inch of D only i as much. Thus, at distances of 1, 2, 3, 
 from a sounding body, the intensities are as 1, i, and ^ ; that is, as 
 we recede from a sounding body, the intensity diminishes in pro- 
 portion to the square of our distance from the body, or more con- 
 
54 
 
 HAND-BOOK OF ACOUSTICS. 
 
 oisely, " The intensity of a sound varies inversely as the square of 
 the distance from its origin. 
 
 Fio. 30. 
 
 It should be clearly noted, however, that the conditions under 
 which the above " law of inverse squares," as it is called, is true, 
 rarely or never obtain. The chief disturbing elements in the 
 application of this law axe echoes. When a ray of light strikes 
 any reflecting surface at right angles, it is reflected back in the 
 direction whence it came. If a ray of light, A 0, fig. 31, does not 
 fall at right angles upon a reflecting surface P Q, it is reflected 
 along a line C B, which is so situated, that the angle B H is equal 
 to the angle A H ; H C being at right angles to P Q. Just so 
 with sound. A person standing at B would hear a sound from A, 
 first as it reaches hiyn in the direction A B, and directly after, 
 along the line B. If the distances A B and A C B were each 
 only a few yards, the two sounds would be indistinguishable, 
 but if there were any considerable difference between these two 
 
INTENSITY OF MUSICAL SOUNDS. 66 
 
 distances, tlie two sounds ■woiild be separately heard, the latter being 
 termed th.e echo of the former. Therefore, when a vibrating body 
 emits a soimd in any room or hall, the waves which proceed from 
 it in aU directions, strike the walls, floor, ceiling, and also the 
 reflecting surfaces of the various objects in the place, and are re- 
 flected again and again from them. Thus the direct and reflected 
 sounds coalesce, and interfere with one another, in the most com- 
 plicated manner, and the simple law of inverse squares is no longer 
 applicable. This is still the case, even in the open air, away from 
 all surrounding objects, for the ground will here present a reflect- 
 ing surface, and other invisible reflectors are found, as Professor 
 Tyndall has shown, in the surfaces which separate bodies of air of 
 different hygrometrio states and of different temperatures. 
 
 This may be put in another way. It is a condition of the truth 
 of the law of inverse squares, as above shown, that the sound 
 shall be able to spread outwards in all directions ; if this is not the 
 case, the law no longer holds good. Now, in a building, this is not 
 the case ; the sound is prevented from spreading by the roof, floor, 
 and walls. If the sound can be entirely prevented from spreading, 
 its intensity wUl not diminish at all. This is the principle of the 
 speaking tube. In this instrument, the vibrations of the air par- 
 ticles are transmitted undiminished, except by friction against the 
 side of the tube, and by that part of the motion which is given up 
 to the substance of the tube itself ; thus sound can be transmitted 
 to great distances in such tubes. Eegnault, experimenting with 
 the sewer conduits of Paris, found that the report of a pistol was 
 audible through them, for a distance of 6 miles. 
 
 The bad acoustical properties of a building are generally due to 
 echoes. A sound from the lips of a speaker, in a building, reaches 
 the ear of the listener directly, and also after one or more reflections 
 from the ceiling, walls, floor, and so on. If the building be of con- 
 
66 SAND-BOOK OF A00U8TIG8. 
 
 siderable dimensions, these echoes may reach, the listener's ear at 
 an appreciable interval of time after the direct sound, or after one 
 another, and wiU then so combine with the succeeding direct sound 
 from the speaker as to make his words quite indistinguishable. 
 The roof is often the chief culprit in this matter, especially when 
 lofty, and constructed of wood, this latter affording an excellent 
 reflecting surface. An obvious remedy is to cover such a 
 surface with some badly-reflecting substance, such as a textile 
 fabric. A sound-board over the speaker's head, will also prevent 
 the sound from passing directly to the roof. The bodies and clothes 
 of the persons forming an audience, are also valuable in preventing 
 echoes. Professor Tyndall, to whose work on sound the student is 
 referred for further information on this subject, says that, having 
 to deliver a lecture in a certain hall, he tried its acoustical proper- 
 ties beforehand, and was startled to find that when he spoke from 
 the platform, a friend he had with him, seated in the body of the 
 empty haU, could not distinguish a word, in consequence of the 
 echoes. Subsequently, when the hall was filled with people, the 
 Professor had no difficulty in making himself distinctly heard in 
 every part. Again, everyone must have noticed the difference 
 between speaking in an empty and imcarpeted room, in which the 
 echoes reinforce the direct sound, and speaking in the same room 
 carpeted, and furnished, the echoes in this case being deadened by 
 the carpets, curtains, &c. 
 
 SUMMART. 
 
 The Intensity of the sound produced by a vibratory body, depends 
 T^jon the amplitude of its vibrations. 
 
 The Intensity of a sound varies inversely as the square of the 
 distance from its origin, only when the sound waves can radiate 
 freely in all directions without interruption. 
 
 Sound is reflected from elastic surfaces in the same way as Ught, 
 thereby producing echoes. 
 
 Sound is well reflected from such surfaces as wood, iron, stone, 
 &c., while cloths, carpets, curtains, and textile fabrics in general, 
 scarcely reflect at all. 
 
57 
 
 CHAPTER VII. 
 
 EesONANCE, Oo-VIBEATION, OB SYMPATHY OF TONES. 
 
 Select two tuning-forks •which are exactly in unison. Having 
 taken one in each hand, strike that in the right hand pretty sharply, 
 and immediately hold it with its prongs parallel, and close to the 
 prongs of the other, but without touching it. After the lapse of 
 not less than one second, on damping the fork in the right hand, 
 that in the left will be found to be giving out a feeble tone. To 
 this phenomenon, the names of Eesouance, Co-vibration, and 
 Sympathy of tones have been given, the first being the one most 
 commonly used in BngUsh works. The explanation of this effect 
 will be better understood after a consideration of the following 
 analogous experiment. 
 
 Let a heavy weight be suspended at the end of a long cord, and 
 to it attach a fibre of silk or cotton. The weight being at rest, pull 
 the fibre gently so as not to break it. The weight wiU thus be 
 pulled forwards through an exceedingly small, perhaps imper- 
 ceptible distance. Now relax the pull on the fibre, till the 
 weight has swung through its original position, and reached 
 the limit of its backward movement. If another gentle pull 
 be then given, the weight wUl swing forward a trifle further 
 than at first. The weight then swings backwards as before, 
 and again a properly-timed pull will still further extend its 
 excursion. By proceeding in this way, after a time, the total effect 
 of these accumulated impulses will have been sufficient to impart 
 to the weight a considerable oscUlation. On examination it wiU. be 
 found that this experiment is analogous to the last one. The 
 regularly timed impulses in the second experiment, correspond to 
 the regularly vibrating fork in the right hand ; the weight to the 
 fork in the left hand, and the fibre to the air between the forks. 
 And here it must be observed, that just as the forks execute their 
 
58 EAND-BOOE OF AOOUSTIGS. 
 
 vibrations in equal times, wlietlier their amplitudes be great or small, 
 so the ■weight performs its swings (as long as they are not too 
 violent) in equal times, whether their range be small or great. 
 Again, the conditions ol success are the same in both cases ; for, in 
 the first place, the impulses in the second experiment must be 
 exactly timed, that is, they must be repeated at an interval of time 
 which is identical with the time taken by the weight to perform a 
 complete swing. In other words, the hand which pulls the fibre 
 must move in perfect unison with the weight. If this were not so, 
 the impxilses would destroy one another's effects. Just so with the 
 forks ; they must be in the most rigorous unison, in order that the 
 effects of the impulses may accumulate. Again, in each experiment 
 a certain lapse of time is necessary to allow the effects of the suc- 
 cessive impulses to accumulate. 
 
 The complete explanation of the experiment with the two tuning- 
 forks is as follows. Let the prong A of the fork in the right hand 
 be supposed to be advancing in the direction of the non- vibrating 
 fork B ; the air between A and B will be compressed, and thus the 
 pressure on this side of the prong B will be greater than that on 
 the other; the latter prong will therefore move through an 
 infinitesimal space away from A. Now suppose the prong A has 
 reached its extreme position and is returning ; then, as both forks 
 execute their vibrations in exactly equal times, whether these be 
 of large or small extent, it follows that B must be returning also ; 
 but as A moves through a greater space than B, the air between 
 the two will become rarefied, and thus the pressure on this side of B 
 will be less than that on the other ; B will therefore receive another 
 impulse, which will slightly increase its amplitude. On its return, 
 it will receive another slight impulse, and thus, by these minute 
 successive additions, the amplitude is soon sufficiently increased to 
 produce an audible sound. 
 
 In the preceding experiment, the exciting fork communicates a 
 small portion of its motion to the air between the forks, and then 
 this latter gives up part of its motion to the other fork. Now, as 
 the density or weight of aii- is so exceedingly small in comparison 
 with that of the steel fork, the ampKtude of the vibrations tiius set 
 up in the latter must necessarily be always very small, that is, its 
 sound wiU be very faint. By using a medium of greater elasti- 
 city, the sound may be obtained of sufficient intensity to be 
 heard by several persons at once. Thus, let one fork be struck 
 sharply, and the end be immediately applied to a sounding board. 
 
BE80NANCE. 59 
 
 on wMoh the end of the non-vibrating fork is already resting; after 
 the lapse of a second or so, the latter will be heard giving forth a 
 sound of considerable intensity, the motion in this case having 
 been transmitted through the board. 
 
 The following ezperiments illustrate the phenomenon of 
 resonance or co-vibration in the case of stretched strings. Press 
 down the loud pedal of a pianoforte, so as to raise the dampers 
 from the strings. Each sound on the pianoforte is generally pro- 
 duced by the vibration of two or three wires tuned in unison. Set one 
 of these vibrating, by plucking it with the finger. After the lapse of 
 a second or so, damp it, and the other wire will be heard vibrating. 
 Again, having raised the dampers of a pianoforte, sing loudly any 
 note of the piano, near and towards the sound-board. On ceasing, 
 the piano will be heard sending back the sound sung into it. The 
 full meaning of this experiment will be explained hereafter. 
 
 The resonance of strings may be visibly demonstrated to an 
 audience in the following manner. Tune two strings on the mono- 
 chord or any sound-board, in perfect unison, and upon one of them 
 place a rider of thin cardboard or paper. On bowing the other 
 string very gently, the rider will be violently agitated, and on in- 
 creasing the force of the bowing, will be thrown ofl. 
 
 In these experiments the sound-board plays an important, or 
 rather an essential part. Thus, in the above experiment with the 
 piano, the soimd waves from the larynx of the singer strike the 
 sound-board of the piano, setting up vibrations in it, which are 
 communicated through the bridges to the wires. It will be found 
 that in these experiments with stretched strings, such perfection of 
 unison as was necessary with the forks is not absolutely essential, 
 and the reason is obvious ; for in the first place, the medium by 
 which the vibrations are communicated in the former, viz., the 
 wood, is much more elastic than in the latter ; and secondly, the 
 light string or wire is much more easily set in vibration than the 
 heavy steel of the fork. A smaller number of impulses is therefore 
 sufficient to excite the string, and consequently such a rigorous 
 unison is not absolutely essential. As we might expect, however, 
 the more exact the imison, the louder is the sound produced. 
 
 In consequence of their small density, masses of enclosed air are 
 very readily thrown into powerful co-vibration. Strike a C tuning- 
 fork, and hold the vibrating prongs over the end of an open tube, 
 about 13 inches long and about an inch in diameter. The sound of 
 the fork, which before was very faint, swells out with considerable 
 
60 HAND-BOOK OF A00U8TI08. 
 
 intensity, Tte material of tlie tube is without influence 'on the 
 result. A sheet of paper rolled up so as to form a tube answers 
 very well. On experimenting with tubes of different lengths, it 
 will be found that the sound of the C fork is most powerfully re- 
 inforced by a tube of a certain length, viz., about 13 inches. 
 A slightly shorter or longer tube will resound to a smaller extent, 
 but Uttle resonance will be obtained, if it differs much from the 
 length given. Before reading the following explanation of this 
 phenomenon, the student should read over again the account given 
 in Chapter II of the propagation of sound. 
 
 Let A B, fig. 32, represent the tube, with the vibrating tuning-fork 
 above it. As the lower prong of the latter descends, it will press upon 
 the air particles beneath it, giving rise to a condensation, AC. 
 The particles in A C being thus crowded together, 
 press upon those below, giving rise to a condensa- 
 tion C D ; the particles in which, in their ttim, 
 press upon those beneath, thus transmitting the 
 wave of condensation to D E. In this way, the 
 condensation passes through the tube, and at length 
 reaches the end, EB. The crowded particles in 
 E B will now press outwards in aU directions, and 
 overshooting the mark, will leave the remainder 
 farther apart than they originally were ; that is, a 
 rarefaction will be formed in EB. But as there is 
 now less pressure in EB than in DE, the particles 
 of air in the latter space will tend to move towards 
 E B, and they themselves will be left wider apart 
 than before; that is, the rarefaction will be trans- 
 mitted from BE to E D, and in like manner wiU 
 pass up the tube tiU it reaches A 0. On arriving 
 here, as the pressure in A will be less than the 
 pressure outside the tube, the air particles will 
 „ ' „„ crowd in from the exterior and give rise to a con- 
 
 densation. Thus, to recapitulate; the downward 
 movement of the prong gives rise to a slight condensation in the 
 tube below; this travels down the tube to B, where it is reflected as 
 a pulse of rarefaction ; this, rushing back, on reaching A is changed 
 again to a pulse of condensation. Now if, while this has been going 
 on, the fork has just made one complete vibration, the lower prong 
 will now be coming down again as at first, and thus wiU cause an 
 increase in the degree of condensation. The same cycle of change 
 will take place as before, and wiU. recur again and again, the degree 
 
 B 
 
hesonanoe. 
 
 61 
 
 of condensation and rarefaction, that is, tlie intensity of tlie sound, 
 rapidly increasing to a maximum. To compare this witli the ex- 
 periment of the suspended weight : — the vibrations of the fork cor- 
 respond to the properly timed impulses, and the air in the tube to- 
 the suspended body : and, just as in that experiment, the essential 
 point was the proper timing of the impulses, so in this case the 
 essential matter is, that the downward journey of the condensation, 
 shall coincide with the downward movement of the prong. In 
 order that this coiucidence may occur each time, it is evident that 
 the wave must travel down and up the tube, in exactly the same 
 time that the fork makes one vibration ; that is, while the fork 
 makes one vibration the sound must travel twice the length of the- 
 tube. Moreover, every -vibration of the fork gives rise to one sound 
 wave ; therefore, in order that a tube open at both ends may give 
 its maximum resonance when excited by a fork, it must be half as- 
 long as the sound wave originated by that fork. 
 
 It -will be seen that a certain amount of resonance is obtained if 
 the tube is twice this length; for in that case, every alternate 
 descent of the prong -will coincide -with a condensation below, and 
 each alternate ascent with a rarefaction ; but such resonance -will 
 evidently be much feebler. For intermediate lengths, the fork -will 
 soon be in opposition to the pulses in the tube, and thus no- 
 resonance can result. 
 
 Tubes closed at one end are termed stopped tubes; with these the 
 case is somewhat different. Let A B, fig 33, represent a stopped 
 tube, the lower prong of the tuning-fork above, 
 being about to descend towards it. As we have 
 already seen, this gives rise to a condensation A 0, 
 which travels do-wn to B D. The air particles in 
 BD, ha-ving no way of escape, save backwards, 
 press upon those in D, and thus the condensation 
 is reflected back to D, and finally to AC. From 
 here some of the condensed particles escape into the 
 external air, leaving the remaining particles slightly 
 ■wider apart ; that is, a slight rarefaction is formed. 
 If while this has been taking place, the prong has 
 reached its lowest position and is just returning, 
 this movement will have the effect of increasing 
 the rarefaction. This latter will then be transmitted 
 down the tube to BD. On reaching this, there 
 will be less pressure in B D than in D 0, and j'jq. 33^ 
 
62 BAm>-BOOK OF ACOUSTICS. 
 
 consequently the air particles in the latter -will crowd into the 
 former, causing a rarefaction in D. In this way the rarefaction 
 is transmitted back to AC. As the pressure of the external air 
 will now be greater than that in A C, air particles from the former 
 will crowd in, forming a condensation iu A C. If at this moment, 
 the prong is a second time begimiing its descent, this condensation 
 will be increased, and the same series of changes will take place as 
 before. It is eyident, therefore, that the sound wave must make 
 two complete journeys up and down the tube, while the fork is 
 executing one vibration; that is, in order that a stopped tube when 
 excited by a fork, may give its maximum resonance, it must be J 
 as long as the sound wave originated by that fork. 
 
 We have already seen, that the length of the sound wave pro- 
 duced by a sounding body, may be ascertaiaed by dividing the 
 velocity of sound by the vibration number of that body; con- 
 sequently it is easy to calculate the length of tube, either open or 
 stopped, which wiU resound to a note of given pitch. The rule 
 evidently is : divide the velocity of sound by the vibration number 
 of the note; haU this quotient will give the length of the open 
 pipe, and one fourth will give the length of the stopped one. It is 
 necessary that the tube should be of moderate diameter, or the 
 rule will not hold good, even approximately. 
 
 The resonance of stopped tubes may easily be illustrated, by 
 means of glass tubes, corked, or otherwise closed at one end. On 
 holding a vibrating tuning-fork over the open end of a sufficiently 
 long tube, held with its mouth upwards, and slowly pouring in 
 water, the sound will swell out when the vibrating column of air is 
 of the requisite length, the water serving the purpose of gradually 
 shortening the column. For small forks, test tubes, such as are 
 used in chemical work, are very convenient. 
 
 It is by no means necessary that the resounding masses of air 
 should be in the form of a cylinder ; this shape was selected for the 
 sake of simpUcity in explanation. Almost any shaped mass of 
 enclosed air will resound to some particular note. Everyone must 
 have noticed, that the air in a gas globe, vase, &o., resounds, when 
 some particular sound is loudly sung near it. The following is an 
 interesting method of optically illustrating this phenomenon. A, 
 fig. 34, is a cylinder 3 or 4 inches in diameter, and 5 or 6 inches 
 long, with an open mouth, B. The other end is covered with an 
 elastic membrane, D, such as sheet india-rubber slightly stretched, 
 thin paper, or membrane. At is fastened a sUk fibre, bearing a 
 
RESONANCE. 
 
 63 
 
 drop of sealing wax, hanging do\ra like a pendulum against the 
 membrane. If now anyone places himself in front of the aperture, 
 and sings up and down the scale, on reaching some particular sound 
 the pendulum will be violently agitated, showing that the mem- 
 brane and the air within the bottle are vibrating in unison with 
 that note. 
 
 Another simple experiment of the same kind can be performed 
 with a common tumbler. Moisten a piece of thin paper with gum, 
 
 and cover the mouth of the tumbler with it, keeping the paper on 
 the stretch. "When dry cut away a part of the paper as seen in fig, 
 35. Put a few grains of sand, or any Kght substance on the cover, 
 
64 HANV-BOOK OF AG0U8TI08. 
 
 and then tilt up the glass, so that the sand wiU nearly, but not 
 quite, roll off. Having fixed the glass in this position, sing loudly 
 up and down the scale. On reaching a certain note, the co-vibration 
 of the air in the tumbler will set the paper and sand into violent 
 vibration. By singing a sound of exactly the same pitch as that 
 to which the air in the tumbler resounds, the sand may be moved 
 when the singer is several yards away. 
 
 The phenomenon of resonance is taken advantage of, in the con- 
 struction of resonating boxes. These are simply boxes (fig. 36), 
 generally made of wood, with either one or two opposite ends open. 
 
 Fig. 36. 
 
 and of such dimensions, that the enclosed mass of air will resound 
 to the tuning-fork to be attached to the box. Such boxes greatly 
 strengthen the sound of the fork, by resonance; the vibration being 
 communicated, through the wood of the box, to the air inside. It 
 may be remarked here, that the sound of a fork attached to a 
 resonance box of proper dimensions, does not last so long as it 
 would, if the fork were held in the hand and struck or bowed with 
 equal force ; for in the former case it has more work to do, in 
 setting the wood and air in vibration, than in the latter, and there- 
 fore its energy is sooner exhausted. 
 
 For forks having the vibration numbers in the first column of 
 the following table, boxes having the internal dimensions given in 
 the 2nd, 3rd, and 4th columns are suitable. The dimensions of the 
 first four are, for boxes open at one end only; those of the last four, 
 for boxes open at both ends. The fork is screwed into the middle 
 of the top of the bos. The dimensions are in inches. 
 
BE80NAN0E. 
 
 65 
 
 Vibration No. 
 
 Length. 
 
 Width. 
 
 Depth. 
 
 128 
 
 22-2 
 
 11-6 
 
 61 
 
 256 
 
 11-5 
 
 3-8 
 
 2 
 
 384 
 
 7-3 
 
 3-2 
 
 1-8 
 
 512 
 
 5-4 
 
 2-7 
 
 1-5 
 
 640 
 
 8-8 
 
 2-7 
 
 1-4 
 
 768 
 
 7-8 
 
 2-3 
 
 1-3 
 
 896 
 
 6-2 
 
 2-1 
 
 1-1 
 
 1024 
 
 5-5 
 
 1-9 
 
 1 
 
 A Resonator is a vessel of varying shape and material, and of 
 such dimensions, that the air contained in it resounds, when a note 
 of a certain definite pitch is sounded near it. Resonators are most 
 commonly constructed of glass, tin, brass, wood, or cardboard. 
 The forms most often met with are the cyhndrioal, spherical, and 
 conical. Their use is to enable the ear to distinguish a sound of a 
 certain pitch, from among a variety of simultaneous sounds, of 
 different pitches. The only essential, therefore, in the construction 
 of a resonator is, that the mass of air which it encloses shall resound 
 to the note which it is intended to detect. The best form for a 
 resonator designed for accurate scientific work is the spherical, as 
 it then reinforces only the simple sound to which it is tuned. The 
 
 Fig. 37. 
 
 spherical resonators employed by Helmholtz in his researches, were 
 of glass, and had two openings as shown in fig. 37. The opening 
 on the left hand serves to receive the sound waves coming from the 
 vibrating body, the other opening is funnel shaped and is to be in- 
 
 r 
 
66 
 
 HAND-BOOK OF AG0USTI08. 
 
 Berted in the ear. Helmholtz caused this nipple to fit closely into 
 the aural passage, by surrounding it with sealing wax, softening the 
 latter by heat, and then gently pressing it into the ear. The 
 resonator when thus used, has practically only one opening. In 
 using these instruments, one ear should be closed, and the nipple of 
 the resonator inserted in the other. On listening thus to simul- 
 taneous sounds of various pitches, most of them will be damped; 
 but whenever a sound occurs of that particular pitch to which the 
 resonator is tuned, it will be wonderfully reinforced by the co- 
 vibration of the air in the resonator. In this way, anyone, even 
 though unpractised in music, will readily be able to pick out that 
 particiilar soimd from a number of others. When, from the faint- 
 ness of the sound to be detected, or from some other cause, any 
 difficulty in hearing it is experienced, it is of advantage to 
 alternately apply the resonator to, and withdraw it from the ear 
 
 A resonator, which is capable of being tuned to any pitch within 
 the compass of rather more than an octave, has been used for some 
 years by the writer. It is composed of three tubes of brass, sliding 
 closely within one another. The innermost, fig. 38a, which is 
 
 JL 
 
 FiQ. 38. 
 about four inches in length, and an inch or more in diameter, is 
 closed at one end by a cap which is screwed on to the tube. In the 
 centre of this cap is an aperture, about half an inch in diameter, 
 which is closed by a perforated cork, through which passes a short 
 piece of glass tube, the end of which is fitted to the ear. The 
 resonator is thus a closed one, and its length can be increased by 
 means of the sliding tubes, b and c (which are each about 4 inches 
 long), from about 4 inches to 12 inches. It is best tuned approxi- 
 mately, by first calculating the length of stopped tube corresponding 
 to a certain note, according to the method explained in the present 
 
BESONANOE. 67 
 
 chapter. Starting with the resonator at this length, by gradually 
 increasing or diminishing its length, while that note ia being 
 sounded on some instrument, a point of maximum resonance wiU 
 soon be obtained. In this way, the length of the resonator for all 
 the notes within its range can be ascertained. The names of these 
 notes may be conveniently written on a slip of wood, or engraved 
 on a strip of metal, each one at a distance from the end equal to 
 the length of the resonator, when tuned to the corresponding note- 
 Then, in order to adjust the resonator to any note, it is only 
 necessary to place the slip inside it, and gradually lengthen or 
 shorten till the open end is coincident with the name of that note, 
 as engraved on the slip. It is of great advantage to have two such 
 resonators, both similarly tuned, and simultaneously apply one to 
 each ear. If only one be used, the other ear should be closed. 
 
 For sounds high in pitch, glass tubes cut to the proper lengths, are 
 very convenient. They may be made to taper at one end for inser- 
 tion in the ear, or in the case of very high notes, left just as they 
 are, and used as open tubes, being held at a small distance from 
 the aural passage. It may be remarked here, that the tube of the 
 ear is itself a resonator. The pitch of the note to which it is tuned, 
 will of course vary in different persons, and may in fact be difierent 
 for the two ears of the same person ; it generally lies between Q-' = 
 3,072 and E^ = 2,360. 
 
 Summary. 
 
 Resonance or Oo-mbration is the name given to the phenomenon 
 of one vibrating body imparting its vibratory movement to another 
 body, previously at rest. 
 
 To obtain the maximum resonance two conditions are essential : 
 
 (1) The two bodies must be in exact unison; that is to say, they 
 must be capable of executing precisely the same number of 
 vibrations in the same time. 
 
 (2) A certain period of time must be allowed for the exciting 
 body to impress its vibrations on the other. 
 
 The phenomenon of resonance may be illustrated by means of 
 tuning-forks, strings, &c., but partially confined masses of air are 
 the most susceptible. 
 
 A Resonance Box is usually constructed of wood ; it may be open 
 at one or both ends, and must be of such dimensions that the en- 
 closed mass of air will vibrate in imison with the tuning-fork to be 
 apphed to it. 
 
68 HAND-BOOK OF ACOUSTICS. 
 
 A Besonator is an open yessel of glass, metal, cardboard, or other 
 material, of suoli dimensions, tliat the mass of air contained in it 
 resounds to a note of a certain pitch. Its use is, to assist the 
 ear in discriminating a sound of this particular pitch, from a 
 number of others at different pitches, all sounding simultaneously. 
 
 In order that a column of air, in a cylindrical tube open at both 
 ends, may -vibrate in unison ynth a given sound, the length of the 
 tube must be approximately one half the length of the correspond- 
 ing sound ■wave. 
 
 If the tube be closed at one end, its length must be one-fourth 
 that of the soimd -wave. 
 
 In both cases the diameter of the tube should not exceed ono- 
 sixth the length. 
 
69 
 
 CHAPTER VIII. 
 
 Ojt the QrAiiiTT OB MusicAi Sounds. 
 
 Hitherto we have treated only of simple sounds, that is to say, each 
 Bomid has been considered to be of some one, and only one particu- 
 lar pitch. This is, however, far from being the case with the great 
 majority of musical sounds we hear. If such sounds are attentively 
 examined, almost all of them will be found to be compound ; that 
 is, each individual sound will be found to really consist of a 
 number of simple sounds of different pitch. Those readers, who 
 are not already practically cognizant of this fact, are strongly 
 recommended to convince themselves of it, by experiment, before 
 proceeding further. Some persons, both musical and unmusical, 
 find great difficulty in distinguishing the simple elementary sounds, 
 that form part of a compound tone. Those who experience any 
 such difficulty, will find it useful to go carefully through the 
 following experiments. 
 
 Strike a note on the lower part of the key-board of a pianoforte, 
 eay C|, in the Bass Clef. As the sound begins to die away, the 
 upper octave of this note may, with a little attention, be readily dis- 
 tinguished. If the listener experiences any difficulty in recognising 
 it, he win find it useful to lightly touch the above (that is, the 
 sound he is listening for) and let it die away before striking the Bass 
 Ci. If he does not then succeed, a resonator tuned to the expected 
 sound, or, better still, two such resonators, one for each ear, shoiild 
 be used. By alternately applying these to, and withdrawing them 
 from, the ears, even the most untrained observer cannot but detect 
 the wished-for sound. Next, strike the same Bass C| as before, but 
 
70 EAND-BOOK OF ACOUSTICS. 
 
 direct the attention to the G in the Treble staff an octave and a 
 fifth above it. This is generally more easily recognised than the 
 preceding, and is usually equally loud. When this has been clearly 
 heard, strike the same note as before, and listen for the C in the 
 Treble staff, two 'octaves above it. More difficulty will perhaps be 
 experienced in detecting this, but by the aid of properly tuned 
 resonators, it will be heard sounding with considerable intensity. 
 The next two sounds are better perceived as the tone is dying away; 
 they are the E', two octaves and a major third above the sound 
 struck, and the G' two octaves and a fifth above. 
 
 The student should vary this experiment by taking other notes, 
 and listening to their constituent elements. These latter will 
 always be found occurring in the above order : thus if the D in the 
 Bass clef be struck the following sounds may be heard : — 
 
 S^ si 
 
 - f 
 
 Key D. d 
 
 No sound intermediate in pitch between any of these will be 
 detected. Further, these sounds are not aural illusions, but have 
 a real objective existence, for they are capable of exciting corre- 
 sponding sounds in other strings, by resonance. Thus, having softly 
 pressed down any key, say (d), without sounding it, so as to raise 
 the damper from the wires, strike sharply the octave below (d|) and 
 after a second or two, raise the finger from tbia latter, so as to damp 
 its wires; the note (d) wiU be plainly heard, the corresponding 
 wires having been set in vibration by resonance. This experiment 
 will be found successful with all the constituent parts given above. 
 For example, press down the PiJ (pii) on the top line of the Treble 
 staff, but without sounding it, and strike sharply the Dj (d|) two 
 octaves and a major third below. Eaise the finger from the latter 
 after a second or two, and the wires of the former will be heard 
 giving forth the Fijf (pi'). 
 
 All the constituent elements of a compound tone given above, 
 are very prominent on an American Organ, and still more soon the 
 Harmonium. They will be found to be in exactly the same order, 
 
ON THE QUALITY OF MUSICAL SOUNDS. 71 
 
 no sound intermediate between those giyen will occur. With the 
 aid of resonators, it will be easy to detect still higher constituents 
 than those mentioned above. It need scarcely be said, that, in the 
 experiments with these instruments, only one reed should be 
 vibrating at a time. 
 
 The constituent elements in the compound tones of the voice are 
 more difi&cult to detect. It is advisable to begin with a good basa 
 voice. All the constituents given above, may be heard after a little 
 practice with the resonators. They are louder in some vowel 
 sounds, as will be seen hereafter, than in others ; the " a" sound 
 as in " father," and the " i" as ia " pine," are favourable ones to 
 experiment with. After a little practice, the ear becomes practised 
 in this analysis of sounds, and the resonators may be dispensed 
 with to a great extent. 
 
 Before proceeding further, it will be best to explain the terms 
 that are used in speaking of these constituents of a compound tone. 
 On one system of nomenclature, the lowest element of a compound 
 tone is termed the Fundamental ; the next one (an octave above), 
 the First Overtone ; the next (a Fifth above that), the Second Over- 
 tone ; the next (a Fourth above that), the Third Overtone ; and so 
 on. The constituent elements are also termed Partials ; the lowest 
 being termed the First Partial ; the next, the Second Partial ; the 
 next, the Third Partial; and so on. Thus, taking (d|) as the 
 Fundamental or First Partial, the others will be. 
 
 6th Overtone 
 
 s> 
 
 6th Partial 
 
 4th 
 
 ni - 
 
 5th „ 
 
 3rd 
 
 di 
 
 4th „ 
 
 2nd 
 
 s 
 
 3rd .. 
 
 1st 
 
 d 
 
 2nd „ 
 
 Fundamental Tone 
 
 di 
 
 1st „ 
 
 Bach of these partials or overtones is a simple tone, that is, a sound 
 of definite pitch, which cannot be resolved into two or more sounds 
 of different pitch. A compound tone is a sound consisting of two 
 or more simple tones. 
 
 By means of resonators, many higher partials than the six 
 abeady mentioned can be detected. The following hst contains the 
 first twenty. The first column gives the order of the partials ; the 
 second and third, their names, calling the fundamental Cg and dg 
 respectively ; and the fourth gives the ratios of their vibrational 
 numbers, to the fundamental, this latter being taken as 1. 
 
72 
 
 HAND-BOOK OF ACOUSTICS. 
 
 Oedee. 
 
 Name. 
 
 Name. 
 
 ViB. 
 
 Eatio. 
 
 XX 
 
 El 
 
 rf 
 
 20 
 
 XTX 
 
 
 
 . . . 
 
 19 
 
 xvm 
 
 Di 
 
 ri 
 
 18 
 
 xvn 
 
 
 
 17 
 
 XVi 
 
 CI 
 
 d' 
 
 16 
 
 XV 
 
 B 
 
 t 
 
 15 
 
 XIV 
 
 
 
 
 14 
 
 XTTT 
 
 
 
 13 
 
 xn 
 
 G 
 
 s 
 
 12 
 
 XI 
 
 
 
 11 
 
 X 
 
 E 
 
 n 
 
 10 
 
 IX 
 
 D 
 
 r 
 
 9 
 
 VTTT 
 
 C 
 
 d 
 
 8 
 
 vn 
 
 
 
 
 7 
 
 VI 
 
 G| 
 
 S| 
 
 6 
 
 V 
 
 E, 
 
 n. 
 
 6 
 
 IV 
 
 C| 
 
 d, 
 
 4 
 
 in 
 
 G, 
 
 8, 
 
 3 
 
 n 
 
 0, 
 
 a. 
 
 2 
 
 I 
 
 C3 
 
 ds 
 
 1 
 
 Those un-named do not coincide exactly with any tone of the modem mnalcal scale 
 vn IB approximately Bb| or tai. 
 
 It will be seen on inspecting the above table, that the partials 
 occur according to a certain fixed law; viz., the vibrational numbers 
 of the partials, commencing at the fundamental, are proportional 
 to the numbers 1, 2, 3, 4, 5, 6, &o. Thus, theoretically, the above 
 table may be indefinitely extended. Practically, the first twelve or 
 more may be verified with a harmonium and a couple of resonators ; 
 those above, are best observed on a long thin metaUio wire, or an 
 instrument of the trumpet class, in which the higher partials are 
 very prominent. 
 
 By experim.ents, similar to those which have been recommended 
 in the case of the piano, it is easy to convince oneself, that nearly 
 all the tones produced by stringed and wind instruments, are com- 
 
ON THE QUALITY OF MUSICAL SOUNDS. 73 
 
 pound, and that the partials of which these compound tones consist, 
 belong to the series giyen above ; that is to say, though any one or 
 more of these partials may be absent, no sound of any other pitch, 
 than those given above, ever makes its appearance. 
 
 Instruments which produce only simple tones are comparatively 
 rare. A tuning-fork, when struck on a hard substance, or when 
 carelessly bowed, gives a compound tone, consisting of a funda- 
 mental and one or two very high overtones. When, however, it is 
 mounted on a resonance box of proper dimensions and carefully 
 bowed, the fundamental tone is so strengthened by resonance, that 
 the resulting sound is practically free from overtones. The tones of 
 flutes and of wide stopped organ pipes gently blown, and the 
 highest notes of the piano, are nearly simple. 
 
 The relative intensities of the partials forming a compound tone 
 vary very greatly in different instruments, and even in different 
 parts of the same instrument ; thus, on the lower part of a piano, 
 the third partial is generally louder than the fundamental, while on 
 the upper part, it is very much softer. In some voices, again, and 
 with some vowel sounds, the third partial is painfully prominent, 
 while in other voices and with other vowel sounds, it is only 
 detected with difficulty. As a general rule, the farther the partial 
 is from the fundamental, the less is its intensity. Taking the sound 
 from a weU-bowed violin as a model of tone, Helmholtz has given 
 the following approximation to the relative intensities of its partials, 
 the intensity of the fundamental being taken as 1. 
 
 Pabtiai. 
 
 iKTElSfSITY. 
 
 VI - 
 
 1 
 
 3S 
 
 V - - 
 
 1 
 25 
 
 IV 
 
 1 
 16 
 
 ni - - 
 
 1 
 
 n . 
 
 1 
 4 
 
 I - - 
 
 1 
 
 Considering the loudness of the partials in many instruments, it 
 may be a matter of surprise to some, that they are not more 
 easily recognised. It should be remembered, however, that, as the 
 partials of a compooind tone all begin together, and usually continue 
 with unvarying relative intensities tiU. the tone ends, when they all 
 
74 HAND-BOOK OF ACOUSTIOB. 
 
 terminate together, the ear has always been acooBtomed to consider 
 it as a whole. Musical people especially, having been in the habit of 
 directing all their attention to a tone as a whole, are often incapable 
 of recognising the constituent parts, until their attention is directly 
 called to them ; just as a person, after having had a clock ticking in 
 his room for some time, ceases to notice the ticking unless some- 
 thing attracts his attention especially to it. Again, when one is 
 thinking deeply, a remark made by another person is often not 
 perceived : the nerves of hearing are doubtless excited, but the 
 attention not being aroused, the sound is not perceived. When 
 anyone has once become accustomed to listen for overtones, there 
 is no difficulty whatever in hearing them ; in fact, they sometimes 
 force themselves upon the ear of the practised listener when not 
 wanted. 
 
 Most of the foregoing facts concerning partials have been 
 known for centuries, but the phenomenon was regarded as little 
 more than a curiosity, trntil Hehnholtz proved that the quality of 
 a musical tone depended upon the occurrence of partials. Before 
 going into this matter, it wiLL be necessary to show more exactly 
 what is meant here by quality. 
 
 Many musical tones are accompanied by more or less noise ; thus, 
 the tone of an organ-pipe is adulterated, as it were, more or less, by 
 the noise of the wind striking the sharp edge of its embouohere ; 
 the tone from a violin is mingled, more or less, according to the 
 skill of the player, with the scraping noise of the bow against the 
 strings ; the tone from the human voice is accompanied, more or 
 less, with the noise of the breath escaping. Again, the sounds of 
 some instruments differ from those of others, in that their intensities 
 vary in different but regular ways. Thus in the piano and harp, 
 the tones, after the wires are struck, immediately decrease regularly 
 in intensity, till they die away ; while on the organ they continue 
 with unvarying intensity, as long as they sound at all. All such 
 peculiarities as the labove are not included under the term quality, 
 as we use it here. The following may be taken as a formal 
 definition of the term quality, as employed below. If two tones 
 perfectly free from noises, of precisely the same pitch, and of equal 
 intensities, differ in any way from one another, then all those 
 respects in which they differ are comprised under the term quality. 
 Using the term quality in this sense, Helmholtz has shown that ; 
 The Quality of a compound tone depends upon the number, order, and 
 relative intensities of its constituent partials. 
 
ON THE QUALITY OF MUSICAL SOUNDS. 75 
 
 Before stating the various methods by which this proposition has 
 been proved, it may be advisable to explain its meaning a little 
 more fully. In the first place, the proposition asserts, that the 
 quality of a tone varies with the number of its component partials ; 
 thus if one tone consists of three partials, another of four, and 
 another of six ; then, each of these three tonfes will have a different 
 quality from the other two. In the second place, the proposition 
 declares that the quality of a tone varies with the order of its con- 
 stituent partials ; for example, suppose we have three tones, the 
 first consisting, say, of the 1st, 2nd, and 3rd partials, the second of 
 the 1st, 3rd, and 5th, and the third of the 1st, 3rd, and 6th, then 
 each of these three tones will have a different quality from the 
 other two. In the above cases, we have supposed the partials to be 
 of the same relative intensities in each case. If, however, the 
 relative intensities vary, the proposition affirms that the quality 
 will vary also. Thus to take a simple case, suppose we have two 
 tones each consisting of the 1st and 2nd partials, and that the two 
 fundamentals are of the same intensity ; then if the second partial 
 of the one differs in intensity from the second partial of the other, 
 the proposition asserts, that the quality of the one tone wiU. differ 
 from that of the other. 
 
 On reading the above propositions the following question at once 
 suggests itself. Is the alleged cause, viz., the variation in the 
 number, order, and relative intensities of the partials, sufficient to 
 account for the observed eflect, viz., the variation in quality ? The 
 variations in quality of tone are infinite ; therefore, if the proposition 
 be true, the variations in the number, order, and relative intensities 
 must be infinite also. Now the number of variations in number 
 and order of partials although very great are practically limited ; 
 but it is obvious that the relative intensities of the partials may 
 vary in an infinite num.ber of ways, and thus the above question 
 must be answered in the affirmative. In the next place, it is easy 
 to see that if the proposition be true, simple tones can have no 
 particular quality at all, they must all resemble one another in this 
 respect, from whatever source they come. On trial this will be 
 found to be the case. We have already observed that these tones 
 can be approximately obtained from tuning-forks mounted on 
 appropriate resonance boxes, wide-stopped organ pipes and flutes 
 gently blown, and the highest notes of the pianoforte. The tones 
 from these four sources caimot be compared together very well, for 
 the reason already referred to : the first mentioned being almost 
 
76 HAND-BOOK OF ACOUSTICS. 
 
 pure, the second and third being accompanied by characteristic 
 noises, and the fourth having its peculiar variation of intensity; but 
 they all agree in being gentle and somewhat dull. Moreover, they 
 can be strictly compared in their own class ; thus, for example, the 
 tones from tuning-forks are all aUie in quality; in selecting a 
 tuning-fork, no one ever thinks of the quality of its tone. 
 
 There are two methods, by which the important proposition now 
 under consideration may be proved, — the analytical, and the 
 synthetical. The process in the former case is to take two sounds, 
 which differ in quality, and by analysing them into their constituent 
 partials (with or without the aid of resonators), show that these 
 latter differ in the two cases, either in number, order, or in their 
 relative intensities. Thus, if the student analyses a tone of rich 
 and full quality, he will find the first six partials tolerably well 
 developed, while on the other hand, in a tone of poor or thin 
 quality, he will find most of them absent, or of much less intensity. 
 Again, the metallic or brassy quality (as it is termed) of instruments 
 of the trumpet class, he will find to be due to the clashing of very 
 high partials, which are very prominent in such instruments, and 
 which, as will be seen by referring to the table on page 72, lie very 
 close together. As another illustration, the peculiar quality of the 
 tones of the clarionet, may be accounted for, by the fact, that only 
 the odd partials, the 1st, 3rd, oth, &c., will be found to be present 
 in the tones of this instrument. The student will find, in the 
 analysis of the vowel sounds, a very instructive series of experi- 
 ments. The differences in these sounds, must be simply difEerenoea 
 in quality, according to our definition ; and thus if the proposition 
 under discussion be true, we ought to find corresponding differences 
 in the number, order, or relative intensities of the partials present 
 in the vowel sounds. On trial, this will be found to be the case. 
 If, for example, the "a" as in " father " be sounded by a good 
 voice, aU the first six partials may be easily heard; but if the same 
 voice gives the " oo " sound, scarcely anything but the fundamental 
 win be detected. 
 
 The general process, in the synthetical method of proof, is, to 
 take simple tones of the relative pitch of the series of partials, and, 
 by combining these together in different numbers and orders, and 
 with different intensities produce different qualities of tone. The fiirst 
 difficulty here, is to procure perfectly simple tones. These are best 
 obtained from tvming-forks fitted wili. suitable resonating boxes. An 
 elementary experiment can be conducted as follows. Select two 
 
ON THE QUALITY OF MUSICAL SOUNDS. 77 
 
 tuning-forks, differing in pitch by an exact octave, and mount 
 them on resonating boxes of the proper dimensions ; set the lower 
 one vibrating, by bowing it with a violin or double bass bow, and 
 note the dull yet gentle effect of the simple tone produced. Now 
 bow both the forks rapidly one after the other; the two simple 
 tones will soon coalesce, and will sound to the ear as one tone, of 
 the pitch of the lower one, but of much brighter quality than 
 before. The effect of the higher fork, that is, of the second partial, 
 will be strikingly seen, by damping it after both have been vibrating 
 a second or two ; the return to the original dull simple tone is very 
 marked. This experiment may be varied very greatly, by the aid 
 of four or five forks tuned to the first four or five partials. The 
 following eight forks form a very serviceable series for these 
 experiments. 
 
 c 
 
 = 1024. 
 
 Bbi 
 
 nearly 896. 
 
 Gi 
 
 = 768. 
 
 El 
 
 = 640. 
 
 CI 
 
 = 512. 
 
 Q 
 
 = 384. 
 
 C 
 
 = 256. 
 
 C| 
 
 = 128. 
 
 Of course, the effect produced by these forks is only an approxi- 
 mation to the effect produced by the real partials, for, in the first 
 place, the forks cannot very easUy be all excited at the same instant ; 
 and again, their intensities can only be regulated in a very rough 
 way. In an arrangement devised by Helmholtz for investigating 
 the vowel sounds, these two difficulties were removed by the use of 
 electro-magnets for exciting the forks, and by employing resonators 
 at different distances, and with moveable openings, to regulate their 
 intensities. 
 
 The way in which a tuning-fork is excited by an electro-magnet 
 will be understood by a reference to fig. 39. Let A and B be the 
 poles of an electro-magnet, and C and I) the ends of the prongs of 
 a tuning-fork between them. If now a current be sent through 
 the electro-magnet, the poles A and B vrill attract C and D. Now 
 if the current be stopped, A and B will cease to attract the prongs, 
 which wiU therefore move towards one another again in consequence 
 of their elasticity. Let the cui-rent again pass, and and D will 
 again be attracted. If we can thus alternately pass and stop the 
 current, every time the prongs move forward and backward, the 
 
78 
 
 HAND-BOOK OF ACOUSTICS. 
 
 fork will continue to yibrate. Thus, if the 
 
 1 ^ I current be intermittent, and the number of 
 I I interruptions per second be the same as 
 
 the vibration number of the fork, the vibra- 
 tion of the latter mil be continuous. If 
 the electro-magnet be powerful enough, 
 the fork will also continue in motion, 
 though the number of interruptions per 
 second be ^, ^, J, &o., of the vibration 
 number of the fork. 
 
 These interruptions of the current can 
 
 be brought about by another fork, the 
 
 X vibration number of which is either the 
 
 Fio. 39. same, or J, ^, ^, &c., of the first one. Let 
 
 (fig. 40) represent this second fork, and 
 
 A, B the poles of an electro-magnet. To the upper prong a small 
 
 wire is fastened which just dips into a little mercury con 
 
 E 
 
 "iz: 
 
 Fio. 40 
 
 tained in a cup D, when the fork is at rest. In this position the 
 current from one end of the battery passes to the cup D, thence 
 through the fork and the wire B to the electro-magnet A B, and 
 then by wire F back to battery. But directly the current passes, 
 the poles A and B attract the prongs, and thus the wire attached 
 to the upper one is lifted out of the mercury in the cup D. The 
 current is thus broken, A and B cease to attract, and the prongs 
 return. But in so doing the wire again comes into contact with the 
 mercury, the current is again set up, and A and B again attract 
 the prongs. This alternate making and breaking of the cii'cuit 
 will thus be kept up, and the motion of the fork is rendered 
 continuous. 
 
 If now the current from B (fig. 40) instead of passing directly 
 back to the battery, be first led thi-ough the electro-magnet of the 
 
ON THE QUALITY OF MUSICAL SOUNDS. 
 
 79 
 
 fork in fig. 39, it mil be seen from what lias been said above, that 
 this fork also will be set in Tibration. Further, the current before 
 returning to the battery may be led through several such electro- 
 magnets, furnished with tuning-forks ; and if the latter have vibra- 
 tion mimbers which are the same, or any multiples of that of the fork 
 in fig. 40, they will be kept in vibration also. Now, the vibration 
 numbers of overtones are multiples of the vibration number of the 
 fundamental, and therefore only one such fork as that of fig, 40 is 
 necessary, in exciting any number of forks such as that of fig. 39, 
 if the latter are tuned to the series of partials and the former is in 
 unison with the fundamental. 
 
 Fig. 41 shows the method by which Hehnholtz obtained variations 
 in intensity in his apparatus. D represents one of the tuning- 
 
 Fig. 41. 
 
 forks, kept in vibration by an electro-magnet, which is not shown 
 in the figure. A is a resonator of suitable dimensions, the aperture 
 of which can be closed by the cap, 0. "When thus closed the 
 sound of the fork is almost inaudible, but, on gradually opening 
 the aperture, the sound comes out with increasing loudness; the 
 maximum being reached when the aperture is quite uncovered. 
 In the figure, the resonator is shown, for the sake of distinctness, 
 at a distai.ce from the fork ; when in use it may be pushed up as 
 close to the fork as desired, by means of the stand working in the 
 groove below. The cap covering the aperture of the resonator is 
 connected, by means of levers and wires as seen in the figure, witt 
 
80 HAND-BOOK OF AGOUSTICU. 
 
 one of the ten keys of a key-board, the other nine of which are in 
 communication with nine other similar resonators each tuned to its 
 own fork. These ten forks are of pitches corresponding to the ten 
 partials of a compound tone. 
 
 Haying thus the power of varying the number, order, and 
 relative intensities of these ten simple tones, compound tones of 
 any quality can be, as it were, built up. 
 
 In Chapter II we found that there are three elements that deter- 
 mine a sound wave, viz., its length, amplitude, and form. We 
 have since found, that it is upon the length of a sound wave that 
 the pitch of the resulting sound depends, and upon the amplitude 
 that its intensity depends. The form of the wave being the only 
 property remaining, it follows that it is upon this element that the 
 quality of the sound depends. We have now to study the connection 
 between these two. 
 
 Fig. 42. 
 
 The simplest vibrational form is that made by a common pen- 
 dulum, and is termed a pendular vibration. Suppose the bob of 
 
ON TEE QUALITY OF MUSICAL SOUNDS. 81 
 
 such a pendulum to be at the higliest point of its swing to the left; 
 as it swings to the right, its rate of motion becomes more and 
 m.ore rapid till it reaches its lowest position; during the latter 
 halt of its swing it gets slower and slower, till it reaches its 
 extreme position on the right, when after a momentary rest, it 
 begins its journey back. The first half of the journey is the 
 exact counterpart of the second, the motion being accelerated in 
 the first half at exactly the same rate that it is retarded in 
 the second. It is easy to construct a pendulum, that shall 
 write a record of its own motion, and thus to obtain a pictorial 
 representation of pendular vibration. Kg. 42 shows a form of the 
 instrument, which the student will have no difiBculty in making for 
 himself. The funnel below which rests in a ring of lead, is flUed 
 with sand. As it swings backwards and forwards, the sand escapes, 
 leaving a straight ridge of sand on the board below, as seen at (06), 
 If, however, the board be at the same time uniformly moved along 
 from A to A', the sand will be deposited along the wavy track seen 
 in the figure. Such a tracing of a pendular vibration is seen on a 
 larger scale in fig. 43. On comparing this tracing with that made 
 
 Fio. 43., 
 
 by a tuning-fork as described in Chapter TV it is found that they 
 are of the same character : that is, a tuning-fork executes pendular 
 vibrations. But a tuning-fork, as we have seen, gives simple tones. 
 It seems, therefore, that simple tones are produced by pendular 
 vibrations. Further experiment and observation confirm this, and 
 we may take it as proved, that simple tones are always the result 
 of pendular vibrations. 
 
 Now a compound tone is made up of partials : and partials are 
 simple tones. Further, simple tones are due to pendular vibrations. 
 It follows, therefore, that compound tones are due to combinations 
 of pendular vibrations. 
 
 How are these pendular vibrations simultaneously conveyed 
 through the air ? Throw a stone into a piece of stiU water, and 
 while the waves to which it gives rise are travelling outwards, 
 throw another stone into the water. One series of waves will be 
 
82 
 
 HAND-BOOK OF ACOUSTICS. 
 
 Been to pass undisturbed through the other series. Let the tracing 
 AaBeO, fig. 44 represent a wave of the first series, and AdBbG 
 one of the second series, and let the dotted straight line, AgBhO, 
 
 Fio. 44. 
 
 represent the surface of stiU water. In the first place consider the 
 motion of a particle of water at g. The first wave would cause the 
 drop to rise to a, and the second if acting alone would raise it to d. 
 According to the fundamental laws of mechanics, each force will 
 have its due eflect and the drop will rise to the height k such that 
 ag ^ dg ^ gk. Again, the drop at h, if under the influence of the 
 first wave alone, would rise to e, but the second wave would depress 
 it to 6. Under these two antagonistic forces it falls to I, such that 
 hi = hh — he. By ascertaining in this way, the motion of each 
 point along the wave, we can, by joining all these points, determine 
 the form of the compound wave made up of these two elementary 
 ones. 
 
 The same mechanical laws apply to sound waves as to water 
 waves. Thus if the two tracings A and B, in fig. 45, be the asso- 
 ciated wave forms of two simple tones at the interval of an octave, 
 then 0, constructed from these, in the way just explained, will be 
 the associated wave form produced by their union ; that is, C is the 
 associated wave form of a compound tone consisting of the first 
 two partials. We have here supposed that A and B commence 
 together, that is, in the same phase. If we suppose the curve B to 
 be moved to the right until the point (1) falls under the point (2), 
 and then compound these waves, we obtain a different resultant 
 wave form, D. If B were displaced a little more to the right, 
 another wave form would result. Hehnholtz has shown experi- 
 mentally that, when two sound waves are compounded in different 
 phases, although waves of different forms are obtained, yet no 
 difference can be detected in the resulting sounds ; that is, the 
 sounds corresponding to the forms C and D would be exactly alike. 
 
ON THE QUALITY OF MUSICAL SOUNDS. 83 
 
 Fio. 45. 
 
 This fact seems to show, that the ear has not the faculty of perceiving 
 compound tones as such, but that it analyses them into their con- 
 stituent partials. If this be the case, it follows that aU compound 
 sounds are formed by the union of tyro or more simple tones. No-w- 
 it has been pro-ved by iPourier, that there is no form of compound 
 vrscve which cannot be compounded out of a number of simple 
 waves, whose lengths are inversely as the numbers 1, 2, 3, 4, 5, &c. 
 Musically, this proposition means, that every compound musical 
 sound may be resolved into a certain number of simple tones, 
 whose relative pitch follows the law of the partial tone series. 
 
 According to the theory of Helmholtz, briefly referred to in 
 Chapter III, this analysis is eflected by the ear as follows: — 
 " When a compound musical tone is presented to the ear, aU. those 
 elastic bodies (that is, the radial fibres of the basilar membrane, 
 and the corresponding arches of Corti) will be excited, which have 
 a, proper pitch corresponding to the various individual simple tones 
 contained in the whole mass of tone, and hence by properly direct- 
 ing attention, aU the individual sensations of the individual simple 
 tones, can be perceived." 
 
 S-DMMAEY. 
 
 A Simple Tone is one that cannot be analysed into two or more 
 Bounds of different pitch. 
 
84 HAND-BOOK OF A00U8TI08. 
 
 A Compound Tone or Clang is a tone which is made up of two or 
 more Simple Tones of difEerent pitch. 
 
 Almost all the sounds employed in modem music are compound. 
 
 The Simple Tones that form part of a Compound Tone are termed 
 Partiah or Partial Tones. The lowest Partial of a Compound Tone 
 is termed the First Partial ; the next aboye, the Second; the next, 
 the Third ; and bo on. 
 
 The First Partial of a Compound Tone is also called the 
 Fundamental Tone, and the others, Overtones; thus the Second 
 Partial is termed the First OTertone ; the Third Partial, the 
 Second Overtone ; and so on. 
 
 In almost aU the Compound Tones used in modem music, the 
 vibration numbers of the Partial Tones, starting with the Funda- 
 mental, are in the ratios of 
 
 1 : 2 : 3 : 4 . 5 : 6 : 7 : &c. 
 
 Any one or more of these paxtials, however, may be absent in 
 any particular tone. Thus, for example, the even numbered 
 partials are absent in the tones of cylindrical stopped pipes. 
 
 The Relative Intensities of the partials of Compound Tones vary 
 almost infinitely. As a general, but by no means xmiversal rule, 
 the higher the order of the partial, the less is its intensity ; that is 
 to say, the first partial is generally louder than the second ; the 
 second louder than the third, and so on. 
 
 Approximately Simple Tones may be obtained from carefully 
 bowed tuning-forks mounted on suitable resonance boxes ; or from 
 flutes and wide stopped organ pipes, gently blown. 
 
 If two pure Musical Tones are of the same pitch and of equal 
 intensities, all those respects in which they yet differ, are included 
 under the term Quality or Timbre. 
 
 The Quality of a Compound Musical Tone depends upon the 
 Number, Order, and Relative Intensities of its constituent partials. 
 Helmholtz has demonstrated this proposition by the Analysis 
 and Synthesis of Compound Tones. 
 
 Just as pitch depends on wave length, and intensity on amplitude ; 
 so the quality of » tone (that is, the number, order, £md relative 
 intensity of its partials) depends on wave form. 
 
 A Simple Tone is the result of pendvlar sound waves or pendular 
 vibrations, that is, of vibrations similar to those of a simple 
 pendulum. 
 
ON THE QUALITY OF MUSICAL SOUNDS. 85 
 
 A Compound Tone is due to the combination of two or more 
 
 pendular vibrations, or waves. 
 
 Every Compound Tone may be resolved into a certain number 
 
 of Simple Tones, wbose relative pitch, follows the law of the partial 
 
 series. 
 
 Similarly, every compound wave maybe resolved into a certain 
 
 number of simple pendular waves, whose lengths are in the 
 
 ratios of 
 
 1 1111, 
 1 - : - : - : - &o. 
 
 2 3 4 5 
 
 The form of the sound wave, therefore, determines the number, 
 order, and relative intensities of the partials, that is, the quality of 
 the resultant sound. On the other hand, given the quality, it is 
 not possible to determine the corresponding wave form, since for 
 any one such quaUty, there is an infinity of wave forms, due to the 
 infinite number of relative positions or phases in which the con- 
 stituent pendular waves may start. 
 
 The ear does not perceive a Compound Tone as such, but analyses 
 it into its constituent pendular waves, each of these latter producing 
 the sensation of a Simple Tone at its own particular pitch and 
 intensity. 
 
86 
 
 CHAPTER IX, 
 
 On the Vibrations or Steinqs. 
 
 A. STEINGED instruineiit consists essentially of three parts, viz : — 
 the string, catgut, or wire, to be set in Tibration ; some means of 
 setting up this vibration; and a sound-boexd, or other resonant 
 body, by means of whioh the vibratory movement is to be trans- 
 mitted to the air. 
 
 The means by which strings are set in vibration vary in different 
 instruments. They may be struck by a hammer, as in the piano- 
 forte ; bowed, as in instruments of the violin class ; set in mofi^on 
 by a current of air, as in the case of the .Solian harp ; or plucked, 
 like the harp and zither. In this last case, the plucking may be 
 done with the finger tips, as in the case of the harp and guitar, or 
 by means of a quill or plectrum, as in the zither and harpsichord. 
 In the case of bowed instruments, the particles of resin with which 
 the bow is rubbed, catch hold of the portion of the string with which 
 they are in contact, and pull it aside; its own elasticity soon sends it 
 back, but being immediately caught up again by the bow, the 
 vibrations are rendered continuous. 
 
 The vibrating string, presenting so small a surface, is capable of 
 transmitting very little of its motion directly to the air. It is 
 necessary that its vibrations should first be communicated to some 
 body, which presents a much larger surface to the air. Thus, in the 
 pianoforte, the vibrations of the wires are first transmitted, by 
 means of the bridge and wrest-pins, to a soimd-board ; in the harp, 
 the motion of the strings is communicated to the massive frame- 
 work. In the violin, the vibratory movement of the strings is 
 communicated by means of the bridge to the " beUy." The bridge 
 stands on two feet, immediately beneath one of which is the "sound 
 post," which transmits the motion to the "back " of the instrument, 
 the whole mass of air between the " back" and the "beUy" thus 
 being set in vibration. 
 
ON THE VIBRATIONS OF STRINGS. 87 
 
 The following simple experiment ■wUl illustrate the important 
 part played by the sound-board or its substitute, in stringed 
 instruments. Fasten one end of a string, 3 or 4 feet in length, to 
 a heavy -weight, and, holding .the other end in one hand, let the 
 weight hang freely. On plucking or bowing the string, scarcely any 
 sound will be heard. Now attach the free end of the string to the 
 peg at the left hand of the Sonometer (fig. 23), and let the weight 
 hang freely over the pulley at the right hand. If the string be 
 now plucked or bowed, a loud sound will be emitted. 
 
 We now proceed to study the conditions which determine pitch, 
 quality, and intensity, in stringed instruments. As the tones pro- 
 duced by such instruments aie rarely or never simple, it will be 
 understood, that in investigating the laws relating to ]pitch, it is the 
 pitch of the fundamental tone alone, that is considered. 
 
 If T denote the tension of a stretched string, and M its mass, it 
 may be shown mathematically, that the velocity V, with which a 
 transverse vibration will travel along it, will be — 
 
 ^ M 
 
 and it L denote the length of the string, it is evident that -=■ is the 
 time required for it to execute one complete vibration. Therefore 
 if N denotes the number of vibrations the string performs in one 
 second 
 
 Substituting the above value of V, we get 
 
 From this formula we may deduce the following laws : — 
 
 (1). The tension of the string remaining the same, N (the 
 vibration number) varies inversely as the length of the string. 
 
 (2). Other things remaining the same, N varies inversely as the 
 diameter of the string. 
 
 (3). Other things remaining constant, N varies directly as the 
 square root of the tension — ^that is, of the stretching force or 
 weight. 
 
 (4). Other things remaining constant, N varies inversely as the 
 square root of the density or weight of the string. 
 
 These statements can also be verified experimentally, without 
 recourse to mathematics, by means of the Sonometer described in 
 
88 
 
 HAND-BOOK OF ACOUSTICS. 
 
 Chapter IV. Thus, to prove the first law, stretch the wire hy 
 attaching any sufficient weight, as shown in fig. 23, and ohserve 
 the pitch of the tone it then gives. Now place the movahle hridge 
 in the centre, pluck the half string and again note its pitch. Do 
 the same with J, J, \, ^, of the wire. It wiU be found that the 
 tones produced are as follows, calling the tone produced by the 
 whole length, d|, 
 
 Whole string ■■ d| •• 1 
 
 d 
 s 
 
 di 
 
 Now we already know, that the ratios of the vibration numbers 
 of these tones are those given in the third column, and we at once 
 Bee that these latter are the inverse of those in the first column. 
 This experiment may be varied in an infinite number of ways. 
 Thus, by placing the movable bridge so that the leng^ths of the 
 string successively cut off, are, 
 
 81323JB^1 
 '■' V 6' 4' 8' 6' 16' i' 
 
 it will be found, that these lengths give the notes of the diatonic 
 scale, and the vibration ratios of the successive intervals of these 
 from the tonic, we already know to be 
 
 96436 15 
 '■' i' 4' 3' 2* 3' ¥' ' 
 
 which numbers are the former series inverted. Illustrations of this 
 law may be seen in musical instruments with fixed tones, like the 
 piano and harp, in which the strings, as every one knows, become 
 shorter and shorter as the notes rise in pitch. In the guitar, 
 violin, and other instruments with movable tones, variation in 
 pitch is obtained, by varying the length of the vibrating portion of 
 the string. 
 
 the second law may be verified, by stretching on the Sonometer, 
 with equal weights, two wires of the same material, the diameter 
 of one of which is, however, twice that of the other. The tones 
 produced will be found to be an octave apart, the smaller wire 
 
ON THE VIBRATIONS OF STRINGS. 89 
 
 giving the higher note, that is, the diameters of the wires being as 
 1 : 2, the vibration numbers of the tones produced are as 2 : 1. 
 Illustrations of this law can be found in many musical instruments; 
 thus, the second string of the violin being of the same length as 
 the first, must be thicker in order that it may give a deeper tone. 
 
 To prove the third law, stretch a string on the Sonometer with a 
 weight of, say 161bs, and note the pitch of the resulting tone. 
 Now stretch the same string with weights of 251bs, 361bs, and 
 64:tt)s successively, and observe the pitch of each tone. Calling the 
 tone produced by the tension of 161bs (d), those produced by the 
 tensions of 25, 36, and 641bs will be foimd to be (n), (s), and (d'), 
 respectively. Now we have already ascertained, that the vibration 
 numbers of d, PI, S, d', are as 4 : 5 : 6 : 8 and these numbers are the 
 square roots of 16, 25, 36, and 64. Examples of the application of 
 this law are to be met with in the tuning of all stringed instruments. 
 The violinist, harpist, or pianoforte tuner stretches his strings still 
 more to sharpen, and relaxes the tension to flatten them. 
 
 The fourth law can be proved by stretching two strings of 
 different densities, but of the same length and thickness, by the 
 same weight. Now, by means of the movable bridge, gradually 
 shorten the vibrating part of the heavier string, till it gives a note 
 of the same pitch as the whole length of the lighter one. Now 
 measure the length Z of the lighter string and the length Z^ of 
 the vibrating portion of the heavier one ; it will be f oimd that 
 
 Z : Z, : : ^D^ : ^D (I) 
 
 jDi and D being the densities of the heavier and lighter string 
 respectively. These densities can be ascertained by weighing equal 
 lengths of the two strings. Let N be the number of vibrations per 
 second performed by the length Z^ of the heavier string, then if ^| 
 be the number performed by the whole length Z of the same string, 
 we know by the first law that 
 
 Z : Z^ : : N : Ni 
 
 therefore from (I) 
 
 Now as N also denotes the number of vibrations per second per- 
 formed by the lighter string, this proves the law. As illustrations 
 of the application of this law to musical instruments, the weighting 
 of the lowest strings of the pianoforte by coiling wire round them, 
 
90 
 
 HAND-BOOK OF ACOUSTICS. 
 
 may be mentioned. The density of the fourth string of the Tiolin 
 is increased in the same way. 
 
 The pitch of a string, stretched between two fixed supports, is 
 materially aflected by heat, especially if the string be of metal. 
 As the metal expands on heating and contracts on cooling, the 
 tension becomes less, and the pitch is lowered in the former case, 
 while the tension becomes greater and the pitch rises, in the latter. 
 Heat also produces a difference in the elasticity of strings which 
 acts in the same direction. Strings of catgut are also aflected by 
 moisture, which by swelling the string laterally, tends to shorten 
 it, thus increasing the tension, and raising the pitch. 
 
 We pass on now, to discuss the conditions, which determine the 
 quality of the tone produced by a stretched string. 
 
 A s c 
 
 Fio. 46. 
 
 fasten one end of an india-rubber tube, about 12 feet long, to 
 the ceiHng of a room, and taking the other end in the hand, gently 
 move it backwards and forwards. It is easy after a few trials, to 
 set the tube Tibrating as a whole (fig. 46A). On moving the hand 
 more quickly the tube will break up into two vibrating segments 
 (fig. 46, B). By still more rapid movements, the tube can be made 
 to vibrate in three (fig. 46, 0), four, five, or m.ore segments. 
 Precisely the same results can be obtained, by fastening the tube 
 at both ends, and agitating some intermediate point. The points 
 
ON THE VIBRATIONS OF STRINGS. 
 
 91 
 
 i 
 
 1 
 
 D, E, F (fig. 46), -which seem to be at rest are termed " nodes " or 
 " nodal points," and the Tibrating portion of the tube BD, CE, or 
 EF, between any two successive nodes is called a "ventral 
 segment." 
 
 To understand how these nodes are 
 
 formed, let ac, fig. 47 (1) represent a 
 
 string similar to that just referred to. 
 
 By jerking the end a, a hump a 6 is 
 
 raised, which travels to the other end. 
 
 In fig. 47 (2) this hump has passed on 
 
 to 6 c. In fig. 47 (3) it has been re- 
 flected, and is returning to the end a, 
 
 but on the opposite side. "While this 
 
 Las been going on, let us suppose 
 
 another impulse to have been given, 
 
 so as to produce the hump a h, fig. 47 
 
 (3). Now the hump 6 c is about to pass 
 
 on to a, and in so doing, the point b 
 
 must move to the left, but the hump 
 
 aZ> is about to travel on to c, and in so 
 
 doing must move the point h to the 
 right. The point J, thus continually 
 urged in contrary directions with equal 
 forces, ■while the humps pass one 
 another, remaias at rest. Suppose that 
 the hump takes one second to travel 
 from a to c and back again: then it 
 is evident that if an impulse is given every half second, the 
 above state of things will be permanent, and the two parts ah, be 
 ■will appear to 'vibrate independently of each other, fig. 47 (4), the 
 point b forming a node. A little reflection ■will show that, on the 
 same supposition, if the imp^ulses follow one another at intervals of 
 one-third of a second, two nodes and three ventral segments ■will 
 be formed, and so on. When therefore a string vibrates iu 2, 3, 4 
 segments, each segment vibrates 2, 3, 4 times as rapidly as the 
 string vibrating as a whole. 
 
 A tuning-fork may be used ■with great advantage, in setting up 
 these segmental ■vibrations. One end of a silk thread is fastened to 
 one of the prongs of the fork, the larger the better ; the other end 
 being either ■wound round a peg, or after passing over a pulley. 
 
 W 
 
 Fia. 47. 
 
 (31 
 
92 HAND-BOOK OF ACOUSTICS. 
 
 attached to a weight. On bowing the fork, the string is set vibrating 
 in one, two, three, or more segments, according to its degree of 
 tension. 
 
 The two ends of a stretched string being at rest, it is evident that 
 the number of ventral segments, into which it can break up, must 
 be a whole number ; it cannot break up into a certain number of 
 ventral segments and a fraction of a segment. Any point of the 
 string capable of being a node, can be made such, by lightly touch- 
 ing that point, so as to keep it at rest, and bowing or plucking at the 
 middle of the corresponding ventral segment. Thus, if the string 
 of the Sonometer be lightly touched at the centre, and bowed about 
 I of its length from the end, it will break up into two (or possibly 
 six) ventral segments, with a node in the centre. Again, if the 
 string be lightly touched at J of its length from the end, and bowed 
 about the middle of this third, it will vibrate in three segments 
 (the other § dividing into two) separated by two nodes. That the 
 larger part of the string, in this experiment, does divide into two 
 segments separated by a node, may be shown, by placing riders on 
 the string, before it is bowed, one in the centre of this part, where 
 the node occurs, and one in the middle of each of the two ventral 
 segments. When the string is now lightly touched at ^ of its 
 length from the end, and bowed as before, the riders in the middle 
 of the ventral segments will be thrown ofE, but that at the node 
 will keep its place (fig. 48). This experiment may be repeated 
 
 ^dL 
 
 
 Fio. 48 
 
 with a larger number of nodes. Thus, suppose four nodes are 
 required. Divide the string into five equal parts, as there will 
 evidently be that number of segments, and at each of the four 
 points of division, place a coloured rider, with a white one 
 equidistant between each pair, and also between the last one and 
 the end. Remove the coloured rider nearest the other end and 
 lightly touch the point where it stood, with the finger. Draw the 
 bow gently across the string, midway between this point and the 
 end, and the white riders will fall ofi, while the coloured ones will 
 remain at rest (fig. 49). 
 
.^ 
 
 ON THE VIBRATIONS OF BTBINGS. 93 
 
 " 
 ::::.^i^:::. 
 
 Fia. 49. 
 
 A stretched string can therefore vibrate as a ■whole, that is, with 
 one Tentral segment, or yrith. two, three, four, five, six, or more 
 ventral segments, but not -with any intermediate fraction. Now if 
 we call the note given forth by a string, when its whole length is 
 vibrating (d|) ; we have already learnt, that when its two halves 
 only are vibrating, we get the octave above, (d) ; if it vibrate in 
 three ventral segments we shall get the twelfth above, (s) ; with four 
 segments, (d') ; with five, (n') 5 S'Hd so on. Only those notes belong- 
 ing to the series 1, 2, 3, 4, 5, &c., can occur; no note intermediate 
 between these can be produced. It will be at once observed, that 
 this is the series of partial tones, and the idea at once suggests 
 itself, that the occurrence of partials in the tones of stringed instru- 
 ments is due to the fact, that a string not only vibrates as a whole, 
 but at the same time in halves, thirds, quarters, &c. , each segment 
 giving rise to a simple sound of its own particxdar pitch. 
 
 The student may convince himself that this is reaUy the case, by 
 a variety of experiments. Thus, while a string is vibrating and 
 giving forth a compound tone in which the fifth partial can be 
 heard, Hghtly touch it with a feather at a point, distant I of its 
 length from the end ; all the partials except the fifth, which has a 
 node at this point, will rapidly die out, but this one will be plainly 
 heard, showing that the string must have been vibrating in five 
 segments. Again, touch the middle of the string in a similar 
 manner, after setting it in vibration, and aU the partials except 
 those which require a node at this point will vanish, that is, only 
 the second, fourth, sixth, &c., wiU remain. Further, bow or pluck 
 the string at its middle point; aU the tones which require a node at 
 this point must then be absent, only the first, third, fifth, seventh, 
 &c., being heard. In this way, the presence or absence of any par- 
 ticular partial of a compound tone may be ensured. 
 
 The occurrence and relative intensities of partials on stringed 
 instruments, depend upon : — 
 
 1st. The nature of the string. 
 
 2nd. The kind of hammer, bowing, plectrum, &o. 
 
 3rd. The place where the string is struck, bowed, or plucked. 
 
94 HAND-BOOK OF ACOUSTICS. 
 
 With regard to tlie string, tlie more flexible it is, the more 
 readily will it break up into vibratory segments, and therefore the 
 greater 'wiU be the number of partials in its tones. Thick stout 
 strings, such as are used for the lower notes of the harp, cannot 
 from their rigidity break up into many segments, and therefore the 
 fundamental will be louder than the other partials. On the other 
 hand, thin strings of catgut, such as first TioUn strings, readily 
 vibrate in many segments, so that their tones contain many 
 partials. Still more is this the case with a long fine metallic wire, 
 in which it is possible to hear some fifteen or twenty partials, the 
 fundamental being very faint or even inaudible. The tinkling 
 metallic quality of tone from such a wire, is due to the prominence 
 of the high partials above the seventh or eighth, which lie at the 
 distance of a tone, or less than a tone, apart. Again, the steel 
 wires which give the highest notes in the pianoforte, being already 
 so short, cannot readily break up into vibrating segments, and 
 hence the highest tones of this instrument are nearly simple. 
 
 As we have seen, the three chief methods of setting strings in 
 vibration are : by a blow from a hammer, by bowing, and by 
 plucking. In the first method (employed, for example, in the 
 pianoforte) the quality of the tone is largely affected by the nature 
 of the hammer. If it is very hard, sharp, and pointed, the part of 
 the string which is struck by it will be affected, and the hammer 
 will have rebounded, before the effect of the blow has time to travel 
 along the length of the wire. Thus small ventral segments will be 
 formed, and prominent upper partials wUl be produced, the lower 
 ones being feeble or absent. On the other hand, if a very soft, 
 rounded hammer be used, the blow being much less sudden, the 
 movement of the wire will have time to spread, and a powerful 
 fundamental may be expected. On the pianoforte, both extremes 
 are avoided, by covering the wooden hammers with felt, so that when 
 they strike the wire, the rebound is not absolutely instantaneous ; 
 nevertheless the time during which the hammer and wire axe in 
 contact is extremely short. Similarly, in plucking; a soft, 
 rounded instrument, such as the finger tip, gives a stronger 
 fundamental and fewer high partials, than the harder and sharper 
 quill, that used to be employed in the harpsichord. 
 
 The quality of the tone, given forth by a stretched string, 
 depends largely upon the point at which it is struck, bowed, or 
 plucked. We have already seen, that the point in question cannot 
 be a node ; it is more likely to become the middle of a ventral 
 segment. All the partials, therefore, that require a node at that 
 
ON TEE VIBRATIONS OF STRINGS. 95 
 
 point, will be absent. Thus if the string be struck at the middle 
 point, only the odd partials will be present in the compound tone 
 produced : if the string be struck at a point, one third of its length 
 from the end, the 3rd, 6th, and 9th partials wiU be absent. Again, 
 if a string be struck at a point, one seventh, one eighth, or one 
 ninth of its length from the end, the 7th, 8th, or 9th partials 
 respectively, will be absent. Now, these are the first three dissonant 
 partials of a compound tone, so that it improves the quality of tone 
 to have them absent; and it is a curious fact, as Helmholtz 
 observes, that pianoforte makers, guided only by their ears, have 
 been led to place their hammers, so as to strike the strings at about 
 this spot. 
 
 With regard to the quality of tone in the pianoforte, it will be 
 found, that in the middle and lower region of these instruments, 
 the tones are chiefly composed of the first six or seven partials, the 
 first three being usually very prominent ; in fact, the second and 
 third are not unfrequently louder than the fundamental. As the 
 first six partials form the tonic chord, the tones that have them 
 well balanced, sound peculiarly rich. The result of pressing down 
 the loud pedal should be noted. The idea usually entertained is, 
 that by keeping the dampers raised from the wires, the tones are 
 prolonged after the fingers are taken off the notes. This is true, 
 but not the whole truth. For as the dampers are raised from all 
 the wires, all the latter which are capable of vibrating in unison 
 with the already vibrating wires, wiU. do so. For example, if the 
 loud pedal be depressed, and the F| in the Bass clef be struck, the P2 
 wires an octave lower will be set vibrating in two halves ; the BI73 
 a fifth below that, in three parts ; the F3, two octaves below the 
 note struck, in four parts ; and so on, each section sounding forth 
 the F|. Again, the wires which were struck will not only vibrate as 
 a whole, giving F,, but in halves giving F, which wiU start the 
 wires of the F digital, and wiU. also set the BI72 wires vibrating in 
 three sections. Further, the original wires will vibrate in three 
 segments producing the partial O, and this will start the wires cor- 
 responding to C and 0, and so on. It is easy to see, therefore, 
 that when the loud pedal is held down, and a low note struck, the 
 number of wires set vibrating is very great, giving an effect of in- 
 creased richness. At the same time, the necessity of raising the 
 pedal at every change of chord is very clearly seen. The result of 
 pressing down the soft pedal is to slide the whole of the hammers 
 along transversely, through a short distance, so that they strike 
 only one of the two or three wires that are allotted to each note. 
 
HAND-BOOK OF ACOUSTICS. 
 
 A great number of partials are usually present in the tones pro- 
 duced by the violin, at least the first eight being nearly always 
 present. The peculiar incisiveness of tone is probably due to the 
 presence of partials above the eighth, which, as will be seen from 
 the table on page 72, lie very closely together. The vioUnhas four 
 strings tuned in fifths, the highest being tuned to E' ; the lower 
 limit is therefore G|. The viola or tenor violin, which is slightly 
 larger than the above, has also four strings tuned in fifths, the 
 highest being A ; the lower limit is consequently C| . The violoncello 
 has also four strings, each of which is tuned an octave lower than 
 the corresponding one in the viola. The lower limit is therefore 
 C2. The double bass usually has only three strings, tuned in 
 fourths, the highest being G2; its deepest tone is thus A3, only two 
 notes below the violoncello, but its larger body of tone makes it 
 seem of a deeper pitch than it actually is. 
 
 SUMMABT. 
 
 The three essentials of a stringed instrument are : (1), the string 
 (2), the means of exciting it ; (3), a sound-hoard or resonator. 
 The vibration number of a stretched string varies 
 Directly as the square root of the tensimi, 
 Inversely ,, ,, ,, ,, density. 
 
 ,, ,, diameter. 
 
 Stringed instruments flatten with rise of temperature, and vice 
 versa. 
 
 Points of rest, or rather of least motion, in a vibrating string 
 are termed nodes. The vibrating part of the string between two 
 consecutive nodes is called a vemtral segment. The middle of a 
 ventral segment is sometimes referred to as an antinode. 
 
 The occurrence of partials in the tone of a stretched string, is 
 due to the fact, that it vibrates, not only as a whole, but simul- 
 taneously also in halves, thirds, quarters, &c. ; each segment 
 producing a simple tone or partial, of a pitch and intensity 
 corresponding to its length and amplitude respectively. 
 
 The occurrence and intensities of these partials depend upon 
 (1). The nature of the string. 
 (2). The nature of the excitation. 
 (3). The position of the point where the string is excited. 
 
ON TEE VIBBATIOm OF STBINOS. 97 
 
 To ensure the absence of any particular overtone, the string 
 should be excited at that point where this overtone requires a node 
 for its formation. 
 
 To favour the production of any particular overtone, the string 
 should be excited at that point where this overtone requires an 
 antinode. 
 
 H 
 
98 
 
 CHAPTER X. 
 
 Fltte-pipes Airo Eeeds. 
 It mil be found on examination, that in all wind instruments, the 
 air contained in the tubes of such instruments is set in vibration, 
 either by blowing against a sharp edge, at or near the mouth of the 
 tube, as in the flute and the flue pipes of the organ ; or by the 
 ■vibration of some solid body placed in a similar position, as in the 
 clarinet and the reed pipes of the organ. We shall proceed first, to 
 investigate the conditions which determine pitch, quality, and 
 intensity in the former class of instruments. 
 
 m 
 
 Fio. 60. 
 
 Fio. 51. 
 
FLUE-PIPES AND REEDS. 
 
 99 
 
 As the type of instruments of tliis class, we may take an ordinary 
 organ pipe. Suoli pipes are constructed either of metal or wood ; 
 the former being an alloy of tin and lead, or for large pipes, zinc ; 
 the latter, pine, cedar, or mahogany. Kg. 50 represents a wooden, 
 and fig. 51a metal pipe, both in general view and in section. They 
 may be closed, or open at the upper end. Fig. 52 shows an enlarged 
 section of the lower part. The air from the wind chest enters at 
 (a) and passes into the chamber (c), the only outlet 
 from which is the linear orifice at (i). The air 
 rushing from {d) in a thin sheet, strikes against the 
 sharp edge (e), and the column of air in the pipe is 
 set in vibration. The precise way in which this sheet 
 of air acts is not quite clear. Helmholtz says " The 
 directed stream of air breaking against the edge, 
 generates a peculiar hissing or rushing noise, which 
 is all we hear when a pipe does not speak, or when we 
 blow against the edges of a hole in a fiat plate 
 instead of a pipe. Such a noise may be considered 
 as a mixture of several inharmonic tones of nearly 
 the same pitch. When the air chamber of the pipe 
 is brought to bear upon these tones, its resonance 
 strengthens such as correspond with the proper tones 
 of that chamber, and makes them predominate over 
 the rest, which this predominance conceals." On the 
 other hand, this thin sheet of air has been compared 
 by Hermann Smith to an ordinary reed, and called 
 by Mm an " aeroplastic reed." His theory is, that 
 in passing across the embouchure {ed) the aeroplastic reed 
 momentarily produces an exhaustive effect tending to rarify the 
 air in the lower part of the pipe. This, by the elasticity of the 
 air, soon sets up a corresponding compression, and these alternate 
 rarefactions and condensations reacting upon the lamina, cause it 
 
 Fio. 52. 
 
 to vibrate, and to communicate its vibrations to the air within the 
 pipe. 
 
 The pitch of the fundamental tone given forth by a pipe, depends 
 upon its length; the longer the pipe the deeper the note. The 
 reason of this has been already fully explained in Chapter VII. 
 To recapitulate what is there stated and proved : The vibration 
 number of the sound produced by an open pipe, may be found, by 
 dividing the velocity of sound by twice the length of the pipe; 
 that of a stopped pipe, by dividing by four times its length. In 
 the latter case the internal length must be measured, as " length 
 
100 HAND-BOOK OF ACOUSTICS. 
 
 of pipe" really means, the length of the vibrating colunm of 
 air. 
 
 The rale just given, although approximately true in the case of 
 narrow pipes, cannot be depended upon, when the diameter of the 
 pipe is any considerable fraction of its length. The following rule 
 quoted from EUis' "History of Musical Pitch" is much more 
 accurate. Divide 20,080 when the dimensions are in inches, and 
 610,000 when the dimensions are in millimetres, by : — 
 
 (1). Three times the length, added to five times the diameter, 
 
 for cylindrical open pipes. 
 (2). Six times the length, added to ten times the diameter, 
 
 for cylindrical stopped pipes. 
 (3). Three times the length, added to six times the depth 
 
 (internal from front to back), for square open pipes. 
 (4). Six times the length, added to twelve times the depth, 
 for square stopped pipes. 
 As a matter of fact, however, the note produced by a stopped 
 pipe is not exactly the octave of an open pipe of the same length r 
 in fact, it varies from it by about a semitone. 
 
 The pitch of a pipe is also aflected by the pressure of the wind. 
 The above rule supposes this pressure to be capable of supporting a; 
 column of water 3^ inches high. If this pressure be reduced to 2J, 
 the vibration number diminishes by about 1 in 300 ; if increased to- 
 4, it rises by about 1 in 440. The pitch is also aSected by the 
 size of the wind slit and the orifice at the foot : by the shape and 
 shading of the embouchure; and by the pressing in or pressing out 
 of the edges of its open end, as by the "tuning cone." 
 
 As already stated, the velocity of sound in air, at 0^ Centigrade, 
 or 32° Fahrenheit, is 1,090 feet per second, increasing about two 
 feet for every rise of temperature of 1° 0. and about one foot for 
 1° P. The velocity of sound at any temperature may be more 
 accurately determined from the formula 
 
 V = 1,090 '/l + ai 
 where t is the centigrade temperature and a :=. ^y%, the coefficient 
 of expansion of gases. Now, as the vibration numbers of the 
 sounds emitted from stopped and open pipes may be approximately 
 found by dividing the velocity of sound by four times and twice 
 their lengths respectively, it is evident that such vibration numbers 
 will vary with the temperature; the higher the temperature, the 
 sharper the pitch, and vice versa. Furthermore, the length of the 
 
FLUE-PIPES AND BEEDS. 
 
 101 
 
 pipe itself varies with change of temperature, increasing with, a 
 rise and shortening again with a fall of temperature. This will 
 ■obTiously have a contrary efleot on the pitch, but to a very much 
 smaller extent; in fact, in wooden pipes the expansion is quite 
 inappreciable. Thus the general efiect of rise of temperature in 
 organ pipes is to sharpen them. 
 
 It is evident, from the above, that the wooden pipes of an organ 
 will sharpen somewhat more than the metal ones, for the same rise 
 of temperature. Furthermore, it is found that small pipes become 
 relatively sharper than large ones, under the same increment of 
 "heat ; and not only is this the case, but the change takes place much 
 more rapidly in small pipes than in large ones, and in open than in 
 ■closed pipes. On the other hand, although metal pipes do not 
 sharpen quite so much as wooden ones, they are afiected much 
 jnore rapidly. According to Perronet Thompson, diminution of 
 .atmospheric pressure sharpens the tones of pipes, and vice versa. 
 He states that a fall of an inch sharpens the tuning C by a 
 comma. 
 
 The lowest note producible in the largest organ is O4 ^ 16, and 
 is obtained from an open pipe about 32 feet long. This pipe 
 together with those giving notes of lower pitch than O3 = 32, are 
 said to belong to the 32 foot octave. O3 =: 32 is produced by an 
 open pipe about 16 feet long ; hence, from 
 
 = 32 to the B3 above, constitutes the 16 foot octave. 
 
 8 
 4 
 2 
 1 
 
 Increase of intensity in the tones of an organ cannot be obtained 
 "by increase of force in blowing : for as we have just seen, a very 
 sUght increase in the wind pressure alters their pitch slightly, and 
 still greater increase, as we shall presently see, would affect their 
 quality also. Hence, increase of intensity on the organ has to be 
 produced by bringing more pipes into action by means of stops, or 
 by enclosing the pipes in a case, which can be opened or closed at 
 -pleasure, as in the sweU organ. 
 
 We have now to turn our attention to the conditions that deter- 
 mine the occurrence of overtones, in the tones of organ pipes. 
 Trocure an ordinary open wooden or metal organ pipe and blow 
 
 C3 
 
 = 
 
 32 
 
 c. 
 
 = 
 
 64 
 
 C| 
 
 =r 
 
 128 
 
 c 
 
 = 
 
 256 
 
 CI 
 
 =zz 
 
 512 
 
 02 
 
 1= 
 
 1024 
 
 03 
 
 = 
 
 2048 
 
 > ^2 
 
 
 > B| 
 
 
 , B 
 
 
 , Bi 
 
 
 , B2 
 
 
 , B3 
 
 
i02 HAND-BOOK OF ACOUSTICS. 
 
 very gently into it. Tlie fundamental tone of tlie pipe wHcli we 
 win call (d|) will be produced. On gradually increasing the 
 strength of the wind, a point will be reached, at which this note 
 will vanish, and a note (d), an octave higher, will be heard. On 
 blowing harder still, this (d) will cease, and a note (s) a fifth above 
 wiU be given forth, and so on. AH these notes d. S, d', n', &c., 
 above the fundamental, which thus apparently make their appear- 
 ance successively, are usually termed the harmonics of the pipe. 
 
 In order to imderstand how these tones are produced, let us turn 
 back to page 60. We saw there how a condensation entering one 
 ^ end (a) of the tube (fig. 53), proceeds to the other end 
 
 (6), and is there reflected as a rarefaction. Now suppose 
 that at the moment this rarefaction starts back towards 
 (o) another rarefaction starts from (a) ; what will happen 
 when they meet in the centre ? The wave from (i), if 
 none other were present, would cause the particles of 
 air in the centre (c) to move upwards; that from (a) 
 would move them with equal force downwards. Under 
 these circumstances the particles in the centre will 
 remain at rest. But, Just as in the case of the string, 
 the two pulses of rarefaction will not interfere with 
 one another; each will pursue its course to the end of 
 the tube, where each wiU be reflected, as formerly 
 explained, as a condensation. Now when these pulses 
 of condensation meet in the centre of the tube, that 
 which comes from (i), if it alone were present, would 
 cause the air particles there to move downwards, while 
 that from (a), would move them in the opposite 
 Fio. 53. direction. The result, as before will be, that the air 
 particles in the centre will remain at rest; and com- 
 paring these pipes with the strings already studied, we see that 
 under these circumstances, the middle of the tube becomes a 
 "node," while the ends, being places of greatest vibration, corre- 
 spond to the middles of "ventral segments." Further, as the 
 impulses enter an open pipe, and are reflected at the ends, these 
 points must always be places of maximum vibration, that is, must 
 always correspond to the middle of ventral segments. But two 
 ventral segments must necessarily be separated by a node : therefore, 
 the above is the simplest way in which the column of air in an 
 open tube can vibrate, and consequentiy this form of vibration 
 must give the fundamental tone of the pipe. It may be represented 
 by fig. 54 (A), in which the straicht line in the centre siowa the 
 
FLUE-PIPES AND REEDS. 
 
 108 
 
 position 
 fonu. 
 
 of the node, and the dotted lines give the associated wave 
 II 
 
 i— 
 
 \J 
 
 \ t 
 
 \ / 
 
 \/ 
 
 \ / 
 
 — J; — 
 / \ 
 
 / \ 
 
 ; \ 
 
 \ \ 
 '\ / 
 
 \/ 
 
 1/ 
 
 /\ 
 
 1 I 
 
 /A 
 
 K, H 
 
 Fig. 54. 
 
 It is easy to show experimentally, that an open pipe ■which is giving 
 forth its fundamental, has a node or place of least vibration in the 
 centre, and two places of maximum vibration, one at each end. Let 
 such a pipe betaken, the frontof which must be of glass. Make alittle 
 tambourine, by stretching a piece of thin membrane over a little 
 hoop. Place a few grains of sand on the membrane, which by 
 means of a cord must then be gently lowered in a horizontal 
 position iato the sounding pipe. On entering it, the sand is at first 
 violently agitated, but as the little tambourine descends, it becomes 
 less and less disturbed, till at the centre, the sand remains quiet; 
 on lowering! it still more, the sand again begins to dance, becoming 
 increasingly agitated as the bottom is approached. 
 
104 
 
 HAND-BOOK OF ACOUSTICS. 
 
 Again, from what has been said above, it will be seen, that at the 
 centre, where the node occurs, the air is alternately compressed and 
 rarefied ; compressed, when two condensations meet, and rarefied, 
 when two rarefactions meet. This can also be experimentally 
 verified ; for if the pipe were pierced at the centre and the hole 
 covered air-tight by a piece of sheet india-rubber, this latter being 
 acted upon by the condensations and rarefactions, would be 
 alternately pressed outwards and inwards. The organ pipe (fig. 55) 
 
 Fig. 55. 
 
 has been thus pierced at the centre B, and also at A and 0, and the 
 membranes covered by three little capsules (a section of each of 
 which on an enlarged scale is shown at the left of the figure), from 
 the cavities of which proceed three little gas jets, the gas being 
 supplied by the three bent tubes which .come from the hollow 
 chamber P, which again is supplied by the tube S. Now on blow 
 
FLUE-PIPES AND REELS. 105 
 
 ing very gently into the pipe, so as to produce its fundamental, all 
 three flames are agitated, but the central one most so. Turning 
 down the gas tiU. the flames are very small, and blowing again, the 
 middle one will be extinguished, while the others remain alight. 
 
 Inasmuch as the two ends of an open pipe must, as we have 
 shown, correspond to the middle of ventral segments, the next 
 simplest way in which such a pipe can vibrate, is, with two nodes, 
 as shown in fig. 54 B. In A there are two half segments, which 
 are equivalent to one ; in B there are two half segments and one 
 whole one, equivalent to two segments ; the rate of vibration in B 
 will therefore be twice as rapid as in A. Accordingly, we find that 
 the next highest tone to the fundamental, which can be produced 
 from an open pipe, is its octave. The occurrence of the two nodes 
 in B can be experimentally proved by the pipe of fig. 55, for if this 
 pipe be blown more sharply, so as to produce the octave of the 
 fundamental, the two flames A and C will be extinguished, while 
 B will remain alight. and D, fig. 54, represent the next simplest 
 forms of vibration with three and four nodes respectively. The 
 rate of vibration in (0) and (D) will obviously be three and four 
 times respectively that in (A). Proceeding in this way, it will be 
 found that the rates of all the possible modes of segmental vibration 
 in an open pipe, will be as 1, 2, 3, 4, 5, &c., and this result, thus 
 theoretically arrived at, is confirmed by practice ; for we have seen 
 that, calling the fundamental tone (d|) ; the harmonics produced 
 from such a pipe are, d, S, d', tn', &c., the vibration numbers of 
 which axe as 2, 3, 4, 5, &c. 
 
 We have hitherto supposed that each of these notes successively 
 appears alone, but this is rarely the case, usually the fundamental 
 is accompanied by one or more of these tones. When they are thus 
 simultaneously produced, it is convenient to term them overtones, 
 or, together with the fundamental, partials, as in the case of 
 stretched strings. In order to explain the simultaneous production 
 of these partials, we simply have to suppose the simultaneous 
 occurrence of the segmental forms represented in fig. 51. We thus 
 see that the notes obtainable simultaneously from an open pipe, are 
 the complete series of partial tones, whose rates of vibration are as 
 the numbers 1, 2, 3, 4, 5, &c. 
 
 Coming now to stopped pipes we have seen in Chap. VII, page 61 , 
 that a pulse of condensation entering a stopped pipe, travels to the 
 closed end, and is there reflected back unchanged. On arriving at 
 the open end, it is reflected back as a pulse of rarefaction, which on 
 
106 
 
 HAND-BOOK OF ACOUSTICS. 
 
 reaoMng the stopped end, is reflfected tmaltered. Now the closed 
 end of a stopped pipe must always be a node, since no longitudinal 
 vibrations of the air particles can occur there ; and as we have seen 
 above, the open end must be the middle of a ventral segment, 
 therefore the simplest form in which the air column in a stopped 
 pipe can vibrate, is that represented in (A), fig. 56. This form of 
 
 
 
 i \ 
 
 
 1 j 
 \ \ 
 
 \ • 
 
 
 
 
 
 \ / 
 
 — ^ — 
 
 
 \ ! 
 
 
 / 
 
 l' 1 
 
 
 
 
 / \ 
 / \ 
 
 
 '\.' 
 
 
 7; 
 
 
 
 J 1 
 
 
 
 
 
 ! 
 ! 
 
 f 
 
 
 1 ; 
 
 
 1 
 
 
 \ 
 
 1 
 
 
 \ 
 
 
 \ 
 
 
 \ / 
 
 '/ 
 
 
 \.' 
 
 
 \ i 
 
 v 
 
 
 V 
 
 Fio. 56. 
 
 vibration must therefore produce the fundamental tone of the pipe. 
 Comparing it with the simplest form in which the air in an open 
 pipe can vibrate, A, fig. 54, it will be seen that the open pipe has 
 two half segments, while the stopped has only one ; consequently, 
 if the pipes be of equal length as represented, the vibrating segment 
 of the latter is twice as long as the former. Hence, as we have 
 already seen, the fundamental tone of a stopped pipe, is an octave 
 lower than that of an open one of the same length. 
 
FLUE-PIPES AND SEEDS. 107 
 
 The next simplest way in which the air in a stopped pipe can 
 vibrate, must be that in which two nodes are formed, and these 
 must necessarily ocoui- as shown in B, fig. 56, where the end of the 
 pipe, as we have seen, forms one node, the place of the other being 
 represented by the vertical line. In order to understand the 
 formation of a node at this point, let fig. 57 represent a stopped 
 pipe, and let ab, Ic, cd, be each one-third of its length. 
 Further, let it be supposed, that the pulses of condensation 
 and rarefaction successively enter the open end, at inter- 
 vals of time, each equal to that required for the pulse to 
 travel from (a) to (c), that is, through two-thirds of the 
 length of the tube. For the sake of simplicity, we will 
 suppose, that the interval of time is one second, although 
 of course it is reaUy but a minute fraction of that period. 
 First let a pulse of condensation C| enter the pipe. After 
 the lapse of a second, that is at the beginning of the 2nd 
 second, 0| will be at (c) and the succeeding pulse of rare- 
 faction E| will be just entering at (a). Neglecting E| for 
 the present, let us see where 0| will be at the beginning 
 of the 3rd second; it will evidently have travelled 
 through {cd) and back, and in fact will be at (c) again, but 
 moving upwards. But by the supposition, another pulse 
 
 of condensation Og is now entering the tube at (a) and 
 
 therefore moving downwards. These two equal pulses of -p^Qg^ 
 condensation wOl meet at (b) and the air particles here 
 being solicited by C| to move upwards, and by O2 to move 
 downwards, will remain at rest. To return now to the pulse 
 of rarefaction B,|, which at the beginning of the 2nd second was 
 entering the tube at (o) : at the beginning of the 3rd second it will 
 be at (c), moving downwards: at the beginning of the 4th second it 
 will be again at (c), but moving upwards. But, by the supposition, 
 another pulse of rarefaction E2 is now entering at {a). These two 
 equal pulses will meet at (b), and the air particles there being 
 solicited by E| to move downwards, and by Eij to move upwards, 
 will remain at rest. Thus the particles at (J) will be permanently 
 at rest, that is, [b) will be a node. It will be noted, that it is per- 
 fectly allocable to consider, as we have done, the pulses of con- 
 densation and rarefaction separately; for we have already seen that 
 two series of waves can cross, without permanently interfering with 
 one another. It will be instructive, however, to consider the com- 
 bined effects of the pulses of condensation 0| and rarefaction E|, at 
 the end of the 2nd second, or what is the same thing, at the 
 
108 EAND-BOOE OF ACOUSTICS. 
 
 commencement of the 3rd. At this instant, as the student will 
 perceive, both C| and E| -will be at (c), C| moving up, and E| down. 
 Now the effect of 0| moving upwards is to swing the particles of 
 air at this point upwards also, with a certain amount of force ; and 
 the effect of E| moving downwards, is to swing the same particles 
 upwards also, with the same amount of force; 0| and E| there- 
 fore, combine their forces to swing the air particles at (c) upwards. 
 At the expiration of another second E| will be back again at (c) but 
 moving now upwards, and Cj will also be at the same point moving 
 downwards. The air particles at (c) will now be swinging down- 
 wards, with the combined forces of E| and C2. Thus it will be seen 
 that (c) is a point of maximum vibration, that is, the middle of a 
 ventral segment. 
 
 The half segment in (B) fig. 56 is seen to be one-third as long as 
 the half segment in (A) ; therefore the length of the sound wave 
 emitted by (B) must be one-third the length of that emitted by (A) ; 
 that is, the note corresponding to the vibrational form (B), has liiree 
 times the vibration number of that corresponding to (A). 
 
 The next simplest way in which the air column in a stopped pipe 
 can vibrate is with three nodes, as represented in (C), fig. 56 ; the 
 next simplest, with four nodes (D), the next with five, and so on. 
 As the length of the half segment in (C) is one-fifth the length of 
 that in (A), the wave length of the note corresponding to the 
 vibrational form (C), must be one-fifth of that corresponding to (A), 
 that is, its vibrational number is five times as great. Similarly in 
 D, it is seven times as great. Summing up, then, we find 
 theoretically that the vibration rates of the tones, which can be pro- 
 duced from a stopped tube, are as the odd numbers 1, 3, 5, 7, &c., 
 no tone intermediate in pitch between these, being possible. This 
 can be easily verified experimentally, by the aid of an ordinary 
 stopped organ pipe. On blowing very gently, the fundamental, 
 which we may call (d|), is heard; on gradually increasing the force 
 of the blast, a point is reached at which this fundamental ceases, 
 and the (s) an octave and a fifth above, springs forth ; stiU further 
 increase the wind pressure, and this gives place to the (n') two 
 octaves and a major 3rd above the fundamental. By no variation 
 in the blowing can any tone intermediate in pitch between these be 
 obtained; and the vibration numbers of these three notes (d|), (s), 
 and (n'), are as 1, 3, and 5, and thus the results obtained above are 
 corroborated experimentally. 
 
 As before observed, sounds thus successively obtained from a pipe, 
 by variation in the wind pressure, may be conveniently termed 
 
FLUE-PIPES AND REEDS. 109 
 
 harmonics; the terms "partials" and "overtones" being used 
 •when they are simultaneously produced. For example, if the air 
 in a stopped pipe 'were simultaneously Tibrating in the forms (A), 
 (B), and (C), fig. 56, we should obtain from it a compound tone con- 
 sisting of the first three odd partials, that is, the 1st, 3rd, and 5th. 
 
 With regard to the open organ pipe, the fundamental is never 
 produced alone; according to the dimensions and shape of the 
 pipe, it is accompanied by, from, two to five, or more overtones. As 
 a rule, the overtones are more prominent in narrow than in wide 
 pipes, and in conical, than in cylindrical ones. The shape of the 
 pipe has a great influence on the production of partials. The 
 conically narrowed pipes found in some organ stops, which have 
 their upper opening about half the diameter of the lower, have the 
 4th, 5th, and 6th overtones proportionally more distinct than their 
 lower ones. Stopped wooden pipes of large diameter, when softly 
 blown, produce sounds which are nearly simple. Such tones are 
 sweet and gentle, but tame and monotonous. A greater pressure 
 of wind, or a reduction in the diameter of the pipe, developes the 
 3rd and 5th partials. 
 
 The great body of tone in the organ is produced by wide open 
 pipes, forming the "principal stops." The tones they produce, 
 owing to the deficiency of upper partial tones, are somewhat dull ; 
 they lack character, richness, and brilliancy. Long before Helm- 
 holtz had shown that richness of tone is due to the occurrence of 
 well-developed upper partial tones, organ builders had learnt how 
 to supply such tones artificially, by means of smaller pipes, tuned 
 to the pitch of these partials, forming what are termed mixture 
 stops. As an example of such mixture stops, the " sesquialtera " 
 may be mentioned, which originally consisted of three pipes to each 
 digital, the smaller two producing tones, a twelfth and a seventeenth, 
 above the fundamental of the larger one, thus reinforcing the 3rd 
 and 6th partials. The sesquialtera is now often made with, from 
 three to six ranks of open metal pipes. The smaller ranks are 
 usually discontinued above middle as they become too shrill and 
 prominent, larger pipes sounding an octave lower, being sometimes 
 substituted. 
 
 In the flute, the tone is produced, as in the organ pipe, by 
 directing a current of air against a thin edge, the edge in this case 
 being the side of a lateral aperture near the end of the tube. In 
 the older form, that of the flageolet, there is an arrangement very 
 similar to that of the ordinary organ pipe, and the air is simply 
 blown in. Variations in pitch are effected, in the first place, by 
 
no 
 
 HAND-BOOK OF ACOUSTICS. 
 
 opening apertures in the side, and thus practically altering the 
 length of the pipe ; and secondly, by so increasing the wind pressure, 
 as to bring out the first harmonio to the exclusion of the funda- 
 mental, all the tones thus springing up an octave. The quality of 
 its tone is sweet but dull, owing to the want of upper partials. 
 "When very softly blown, it gives tones that are all but simple. 
 
 Eeed Instrtjments. 
 Two kinds of reed are used in musical instruments, the free reed 
 and the beating reed. Fig. 58 shows the construction of the former. 
 
 Fio. S8. 
 It consists of a, thin narrow strip of metal called a "tongue" 
 fastened by one end to a brass plate, the rest of the tongue being 
 free. Immediately below the tongue, there is an aperture in the 
 brass plate, of the same shape, and very slightly larger than the 
 tongue itself. Thus the tongue forms the door of the aperture, 
 capable of swinging backwards and forwards in it. If a current of 
 air be driven upon the free end of the tongue, the latter is set 
 vibrating to and fro in the aperture between its limiting positions 
 A and B, fig. 59. When in the position A, the current of wind 
 
 Fig. 59. 
 
 passes through, but when the tongue reaches the position B, the 
 current is suddenly shut off; only when the tongue resumes the 
 position A, can the air again pass. As the vibrations of the tongue 
 are periodic, a regular succession of air pulses are thus produced, 
 giving rise to a musical sound, precisely as in the case of the Syren. 
 
 The action of the beating reed is similar to that of the free reed ; 
 in fact, the beating reed only differs from the free reed, in having 
 its tongue slightly larger than the aperture, so that it beats against 
 the plate, in closing the aperture, instead of passing into it. 
 
 The reed is used in its simplest form in the harmonium, American 
 organ, and concertina. In the harmonium and concertina, the 
 
FLUE-PIPES AND HEEDS. Ill 
 
 current of air is forced through the reeds by means of a bellows. lu 
 the American organ, the bellows so act, as to form a partial vacuum 
 below the reeds, the external air being thus drawn through them. 
 
 The pitch of the sounds, obtained from such reeds evidently 
 depends upon the vibration rate of the reed itself. This again 
 depends upon the size and thickness of the reed, and the elasticity 
 of the material of which it is composed. Harmonium reeds are 
 usually sharpened by gently filing or scraping the free end, and 
 flattened by applying the same operation to the part of the tongue 
 near the fixed end. A rise of temperature, diminishes the rate of 
 vibration, as the tongue expands and its elasticity is diminished. 
 The pitch is also somewhat aSected by the force of the wind. 
 
 The tones obtained from reeds such as the above, are very rich in 
 overtones. All the series of partials up to the sixteenth, or even 
 higher, may be distinctly recognised in any of the lower notes of 
 the harmonium; in fact, the undue prominence of the higher 
 partials is one of the drawbacks of this instrument. 
 
 In order to understand this wealth of partials in reed tones, we 
 must turn back to Chap. VIII. We saw there, that Fourier has 
 proved mathematically, that every form of wave may be analysed 
 into a number of simple waves, whose lengths are inversely as the 
 numbers 1,2, 3, 4, 5, &c. Now it is plain, that the more abrupt or 
 discontinuous the compound wave, the greater will be the number 
 of its constituent simple waves. The compound sound wave 
 resulting from the vibration of a reed, is highly discontinuous ; 
 since the individual pulses must be separated by complete pauses 
 during the closing of the apertures. Hence the number of its con- 
 stituent simple waves will be correspondingly large, that is, the 
 compoTind tone produced by a vibrating reed is made up of a very 
 large number of partial tones. The harder and more unyielding 
 the tongue, and the more perfectly it fits its aperture, the 
 more discontinuous will be the pulses, and consequently the more 
 intense and numerous the overtones. 
 
 The compound tone of the reed, being thus overburdened by the 
 intensities of its upper partials, it becomes an advantage to soften 
 these latter, or what comes to the same thing, to strengthen the 
 fundamental, without at the same time strengthening the overtones. 
 This can be done, by placing the reed at the mouth of an open pipe, 
 the fundamental tone of which is of the same pitch as the 
 fundamental tone of the reed. This latter tone will then be greatly 
 reinforced by the resonance of the pipe. The other partial tones of 
 
112 
 
 HAND-BOOK OF ACOUSTICS. 
 
 the reed mil also be strengthened, but to a muoh less extent; 
 for the force necessary to produce segmental vibration, increases 
 rapidly as the number of segments increases. The higher partials 
 of the reed are therefore practically unsupported by the associated 
 pipe. It is evident that the pipe associated with a reed, may be 
 selected to resound to one of the overtones of the reed, instead of 
 to the fundamental, the resulting tone being in this case of quite 
 different quality to the above. The form of the pipe may also vary, 
 producing other changes in the quality of tone produced. It is 
 thus that the varieties of reed pipes in the organ are obtained. 
 Fig. 60 shows how the reed is inserted in 
 the organ pipe. V is the socket in which 
 the lower end of the pipe is fixed, I is the 
 beating reed, which is tuned by increasing 
 or diminishing its eflective length, by 
 means of the movable wire d, sliding in the 
 block, 8. 
 
 The reed instruments in use in the 
 orchestra, may be classified into the wood 
 wind instruments, which have wooden reeds, 
 and the brass wind instruments, which have 
 cupped mouth-pieces. The chief instru- 
 ments of the former class are the Clarinet, 
 the Hautbois or Oboe, and the Bassoon. 
 In these instruments the proper tones of 
 the reeds themselves are not used at all, 
 being too high and of a, shrill or screaming 
 quality; the tones employed axe those de- 
 pending on the length of the column of air 
 in the tube, as determined by the opening 
 or closing of the apertures. The vibration 
 of the air column thus controls the yielding 
 reed, which is compelled to vibrate in 
 sympathy with it. 
 
 The Clarionet or Clarinet has a cylindrical 
 tube terminating at one end in a bell. At 
 the other end is the mouth piece, which is 
 of a conical shape, and flattened at one side 
 so as to form a kind of table for the reed, 
 the opposite side being thinned to a chisel 
 edge. The bore of the instrument passes 
 through the table just mentioned, which 
 
 FiQ. 60. 
 
FLUE-PIPES AND REEDS. 113 
 
 moreover is not quite flattened, but slightly curred away from 
 the reed, so as to leave a thin gap between the end of the reed 
 and the mouth piece. The Clarinet has thus only a single reed, and 
 that a beating one. 
 
 The tube is pierced with eighteen holes, half of which are closed 
 by the fingers, and half by keys. The lowest note is produced by 
 closing all the apertxires and blowing gently. By opening succes- 
 sively the eighteen apertures, eighteen other notes may be obtained 
 at intervals of a semitone ; and thus the lower scale, of one-and-a- 
 half octaves, is obtained. By increase of wind pressure, or by 
 opening an aperture at the back of the tube, the pitch of the tube 
 is raised a twelfth ; in fact the instrument acts like a stopped tube, 
 increased wind pressure bringing out, not the second, but the third 
 of the ordinary harmonic series. 
 
 The quality of tone on the Clarinet is very characteristic, and is 
 due to the fact that only the odd partials, 1, 3, 5, 7, &c., are present 
 in its tones; just as in the case of stopped organ pipes. In fact the 
 Clarinet must be considered as such a pipe, stopped at the end 
 where the reed is placed ; for it is here that the greatest alternations 
 of pressure occur ; that is, as we have seen above, this point must 
 be a node. 
 
 The Oboe and Bassoon have conical tubes expanding into bells. 
 The reed in each is double and formed of two thin broad spatula- 
 shaped plates of cane in close approximation to one another. 
 Variations in pitch are obtained as in the flute, by varying the 
 eflective length of the tube, by means of apertures closed by the 
 fingers or keys. Like the flute also, the first harmonic is the 
 octave, so that increase of wind pressure raises the pitch by that 
 interval. The partials present in the tones of these instruments, 
 are those of the complete series, 1, 2, 3, 4, 5, &c. 
 
 In instruments with cupped mouth pieces, the lips of the player, 
 which form the reed, are capable of vibrating at very different 
 rates, according to their tension, form, &c. A very simple type of 
 this class of instrument may be obtained, by placing a common 
 glass funnel into (one end of a piece of glass tubing, a few feet 
 long, and half an inch or so in diameter. The tones, which can be 
 obtained from such a tube, by varying the tension and form of the 
 lips and the force of the wind, are those of the complete partial 
 series ; the lowest ones are, however, very difficult to obtain. Thus 
 no note can be produced on such an instrument, but such as belong 
 to the series of partials, or harmonic scale, as it is sometimes termed. 
 
 I 
 
114 HAND-BOOK OF ACOUSTICS. 
 
 All instruments witli cupped mouth-pieces are constracted on the 
 same principle as this primitive instrument ; that is, they are tubes 
 ■without lateral apertures, the notes producihle upon them being 
 the harmonics of the tube. Now, as will be seen from the table on 
 page 72, there are various gaps in this harmonic scale, as compared 
 with the diatonic and chromatic scale, and accordingly it will be 
 found that the most important departures of brass instruments from 
 the rude type selected above, have been made for the purpose of 
 supplying these missing notes. 
 
 The chief instruments of this class are the French Horn, Trum- 
 pet, and Trombone. The French Horn consists of a conical 
 twisted tube of great length, expanding at the larger end into a bell. 
 The fundamental, which is a very deep tone, is not used. As will 
 be seen on reference to the table on page 72, the higher harmonics 
 (for example, those from the seventh upwards) form an almost un- 
 broken scale. To supply the missing notes, the hand is thrust into 
 the beU to a greater or less extent, thus lowering the pitch of the 
 note which is being produced at the time. These instnunents are 
 also frequently supplied with keys, which vary the effective length 
 ■of the tube, and thus produce the missing tones, but at some 
 expense of the quality of tone. 
 
 The Trumpet supplies the notes which are wanting to complete 
 its scale in a much more effective manner. An U shaped portion 
 of the tube is made to slide with gentle friction, upon the body of 
 the instrument, so that the tube can thus be lengthened or shortened, 
 within certain limits by the player. 
 
 The Trombone is simply a bass trumpet, and in principle is the 
 same as the above. In these brass instruments, the tension, &c., 
 of the lips only determines which of the proper tones shall speak. 
 the actual pitch of the tone being almost entirely independent of 
 the tension itself. 
 
 The vocal organ, or larynx, is essentially a reed instrument. 
 The reed itself is a double one, and consists of two elastic bands, 
 called the Vocal Chords, or Ligaments, which stretch from front to 
 back across the larynx. When they are not in action, these 
 ligaments are separated by a considerable aperture. By means of 
 muscles inserted in the cartilages to which the vocal ligaments are 
 attached, these latter can be brought close together with their edges 
 parallel. The air from the lungs acting upon them, while in this 
 position, sets them in vibration, in the same way as the air from a 
 bellows operates upon a reed. Variations in pitch are effected by 
 
FLUE-PIPES AND REEDS. 115 
 
 varying tlie tension of the vocal ligaments. This is effected by the 
 contraction of certain muscles, which act on the cartilages to which 
 the ligaments are attached in such a way as to stretch these latter 
 to a greater or less extent. The density of the vocal ligaments also 
 seems to be variable. According to Helmholtz " much soft watery 
 inelastic tissue lies underneath the proper elastic fibres and 
 muscular fibres of the vocal chords, and in the breast voice this 
 probably acts to weight them and retard their vibrations. The head 
 voice is probably produced by drawing aside the mucous coat below 
 the chords, thus rendering the edge of the chords sharper and the 
 weight of the vibrating part less, while the elasticity is unaltered." 
 
 As in other reed instruments, the tones of the human voice are 
 very rich in overtones. In a sonorous bass voice, it is easy to 
 detect the first seven or eight, and by the aid of resonators even 
 more. When a body of voices are heard together, close at hand, 
 singing /orie, the shrill overtones are only too prominent. These 
 overtones are very largely modified in intensity by the size and 
 shape of the nasal cavity and the pharynx, also by the varying size 
 and shape of the mouth and position of the tongue. Hence, when 
 the vocal ligaments have origiriated a compound tone rich in 
 partials, the varying features just mentioned, may reinforce now 
 one set of partials, and now another, in very many diiJerent ways, 
 thus producing the endless variety of qualities found in the human 
 voice. 
 
 The following is a very instructive experiment in connection with 
 this subject, showing how the reinforcement of particular partials 
 by the resonance of the mouth cavity modifies the quality. Strike 
 an ordinary tuning-fork, and hold the ends of the vibrating 
 prongs close to the open mouth, keeping the latter in the position 
 required for singing " ah." Notice how the quality of the fork is 
 afiected. Now do the same again, but put the mouth in the 
 position required for sounding " oo." Observe the change of 
 quality. A looking glass will be required in order to see that the 
 fork is in the right position. Once more repeat the experiment ; 
 but this time, while the vibrating fork is held in position, move the 
 mouth from the position "ah" to the " oo " position, at first 
 gradually, and then rapidly; the corresponding change in the 
 quality of the tone of the fork is very striking, For a detailed 
 account of the Larynx and of Voice production, the student is 
 referred to Behnke's " Mechanism of the Human Voice." 
 
116 EANB-BOOK OF ACOUSTICS. 
 
 Summary. 
 To find approximately the vibration number of any given flue- 
 pipe ; divide the velocity of sound by twice the length of the pipe 
 for open, and by four times its length for stopped pipes. 
 
 The pitch of an open pipe is not exactly the same as that of a 
 stopped pipe of hall its length. 
 
 The pitch of a flue pipe is sharpened by a rise of temperature. 
 
 Wooden pipes sharpen more than metal ones for the same increase 
 of temperature. 
 
 Nodes are produced in flue pipes by the meeting of two rarefactions 
 or of two condensations travelling in opposite directions : con- 
 sequently it is at the nodes that the greatest variations in density 
 occur. 
 
 The open end of a pipe is always an antinode, 
 „ closed ,, ,, „ ,, a node. 
 
 When the air column in an op^^ pipe vibrates with one node only, 
 that is as a whole, the fundamental (say d|) is produced ; when with 
 two nodes only, that is in two halves, the 1st Harmonic (d) ; when 
 with three nodes only, the 2nd Harmonic (s) ; and so on. 
 
 When the air column in a stopped pipe vibrates with one node 
 only, that is as a whole, it gives the fundamental (say d|) ; when 
 with two nodes, that is in three thirds, its 1st Harmonic (s) ; when 
 with three nodes, its 2nd Harmonic (n') ; and so on. 
 
 The tones produced by flue pipes are compound, because of the 
 fact, that the air column vibrates simultaneously, as a whole and 
 in aliquot parts, each part producing an overtone of a pitch cor- 
 responding to its length. 
 
 Stopped pipes only give the partials of the odd series, 1, 3, 5, &o. 
 
 The flute is a flue pipe of variable length. 
 
 The pitch of a reed is lowered by a rise of temperature. 
 
 The sounds produced by reeds are rich in overtones. 
 
 The fundamental tone of a reed-clang may be strengthened 
 relatively to its overtones, by placing over the reed, a pipe which is 
 in unison with that fundamental. 
 
 The Clarinet, Ohoe, and Bassoon are stopped pipes, in which the 
 pipe governs the reed, that is to say, the tones produced depend on 
 the varying length of the pipe, and not upon the reed. The 
 
FLUE-PIPES AND BEEDS. 117 
 
 Clarinet is a cylindrical stopped pipe, and therefore its tones consist 
 of only tlie odd partials. 
 
 The Oboe and Bassoon are conical stopped pipes, and their partials 
 foUow the ordinary series. 
 
 In instruments, with cupped mouth pieces, such as the French 
 Horn, Trumpet, &c., the lips of the player form the reed. Changes 
 of pitch were originally brought about, by successively developing 
 their different harmonics. Their partials belong to the complete 
 series. 
 
 The Human Voice is essentially a reed instrument (the Vocal 
 Ligaments or Chords) ■with a resonator (the Mouth, Pharynx, and 
 Nasal Cavities) attached. The Vocal Ligaments originate a com- 
 pound tone, rich in partials, the relative intensities of which are 
 profouLdly modified by the ever varying resonator, thus producing 
 the almost infinite variety in quality of tone, which is characteristic 
 of the Human Voice. 
 
118 
 
 CHAPTER XI. 
 
 On the Vibrations of Rods, Plates, &o 
 
 As the musical instruments treated of in tlie present chapter are 
 of comparatively less importance than those already studied, the 
 principles which they involve will be more briefly touched on. 
 We shall first consider the 
 
 Vibrations or Bods or Bars. 
 A Bod is capable of vibrating in three ways (the last however 
 being of little importance, musically speaking), viz. 
 
 1. Longitudinally. 
 
 2. Laterally. 
 
 3. Torsionally. 
 
 1. Longitudinal vibrations again may be classified according as 
 the rod is 
 
 (a) Fixed at both ends. 
 
 (b) Fixed at one end only, 
 (e) Free at both ends. 
 
 (o) The Longitudinal vibrations of a rod or wire, fixed at both 
 ends, may be studied on the monochord, by passing briskly along 
 the wire, a cloth, which has been dusted with powdered resin. The 
 sound produced is much higher in pitch than that obtained by 
 causing the wire to vibrate transversely. On stopping the wire at 
 the centre, and rubbing one of the halves, the upper octave of 
 the sound first heard, is emitted. When the wire is stopped at one 
 third its length, and this third excited, the fifth above the last is 
 heard ; and so on. Thus, as in the case of transverse vibrations, 
 the vibration number varies inversely as the length of the wire. 
 On altering the tension, the pitch will not be found to have 
 varied ; that is, the pitch is independent of the tension. 
 
ON THE VIBRATIONS OF SOBS, PLATES, ETC. 119 
 
 The longitudinal vibrations of a wire fixed at both ends, some- 
 wbat resemble those that take place in an open organ pipe. In 
 both cases, the time of a complete vibration is the time taken by a 
 pulse to move through the length of the wire or pipe, and back again. 
 In the case of the latter, we have seen, that the vibration number 
 of the note produced by any given pipe, may be ascertained by 
 dividing the velocity of sound in air, by twice the length of the 
 pipe. Conversely, if we know the vibration number of the pipe, 
 we can ascertain the velocity of sound in air, by multiplying this 
 number by twice the length of the pipe. This principle may be 
 employed to determine the velocity of sound in other gases. Thus, 
 fill and blow the pipe with hydrogen, instead of air, and ascertain 
 the pitch of the note produced : . its vibration number multiplied 
 by twice the length of the pipe, will give the velocity of 
 sound in hydrogen. Or we may proceed thus ; blow one pipe with 
 air, and another with hydrogen, the latter pipe being furnished 
 with telescopic sliders, so that its length can be altered at pleasure. 
 Now while both pipes are sounding, gradually lengthen the pipe, 
 till both are in unison. When this is the case, let I denote the 
 length of the air sounding pipe, and l\ the length of the other : 
 then if F be the velocity of sound in air, and F| its velocity in 
 hydrogen, it is evident that — 
 
 F| _ 2i 
 F - 1 
 from which F| may be readily calcxilated. In this way, the velocity 
 of sound in the various gases has been ascertained. 
 
 Now from what has been said above, it will be seen, that this 
 same method may be applied, in order to ascertain the velocity of 
 sound in solids. For example, suppose we wish to ascertain the 
 velocity of sound in iron. Stretch some twenty feet of stout iron 
 vrire between two fixed points, one of which is movable : a vice, 
 the jaws of which are lined with lead answers very well. Eub the 
 wire with a resined piece of leather, and gradually shorten it till 
 the sound produced is in unison with a tuning-fork, the vibration 
 number of which is, say, 512. When the unison point is reached, 
 measure the length of wire : say it is 16f feet. Then the time of 
 a complete vibration is the time required for the pulse to run 
 through 2 X 16^ = 33 feet of the wire. But there are 512 
 vibrations or pulses per second : therefore sound travels along the 
 iron wire at the rate of 512 x 33 = 16,896 feet in a second. 
 Generally, let I denote the len°-fch. in feet, of a wire or rod fixed at 
 
120 HAND-BOOK OF ACOUSTICS. 
 
 both ends, and n the -vibration number of the note it emits when 
 vibrating longitudinally ; then, if V denote the velocity of sound in 
 the substance of -which the rod or -wire is composed. 
 
 The overtones of a -wire fixed at both ends foUo-w the ordinary 
 series, 1, 2, 3, 4, 5, &c., the -wire vibrating in two segments, -with a 
 node in the centre to produce the first overtone, and so on. 
 
 (i) The longitudinal vibrations of rods fixed at one end, present 
 considerable analogy -with those in stopped organ pipes. Thus the 
 -vdbration number varies inversely as the length of the rod, as may 
 be easily sho-wn by fixing varying lengths of rod in a -vice, and 
 exciting them -with a resined cloth. Again, the time required for a 
 complete vibration, is the time during -which a pulse makes two 
 complete journeys up and do-wn the rod. Thus, these vibrations 
 may be used to ascertain the velocity of sound in any substance, 
 the method of proceeding being similar to that explained above, but 
 the formula -will be 
 
 V = iln. 
 
 The partials obtainable from these rods, are, like those of a 
 stopped organ pipe, the odd partials of the complete series, 1, 3, 5, 
 7, &o. ; the first overtone requiring a node, at a point one-third the 
 length of the rod from the free end ; the second at one-fifth of the 
 length, and so on. 
 
 The only musical instrument in which this kind of rod vibration 
 is utilized is Marloye's harp. It consists of a series of wooden rods 
 of varying lengths, vertically fixed on a sound-board below. The 
 rods are excited, by rubbing up and down with the resined fingers. 
 
 (c) In rods or tubes free at both ends, the simplest longitudinal 
 -vibrations are set up, when the tube is clasped or clamped at the 
 centre, and excited by longitudinally rubbing either half : the 
 simplest form of -vibration is therefore, -with one node in the centre, 
 iiods so treated are analogous -with open organ pipes. For example, 
 the vibration number varies inversely as the length of the rod ; and 
 the time of a complete -vibration is the same as that required for a 
 pulse to run to and fro over the rod; so that here again the velocity 
 of sound in the substance of which the rod is composed, may be 
 ascertained by multiplying the vibration number of the note 
 produced, by t-wice the length of the rod. 
 
 As just stated, the simplest form in which these rods can -vibrate, 
 is with one node in the centre ; the next simplest, as in the case of 
 
ON THE VIBBA TIONS OF BODS, PLATES, ETG. 121 
 
 the open organ pipe, is with two nodes ; the next, vidth three, and 
 so on; the partials produced being those of the complete series, 
 1, 2, 3, 4, 5, &c. 
 
 2. Coming now to the lateral vibrations of rods, we find these 
 may also be classified according as the rods are, — 
 
 (a) Fixed at both ends, 
 
 (6) Fixed at one end only, 
 
 (c) Free at both ends, 
 (o) A rod fixed at both ends, -vibrates laterally in exactly the 
 same manner as a string ; that is, it may vibrate as a whole, form- 
 ing one vential segment ; or with a node ia the centre, and two 
 ventral segments ; or, with two nodes, and three ventral segments, 
 &c. The relative vibration rates are, however, very difEerent, as 
 may be seen from the following table : — 
 
 segments 1, 
 
 2, 
 
 3. 
 
 4. 
 
 vibration rates 9 
 
 : 25 
 
 : 49 
 
 : 81 
 
 or 32 
 
 : 52 ; 
 
 ; 7ii ; 
 
 ; 92, 
 
 (J) In laterally vibrating rods fixed at one end, the vibration 
 number varies inversely as the square of the length. Chladni 
 endeavoured to utilize this fact, in ascertaining the vibration 
 number of a musical sound. He fiist obtained a strip of metal of such 
 a length, that its vibrations were slow enough to be counted. 
 Suppose, for example, a strip is taken, 36 inches long, and that it 
 vibrates once in a second. Eeducing it to one-third that length, 
 according to the above law, it will vibrate nine times per second. 
 Eeducing it to six inches, it will make 36 vibrations per second; to 
 three inches, 144 vibrations; to one inch, 1,296 vibrations. By a 
 little calculation, it is easy to find the vibration number of any 
 intermediate length. 
 
 The relative rates of vibration of the partial tones of such rods, 
 are very complex ; the second partial is more than two octaves 
 above the fundamental, and the others are correspondingly distant 
 from one another. Examples of instruments, in which these 
 lateral vibrations of rods fixed at one end are utilized, may be 
 found in the musical box and the bell piano. 
 
 (c) The simplest mode in which a rod free at both ends can 
 vibrate laterally, may be experimentally observed by grasping a 
 lath some six feet long, with both hands, at about one foot from 
 either end, and striking or shaking it in the centre. It wiU.be 
 found that there are two nodes, as shown in fig. 61, A- The next 
 
122 
 
 EAND-BOOE OF ACOUSTICS. 
 
 I'lO. 61. 
 
 simplest is with three nodes, fig. 61, B. The tones, corresponding 
 to these divisions, rise very rapidly in pitch, thus : 
 
 Numher of nodes 2, 
 
 3, 
 
 4, 
 
 5, 
 
 6, 
 
 7 
 
 Approximate vib. rates 9 
 
 25 
 
 49 
 
 81 : 
 
 121 : 
 
 169 
 
 or 3'^ 
 
 : 5^ 
 
 : 7^ 
 
 : 9-^ 
 
 : 11^ 
 
 : 13^ 
 
 The Harmonicon is an example of an instrument, in which the 
 lateral vibrations of rods free at both ends are utilized ; but the 
 most important member of this class is the Tuning-fork. This in- 
 strument, generally constructed of steel, may be considered as 
 derived from a straight bar, such as that depicted at the lower part 
 of fig. 62, by folding it in two, at the middle. The tone of the bent 
 
 N 
 
 Fia. 62. 
 
 bar wUl be somewhat flatter than the original straight one, and the 
 nodes, which in the straight bar were near the two ends, will have 
 approached very close together in the bent one. Fig. 62 shows by 
 the short marks, this gradual approach of the nodes, as the bar is 
 more and more bent; and fig. 63, by its thin and dotted lines, 
 represents the two extreme positions of the fork, while sounding its 
 fundamental. When the prongs are at their extreme outward 
 position h m, the portion between the nodes p and q rises ; when 
 they are closest together, at n f, this same portion descends. Thus 
 
ON THE VIBRATIONS OF BOBS, PLATES, ETC. 123 
 
 while tie prongs move horizontally the portion between p and q 
 vibrates vertically. To this portion there is usually welded or 
 screwed, an elongated piece of steel, which shares this vertical 
 motion, and does duty as a handle. "When this handle is placed 
 upon a sound-board, its vertical vibrations are commim.icated to it ; 
 a larger body of air is set in motion, and thus the sound of the fork 
 augmented. A tuning-fork does not divide like a straight bar into 
 four vibrating segments with three nodes; its second complete form 
 of vibration, which corresponds to the first overtone, is with four 
 nodes, two at the bottom and one on each prong. In some forks, 
 examined by Helmholtz, the relative rates of the fundamental and 
 first overtone, varied from 1 : 5'8 to 1 : 6'6. The overtones of 
 tuning-forks are consequently very distant from the fundamental 
 and from one another; the first overtone, as we see from the above, 
 being more than two octaves above the fundamental. The rates of 
 vibration of the whole series of overtones, starting with the first 
 overtone, are approximately as 9, 25, 49, 81, &c., that is, as the 
 squares of the odd numbers, 3, 5, 7, 9, &o. 
 
 These high overtones are very evanescent, and soon leave the 
 fundamental tone pure and simple. This is especially the case, as 
 already observed, when the fork is mounted on a resonance 
 chamber, tuned to its fundamental. The fork should either be 
 struck with a soft hammer, or carefully bowed. Striking with a 
 hard metallic substance, favours the production of the higher 
 partials, for the reason given in the case of pianoforte strings. 
 Large forks, when too rapidly bowed, produce very powerful over- 
 tones. The best method of keeping tuning-forks in continuous 
 vibration, is by means of electro-magnets, as already described in 
 Chap. Vin. 
 
 The pitch of a tuning-fork is only very slightly affected by heat. 
 The eflect of increase of temperature on a fork, is to slightly flatten 
 it; for the fork itself expands, and its modulus of elasticity is 
 lowered on heating ; both of these causes combining to lower the 
 pitch. The variation with temperature is only about one vibration 
 in 21,000, for each degree Fahrenheit. Porks are also little affected 
 by ill usage. A slight amount of rust is imperceptible in its 
 influence on pitch ; and with a very large amount, such as could 
 only occur through great carelessness, the error is never likely to 
 exceed 1 in 250. Bust about the bend has a much greater influence 
 over the pitch, than at the ends. Tuniug-forks are perhaps most 
 injured, by wrenching or twisting of the prongs, such as might 
 
124 
 
 HAND-BOOK OF ACOUSTICS. 
 
 occur tbrougli a fall, or by screwing or unscrewing tliem in and 
 out of resonance boxes. 
 
 ViBEATIONS OF PLATES. 
 
 Though not of much importance in reference to music, these 
 vibrations are of much interest, on account of the beautiful method 
 by which their forms are analysed. The plates usually employed 
 are constructed of either metal or glass, the metal being usually 
 brass. Any regular shape may be adopted, the most common being 
 the circular and square forms. The plate is firmly fastened at the 
 centre or some other point, to a stand ; and the vibrations are best 
 set up, by bowing the edge of the plate with a double-bass bow. 
 The rate of vibration of a circular plate is directly proportional to 
 the thickness, and inversely proportional to the square of the 
 diameter. 
 
 A node can be formed at any desired point, by touching that 
 point firmly, while bowing. By thus successively touching various 
 parts of the plate, a variety of notes of different pitches, corre- 
 sponding to its overtones, may be obtained, the plate vibrating 
 
 differently for each note. About 100 years ago, Chladni discovered 
 the method of rendering these different vibration forms visible, by 
 
ON THE VIBRATIONS OF BOSS, PLATES, ETO. 125 
 
 strewing sand lightly and evenly over the plate before bowing. 
 When a plate, thus treated, vibrates, the sand being violently 
 agitated over the vibrating segments, is rapidly jerked away from 
 these parts, and arranges itself along the nodal lines (fig. 64). 
 
 The simplest way in which a circular plate can vibrate, is in four 
 segments (fig. 65, A) ; the next simplest in six segments (fig. 65, B); 
 
 the next in eight (fig. 65, C); and so on. Much more complicated 
 figures, with nodal circles, may be obtained by stopping the plate at 
 appropriate points and bowing accordingly. 
 
 Pigure (66, A) shows the simplest way in which a square plate 
 can vibrate, and (fig. 66, B) gives the next simplest form ; the note 
 
 produced in the latter case being the fifth above that produced in 
 the former. The sand figures become very complicated and 
 beautiful as the tones rise in pitch ; (fig. 66, 0) representing one of 
 the least complex. Adjacent segments are always in diflerent 
 phases; that is, while one is above its ordinary position, the adjacent 
 ones are helow it. This can be proved experimentally, as will be 
 subsequently shown. 
 
 Bells. 
 Theoretically, a bell vibrates in the same way as a plate fixed at 
 the centre. The simplest way in which it can vibrate is with four 
 nodal lines, the tone thus produced being the fundamental. The 
 
126 HAND-BOOK OF AOOUSTIOS. 
 
 next simplest form of vibration is vnth. six nodes, tlie next with 
 eight, and so on ; an odd number of nodes never being produced. 
 The corresponding vibration rates are as follows : — 
 
 Number of nodes 4, 6, 8, 10, 12, 
 Eelative vibration rates ■^ : 9 : 16 : 25 : 36, 
 or 22 : 32 : 42 : 52 : 62. 
 
 Practically, however, owing among other things, to unavoidable 
 irregularities in the casting, no church bell ever has a single 
 fundamental, or only one series of overtones. To this fact is due, 
 the well-known difSculty in ascertaining the precise pitch of such 
 a bell ; the discords and throbbings that are heard, even in the best 
 sounding bells, when the listener is close to them, may be put down 
 to the same cause. There are no absolute points of rest in a 
 vibrating bell, for the fundamental is never produced alone ; but it 
 is easy to explore the surface of the sounding bell with a light ball 
 suspended from a thread, and thus find the places of least and 
 greatest motion, the ball being violently dashed away from the 
 latter. 
 
 Membranes. 
 
 These, in the form of side drums, bass drums, and kettle drums, 
 are used in the orchestra rather to mark the rhythm, than to 
 produce a musical sound; although it is true, the last mentioned are 
 approximately tuned to two or three notes of the 16 foot octave. 
 
 They may be studied by exciting them sympathetically, by means 
 of organ pipes, and analysing the vibration forms produced, by 
 scattering sand over them, as in the case of plates. The nodal 
 lines are circles and diameters, or combinations of these. 
 
 SUMMAHT. 
 
 Bods vibrating longitudinally and (1) free or (2) fixed at both ends 
 are analogous with open organ pipes : their vibration numbers are 
 inversely as their lengths, and they give the complete series of 
 partial tones. 
 
 When (3) fixed at one end only, they are analogous with stopped 
 organ pipes and give only the odd series of partials. 
 
 In (1) and (2) the time required for a complete vibration is the 
 same as the time taken by a pulse to move along the whole length 
 of the rod, and back again ; in (3) the time required for a complete 
 vibration is twice this time. -These facts being known, it is possible 
 to determine the velocities of sound in various solids, just as a 
 
ON THE VIBHATIONS' OF BODS, PLATES, ETO. 127 
 
 knowledge of the similar fact in tte analogous case of organ pipes, 
 renders it easy to ascertain the yelooities of sound in different 
 
 Eods yibrating laterally and (1) fixed at both ends may vitrate in 
 1, 2, 3, 4, 5, &c., segments. If (2) free at both ends they can 
 vibrate only in 3, 4, 5, 6, &o., segments. Beckoning the two fixed 
 ends in the former case as nodes, the vibration rates of the segments 
 are as follows : 
 
 Number of nodes 2, 3, 4, 5, 6, &c. 
 Vibration rates 32, 52, 72, 92, 112. 
 
 When fixed at one end only, the vibration number varies inversely 
 as the square of the length, and the overtones are very distant from 
 one another and from the fundamental. 
 
 The 1st overtone of the tuning-fork is more than two octaves 
 above the fundamental, and the 2nd overtone more than an octave 
 above the 1st. 
 
 The pitch of the tuning-fork is very slightly afleoted by ordinary 
 changes of temperature. 
 
I2& 
 
 CHAPTER XII. 
 
 Combination Tones. 
 In the preceding chapters, musical soimds, whether simple or 
 compound, have been considered singly, and the phenomena they 
 present, so studied. When two or more such sounds are heard 
 simultaneously, other phenomena usually occur. In the present 
 chapter, we proceed to study one of these. 
 
 When two musical tones, either simple or compound, are sounded 
 together, new tones are often heard, which cannot be detected when 
 either of the two tones is sounded bv itself. For example : press 
 
 down the keys corresponding to the notes C and 
 
 ^'■i 
 
 ^ 
 
 on the harmonium, and blow vigorously. On listening attentively, 
 a tone may usually be heard, nearly coinciding with F|, v^ 
 
 which will not be heard at all, when either of the two notes above 
 is separately sounded. Again, sound the two notesBt'^andPi 'S^'lj ^ 
 loudly on the same instrument. With attention, a tone will be 
 
 heard almost exactly coinciding with D, es: 
 
 f^ 
 
 Baise either of 
 
 the fingers, and this tone will vanish. 
 
 These tones, which make their appearance when two independent 
 tones are simultaneously sounded, have been termed by some 
 authors. Resultant Tones, by others Combination Tones. The inde- 
 pendent tones, which give rise to a combination tone, may con- 
 veniently be termed its generators. 
 
 Two varieties of combination tone are met with : in the one, the 
 vibration number of the combination tone is equal to the difference 
 
COMBINATION TONES. 129 
 
 between the vibration numbers of its generators ; in the other, it is 
 equal to their sum. The former are consequently termed Difference 
 or Differential Tones, and the latter, Summation Tones. 
 
 DlTFEEENTIAI, TONES, 
 
 These tones have been known to musicians for more than a 
 century. They appear to have been first noticed in 1740 by Sorge, 
 a German organist. Subsequently, attention was drawn to them 
 by Tartini, who called them " grave harmonics," and endeavoured 
 to make them the foundation of a system of harmony. 
 
 As already stated, the vibration number of a diflerential tone, is 
 the difference between the vibration numbers of its generators. It 
 is easy therefore to calculate what differential any two given 
 generators will produce. For example, two tones, having the 
 vibration numbers 256 and 412 respectively are sounded simul- 
 taneously, what wiU be the vibration number of the differential 
 tone produced ? Evidently 412 — 256, that is 156. 
 
 Further, if the two generators form any definite musical interval, 
 the differential tone may be easily ascertained, though their vibra- 
 tion numbers may be unknown. For example, what differential 
 will be produced by two generators at the interval of an octave ? 
 Whatever the actual vibration numbers of the generators, they 
 must be in the ratio of 2 : 1. Therefore the difference between 
 them must be the same as the vibration number of the lower of the 
 two generators ; that is, the differential will coincide with the lower 
 generator. Or shortly it may be put thus : — 
 
 generators J ? , 
 
 Differential Tone, d = 1 
 
 Again, what differential will be produced by two tones at the 
 interval of a Fifth ? The vibration numbers of two tones at the 
 interval of a Fifth are as 2 : 3, difference = 3 — 2^1. Therefore 
 the vibration number of the differential will be to the vibration 
 number of the lower of the two tones as 1 : 2; that is, the 
 differential will be an octave below the lower generator, or briefly, 
 
 generators | || ~ g 
 
 Differential Tone, d = 1 
 
 In the following Table, the last column shows the Differentials 
 produced by the generators given in the second column. 
 
130 
 
 HAND-BOOK OF ACOUSTICS. 
 
 Ikteeval. 
 
 GEKEEAT0R8. 
 
 Relative 
 Tib. Bates. 
 
 DiFFEEBNCE. 
 
 DiFFL. Tones. 
 
 Octave 
 
 
 2 
 1 
 
 
 di 
 
 Fiftli 
 
 is' 
 
 3 
 
 2 
 
 
 d 
 
 Fourth 
 
 (di 
 s 
 
 4 
 3 
 
 
 d, 
 
 Major Third 
 
 U' 
 
 5 
 
 4 
 
 
 di 
 
 Minor Third 
 
 (si 
 
 6 
 
 5 
 
 
 d, 
 
 Major Sixth 
 
 (1' 
 
 5 
 3 
 
 2 
 
 f 
 
 Minor Sixth 
 
 
 8 
 5 
 
 3 
 
 s 
 
 Tone 
 
 [t 
 
 9 
 
 8 
 
 1 
 
 d^ 
 
 Semitone.. . . 
 
 {% 
 
 16 
 15 
 
 1 
 
 d. 
 
 It is evident from the above, that when the two generators are 
 at a greater interval apart than an octave, the differential tone lies 
 between them ; when they are at the exact interval of an octave, 
 it coincides with the lower generating tone; but when at a less 
 interval than an octave it lies below this latter, and the smaller the 
 interval, the lower relatively will be the differential. 
 
 Any of the partials of compound tones may act as generators, if 
 sufficiently powerful. Thus, if { ^ be two compound generators, 
 we see from the above table that the fundamentals d' and n may 
 give rise to the differential tone Sj; but the 2nd partials d' and n', if 
 sufficiently powerful, may also generate the differential s ; or the 
 partial n' and the fundamental d' may produce differential d|; and so 
 on. It is only in rare cases, however, that the overtones wiU be 
 strong enough to produce audible differentials. 
 
COMBINATION TONES. 131 
 
 The one condition necessary for the production of diflerential 
 tones, is, that the same mass of air be simultaneously and power- 
 fully agitated by two tones ; that is, the tones must be sufficiently 
 loud. As the intensity of the generators increases, so does that of 
 the differential, but in a greater ratio. The condition just referred 
 to, is best satisfied on the Double Syren of Helmholtz (fig. 22), two 
 circles of holes in the same chamber being open. The differentials 
 produced by this instrument are exceedingly powerful. 
 
 Two flageolet fifes, blown simultaneously by two persons, also 
 give very powerful differentials. The latter may be approximately 
 ascertained from the table given above, but allowance must be 
 made for the tempered intervals. Thus, if the tones G^ and F^ be 
 loudly blown, the differential produced will be very nearly that 
 given ia the table, viz., P, three octaves below. 
 
 Differential Tones are very conspicuous on the English 
 Concertina : in fact, so prominent are they, that their occurrence 
 forms a serious drawback to the instrument. They may be plainly 
 heard also on the Harmoniuna and American organ : especially 
 when playing in thirds on the higher notes. Two soprano voices 
 singing loudly, will produce very audible differentials. Owing to 
 the evanescent character of its tones, it is difficult to hear differ- 
 entials on the pianoforte, but they can be detected even on this 
 instrument by a practised ear. Differential tones may be easUy 
 obtained also from two large tuning-forks, which should be struck 
 sharply. Two singing flames are also well adapted for producing 
 these tones. 
 
 Not only do two generating tones give rise to a differential, but 
 this differential may itself act as a generating tone, together with 
 either of its generators, to produce a second differential tone ; and 
 this again may in its turn act as a generator in combination 
 with one of the original generators, or with a differential, to 
 produce a third ; and so on. 
 
 The differential tone z^ which is generated by two simple or 
 compound tones x and y, is termed a differential of the first order. 
 If X and Z| or y and Z| generate a differential z^, this is said to be of 
 the second order ; and so on. Differential tones of the second order 
 are usually veiy faint, and it requires exceedingly powerful tones 
 to make differential tones of the third order audible : in fact, the 
 latter are only heard imder very exceptional circumstances. 
 
 To determine what differentials of the second and third order can 
 be present, when two tones at any definite interval are loudly 
 
132 
 
 HAND-BOOK OF ACOUSTICS. 
 
 two 
 the 
 
 sounded, -we proceed as before. For example, let the 
 generators be I 5f'. The relative rates of vibration being j ® , 
 relative vibration rate of the difEerential of the first order 
 = 5 — 4 =: 1. Subtracting this from the generators 5 and 4 we 
 obtain the relative vibration rates of the differentials of the second 
 order viz. 5 — 1 = 4 and 4 — 1^3, this latter only being a new 
 tone. Again, subtracting this 3 from the higher generator, we get 
 another new tone 5 — 3 :^ 2, a difEerential of the third order. 
 Thus, omitting duplicate tones we have — 
 
 generators | ^i = 4 
 Differential of 1st order, d| = 1 (= 5 — 4) 
 2nd „ s =3(=4 — 1) 
 3rd „ d = 2 (= 5 — 3) 
 The 2nd, 3rd, 4th, and 5th columns of the following table show the 
 differentials of the 1st, 2nd, 3rd, and 4th order which may be pro- 
 duced by the tones in the 1st column. 
 
 Intebval. 
 
 DiFF. OP let. 
 
 OEDEE. 
 
 2nx> oedeb. 
 
 3rd oedeb. 
 
 4th obdbb. 
 
 Fourth \^^—t 
 
 (.0 
 
 i = d, 
 
 2 = d 
 
 
 
 Major 3rd (J = ° 
 
 i = d, 
 
 3 = s 
 
 2 = d 
 
 
 Minor 3rd ( ^1 = ^ 
 
 i=d, 
 
 4 = di 
 
 2 = d 
 
 3 = s 
 
 
 Major 6th [^'1^3 
 
 2 = f 
 
 l=f, 
 
 4 = fl 
 
 
 Minor 6th ( ^1 = ? 
 ( n= 
 
 3 = s 
 
 2 = d 
 
 6 = si 
 1 =d, 
 
 7 = ta' 
 4=di 
 
 ^-(5: = ^ 
 
 l=d, 
 
 7 = t*a 
 
 2 = d, 
 6 = s 
 
 5 = n 
 4 = d 
 
 3= s, 
 
 •Neai'ly, 
 
 
 
 
 
COMBINATION TONES. 133 
 
 It "will te seen from the aboye, that in general a complete series 
 of tones may be produced, correspondiag to the complete series of 
 partial tones, 1, 2, 3, 4, &c., up to the generators. It will be 
 noticed also, that the same tones may occur with compound tones, 
 as differential tones of their upper partials. 
 
 Though combination tones are generally subjective phenomena, 
 yet on some instruments, as for example, the Double Syren and 
 the Harmonium, they are objective, or at any rate partly so. As a 
 proof of this fact, it is found that difierential tones on these 
 instruments, maybe strengthened by resonance. Thus, sound loudly 
 Gi and D* on a harmonium, and tune a resonator to the differential G. 
 By alternately applying the resonator to, and withdrawing it from 
 the ear, while the generating notes are being sounded, it is easy to 
 appreciate the alternate reinforcement and falling ofi of the G. 
 
 It was formerly thought, that differential tones were formed by 
 the coalescence of beats (see next Chap.), a supposition which was 
 supported by the fact, that the number of beats generated by two 
 tones in a second, is identical with the vibration number of the 
 differential tone they generate. That this is not the cause of 
 Differential Tones, will be seen from the following considerations : 
 1st. Under favourable circumstances, the rattle of the beats 
 
 and the differential tone may be heard simultaneously. 
 2nd. Beats are audible, when the generating tones are very 
 faint, in fact, they may be heard even when the generating 
 tones are inaudible. Differentials, on the other hand, 
 invariably require tolerably loud generators. 
 3rd. This supposition offers no explanation of the origin of 
 the analogous phenomenon of Summation Tones. 
 Pinally, Helmholtz has offered a theory of the origin of Differential 
 Tones, which satisfactorily explains all the phenomena of both 
 Differential and Summation tones. This theory is difficult to 
 explain, without such recourse to mathematics, as would be unsuit- 
 able to a work like this. We must be content, therefore, to state 
 it in general terms as follows : 
 
 When two series of sound-waves simultaneously traverse the 
 same mass of air, it is generally assumed that the resultant motion 
 of the air particles is equal to the algebraic sum of the motions, 
 that the air particles would have had, if the two series had 
 traversed the mass of air, independently of one another. This, 
 however, is only strictly true, when the amplitudes of the sound- 
 waves are very small, that is, when the air particles oscillate only 
 through very small spaces. When the amplitudes of the waves are 
 
134 
 
 HAND-BOOK OF ACOUSTICS. 
 
 at all considerable in proportion to their length, secondary waves 
 are set up, which on reaching the ear give rise to Combination 
 Tones. The higher octave of the fundamental tone, which may be 
 frequently heard, when a tuning-fork is sharply struck, has a 
 similar origin. 
 
 Summation Tones. 
 
 Helmholtz, as already mentioned, worked out the theory just 
 referred to, mathematically, and proved that two tones with given 
 vibration numbers, may not only produce a third tone, having its 
 vibration number equal to their difference, but also another tone 
 equal to their sum. To this latter sound, the term " Summation 
 Tone " is applied. 
 
 It is not difficult to satisfy oneself experimentally of the reality 
 of the summation tones on such an instrument as the Harmonium 
 or American Organ ; indeed, these tones are much louder than is 
 generally supposed. Thus if Pg ^"^^ C| be selected, the summation 
 tone will be Aj, which with careful attention may generally be 
 detected; the following table gives in the last column the 
 summation tones that may be produced by the generators in the 
 second column. 
 
 Inteeval. 
 
 Genbbatobs. 
 
 Relative 
 ViB. Bates. 
 
 Sum. 
 
 Sdhmatioh 
 Tones. 
 
 Octave 
 
 {d, 
 
 2 
 1 
 
 3 
 
 S 
 
 Fifth 
 
 fs: 
 
 3 
 2 
 
 5 
 
 n 
 
 Fourth 
 
 (d 
 
 4 
 3 
 
 7 
 
 ta* 
 
 Major 3rd. . 
 
 is; 
 
 5 
 
 4 
 
 9 
 
 r 
 
 Minor 3rd . . 
 
 (s, 
 (n. 
 
 6 
 5 
 
 11 
 
 f 
 
 Major 6th. . 
 
 (Si 
 
 5 
 3 
 
 8 
 
 f 
 
 Minor 6th . . 
 
 
 8 
 6 
 
 13 
 
 1* 
 
 •Approximatdy. 
 
COMBINATION TONES. 135 
 
 Summary. 
 A Oomlination or Resultant Tone is a tMrd soimd, •wHoh. may be 
 heard, when two tones of different pitch, are simultaneously sounded, 
 and which is not heard, when either of these two tones is sounded 
 alone. 
 
 The two tones which give rise to a Combination Tone are termed 
 its generators. 
 
 There are two kinds of Combination Tone — 
 
 (1). The Differential Tone : the vibration number of which is 
 the difference of the vibration numbers of its generators ; 
 (2) the Summation Tone: the vibration niuaber of which is the 
 sum of these vibration numbers. 
 Difierential Tones may be of various orders. 
 A Differential of the 1st order is that which is produced by two 
 independent tones or generators. 
 
 A Differential of the 2nd order is that which is produced by the 
 Difierential of the 1st order, and either of the generators. 
 
 A Differential of the 3rd order is that which is produced by the 
 Difierential of the 2nd order, and either of the foregoing tones; i.e., 
 either the Difierential of the 1st order, or one of the generators. 
 
 A Differential of the 4th order is that which is produced by the 
 Differential of the 3rd order and either of the foregoing tones ; and 
 so on. 
 Differential Tones are not the result of the coalescence of beats. 
 
136 
 
 CHAPTER XIII. 
 
 On Lstteetebenoe. 
 We have now to consider another of the phenomena whidi may 
 occur, when two musical sounds are heard simultaneously ; and in 
 the present chapter, we shall suppose the two sounds in question to 
 be simple tones. 
 
 Let the horizontal dotted straight lines in fig. 67, represent 
 surfaces of still water ; and let two series of waves of equal length 
 
 Fio. 67. 
 
 and amplitude be, at the same moment, passing from left to right. 
 Let the curved liae (1) represent in section the form, that the waves 
 would have, if those of the first series alone were present; 
 and let (2) represent, in the same way, the form, that the water 
 would assume, if the second series of waves alone were passing. 
 Let us also, in the first place, suppose the two series of waves to 
 coincide, so that crest falls on crest, and trough on trough ; that is, 
 let them both be in the same pJiase, as represented in (1) and (2). 
 Under these circumstances, each series will produce its full efFeot, 
 
ON INTEBFEBENCE. 
 
 137 
 
 independently of the other, as explained in Chap. Vlil, pp. 82 & 83; 
 crest being added to crest, and trough to trough, to produce a wave 
 (3) of the same length as each of the coincident waves, but of twice 
 the amplitude of either. 
 
 Now, let us suppose, that these two series of waves come together 
 in such a way, that the crests of one exactly coincide with the 
 troughs of the other : in other words, let them be in opposite phase 
 as represented in fig. 68 (4) (5). In this case, by the use of the 
 
 (6)- 
 
 FiG. 68. 
 
 same kind of reasoning as employed in Chap. VIII, we find that, as 
 the drops of water are solicited in opposite directions, by equal 
 forces, at the same time, the result is no wave at all, fig. 68 (6). 
 We have supposed here, that the waves in both series have the 
 same amplitude. If they have different amplitudes, it is evident 
 from the above, that, 1st, when the two series are in the same phase, 
 the amplitude of the resultant wave is equal to the sum of the 
 amplitudes of the constituents ; 2nd, when the two series are in 
 opposite phases, the amplitude of the resultant wave is equal to the 
 difference of the amplitudes of the constituents. Further, it is 
 evident, that if they are neither in the same nor opposite phases, 
 the amplitude of the resultant wave will be intermediate between 
 these two limits. 
 
 Now we may take the curves (1) (2) (4) (5) of figs. 67 and 68, as 
 the associated waves of two simple sounds, arid therefore at once 
 deduce the following results. 1st, Two sound waves of the same 
 length and amplitude, and in the same phase, produce a resultant 
 wave of the same length, but twice the amplitude of either wave. 2nd, 
 Two sound waves of the same length and amplitude, but in opposite 
 phase, destroy one another's eflects, and no wave is produced. 
 3rd, Two sound waves of the same length but diflerent amplitudes, 
 will produce a wave of the same length as either wave, but having 
 an amplitude equal to the sum or difference of their amplitudes, 
 
138 HAND-BOOK OF ACOUSTICS. 
 
 according as the waves are in tte same or opposite phase. 4th, 
 If the two sound waves are not exactly in the same or opposite 
 phase, the amplitude of the resultant wave will be intermediate 
 between these hmits. 
 
 If one sound wave have twice the amplitude of another, the 
 intensity of the tone produced by the one will be four times that 
 produced by the other, since intensity varies as the square of the 
 amplitude. It follows therefore from the above, that when two 
 simple tones of the same pitch and intensity are sounded together, 
 the two may so combine as to produce 1st, a simple tone of the 
 same pitch, but of four times the intensity of either of them ; 2nd, 
 silence; or 3rd, a simple tone of the same pitch, but intermediate in 
 intensity between these two limits ; according as their sound waves 
 come together in the same, opposite, or intermediate phases. 
 
 The fact that two sounds may so interfere with one another as to 
 produce silence, strange as it may seem at first, can be demonstrated 
 experimentally, and is a special case of the general phenomenon of 
 " Interference oJ waves." The only di&culty in the experimental 
 proof, is to obtain sound waves of equal length and intensity, and 
 in exactly opposite phase. Before explaining the way in which 
 this difficulty can be overcome, we shall take the following 
 supposititious case. 
 
 Let A and B (fig. 69) be two tuning-forks of the same pitch, and 
 let us consider only the right hand prongs A and B. Now if these 
 
 Fig. 69. 
 
 prongs are in the same phase, that is, both swinging to the right and 
 left, at exactly the same times, and if they are exactly a wave- 
 length apart, it is evident that the two series of waves passing 
 along A C, originated by their oscillation, will exactly coincide, 
 condensation with condensation, and rarefaction with rarefaction, 
 as represented by the dark and light shading. The same thing will 
 occur, if the distance between A and B be two, three, four, or any 
 whole number of wave-lengths. But suppose the distance from A 
 to B were only half a wave-length, as represented in fig. 70 ; 
 evidently, the condensations from the one fork will coincide with 
 the rarefactions from the other, and thus the air to the right of B 
 
ON INTEBFEBENGE. 
 
 139 
 
 Fig. 70. 
 
 ■will be at rest, as indicated by tbe imifonn shading. Precisely the 
 same thing would occur, if A and B were three or any number of 
 half wave-lengths apart. 
 
 Sir John Hersohel made use of this principle in the construction 
 of the apparatus shown in fig. "71. The tube of, which should be 
 
 A 
 
 y ^ 
 
 ^ /7=^ 
 
 Y 
 
 Fio. 71. 
 
 longer than represented in the figure, divides into two at/, the one 
 branch being carried round m, and the other round n. These two 
 branches again unite at g, to form the tube gp. The U shaped 
 portion n I, which slides air-tight by telescopic joints over the main 
 tube a h, can be drawn out, as shown in the figure. When a 
 vibrating fork is held at o, the sound waves produced, divide at/, 
 and pass along the two branches, reuniting at g, before reaching the 
 ear of the observer at p. Now if the TJ shaped portion is pushed 
 home to a, the waves through both branches reach the ear together; 
 but if it be gradually puUed out, a point is reached at which the 
 sound disappears altogether. From what has been said above, it 
 
140 HAND-BOOK OF ACOUSTICS. 
 
 ■will be seen, that tiiis is the case, when the right hand branch is 
 half a wave-length longer than the left hand branch, that is, when 
 « 6 is equal to one fourth of a wave-length. Thus, this instrument 
 may be used, not only to demonstrate the phenomenon of Inter- 
 ference, but also for roughly ascertaining the wave-length, and hence 
 the pitch of a simple tone. 
 
 The vibrating plate (fig. 72) is a very convenient instrument with 
 which to iUustrate the phenomenon of interference. In the brief 
 
 Fig. 72. 
 
 description of this instrument given in Chap. XI, it was stated that 
 adjacent sectors are always in opposite phase; that is, while one 
 sector is moving upwards, the adjacent ones are moving downwards. 
 If this be the case, it follows, that the sound waves originated 
 above two adjacent sectors are in opposite phase, and thus 
 interfere with one another, to diminish the resultant sound. 
 Accordingly, if the hand be placed above any vibrating sector, the 
 sound is not diminished, but increased. Still more is this the case 
 if the cardboard or wooden sectors, on the right of fig. 72, be held 
 over the segments of the plate when vibrating as shown in the figure ; 
 interference being then completely abolished, the remaining 
 A segments sound much more loudly. Thus, by sacri- 
 
 ficing a part of the vibrations, the remainder are 
 rendered more effective. 
 
 The effect of the interference of adjacent sectors 
 may be rendered visible, by the additional apparatus 
 shown in fig. 73. A B is a tube which branches into 
 two at the bottom, and is closed at A by a mem- 
 brane, upon which a few grains of sand are scattered. 
 Holding the ends of the branches over adjacent 
 Fio. 73. segments, the membrane is unaffected, and the sand 
 
ON INTEBFEBENOE. 
 
 141 
 
 remains at rest, for the condensation which enters at one branch 
 and the rarefaction which simultaneously enters at the other, unite 
 at B to neutralize one another's effects. When, however, the 
 ends are held over alternate sectors, the sand is violently agitated, 
 showing that they are in the same phase. 
 
 It is easy to illustrate the phenomenon of Interference, with no 
 other apparatus -than an ordinary tuning-fork. Let 0,0, fig. 74 
 
 Fig. 74. 
 
 represent the ends of«the prongs of such a fork, looked down upon, 
 as it stands upright. In the first place, let it be supposed, that 
 these prongs are moving towards one another. In this case, the 
 particles of air between the prongs wiU become more closely packed 
 together, and consequently will crowd out both above and below, 
 giving rise to condensations both in c and d. At the same time, in 
 consequence of the inward swing of the prongs, the air particles 
 to the left and right, sharing this movement, wiU be left wider 
 apart than at first ; that is, rarefactions wiU be formed at a and 6. 
 Now, let it be supposed that the prongs are making an outward 
 journey ; a partial vacuum wiU. then be formed between them, and 
 the air rushing in from without, vriU cause rarefactions at c and d ; 
 while and 0, pressing on the air at either side, wiU at the same 
 time, give rise to condensations at a and b. Thus, we see,' that as 
 long as a tuning-fork is vibrating, four sets of waves are proceeding 
 from it, two issuing in directly opposite directions from beWeen 
 the prongs, and two, also in directly opposite directions, at right 
 
142 EAND-BOOE FO A00U8TIGS. 
 
 angles to the first mentioned ; and as we have just seen, the waves 
 that issue from between the prongs are in the opposite phase, to 
 those that proceed at right angles to them ; that is, whenever there 
 are condensations at c and d, there are rarefactions at a and b, and 
 vice versa. Now, each of these four sets of waves, in passing 
 outwards from its source, will of course spread in aU directions; 
 and therefore, the adjacent waves will meet along four planes, 
 represented by the dotted lines in the figure. Along these lines, 
 therefore, the interference must be total ; that is, any air particle in 
 any one of them, which is urged in any direction by the waves in c 
 or d, will be urged in the opposite direction, with precisely equal 
 force, by the waves from a or 6; that is, it will remain at rest. 
 Consequently the dotted lines are lines of silence, the maximum of 
 sound being midway between any two of them. If the vibrating 
 fork were large enough, and a person were to walk round it in a 
 circle, starting from one of these points of maximum intensity, he 
 would find, that the sound gradually diminished as he approached 
 the dotted line, where it would be nil. Alter passing this point, 
 the sound would increase to the maximum, then diTniTn'gTi again, 
 and so on ; four points of maximum, and four of miniTnnTn intensity 
 occurring during the circuit. 
 
 To verify aU this, strike a tuning-fork, and then hold it with the 
 prongs vertical, and with the back of one of them parallel to the 
 ear. Note the intensity of the sound, and then quickly revolve 
 the fork half-way or a quarter- way round : the intensity is 
 unaltered. Now strike the fork again, and after holding it as at 
 first, turn it one-eighth round, so that it is presented comer- wise to 
 the ear ; the sound will be aU but extinguished. Again strike the 
 fork, and holding it to the ear as at first, revolve it slowly: the four 
 positions of greatest intensity and the four interference positions 
 axe readily perceived. To vary the experiment, again strike the 
 fork, and rotate it rapidly before the ear : the effect is very similar 
 to the beats, to be studied presently. 
 
 These experiments are much more effective, and the results can be 
 demonstrated to several persons at once, when a resonator is used. 
 For an ordinary 0> tuning-fork, a glass cylinder closed at one end, 
 about f inch in diameter, and between six and seven inches long, is 
 very convenient. If not of the exact length to resound to the fork, 
 a little water may be gradually poured in, as described in Chap. VH. 
 "When the vibrating fork is held with the back of one prong 
 parallel to the top of the resonator, or at right angles to this 
 portion, the sound of the fork is much intensified ; but when held 
 
ON INTEBFEBENOE. 
 
 143 
 
 ■with, the corner of the prong towards the resonator, the sound dies 
 out. The efiects produced when the fork is revolved, are precisely 
 the same as those above mentioned, but much intensified by the 
 resonance of the tube. 
 
 These experiments with resonators may be varied in many ways. 
 Thus, first hold the vibrating fork with the edge of one of the 
 prongs towards the resonator ; no sound is heard. Now, keeping 
 the fork stiU in this position, move it along horizontally for a short 
 distance, so that only the lower prong is over the resonator ; the 
 sound will now buxst forth, for the side of the resonator cuts off the 
 "waves issuing from between the prongs, which before interfered 
 with those from the outside of the lower prong. 
 
 Again, tune two such resonators as the above to any tuning-fork, 
 and arrange them at right angles to one another, as represented in 
 fig. 75. Now hold the vibrating tuning-fork in such a position, 
 
 Fio. 75. 
 
 that, while the back of one of the prongs is presented to one 
 resonator, the space between them is presented to the other. Under 
 these circumstances, very little sound will be heard, for, from what 
 has been already said, it will be seen, that the waves proceeding 
 from the two resonators wiU always be in opposite phase, and thus 
 will neutralize one another's efieots. If, however, while the fork is 
 vibrating, we slide a card over the mouth of one of the resonators, 
 the other resonator will produce its due efiect, and the sound will 
 burst forth. 
 
 It is found, that two similar organ pipes placed together on 
 the same wind chest, interfere with one another ; the motion of the 
 air in the two pipes taking place in such a manner, that as the wave 
 streams out of one, it streams into the other and hence an observer 
 at a distance hears no tone, but at most the rustling of the air. 
 For this reason, no reinforcement of tone can be efieoted in an 
 
144 HAND-BOOK OF AGOUSTIGS. 
 
 organ, by combining pipes of tbe same kind, under the conditions 
 just referred to. 
 
 We pass on to consider the case of the interference of two simple 
 tones, which differ slightly in pitch. Let two tuning-forks, stand- 
 ing close together, side by side, be supposed to commence yibrating 
 together in exactly the same phase ; and for the sake of simplicity, 
 we will suppose their vibration numbers to be very small, viz., 15 
 and 16 respectively. Now, although these two forks may start in 
 exactly the same phase, that is, the prongs of each may begin to 
 move inward or outward together, this coincidence can evidently 
 not be maintained, since their vibration rates are different. The 
 flatter fork wiU. gradually lag behind the other, till, in halt a second, 
 it will be just half a vibration behind, having performed only 74 
 vibrations while the other fork has performed 8. At the end of half 
 a second, therefore, the two forks will be in com^plete opposition ; 
 the prongs of the one fork moving one way, while those of the 
 other fork are moving in the opposite. After the lapse of another 
 half second, the flatter of the two forks wiU be exactly one com- 
 plete vibration behind the other, and consequently the forks wiU be 
 in exact accordance again, as they were at first. These changes 
 wiU evidently recur regularly every second. Thus, assuming as we 
 have done, that the forks are in exactly the same phase at the 
 commencement, we find that, at the beginning of each successive 
 second, the sound-waves from the two forks coincide, condensation 
 with condensation, or rarefaction with rarefaction, to produce a 
 sound-wave of greater amplitude than either ; but at the half 
 seconds, the two series of sound waves will interfere, the conden- 
 sation of one with the rarefaction of the other, to produce a sound- 
 wave of less amplitude, or even, if the amplitudes of the two waves 
 are equal, to produce mom.entarily, no sound wave at all. These 
 changes in the amplitude of the resultant wave will evidentiy be 
 gradual, so that the effect on the ear will be as follows : at the 
 commencement, a sound of considerable intensity will be heard; 
 during the first half second, its intensity will diminish, tiU. at the 
 exact half second, it is at a TniniTTmrn , or may even be nil ; during 
 the next half second the intensity wiU increase, till at the beginning 
 of the next second the sound has the same intensity as at first. 
 Precisely the same changes will occur during each successive second; 
 BO that a series of cresoendos and diminuendos, or swells, will be 
 heard, one crescendo and one diminuendo being produced in the 
 present supposed case, every second. 
 
ON INTEBFBBENOE. 
 
 145 
 
 These alternations of intensity, which are perceived whenever 
 two tones of nearly the same pitch are sounded together, are 
 commonly termed Beats. 
 
146 EAND-BOOK OF A00V8TI08. 
 
 In order to obtain a clearer insight into this matter, let us suppose 
 the forks in the above case to commence vibrating, as before, in 
 exactly the same phase, and let us consider the waves produced 
 during the first haU second. As their vibration numbers are 
 assumed to be 16 and 15, the sharper fork vnM have originated 
 exactly 8, and the flatter fork exactly 7J waves during this period. 
 Let the 8 equal associated waves of fig. 76 B, represent the former, 
 as if they alone were present : and let the 7J associated waves of 
 fig. 76 A, represent the latter, on the same supposition. The two 
 series are placed one above the other, instead of being superposed, 
 for the sake of distinctness. The forks are supposed to be at the 
 right hand side of the figure, the waves travelling towards the left; 
 thus the first pair of waves originated, are now on the extreme left, 
 the next pair immediately behind these, and so on. In accordance 
 with the supposition, the two series of waves (which, it may be 
 noted, are not represented as of equal amplitude) commence in 
 exactly the same phase, but in consequence of their diSerence in 
 length, this exact accordance becomes less and less in succeeding 
 waves, till at length, those on the extreme right are in exactly 
 opposite phase. Now when the two forks are simultaneously 
 sounding, their sound-waves combine or interfere, to produce a 
 resultant wave, the associated wave form of which we can obtain, 
 by compounding the two associated wave forms, A & B, in the manner 
 before described. The thick curved line of fig. 76 has been thus 
 obtained ; and we see from it, that the two original sound-waves 
 coalesce, to produce a resultant sound-wave, which at first has an 
 amplitude equal to the sum of the amplitudes of its constituent 
 waves, but that the amplitude gradually diminishes, till in half a 
 second, it is only equal to the diSerence of the amplitudes of its 
 constituents. It is easy to see from the figure, that during the next 
 half second, the amplitude of the resultant wave will gradually 
 increase, till at the beginning of the next second, it will again have 
 reached its maximum. These alternations in the amplitude of the 
 resultant waves, produce of course in the resultant sound, corre- 
 sponding alternations of intensity, which, as already mentioned, 
 are termed Beats, and which may be represented in the ordinary 
 musical way by crescendo and diminuendo marks — 
 
 It is evident from fig. 76, that half a beat is formed by the 
 interference of the waves there represented, that is, in half a second. 
 Therefore when two sounds, the vibration numbers of which are 16 
 
ON INTERFERENCE. 
 
 147 
 
 and 16 respectively, are heard togetlier, 16 — 16 or 1 beat per second 
 will bo heard; that is, the number of beats per second, is equal to 
 the difference of the vibration numbers. It is true, that 16 
 vibrations per second would not produce a musical sound, but that 
 in no way affects the above reasoning. For suppose the vibration 
 numbers of the forks to have been 160 and 150; the figure will 
 represent the waves originated in one-twentieth of a second. Con- 
 sequently in this case half a beat will be formed in one-twentieth 
 of a second, or one beat in one-tenth of a second; that is, 10 = (160 — 
 150) beats per second. It is evident, therefore, that the nwniber of 
 leafs per second, dvs to two simple tones, is equal to the difference of 
 their respective vibration numbers. 
 
 For purposes of experimental study, wide stopped organ pipes are 
 well adapted for the production of beats between simple tones ; for 
 when such pipes are gently blown, the fundamentals only are 
 heard, or at most, accompanied by very faint third partials. If 
 two exactly similar pipes be used, the tones produced will of course 
 be in unison. To obtain beats, the pitcli of one may be slightly 
 lowered by shading the embouchure ; or better still, one of the pipes 
 instead of being permanently closed at the top, may be stopped by 
 a movable wooden piston, or plug, working air-tight in the pipe. 
 After the pipes have been brought into unison, the pitch of the one 
 may be varied to any desired extent, by moving the wooden piston, 
 which alters the length of the vibrating air 
 colunm. If the plug be moved very slightly 
 from its unison position, very slow beats 
 may be obtained, each beat lasting for a 
 second or more. The crescendo and sub- 
 sequent diminuendo of the beat is then 
 very perceptible. By gradually moving 
 the plug farther and farther from its unison 
 position, the beats follow one another more 
 and more rapidly, tiU at last they cease to 
 be separately distinguishable. 
 
 The interference of two such organ pipes 
 as the above, may be rendered visible by 
 the use of the manometric flame apparatus 
 shown in fig. 55. Instead of each tympanum 
 having its own flame however, the outlet 
 pipes from the two tympana unite into one 
 (fig. 77), with a single flame at the end. 
 
 
 
 □ 
 
 Fig. 77. 
 
148 
 
 HAND-BOOK OF A00USTI08. 
 
 Thus, -when the vibrating air columns are in the same phase, the 
 india-rubber membranes ■will vibrate simultaneously in the same 
 direction, so as to expel the gas with greater force, and thus pro- 
 duce a very elongated flame. On the other hand, when they are in 
 opposite phase the membranes will move simultaneously in opposite 
 directions, and thus, neutralizing one another's effects, their move- 
 ments will be without influence on the flame. The latter will 
 therefore rise and fall with the beats, of which indeed, they are the 
 optical expression. By the aid of a rotating mirror, the separate 
 vibrations of the flame may also be observed as explained on 
 page 5. 
 
 Instead of the organ pipes referred to 
 above, two of the singing flames described 
 in Chap. I, fig. 3, may be used; but in 
 tbis case, beats from overtones, as well as 
 from the fundamentals, will, in all proba- 
 bility, be heard. In order to vary the 
 pitch, one pipe should be supplied with a 
 sliding tube, as shown in the left hand 
 pipe of fig. 78. 
 
 Two unison tuning-forks may be used 
 to produce beats of varying rapidity. The 
 pitch of one may be lowered by attaching 
 pieces of bees-wax to its prongs, or, if the 
 forks be large, by fastening a threepenny 
 piece, by means of wax, to each prong. 
 With large forks, these beats also may be 
 optically expressed. A pencil of light 
 from the lamp L (fig. 79), passes through 
 the lens I, and then strikes against a little 
 • concave mirror fastened to one prong of 
 
 the fork T. From this mirror it is then reflected to a similar 
 mirror attached to the fork T', and is finally received on the 
 screen A. When the forks are at rest, only a spot of light appears 
 on the screen ; but if one fork is set vibrating, this spot lengthens 
 out, to form a vertical line of light. Now let both forks vibrate 
 together with equal ampKtudes, and in the firstplace suppose them to 
 be in unison : if they are in exactly the same phase, the line of light 
 will be twice as long as at first ; if in opposite phase, the Une will 
 be reduced to a spot : in any intermediate phase, the line will have 
 a length intermediate between these two extremes. In the next 
 
ON INTEBFEBENGE. 
 
 149 
 
 place, suppose the forks are not in exact unison ; as we have already 
 seen, they wiU he at one moment in the same phase, then gradually 
 diverge till in opposite phase, and again gradually converge to the 
 same phase. The Kne of light will vary coincidentally ; at one 
 moment being of considerable length, then gradually shortening 
 till but a mere spot, and then lengthening again. The beats, of 
 which this alternate lengthening and shortening is the optical 
 expression, will at the same time be heard. 
 
 If the beam of light in the above, instead of faUing on a screen, 
 be received on the revolving mirror of fig. 3, the separate 
 vibrations will, as it were, be visible, and will appear as represented 
 in fig. 79 0^, in which the varying amplitude of the sinuosities 
 corresponds to the varying intensity of the resultant sound. 
 
 If no better apparatus be at hand, beats may be studied on the 
 pianoforte, by loading one of the two wires of a note with wax, and 
 then striking the corresponding key ; or they may be observed by 
 stretching two similar strings on a violin, and after bringing them 
 into imison, throwing one more or less out of tune ; but in these 
 cases, as the tones are compound, the matter is complicated by the 
 beats of the overtones. 
 
 If two tuning-forks are nearly, but not quite in unison, and the 
 vibration number of one of them is known, it is easy to ascertain 
 the vibration number of the other, by counting the beats between 
 them, provided we know which is the sharper of the two. For 
 example, suppose we have a standard fork producing exactly 512 
 vibrations per second, and on sounding it with another fork, we 
 find that in half a minute, 90 beats are counted. Now 90 beats per 
 half minute, is at the rate of three beats per second ; but we know 
 
150 EANB-BOOK OF A00USTI08. 
 
 that the mimlier of beats geBorated by two sounds, is equal to the 
 difference of their vibration numbers; therefore the vibration 
 number of the fork imder trial, must be either 512 + 3 or 512 — 3; 
 that is, either 515 or 509, acoordiag as it is sharper or flatter than 
 the standard, — a matter, which the ear of the musician can easily 
 decide. 
 
 It is found by experience, that beats which occur at the rate of 
 from 2 to 5 per second, are the most easily counted. Beyond five 
 beats in a second, there is considerable difficulty in counting, owing 
 to their rapidity ; and below two beats in a second, there is also a 
 difficulty, owing to the length of time occupied by each loudness. 
 For ascertaining the pitch of instruments in the way just described, 
 cases of tuning-forks are constructed consisting each of twelve 
 forks, the vibration numbers of which increase by four vibrations 
 per second, from 412 to 456 for A, and from 500 to 544 for C. To 
 show the method of using them, we will take the following case. 
 It was desired to ascertain the pitch of a certain piano. In a pre- 
 liminary trial, by sounding the C with each of the forks, it 
 was found that it produced with the 536 fork, from 2 to 3 beats per 
 second, and with the 540 fork, beats at a somewhat slower rate. 
 The former was first taken, and the beats produced by it with the 
 pianoforte C, carefully counted for 30 seconds. The number was 
 found to be 75, which is at the rate of 44 = 2^ beats per second. 
 Therefore the vibration number of the note in question was 
 536 + 21 = 538J. To verify this, the 540 fork was sounded with 
 the pianoforte C; 44 beats were now counted in 30 seconds, that is 
 ^ ^ IJ beats per second, nearly. This gives the same result as 
 before, viz., 540 — 1^ = 538^. 
 
 It is possible, however, to ascertain the vibration number of a 
 musical sound by means of beats, independently of any previously 
 ascertained standard. This will be seen from the following con- 
 siderations. Suppose we have two forks, one of which gives the 
 exact octave of the other. Let us further suppose, that it is possible 
 to count the number of beats per second produced, when they are 
 sounded together, and let the number be, say 100. What wiH be 
 the vibration numbers of the forks ? Now, in the first place, it is 
 evident that, whatever they are, the difference between tiiem must 
 be 100 ; since the number of beats per second, produced by two 
 sounds, is equal to the difference of their vibration numbers. In 
 the second place, the vibration number of the higher fork must be 
 twice that of the lower, since they are an octave apart. Thus the 
 
ON INTEBFERBNOE. 151 
 
 problem reduces itself to finding two numbers, one of wliicli is 
 double the other, the difference between them being 100. Now 
 200 and 100 are the only numbers which satisfy these conditions, 
 and therefore the vibration numbers of the forks will be 200, and 
 100, respectively. 
 
 To put this in a general way 
 
 Let X denote the vibration number of the lower fork, then 2a! 
 wiU denote the vibration number of the higher fork, therefore if n 
 denote the number of beats per second produced by them 
 2a! — X ^ n 
 that is X := n 
 
 Therefore, if two sounds are exactly an octave apart, the number 
 of beats they generate per second, will be the vibration number of 
 the lower sound. 
 
 But when two sounds, at the interval of an octave, are heard 
 together, no beats at all are perceived. How is this difficulty to be 
 overcome ? Let us suppose we have two forks A and Z, an octave 
 apart, A beiog the lower one. Tune another fork B slightly 
 sharper than A, so that it produces with it, not more than 4 beats 
 per second ; tune another fork C sharper than B, and making with 
 it about 4 beats per second ; tune another fork D in the same 
 manner, to beat with C ; and so on, till we get a fork within 4 
 beats of Z. Now count accurately the number of beats between 
 A and B, B and 0, and D, and so on up to Z ; add these aU 
 together, and the total will evidently be the number of beats between 
 A and Z. 
 
 Listruments constructed on the above principle are called 
 Tonometers, of which there are two varieties: the Tuning-fork 
 Tonometer and the Eeed Tonometer. 
 
 ■ The Tuning-fork Tonometer was invented by Scheibler, who died 
 in 1837. One of his instruments, which still exists, consists of 56 
 forks, each of which produces four beats per second with the 
 succeeding one. Therefore, between the lowest and the highest 
 forks, there are 55 sets of four beats ; that is, 55 X 4 = 220, which, 
 by the above, must be the vibration number of the lowest fork, 440 
 being that of the higher one. 
 
 In Appim's Tonometer, the tuning-forks are replaced by reeds. 
 Although better adapted to all pxirposes of lecture illustration than 
 the Tumng-fork Tonometer, the Eeed Tonometer has two serious 
 drawbacks, viz. : the reeds do not retain their pitch with accuracy, 
 and their variation with temperature is unknown. 
 
152 BAND-BOOK OF ACOUSTICS. 
 
 The method of using the Tonometer is similar to that above 
 described, in the case of the standard forks. In the experienced 
 hands of Mr. Ellis, the tuning-fork Tonometer has given results at 
 least equal in accuracy to those obtained by means of any other 
 counting instrument (see table on page 39). 
 
 Stjmmabt. 
 
 When two series of sound waves of the same lengths and 
 amplitudes, traverse simultaneously the same mass of air : 
 
 (1) If the waves of the one series are in exactly the satne phase 
 as those of the other, resultant waves are produced of the 
 same length, but of douhle the amplitude ; 
 
 (2) If the waves of the one series are in exactly the opposite 
 phase to those of the other, the result is, — no wave ; 
 
 (3) If the waves of the one series are neither in the sam.6 phase 
 as, nor in opposite phase to, those of the other, the amplitude 
 of the resultant waves will be intermediate between the two 
 limits given above, viz., no amplitude at all, i.e., silence, and 
 twice the amplitude of the constituent waves. 
 
 When two simple sounds of the same pitch and intensity are 
 simultaneously produced the result is 
 
 (1) Silence; or 
 
 (2) A sound of the same pitch as, but of four times the intensity 
 of, either ; or 
 
 (3) A sound, intermediate in intensity between these two 
 limits, 
 
 according as the corresponding sound waves are in (1) opposite 
 phase, (2) the same phase, or (3) any relative position intermediate 
 between these two. 
 
 When two sounds differing slightly in pitch are simultaneously 
 produced, the flow of sound is disturbed by regular recurring throbs 
 or alternations in intensity, termed heats. These beats are due to 
 the alternate coincidence and interference of the two systems of 
 waves. 
 
 If the two tones be of equal intensity, the maximum intensity of 
 the beat, will be four tim/es that of either sound heard separately, the 
 miniTmiTn intensity being zero. 
 
ON INTERFEBBNGE. 153 
 
 The number of beats per second, due to simple tones, is equal to 
 tbe difference of their vibration numbers. 
 
 This fact is the principle of the Tonometer, of which there are 
 two varieties, 
 
 (1) The 2'Mm7i5r-/orfe Tonometer; 
 
 (2) The Seed Tonometer. 
 
134 
 
 CHAPTER XIV. 
 
 On Dissonance. 
 
 Having studied, in the preceding Chapter, the causes and 
 characteristics of beats, we now proceed to inquire into the effects 
 they produce, as they become more and more rapid. 
 
 Slow beats in music are not altogether unpleasant ; in low tones, 
 and in long sustained chords they often produce a solemn effect : 
 in higher tones, they impart a tremulous or agitating expression ; 
 accordingly, modem organs and harmoniums usually have a stop, 
 which, when drawn, brings into play a set of pipes or reeds, so 
 tuned, as to beat with another set, thus imitating the trembling of 
 the human voice and of violuis. 
 
 When, however, the beats are more rapid, they become 
 unpleasant to the ear. In studying this matter, it will be best to 
 begin with simple tones. Select two Ci tuning-forks, and gradually 
 throw them more and more out of tune, by sticking wax on the 
 prongs of one of them, as described in the last Chapter. Sound 
 the forks together after each addition of wax, and note the effect of 
 the increasing rapidity of the beats. It will be f oimd, that when 
 they number five or six per second, the effect begins to be 
 unpleasant, and becomes harsher and more jarring, as they grow 
 more and more rapid. Of course the beats soon become too rapid to be 
 counted by the unaided ear, but their rate can easily be ascertained 
 by subtracting the vibration numbers of their generators. "When 
 the beats amount to about 32 per second,, though they are too 
 rapid to be individually discriminated, yet the resultant sound has 
 the same harsh jarring intermittent character, that it has had all 
 along, only much more disagreeable. The two tones are now at 
 the interval of a semitone, about the worst discord in music, and no 
 one, who tries the above experiment, and notes carefully the effect 
 
ON DISSONANCE. 155 
 
 of the beats, as the interval between the forks increases from 
 imison to a semitone, can doubt that the discord here arises from 
 these beats. 
 
 Now, gradually increase the interval between the forks still 
 more ; the rapidity of the beats of course increases, but the 
 resultant sound becomes less and less harsh ; till finally, when the 
 beats number about V8 per second, all the harshness vanishes. At 
 this point, the interval between the forks is rather less than a minor 
 third. The interval at which the dissonance thus disappears, has 
 been termed the Beating Diatcmce. 
 
 The fact just alluded to, — that aU Discord or Dissonance between 
 musical tones arises from beats, — is one of Helmholtz's most 
 important discoveries. In order to thoroughly convince himself of 
 its truth, the student must proceed step by step. In the first place, 
 as we have seen, beats are reinforcements and diminutions of 
 intensity, which are due to the interference of two separate sound 
 waves. Now this being the case, i£ such reinforcements and 
 diminutions can be made to occur in the case of a single sound, 
 then, not only should beats be heard, but the harsh jarring we call 
 discord, which is supposed to be due to beats, should be heard also. 
 
 This was put to the test of experiment by Helmholtz, in the 
 following way. A little reed pipe was substituted for the wind 
 conduit of the upper box of his Syren (see page 33), and wind 
 driven through this reed pipe. The tone of this pipe could be 
 heard externally, only when the revolution of the disc brought its 
 holes before the holes of the box, and so opened an exit for the air. 
 Hence, allowing the disc to revolve, while air was being driven 
 through the pipe, an intermittent sound was obtained, which sounded 
 exactly like the beats arising from two tones sounded at once. 
 
 By means of a perforated disc and multiplying wheel, similar to 
 that shown in fig. 19, the same thing may be still more easily 
 demonstrated. One circle of holes on the disc is sufficient, but 
 they should be larger than shown in the figure. One end of the 
 india-rubber tube is held opposite to the circle of holes, just as in 
 the figure, but the other end is to be applied to the ear. On the 
 other side of the disc and opposite to the end of the india-rubber tube, 
 a vibrating tuning-fork is held, the necessary intermittence of tone 
 being brought about by the revolution of the disc. 
 
 In either of the above ways, intermittent tones may be obtained, 
 and this intermission gives them all exactly the same kind of 
 roughness, that is produced by two tones which beat rapidly 
 
156 HAND-BOOK OF ACOUSTICS. 
 
 together. Beats and intermissions are thus identical, and both, 
 when succeeding each other fast enoiigh, produce a harsh discordant 
 jar, or rattle. 
 
 Two questions now suggest themselves ; first, why should such 
 an intermittent sound — why should rapid beats — be unpleasant ? 
 and secondly, why should beats cease to be unpleasant when they 
 become sufBciently rapid ? 
 
 "With regard to the first question, beats produce intermittent 
 excitement of certain auditory nerve fibres. Now any excitement 
 of a nerve fibre deadens its sensibility, and thus during a 
 continuance of the excitement, the excitement itself deadens the 
 sensibility of the nerve, and in this way protects it against too long 
 and too violent excitement. But during an interval of rest, the 
 sensibility of the nerve is quickly restored. Therefore if the 
 excitement instead of being continuous is intermittent, the nerve 
 has tune to regain its sensibility more or less, during the intervals 
 of rest ; thus the excitement acts much more intensely than if it 
 had been continuous, and of the same uniform strength. 
 
 In the analogous case of light, for example, every one must have 
 experienced the unpleasant sensation of walking along the shady 
 side of a high picket fence, with the evening sun shining through. 
 Here the fibres of the optic nerve are alternately excited and at 
 rest. During the short intervals of rest, the nerve regains more or 
 less its sensibility, and thus the excitements due to the sunlight are 
 much more intense than they would have been, had the irritation 
 been continuous ; for in this case, the continuous irritation would 
 have produced a continuous diminution in the sensibility of the 
 nerve. It is precisely the same cause, which renders the flickering 
 of a gas jet, when water has got into the pipe, so unpleasant. 
 
 An intermittent tone is to the nerves of hearing, what a flickering 
 light is to the nerves of sight, or scratching to the nerves of touch. 
 A much more unpleasant and intense excitement is produced than 
 would be occasioned by a continuous tone. The following simple 
 experiment is instructive on this point. Strike a tuning-fork and 
 hold it farther and farther from the ear, tiU. its tone can Just not 
 be heard. Now if the fork, while still faintly vibrating, be 
 revolved, it will become audible. For as we have seen, during its 
 revolution, it is brought into positions such, that it alternately can 
 and cannot transmit its soTind to the ear, and this alternation of 
 strength is immediately perceptible to the ear. As Helmholtz has 
 pointed out, this fact supplies us with a delicate means of deteotins 
 
ON DISSONANCE. 157 
 
 very faint tones. For if another tone of about the same intensity, 
 but differing .very slightly in pitch, be sounded with it, the intensity 
 of the resulting sound, as -we saw in the last chapter, -will alternate 
 between silence and four times the intensity of the original sound, 
 and this increase of intensity will combine with the alternation to 
 render it audible. 
 
 With regard to the second question, " why should the beats cease 
 to be unpleasant, when they become sufficiently rapid '' ? we must 
 again have recourse to the analogous phenomenon of light. If a 
 carriage wheel be revolved slowly, we can see each of the spokes 
 separately ; on revolving more quickly, they merge together into a 
 shadowy circle. Again the singing flame of fig. 3 is all but 
 extinguished two or three hundred times per second, but to the 
 unaided eye it appears stationary. When the alternations between 
 irritation and rest follow one another too quickly, they cease to be 
 perceived, and the sensation becomes continuous. So in the case of 
 sound, after the exciting cause has ceased to act, a certain minute 
 interval of time is necessary for the excited nerve to lose its 
 excitement ; and, when the beats succeed one another so rapidly, 
 that there is not this interval between them, then the cessations 
 and reinforcements, that is, the beats, become imperceptible. 
 
 In our first experiment, we began with two C forks in unison, 
 and on gradually increasing the interval between them, we found 
 that the harshest discord was obtained, when they produced about 
 32 beats per second, and that, when their vibration numbers 
 differed by about 78, the two tones were just beyond beating 
 distance : that is the 78 beats so coalesced as to be imperceptible. 
 Now these numerical results apply only to this region of pitch. If 
 we select another pair of tones in a different part of the musical 
 realm, the general result will be the same, but the numbers will not 
 be those above : that is to say, the discord wiU become harsher and 
 harsher as the beats increase up to a certain point, but the number 
 of beats per second at this point will not be 32 ; and s imil arly, the 
 discord will become less and less after this, and finally vanish, but 
 the number of beats per second at the beating distance, will not 
 be 78. 
 
 The harshness of a dissonance therefore, does not depend upon 
 the rapidity of beats alone : it depends also upon the position of 
 the beating tones in the musical scale. This will be evident from 
 the following examples — 
 
158 
 
 HAND-BOOK OF AOOUSTIOS. 
 
 Inteeval. 
 
 Tones. 
 
 ViB. Nos. 
 
 No. OF Beats 
 
 PEK SEC. 
 
 Semitone.. .. 
 
 IS' 
 
 (512 
 (480 
 
 32 
 
 Tone 
 
 i? 
 
 (288 
 ( 256 
 
 32 
 
 Major Third 
 
 1?; 
 
 (160 
 { 128 
 
 32 
 
 Fifth 
 
 (1- 
 
 (96 
 (64 
 
 32 
 
 The number of beats produced in each of these four intervals is 
 32 per second, and therefore if harshness of discord depended on 
 rapidity of beats alone, these intervals should be equally discordant. 
 But as every one knows, they are not ; in fact, if, as we suppose, 
 the tones are simple, the last two will have no trace of harshness 
 whatever. Thus the number of beats per second, necessary to 
 produce a certain degree of discord, varies in different parts of the 
 scale of musical pitch, diminishing as we descend, and increasing 
 as we ascend. Similarly the Beating Distance becomes greater, as 
 we get lower in pitch, and contracts as we go higher. 
 
 The following are Mayer's determinations of the beating 
 distance between Simple Tones, in various parts of the musical 
 scale. The first column gives the name of the Simple Tone ; the 
 second, its vibration number; the third, the number of beats 
 generated between the simple tone given in the first column, and 
 another simple tone at Beating Distance ; the fourth, the Beating 
 Distance approximately expressed in musical language, — in other 
 words, this column shows the smallest consonant interval in the 
 region of the tone given in the first column. It is difficxilt to fix the 
 points of greatest discord, but we should probably be not far wrong 
 in placing it at somewhat less than half the Beating Distance ; or 
 throughout the greater part of the scale, at about a semitone. 
 
ON DISSONANCE. 
 
 159 
 
 TOHE. 
 
 ViB. No. 
 
 Beating 
 
 DiST. 
 
 NsABSaT COHSOHAHT IhTEBTAL. 
 
 Duration of 
 
 Sensation. 
 
 0. 
 
 64 
 
 16 
 
 Major 3rd. 
 
 ■jL. of a sec. 
 
 0, 
 
 128 
 
 26 
 
 Minor 3rd. 
 
 aV >> 
 
 
 
 256 
 
 47 
 
 Minor 3rd, less J Semitone. 
 
 *v .. 
 
 G 
 
 384 
 
 60 
 
 
 Fo l» 
 
 01 
 
 612 
 
 78 
 
 Miaor 3rd, less § Semitone. 
 
 78 » 
 
 B' 
 
 640 
 
 90 
 
 
 9"0 »l 
 
 G' 
 
 768 
 
 109 
 
 
 109 » 
 
 O 
 
 1024 
 
 135 
 
 Tone or Second 
 
 iJT " 
 
 Two other points may be noticed before leaving this table. In the 
 first place, it appears, that a very high number of beats — ^more 
 than 100 per second — ^may be appreciable to the ear without 
 coalescing. In order to convince oneself that this is true, it is 
 only necessary to hear the following four intervals successively 
 between simple tones, and note that, though it soon becom.es 
 impossible to discriminate the separate beats, yet, the harsh Jarring 
 effect is the same throughout. 
 
160 
 
 HAND-BOOK OF ACOUSTICS. 
 
 Simple Tones. 
 
 Vibration Nos. 
 
 Beats peb Sec. 
 
 
 (256 
 (240 
 
 16 
 
 IB 
 
 C512 
 <480 
 
 32 
 
 
 (1024 
 ( 960 
 
 64 
 
 03 
 
 12048 
 f 1920 
 
 128 
 
 Secondly, the time diniiig which a sensation of soimd -will endure, 
 after its cause has ceased to act, varies for sounds of low and high 
 pitch. For since 16 beats per second in the region of Oj coalesce, 
 it is only reasonable to conclude that the sensation of each of 
 these beats remains for -Jg- of a second. Similarly the duration of 
 sound in the region of C| is ^ of a second, and so on, as given in 
 the last column of the above table. If this conclusion be correct, 
 it seems to afford an explanation of the fact, that the Beating 
 Distance becomes greater as we descend in the scale. 
 
 To sum up, then, as far as we have gone : Dissonajioe between 
 two simple tones, is due to Beats : taking two Simple Tones in 
 unison with one another, and gradually altering the pitch of one of 
 them, the harshness of the dissonance increases with the rapidity of 
 the beats, up to a certain point ; beyond that point it diminishes, 
 until finally, all harshness — all dissonance— vanishes when the two 
 tones are at a certain distance apart : and finally, the number of 
 beats per second which produces the greatest dissonance, and the 
 Beating Distance both vary as we ascend and descend in the 
 musical scale. 
 
 It would seem from the above, that however much we widen the 
 interval between two simple tones beyond the beating distance, they 
 never again become dissonant, for being now beyond that distance, 
 it is plain they can no longer beat. On putting the matter to the 
 test of experiment, however, it is found that this is not the case ; 
 there are certain intervals, beyond the beating distance, which do 
 beat. For example, if two forks b" t'lried one +" f^ btI t.lm nf.ViBT 
 
ON DISSONANCE. 161 
 
 to B, or to Ci# or thereabouts, beats will be heard when they axe 
 sounded together, although they are far beyond the Beating 
 Distance. 
 
 This fact, though at first sight inconsistent with the foregoing, is, 
 in reality, not so ; for the beats in question are not produced by 
 the two simple tones, but by one of them and a differential tone 
 generated by them. The following figures show this — 
 
 B = 480 \ / Gift = 540 
 
 C = 256 J I = 256 
 
 > or < 
 
 Differential Tone, 224 \ I 284 Diffl. Tone, 
 
 and 256 — 224 — 32 beats.j \ and 284 — 256 = 28 beats. 
 
 C and B generate a Differential, the vibration number of which is 
 224, and this with the tone will produce 256 — 224 = 32 beats 
 per second; similarly C and CJ generate the Differential 284, 
 which with C gives 284 — 256 = 28 beats per second ; and on 
 reference to the table on page 159, we see that both 32 and 28 
 beats per second, are well within beating distance at this part of 
 the musical scale. 
 
 Again, if we sound together two forks, one tuned to and the 
 other tuned only approximately to Gr, beats may be heard, but only 
 when the forks are vigorously excited. Thus taking = 256, G 
 should be 384 : let the Q- fork, however, be mistuned to 380, then 
 380 — 256 = 124 Differential of 1st order 
 256 — 124 = 132 „ 2nd „ 
 
 and these two differential tones will produce 132 — 124 == 8 beats 
 per second. These beats, however, will be faint, inasmuch as the 
 differential tone of the second order is itself very weak. 
 
 With other intervals beyond the beating distance, no dissonance 
 wiU be heard between simple tones. Two forks, forming any 
 interval between a minor and a major third for example, in the 
 middle or upper part of the musical scale, produce no roughness 
 when sounded together; the interval may sound strange to 
 musical ears, but there is no trace of dissonance. 
 
 To sum up, therefore : if the interval between two simple tones be 
 gradually increased beyond the beating distance, no roughness or 
 dissonance will be heard, till we are approaching the Hfth ; and 
 only then, if the tones are sufficiently loud to produce a Differential 
 of the second order : on stiU further widening the interval, 
 beats may be heard in the neighbourhood of the octave, due to a 
 Differential of the first order, ' 
 
 M 
 
162 HAND-BOOK OF ACOUSTICS. 
 
 Thus all diasonance between simple tones ■wiU be found on 
 examination to be due to beats, generated, either by the simple 
 tones themselves, by one of the simple tones and a DLSerential, or 
 by two Differentials. 
 
 Before inquiring into the causes of dissonance 
 between Compound Tones, it will be as well to 
 call to mind the fact, that a single compound 
 tone may and often does contain dissonant 
 elements in itself. Let us take the compound 
 tone O2, for example : Inasmuch as its funda- 
 mental has the vibration number 64, the 
 difference between the vibration numbers of any 
 two successive partials must be 64. By reference 
 to the accompanying table of partials, and to the 
 table on page 159, we see that the intervals 
 between the first 7 partials are greater than the 
 Beating Distance, but that the intervals between 
 the partials above the 7th are less than the 
 Beating Distance. For, take the 8th and 9th 
 partials, which are C and D' respectively, the 
 number of beats produced by these two simple 
 tones is 64 and we know by the table on page 159 
 that the number of beats necessary to concord, 
 in this part of the musical scale is 78; therefore 
 a certain amount of roughness, due to these 
 64 beats wiU. result. The dissonance gets worse as we ascend; 
 for example, the number of beats per second between the 
 15th partial, B', and the 16th, C^ is of course 64, which forms a 
 very harsh dissonance in this part of the scale. As we have already 
 seen, the partials of the tones of most instruments, become weaker 
 and weaker, the farther they are from the fundamental ; so that in 
 general, these very high partials are not strong enough to produce 
 any appreciable roughness, but this is by no means always the case. 
 If the note 0^ be sounded on the Harmonium or American Organ, 
 especially with such a stop as the bassoon, it is quite easy to detect 
 the jarring of these higher partials, and by means of a resonator 
 tuned to a note intermediate between any two of them, the beating 
 of those two is perceptibly increased. The same jarring effect may 
 be readily perceived in the tones of the Trombone and Trumpet ; in 
 fact, it is this beating that gives to the tones of these instruments, 
 their peculiar penetrating or braying character; a discontinuous 
 sensation, as before obser^a'' -nrnduciner u, much more intense 
 
 XVi 
 
 C? 
 
 xv 
 
 . B' 
 
 XiV 
 
 « 
 
 xin 
 
 « 
 
 xn 
 
 Gi 
 
 XI 
 
 * 
 
 X 
 
 El 
 
 TX 
 
 D' 
 
 VIII 
 
 CI 
 
 VU 
 
 * 
 
 VI 
 
 G 
 
 V 
 
 E 
 
 IV 
 
 
 
 m 
 
 G. 
 
 n 
 
 c, 
 
 I 
 
 Cj = 64 
 
 •Out of Scale. 
 
ON DISSONANCE. 
 
 163 
 
 effect than a continuous one of equal strength. For precisely 
 the same reason, the tones of a powerful bass voice are apt to 
 partake of this strident quality. 
 
 Coming now to the subject of dissonance between two compound 
 tones, we shall find that beats may arise ; 
 
 (1) Between the Fundamentals themselves ; 
 
 (2) Between the Fundamental of one Tone and an overtone of 
 the other ; 
 
 (3) Between overtones ; 
 
 (4) From the occurrence of Differentials ; 
 
 (5) From the occurrence of Summation Tones. 
 To take these causes of beats one at a time ; 
 
 (1) Beats arising between Fundamental Tones. 
 Inasmuch as these Fundamental tones are simple, all the 
 conclusions above as to simple tones, at once apply to them. But 
 when such beats arise between the fundamentals of two compound 
 tones, the dissonance will in general be harsher, than between two 
 simple tones of the same pitch, for in the former case each pair of 
 overtones may beat also. Supposing for example, the two funda- 
 mentals to be B| and C, the following diagram shows the dissonant 
 overtones. 
 
 &o. 
 
 Bi 
 
 m 
 
 B 
 
 &0. 
 
 c» 
 
 G' 
 
 C 
 
 B. 
 
 C 
 
164 
 
 HAND-BOOK OF ACOUSTICS. 
 
 The harshness of the beats between each pair of overtones in the 
 above, must be estimated, from the conclusions we arrived at 
 before, in the case of simple tones, for these overtones are simple 
 tones; but in estimating the total harshness of the whole 
 combination, it should be remembered that for ordinary qualities of 
 tone, the intensity of the partials becomes less and less, as we go 
 farther from the Fundamentals (a fact roughly indicated in the 
 above by the use of smaller type for the upper partials) ; and 
 therefore the intensity of the beats in the above, will become less 
 and less as we ascend. 
 
 (2) Beats arising between the Fundamental of one tone and an 
 overtone of the other. As an example, we may take the common 
 dissonance — 
 
 '8, 
 
 or 
 
 F 
 
 a, 
 
 This interval, when sounded between simple tones, is quite free 
 from harshness ; the tones are far beyond beating distance, and no 
 differential is near enough to produce beats. When, however, it is 
 sounded between ordinary compound tones, beats axe generated by 
 the fundamental f and the 2nd partial of S|, thus : — 
 
 \ 
 
 or 
 
 
 G. 
 
 The following dissonances, between compound tones, although 
 often called by the same name, are very different indeed in their 
 degree of dissonance. 
 
 No. 1. 
 
 No. 2. 
 
 No. 3. 
 
 
 No. 4. 
 pi 
 
 To render this evident, it is only necessary to set forth the partials 
 of each tone, thus : — 
 
ON DISSONANCE. 
 
 165 
 
 \ 
 
 \ 
 
 / 
 \ 
 
 \ 
 
 S| 
 
 \ 
 
 \ 
 
 \ 
 \ 
 
 \ 
 
 \ 
 
 S2 
 
 Sj 
 
 No. 1, Primary. No. 2, Seamdary. No. 3, Tertiary. No. 4, Quartemary, 
 
 In. setting out the above, we do not go above the 6th partial, 
 inasmuch as the partials above this point are in general too weak 
 to have any influence on the subject xmder discussion. 
 
 In No. 1, not only do the fundamentals beat, but every pair of 
 overtones also, while above the 3rd pair, there is a perfect galaxy of 
 dissonances. In No. 2, the Fundamentals are beyond beating 
 distance, but there are beats between one of them (f |) and the 1st 
 
166 
 
 HAND-BOOK OF ACOUSTICS. 
 
 overtone of the other (S|). The harshness of this dissonance ■will 
 consequently chiefly depend on the intensity of this overtone, which 
 ■will vary in different iostruments, and even in different parts of 
 the same instrument. Thus, in the lower notes of the piano, the 1st 
 overtone is not unfrequently louder than the fundamental itself. 
 On other instruments, however, and in general, its intensity is not 
 so great ; in a well bowed violin, for example, it is only about one 
 fourth as loud. Further, the beating between the 3rd pair of partials, 
 and between the 5th pair of No. 1, is wanting in No. 2. Thus, on 
 the whole, this latter dissonance is much less harsh than No. 1. 
 The dissonance in No. 3 is of a very mild character, for the 
 Fundamental (f) beats only against the 4th partial (s) and as a 
 general rule, the 4th partial is comparatively weak. In No. 4 there 
 is no beating whatever, unless the 7th or 8th partial is audible, and 
 even then it would be very slight. 
 
 The late Mr. Ourwen proposed to distinguish dissonances such as 
 Nos. 1, 2, 3, and 4 above, by terming them respectively Primary, 
 Secondary, Tertiary, and Quaternary dissonances. Thus, in 
 Primary dissonances the fundamentals themselves beat, while in 
 Secondary, Tertiary, and Quaternary dissonances, the Fundamental 
 of the one tone beats respectively with the 
 2nd, 4th, and 8th partials of the other. 
 
 The above conclusions must of course be 
 modified for the tones of instruments, which 
 have not the complete series of partials up to 
 the 6th. For example, the tones of stopped 
 organ pipes, and of clarionets are wanting in 
 the even partials, and therefore a secondary 
 dissonance between such tones, is of a very 
 mild character indeed, the only beating which 
 occurs, arising from a 6th and a third partial, 
 as shown in the accompanying sketch. 
 
 (3) Beats between the overtones of Compound 
 Tones. In studying these beats, we shall for 
 the reasons stated above, take into considera- 
 tion, the first six, and only the first six 
 partials; and the student must continually 
 bear in mind the fact, that, in general, the 
 intensity of these partials rapidly diminishes, 
 as we go farther and farther from the fonda- 
 mental. 
 
 V t 
 
 IV 
 
 ^ 
 
 d' m 
 
 m 
 
 f, 
 
ON DISSONANCE. 
 
 167 
 
 If we limit ourselves to intervals not greater than an octave, we 
 ahall find, that the only interval entirely free from these partial 
 beats, is the Octave itself, thus : — 
 
 d' d' 
 
 d d 
 
 In aU intervals smaller than the Octave, it will be found that two 
 or more of the first six partials beat with one another. To take a 
 couple of exam.ples : In the Perfect Fifth a 3rd partial beats against 
 4th and 5th partials ; and in the Diminished Fifth 2nd, 3rd and 4th 
 partials come within beating distance of 3rd, 4th, 5th and 6th.; 
 thus : — 
 
i68 
 
 HAND-BOOK OF ACOUSTICS. 
 
 
 ../ 
 
 <f^ 
 
 fe 
 
 Perfect 5tli 
 
 Diminished 5th 
 
 Ab other examples we in&y take the interval i gg or ! q«, and its in- 
 version j |®| or j S*'. It ■will be more convenient in this case to 
 give the vibration niimbers of the partials, rather than to express 
 them in musical notation. Taking C or d ^ 256, the vibration 
 number of Gj(| or sei = 200, and the partials are as follows : 
 
 Partials of = 256 , 512 , 768 , 1024 , 1280 
 
 „ G#| = 200 , 400 , 600 , 800 , 1000 , 1200 
 
 32 
 
 24 
 
 80 
 
 Partials of G#| = 200 , 400 , 600 , 800 
 
 „ 0| = 128 , 256 , 384 , 512 , 640 , 768 
 
 16 
 
 40 
 
 32 
 
ON DISSONANCE. 169 
 
 On reference to the table on page 159, it ■will be found, that the 
 beats that are not calculated above, are beyond beating distance ; 
 but the 32, 24, and 80 beats per second of the first interval, and the 
 16, 40, and 32 of the second, form very harsh dissonances; and 
 moreover in the second interval, the 16 beats per second will usually 
 be very prominent, as they are between a 2nd and a 3rd partial. 
 
 The Interval J ^^' or J p*' and its inversion j '' or ) (S, are of 
 frequent use in music. Taking G#| = 200 as before, the vibration 
 number of F will be 170J ; therefore 
 
 Partials of Gt, = 200 , 400 , 600 , 800 , 1000 
 
 .. F| = 170f , 341i , 512 , 682| 853J , 1024 
 
 J9i 58| _82| _53J 24 
 
 Partials of F, == 170f , 341^ , 512 , 682J 
 
 „ G#j = 100 , 200 , 300 , 400 , 500 , 600 
 
 J£i Jlk -iH. 82g 
 
 From the table on page 159 we see that the 29 J, 58f , and 82| beats 
 per second are slightly within or just on the verge of the beating 
 distance. In the first Interval, the 53 J beats per second will produce 
 a harsh discord, as will also the 24 per second, though as they only 
 arise between soft 5th and 6th partials, they have no very great 
 intensity. The inversion is the worse of the two, the 41J beats per 
 second between a 3rd and a 2nd partial, forming a bad and some- 
 what prominent dissonance, which is made stiU worse by the 12 and 
 and 82f beats per second higher up. 
 
 As already remarked, the above results must be modified for 
 Compound Tones, which do not possess all the first sis partials. 
 
 (4) Beats arising between Compound Tones, through the occurrence 
 of Differentials. The Fundamentals being Simple Tones, the con- 
 clusions we arrived at concerning the beats due to the Differentials, 
 generated by simple tones, in the former part of this chapter, at 
 once apply to them. It only remains to ascertain the efEects, due 
 to the DifEerehtials, generated by the Overtones. 
 
 Differential Tones are only produced when the generators are 
 pretty loud, therefore we shall not go beyond the 2nd or 3rd partial, 
 as those above rarely have any considerable intensity. Moreover 
 it will not be necessary to consider any differentials except those of 
 
170 
 
 HAND-BOOK OF ACOUSTICS. 
 
 the 1st order ; differentials of the 2iid order seldom or never occur- 
 ing between overtones. Let us consider the differentials of the Ist 
 order generated between the partials of two compound tones, the 
 fundamentals of which have, as we will suppose, the vibration 
 numbers 200 and 304. Then the numbers in the first horizontal 
 line of the following table are the partials of the former tone, and 
 those in the first vertical column those of the latter. At their 
 intersections are found the differences of these numbers ; that is, 
 the vibration numbers of the differentials due to them. 
 
 
 200 
 
 400 
 
 600 
 
 304 
 608 
 
 104 
 408 
 
 96 
 208 
 
 296 
 
 8 
 
 If we arrange these tones in the order of their pitch, omitting the 
 8, which of course produces no tone at all, we have the groups -.^ 
 
 104 208 304 408 
 
 96 200 296 400 
 
 8 8 8 8 
 
 The difference between each pair is 8, and this is the only number 
 of beats per second, which wiU be produced by all these differential 
 tones; for the difference between any other two of the above 
 numbers, gives too great a num.ber of beats per second to be 
 perceptible at the pitch of the tones that produce them. 
 
 Now let us ascertain the number of beats per second that will be 
 generated by direct action between these same partials : 
 
 Partials of lower tone 
 „ higher „ 
 
 Beats per second 
 
 200 
 
 400 600 
 
 304 608 
 
 96 8 
 
 Ninety-six beats per second, at the pitch of 400, is far beyond 
 beating distance ; 8 beats per second therefore is the number due 
 to the direct action of overtones, that is, the same number which 
 we found to arise from the action of the differentials. A like result 
 will be obtained whatever numbers are selected for the funda- 
 mentals, so that in general, " dissonance due to combination tones 
 
ON DISSONANCE. 171 
 
 produced between overtones, never exists, except where it is already- 
 present by virtue of direct action among the overtones themselves." 
 Thus we may pass on to the last cause of beats between compound 
 tones, viz : — 
 
 (5) Beats due to Summation Tones. Summation Tones, though 
 certainly not very loud, are much louder than they are commoidy 
 supposed to be. On the Harmonium, and American Organ, 
 Summation Tones, generated by any pair of tones on the lower 
 half of the key-board, may be readily heard without the use of 
 resonators, and therefore cannot but have some effect on the result- 
 ing sound. 
 
 From the table at the end of Chap. XII, page 134, we see that the 
 fundamentals j d, generate the Summation Tone (s) which coincides 
 with the 3rd partial of the lower tone, so that in the case of the 
 Octave, no new element is introduced. 
 
 In the Fifth ! |[, the Summation Tone (pi) is a new tone introduced 
 between the 2nd and 3rd partials of the (d|), but it forms no 
 dissonance with them. 
 
 In the Fourth ) f,, the Summation Tone (approximately ta) 
 comes very near the Beating Distance with the 2nd partial of d. 
 For, taking d = 400, then S| = 300, the Summation lone ^ 300 
 + 400 = 700, and 2nd partial of d' = 800; therefore the Summation 
 Tone and (d') will produce 800 — 700 = 100 beats per second, which, 
 as may be seen from the table on page 159, is just beyond beating 
 distance at this pitch. 
 
 In the Major Third ) ^1' > the Summation Tone (r) wiU dissonate 
 with the 2nd partials (d) and (pi) of both tones ; the same is true in 
 the case of the Minor Third, but the dissonance is harsher; for 
 take S| = 300 ; then r\\ = 250, the Summation Tone is 300 + 250 
 = 550, and the 2nd partials (s) and (pi) are 600 and 600 respectively. 
 Thus the number of beats per second is 600 — 550 = 50, and 550 
 — 500 = 50, which, at this pitch, is less than the nimiber due to the 
 whole tone. 
 
 In the Major Sixth, j ^J , the Summation Tone (f ) dissonates at the 
 mterval of a tone, with the 3rd partial of the (d|). The Minor 
 Sixth, I m, > is better in this respect, for take d = 400, then rii = 
 250, Summation Tone = 650, which with the 3rd partial t (== 750) 
 will produce 750 — 630 = 100 beats per second, which at this pitch 
 is only just on the borders of the Beating Distance. 
 
172 HANB-BOOK OF ACOUSTICS. 
 
 The Summation Tone, when present, renders a primary dissonance 
 between Compound Tones harsher than it otherwise would be. 
 Take ! J for example : let d| = 80 then ri = 90 and the Summation 
 Tone will be 80 + 90 = 170, a tone about midway between the 2nd 
 partials, d (= 160) and r (=180). 
 
 STJMMAaY. 
 
 Beats are the source of all discord in music. 
 
 Starting with two simple tones in unison; if one of them be put 
 slightly out of tune, slow beats will be heard, which are not very 
 unpleasant, as long as they do not exceed one or two per second. 
 On increasing the interval between the two tones, the beats gradually 
 become more and more rapid, and at length form a harsh dissonance. 
 If this interval be gradually increased, a point is finally reached, 
 where all dissonance vanishes. The interval at which the dissonance 
 ytM< disappears, is termed the Beating Distance. 
 
 The harshness of any particular dissonance, depends jaartZy upon 
 
 (1) the rapidity of the heats, and partly upon 
 
 (2) the region of Pitch in which the dissonance lies. 
 Similarly, the Beating Distance for Simple Tones varies in 
 
 different parts of the realm of pitch, from a Tone at C^ = 1024 to a 
 Major Third at 0^ = 64. 
 
 Dissonance may arise between Simple Tones beyond Beating 
 Distance, from the occurrence of Differentials. 
 
 A Compound Tone may be dissonant or harsh in itself, if it 
 contain very high and loud partials. 
 
 Dissonance between Compound Tones may arise, 
 
 (1) From beats between fundamentals, 
 
 (2) ,, ,, the fundamental of one tone and an 
 overtone of the other, 
 
 (3) From beats between overtones only, 
 
 (4) From beats due to Differential Tones, 
 
 (5) „ „ ,, Su^mmation Tones. 
 
173 
 
 CHAPTER XV. 
 
 The Definition of the Consonant Intertais. 
 We have seen in Chap. V, how, by means of the Double Syren, it 
 may be proved, that, for two sounds to be at the exact interval 
 given in the first column below, their vibration numbers must be in 
 the exact ratio of the numbers given in the second column. 
 Interval. Ratio. 
 
 Octave . . . . . . 2:1 
 
 Fifth 
 Fourth 
 Major Third 
 Minor Third 
 
 3 : 2 
 
 4 : 3 
 
 5 : 4 
 
 6 : 5 
 
 &c. &c. 
 
 If the vibration numbers are not in the exact ratio given above, the 
 interval wiU be perceptibly out of tune. This fact had been 
 ascertained long before the instrument just referred to was 
 invented, by the actual counting of the vibration numbers. 
 Ingenious, but unsatisfactory theories, of a more or less meta- 
 physical nature (among which, that of Euler held sway for many 
 years), were devised to account for this reidarkable fact. Its true 
 explanation, as given below, is due to Hehnholtz. 
 
 We commence as usual with Siinple Tones, and first with the 
 Octave. Let two Simple Tones be sounded together, the vibration 
 numbers of which are in the I'atio of 2 : 1, say 200 and 100 
 respectively. They will generate a Differential Tone, the vibration 
 number of which will be 200 — 100 = 100, which Differential 
 Tone will therefore coalesce and be indistinguishable from the 
 lower of the two Simple Tones. This identity in pitch, of the 
 DiSerential, and the lower of the Simple Tones will always occur, 
 provided the ratio of the two tones is as 2 : 1 ; for 
 
 let 2» be the vibration number of the upper tone, 
 then ra will be ,, ,, ,, lower ,, 
 
 consequently 2ra — n=:n ,, ,, ,, Differential „ 
 
174 HAND-BOOK OF ACOUSTICS. 
 
 If, however, the exact ratio be not preserved, the lower tone and the 
 Differential will not coincide, and beats will be heard between them. 
 For example, let the vibration niimbers of the two tones be 200 
 and 99 respectively ; then 
 
 Vib. No. of upper tone = 200 
 ,, ,, lower ,, = 99 
 
 „ Diffl. „ =101 
 and therefore 101 — 99 == 2 beats per second wiE be heard. 
 
 We might therefore define an Octave between two Simple Tones, 
 as that Interval at which the Differential generated by them 
 coincides in pitch with the lower of the two tones ; and we see that 
 this perfect coincidence can only occur, when the ratio between the 
 vibration numbers of the two tones is exactly 2:1. 
 
 In the example given above, if we had taken 200 and 98 as the 
 respective vibration numbers, that of the Differential would have 
 been 200 — 98 = 102, which would have given 4 beats per second 
 with the lower tone ; from which it is evident, that the more the 
 interval is out of tune, the greater is the number of beats produced. 
 Thus in order to time two Simple Tones to an exact Octave, after 
 tuning them approximately, one of them, must be sharpened or 
 flattened more and more, till the beats becoming less and less, 
 finally vanish. This is an entirely mechanical operation and does 
 not even need a musical ear. For suppose two forks give a. false 
 octave, producing beats, and it is required to tune the upper one to 
 a true octave with the lower. Sharpen the former slightly and 
 sound them again ; if the beats are more rapid than before, then 
 the higher fork was already too sharp and must be flattened 
 gradually till the beats disappear ; it on the other hand they are 
 slower, the fork is too flat, and must be sharpened in a similar 
 manner. 
 
 Fifth. Let 3n and 2» be the vibration numbers of two Simple 
 Tones at this interval. Then 
 
 Zn — 2ji = ji vib. no. of Differential of 1st order. 
 
 and2ji — TC = n „ „ „ 2nd „ 
 
 Thus a Fifth between Simple Tones is defined by the coincidence of 
 Differentials of the 1st and 2nd order; and this coincidence can 
 evidently only occur, when the ratio of the vibration numbers of 
 the Simple Tones is as 3 : 2. Differentials of the 2nd order are, 
 however, generally weak, so that this interval between Simple Tones 
 is by no means well defined. 
 
DEFINITION OF THE CONSONANT INTERVALS. IVo 
 
 If the ratio is not exactly that of 3 : 2, beats are generated. 
 For instance, let the vibration numbers of the two tones be 300 
 and 201 respectively, then 
 
 300 — 201 = 99 . . Differential of 1st order. 
 201— 99 = 102 .. „ 2nd „ 
 
 102 — 99 =: 3 beats per second being produced. The more the 
 tones are out of tune, the greater the rapidity of the beats; so that 
 to tune the interval, one tone must be sharpened or flattened 
 gradually, as the rapidity of the beats decreases, until they vanish 
 altogether. 
 
 Fourth. Let in and Zn be the vibration numbers of two Simple 
 Tones at this interval. Then 
 
 4n — Zn = n .. vib. no. of DiEEerential of 1st order 
 Zn — n=z2n .. „ „ „ 2nd „ 
 
 in — 2nz=2n\ „ •, 
 
 2n— n= n) ■■ " " .. -Jra „ 
 
 A Fourth between Simple Tones, therefore, is only defined by the 
 coiaoidence of Differentials of the 1st and 3rd, and of the 2nd and 
 3rd order. Inasmuch, however, as a 3rd Differential can only be 
 heard under extremely favourable circumstances, this interval can 
 scarcely be said to be defined at all. This is still more the case 
 with the Thirds, the definition of which, in the case of Simple 
 Tones, depends upon the existence of Differentials of the 4th order. 
 Accordingly it is found, as stated before, that in the case of Simple 
 Tones, intervals of any magnitude intermediate between a Minor 
 Third and a Fourth, are usually of equal smoothness. For the 
 same reason, it is impossible without extraneous aid to tune two 
 Simple Tones to the exact interval of a Third, either Major or 
 Minor ; there is no check : they have no definition. 
 
 If, however, more than two Simple tones be employed, it becomes 
 easy to tune these intervals. Indeed, it is better to tune the Fifth 
 also by the aid of a third Tone ; for, as we have seen, the interval 
 of the Fifth alone, is only guarded by a Differential of the 2nd 
 order; while if the Octave of the lower tone be present, a 
 Differential of the 1st order becomes available. Suppose for example 
 the vibration numbers of three Simple Tones be 200, 301, and 400 
 respectively, the 5th, 301, being mistuned, then 
 
 301 — 200 = 101 . . Differential of 1st order, 
 400 — 301 = 99 . . „ „ 1st „ 
 
 101 — 99 = 2 beats per second being thus produced. Thus by 
 
176 HAND-BOOK OF ACOUSTICS. 
 
 flattening the middle note, till these beats vanish, we may obtain a 
 perfect flfth. 
 
 Similarly, to rectify a mistuned Fourth, 301 and 400, for example, 
 ■we may take a tone, 200, an octa-vre below the higher one, and 
 proceed as in the above case. 
 
 Again, in the case of a false Major Third, say 400 and 501, tune 
 a Simple Tone 600 a perfect Fifth from the lower tone. Then 
 501 — 400 = 101 . . DiEt'erential of 1st order, 
 600 — 501 = 99 . . „ „ 1st „ 
 
 101 — 99 = 2 beats being heard between Differentials of 1st order. 
 Tune as before till tbe beats disappear. 
 
 Similarly in the case of the mistuned Minor Third, 600 and 501, 
 take a third tone 400, a true Fifth below the higher tone, and 
 proceed as above. 
 
 We come now to the definition of Intervals between Compound 
 Tones, and in the first place we shall assume the Compound Tones 
 in question to be such as axe produced by the Human Voice, 
 Harmonium, Piano, and stringed instruments in general; that is 
 to say, we shall suppose them to consist of, at least, the first 
 six partials. 
 
 Octave. Let the vibration numbers of the fundamentals of two 
 Compound Tones, at the interval of an octave, be n and 2n 
 respectively. Then the 2nd partial of the former will be 2n which 
 win thus coincide with the other fundamental; or, in muaioal 
 language 
 
 2nd partial . . d' d' ■ • Fundamental. 
 
 Fundamental. . d 
 If the ratio of the vibration numbers be not exactly as 2 : 1, beats 
 will be heard between the 2nd partial of one tone and the funda- 
 mental of the other. Suppose, for example, that the vibration 
 numbers are 200 and 99. Then 
 
 2nd partial . . 198 200 . . Fundamental 
 
 Fundamental . . 99 
 and thus 200 — 198 = 2 beats per second will be heard. 
 
DEFINITION OF THE CONSONANT INTERVALS. 177 
 
 Inasmuch as these Fundamentals are Simple Tones, all that has 
 been said above about the latter apply to the former ; moreover, it 
 ■will be noted, that in the case just taken, the number of beats due 
 to the 2nd partial, viz. 2, is the same as that due to the Combination 
 Tone of the 1st order (see page 1 74), and it is evident that this must 
 always be the case. 
 
 An Octave between Compound Tones, therefore is defined, 
 
 Ist, by the coincidence of the Difierential Tone, generated 
 between their two Fundamentals, with the lower of the 
 Fundamentals; and 
 
 2nd, by the coincidence of one of the Fundamental Tones with 
 the 2nd partial of the other. 
 
 These coincidences it is plain can only occur when the vibration 
 numbers of the Fundamentals are in the exact ratio of 2 : 1. 
 Consequently, this explaias why this exact ratio is necessary to the 
 perfection of this interval. 
 
 To tune the Octave is thus a very easy matter: the mere 
 mechanical process, of altering the pitch of one tone, till all beats 
 vanish. As this interval is so well defined, great accuracy in its 
 tuning is necessary, the slightest error becoming evident to the ear 
 in the form of beats. 
 
 Fifth. Let the vibration numbers of two Compound Tones at 
 this interval be 3n and 2ji respectively. Then 
 
 3rd partial. . . .6n 6» 2nd partial 
 
 2nd 4n 
 
 Fundamental 2n 
 
 3n Fundamental 
 
 the 2nd partial of the former will exactly coincide in pitch with the* 
 3rd of the latter ; or musically 
 
 N 
 
178 
 
 HAND-BOOK OF ACOUSTICS. 
 
 3rd partial. . . .8- 
 
 -s. . . .2nd partial 
 
 2nd 
 
 ....a 
 
 g . . . .Fundamental 
 
 Fundamental. . . .d| 
 
 A Fifth between Compound Tones, therefore, though also guarded 
 by Differentials, is chiefly defined by the coincidence of the 3rd 
 partial of the lower, with the 2nd of the upper tone, and this coin- 
 cidence can evidently only happen, when the vibration numbers of 
 the Fundamentals are in the ratio of 2 : 3. If the vibration 
 nimLbers vary from this ratio, beats wiU be heard between these 
 partials. For example, let the vibration numbers be 201 and 300 
 respectively, then 
 
 3rd partial. . . . eoa- 
 
 2nd 
 
 -600 .... 2nd partial 
 
 .402 
 
 300 Fundamental 
 
 Fundamental 201 
 
 and 603 — 600 = 3 beats per second wiU be heard. To tune a false 
 Fifth, therefore, one of the tones must be altered, till these beats 
 vanish. 
 
 Inasmuch as the definition of a Fifth depends upon the 
 coincidence of 2nd and 3rd partials, while the definition of an 
 Octave depends upon that of 1st and 2nd partials, we see that beats 
 from a mistuned Fifth will not usually be so powerful as those 
 from a mistuned Octave ; that is to say, the same rigorous exacti- 
 tude in tuning, which the octave demands, is not so essential in the 
 case of the Fifth. As an illustration of this fact, it may be 
 mentioned, that while the Octave is preserved intact, in all systems 
 of temperament, the Fifth is always more or less tampered with. 
 
DEFINITION OF TEE CONSONANT INTERVALS. 179 
 
 Fourth. Let the vibration numbers of two Compound Tones at 
 this interval be 3ra and in respectively. Then, the 4th partial of 
 the former mU exactly coincide with the 3rd partial of the latter, 
 thus, 
 
 4th partial. . . . i2n- 
 3rd „ 9n 
 
 2nd 6» 
 
 Fundamental. . .3w 
 
 _i2n 3rd partial 
 
 8»i 2nd „ 
 
 4» Fundamental 
 
 or in musical language, calling the Fundamentals d| and f | 
 
 4th partial d'- d' 3rd partial 
 
 3rd „ s 
 
 2nd d 
 
 Fundamental d| 
 
 f ....2nd 
 
 f Fimdamental 
 
 Thus a Fourth, between Compotind Tones, is defined by the 
 coincidence of 3rd and 4th partials, and for exact coincidence, it is 
 obvious that the vibration numbers of the Fundamentals must be 
 in the ratio 4:3. If they are not exactly in this ratio the 
 inaccuracy will manifest itself in the form of beats. Let them be 
 400 and 301 for example : then 
 
180 
 
 HAND-BOOK OF ACOUSTICS. 
 
 4th partial 1204 ^1200 . • • • 3rd partial 
 
 3rd „ ....903 
 
 2ad , 602 
 
 800....2iid 
 
 400. . . .Fundamental 
 
 Fundamental 301 
 
 4 beats per second will be produced. 
 
 A Fourth is not so well defined as a Fifth, for not only are the 
 coincident partials of a higher order, and therefore not so prominent 
 in the former case, but also the dissonance between the 3rd and 
 2nd partials (s and f in the abore) masts, to a certaiu extent, the 
 beats between the 3rd and 4th partials of this interval, when not 
 exactly in tune. 
 
 Major Third. Let d| ^^ 47i and pii = 5ra be the Tibration numbers 
 of two Compound Tones at this interval ; then. 
 
 (V) 20n- 
 
 -20n (IV) 
 
 (IV) 16n 
 
 (HI) 12n 
 
 m Bra 
 
 m 4w 
 
 15n (HI) 
 
 lOn (H) 
 
 6» m 
 
 (V) 
 
 ffv) d' 
 
 (in) s 
 
 (H) cl 
 
 (I) d 
 
 -m' (IV) 
 
 t cm) 
 
 n m 
 
 m, m 
 
 the 5th partial of the lower tone will exactly coincide with the 4th 
 of the higher one. Thus a Major Third is even more ill-defined 
 
DEFINITION OF TEE CONSONANT INTERVALS. 181 
 
 than a Fourth, the coincidence beingibetween higher, and therefore 
 usually weaker partials, and being masked more or less by the harsh 
 dissonance of a semitone between the more powerful 3rd and 4th 
 partials (d' and t above). 
 
 Minor Third. Let S| = 6n and rii = 5w be the vibration numbers 
 of two Compound Tones at this interval : then, 
 
 (VI) SOn- 
 (VJ 25>i 
 
 (IVJ 20n 
 
 (HI) \5n 
 
 miow 
 
 -S0» (V) 
 
 (VI) t<- 
 
 -V (V) 
 
 24» (IV) 
 
 18» (HO 
 
 12» m 
 
 (V) ee- 
 
 (IV) m' 
 
 pn) t 
 
 PD n 
 
 6« m 
 
 W5w 
 
 a' (IV) 
 
 r" (m) 
 
 s (H) 
 
 m 
 
 (D m, 
 
 the 6th partial of the lower tone will coincide with the 5th of the 
 higher. The Minor Third is still less defined, therefore, than the 
 Major Third ; the coincident partials being of a higher order, and 
 obscured not only by the semitone dissonance between the 4th and 
 6th partials (gi and se' above) but by the tone dissonance between 
 the 3rd and 4th (r' and n'). 
 
 For a given departure from the exact ratios, the beats are more 
 rapid in the case of the Thirds, than in the preceding intervals; for 
 example, let 401 and 500, and 601 and 600 be the vibration numbers 
 of the fundamentals of a Major and Minor Third respectively: then. 
 
182 
 
 HAND-BOOK OF ACOUSTICS. 
 
 8006- 
 
 -3000 
 
 200S 200O 
 
 1500 
 
 1604 
 
 1203 
 
 802 
 
 401 
 
 1000 
 
 500 
 
 2S05 
 
 2004 
 
 1503 
 
 1002 
 
 501 
 
 2400 
 
 1800 
 
 1200 
 
 600 
 
 in the former case 5 beats and in the latter 6 beats per second will 
 be produced. 
 
 As the intonation of the Thirds is guarded by such high, and 
 therefore weak partials, a slight error in their tuning is much less 
 evident, than in the case of the Fifth. Thus Thirds tuned in equal 
 temperament are, as vre shall see later on, mistimed to an extent, 
 which if adopted with the Fifth, would render this latter interval 
 unbearable. 
 
 Major and Minor Sixths. By pursuing the method adopted above, 
 the student will find that the former of these two intervals between 
 Oompoimd Tones, is defined by the coincidence of the 3rd and 5th, 
 and the latter by the coincidence of 5th and 8th partials. Inasmuch 
 as the 8th partial is generally exceedingly weak, the Minor Sixth 
 can scarcely be said to be defined at all. 
 
 In aU the above intervals, we have only considered the lowest pair 
 of coincident partials, as these are by far the most important; but 
 it must not be forgotten, especially in the case of the Octave, and 
 Fifth, that there are coincident pairs above those given. If the 
 interval be not quite true, not only wiU beats be produced by this 
 lowest pair, but by the higher also, and at a more rapid rate. Thus, 
 let di = 101 and d = 200 : then. 
 
DEFINITION OF THE CONSONANT INTERVALS. 183 
 
 806- 
 
 600 ^ (6) 
 
 404- 
 
 S03 
 
 .400 = (4) 
 
 202 200 = P) 
 
 101 
 
 2 beats per second ■will be lieard from the pair marked (2), 4 beats 
 per second from tbat marked (4), and 6 from that marked (6}. 
 
 In tuning Fifths, Thirds, &c., between Compound Tones -with 
 perfect exactness, a resonator tuned to the pitch of the coincident 
 partials -will be found of great service; for these partials being 
 thus reinforced, it ■will be easy to discriminate any beats 
 between them, from the beats of other partials ; and furthermore 
 the disturbing eflect of any dissonating partials ■which may be 
 present, ■will be much lessened. 
 
 It ■will be seen from the above, that the particular partials ■which 
 coincide in any interval are given by the figures ■which denote its 
 ■vibration ratio. Thus, the vibration ratio of the octave is 2 : 1, and 
 the coincident partials axe the 2nd and 1st ; the vibration ratio of 
 of the Fifth is 3 : 2, and the coincident partials are the 3rd and 2nd, 
 and so on. 
 
 Furthermore, the preceding illustrations sho^w, that in any 
 particular interval, if the lower of the two tones is one vibration 
 too sharp or too flat, the number of beats produced by the lo^west 
 pair of coincident partials is the same as the greater of the two 
 numbers which denote its ■vibration ratio. Thus, in the case of the 
 Major Third taken above, — 401 and 500, — ^we found the number of 
 beats per second to be 5, and that is the greater of the two numbers 
 5 : 4 which give its vibration ratio. Similarly, if the higher of 
 the two tones of an interval be one ■vibration too sharp or too flat, 
 the number of beats per second ■will be the smaller of the two 
 numbers which denote its ■vibration ratio. For example, let the 
 ■vibration numbers of a mistuned Fifth be 200 and 301 : Then 
 
184 
 
 HAND-BOOK OF ACOUSTICS. 
 
 600- 
 
 -602 
 
 400 
 
 200 
 
 301 
 
 tke number of beats per second ■wiU be 602 — 600 = 2, ■which is 
 the smaller of the two numbers, in the ratio, 3 : 2. 
 
 When the two Compound Tones forming an interval do not 
 possess all the first six partial Tones, the above results require to be 
 modified. 
 
 Thus for example, in •wide open organ pipes, the tones of which 
 consist of only the first two partials, the Octave is the only 
 Interval which is defined by the coincidence of partials ; the other 
 Intervals being guarded merely by Differentials. 
 
 Again, in stopped organ pipes the tones of which only consist 
 of the 1st and 3rd partials, the Twelfth is the only interval defined 
 by the coincidence of the partials ; the other intervals, even the 
 Octave, being guarded by DifEerentials onij . 
 
 In such cases as these, however, the definition is better than it 
 would be if the tones were simple, more DifEerentials and those of 
 a higher order being produced. For example, take the mistuned 
 Major Third d = 400 and n ^ 501, and suppose each of these 
 tones to consist of the first three partials only. Then the 1st 
 horizontal line of the following table shows the partials of the one 
 tone, and the 1st vertical line those of the other, the diflerentials of 
 the 1st order being at the intersections. 
 
 
 400 
 
 800 
 
 1200 
 
 501 
 1002 
 1503 
 
 101 
 
 602 
 
 1103 
 
 299 
 202 
 703 
 
 699 
 198 
 303 
 
DEFINITION OF THE CONSONANT INTERVALS. 185 
 
 The following beats -will be generated between these DifEerentials, 
 202 — 198 : 
 
 :4N 
 
 303 — 299 = 4 V beats per second. 
 703 — 699 = 4 ) 
 
 The Major Third between tones consisting of the first three partials 
 is guarded therefore by three sets of Differential Tones of the 1st 
 order. 
 
 SUMMABT OF DEFIOTTION OF ISTEEVAIS. 
 
 Simple Tones. 
 
 Octave. 1st Differential in unison with lower tone. 
 
 Fifth, ,1st „ „ „ 2nd Differential. 
 
 FovHh. 1st ,, ,, ,, 3rd ,, 
 
 Any departure from true intonation produces beats between these 
 unisons. 
 Other intervals practically undefined. 
 
 Ordinary Compound Tones. 
 
 The Octave, Fifth and Fourth defined as above, but also and 
 chiefly as follows, 
 
 Octave. 2nd partial of lower tone unisonant with 1st partial of higher. 
 Fifth. 3rd „ „ „ „ „ 2nd „ 
 
 Fourth. 4th ,, ,, „ „ ,, 3rd ,, ,, 
 
 Major Third. 5th „ „ „ ,, 4th ,, ,, 
 
 Minor Third. 6th ,, ,, „ ,, 5th „ „ 
 
 and generally, in any interval the unisonant or defining partials are 
 given by the numbers which denote its vibration ratio. 
 
 If the lower of the two tones of any interval be out of tune by 1 
 vibration per second, the number of beats generated (by lowest 
 pair of defining partials) is the same as the greater of the two 
 numbers which denote its vibration ratio ; it the higher tone be out 
 of tune by the same amount, the niunber of beats is the smaller of 
 these two numbers. 
 
186 
 
 CHAPTER XVI. 
 
 On the Eelative Habmonioxtsness of the Consonant 
 Intekvals. 
 
 We have now to examine into the causes of the relative smoothness 
 of those intervals -which are usually called consonant. 
 
 With regard to perfectly simple tones, there is, as we have already 
 seen, no element of roughness in any of these intervals, except in 
 the case of the Thirds, and in these only when very low in pitch ; 
 consequently, there is found to be little or no difference in 
 smoothness, between any of these intervals, when strictly Simple 
 Tones are employed, and when the tones in question are in perfect 
 tune. 
 
 With Compound Tones, however, the case is very different : not 
 only do these intervals vary in smoothness — in harmoniousness — 
 one with another, but the smoothness of any one particular interval 
 varies according to the constitution, that is the quality, of its 
 Compound Tones. 
 
 In the first instance, we shall consider these intervals as formed 
 between Compound Tones, each consisting of the first six partials ; 
 and as before, we shall suppose, as is generally the case, that the 
 intensity of these partials rapidly diminishes as we ascend in the 
 series. In fig. 80 we have the ordinary Consonant Intervals, 
 together with a few others, drawn out so as to show the first five or 
 six partials of each tone. To facilitate comparison, the lower of the 
 two tones in each interval, is supposed to be of the same pitch 
 throughout, so that tones on the same horizontal lines are of the 
 same pitch. The symbols for the partials diminish in size, as they rise 
 above the fundamental, in order to represent roughly their diminution 
 in intensity. As before, partials forming a tone dissonance are 
 connected by a single line, those that dissonate at a semitone are 
 joined by a double one. In comparing the intervals of the figure, 
 it must be borne in mind, not only that the beats of the semitone 
 
HARMONIOUSNESS OF CONSONANT INTERVALS. 187 
 
 are much worse than those of the tone, but also that these vary in 
 themselves — the beats of the f tone ( j 5 for example) not being so 
 harsh as those of the y tone ( j ™), nor those of the 44 semitone 
 ( I ^'), usually so discordant, as those of the ||- semitone ( j ^°). 
 The small letters or asterisks in curved brackets show the positions 
 of the Summation Tones generated by the fundamentals. 
 
 The facts thus summarized in fig. 80, will be found, on caieful 
 examination, to throw light on several fundamental phenomena in 
 harmony relating to these intervals. 
 
 In the first place, it will be at once seen that with regard to 
 Compound Tones such as those depicted, the Octave is the only 
 perfectly consonant interval, that is, the only one absolutely free 
 from roughness. Moreover, the student will readily perceive, that 
 no roughness, except such as may be inherent in the tones 
 themselves, can ever occur between two Compound Tones, at this 
 interval, no matter what their constitution may be ; for the higher 
 of the two tones only adds to the lower one, elements which are 
 already present. 
 
 The fact that the Octave is the only Interval devoid of all 
 roughness, explains why this interval is the only one that can be 
 used in aU regions of the musical scale, on all instruments. Again 
 the fact, that the Compoim.d upper tone of the Octave, adds nothing 
 new, but simply reinforces tones already present in the Compound 
 lower tone, explains the similarity in effect of the two tones forming 
 an Octave. We can thus understand, how it is that a company of 
 men and women totally unskilled in music, and utterly unable to 
 sing in Thirds, &o., yet experience no diffloulty in singing together 
 a tune in Octaves, and indeed when doing so usually consider 
 themselves to be singing tones of the same pitch; in fact, such 
 singing is called, even by musicians, unison singing. 
 
 Again, we see why a part in music for the Pianoforte, Harmonium, 
 &c., may be doubled with impimity ; for such addition adds nothing 
 absolutely new ; it simply reinforces the upper partials of tones 
 already present, thus producing a brighter eSect. 
 
 The Fifth as constituted in fig. 80 is not always a perfectly 
 Consonant Interval, for as the figure shows the 3rd partial of the 
 upper compound tone, dissonates with both the 4th and 5th of the 
 lower one. The degree of roughness thus produced, will depend 
 upon the intensity of these partials, and inasmuch as they are 
 usually faint, the roughness wiU. be but slight. Other things being 
 
188 
 
 HAND-BOOK OF ACOUSTICS. 
 
 I I I I. 
 
 \ 
 
 d' d' d' 
 
 s(.) 
 
 d d d 
 
 1,/ 
 \ 
 
 S 8 8 
 
 (n) 
 
 \ 
 
 f. 
 
 w 
 
 I . 
 
 n' n' m' 
 
 \ 
 
 (•) 
 
 m, 
 
 U M |d| \ 
 
 n, 
 di 
 
 Hi 
 
 Fia. 80. 
 
EABMOmOUSNESa OF CONSONANT INTERVALS. 189 
 
 1./ 
 
 1,-^*'' J. 
 
 n' m' le' 
 
 / 
 
 le' 
 
 / 
 
 ^' 
 
 h-^ 
 
 (') 
 
 .di 
 
 (•) 
 
 fe. 
 
 (•) 
 
 di 
 
 (*) 
 
 m, 
 
 f, 
 
 |d. [n, (t, \ 
 
 Fio. 80. 
 
190 HAND-BOOK OF ACOUSTICS. 
 
 equal, the roughness of this interval will depend upon its position 
 in the musical scale; such roughness beconung greater as we 
 descend, and less as we ascend. Two reasons may be assigned for 
 this ; in the first place, the upper partials of low tones are usually 
 stronger than those of higher ones, and consequently, when they 
 beat with one another the beats are more intense, thus producing a 
 harsher effect ; secondly, partials that beat with one another in. the 
 lower part of the musical scale may be beyond beating distance in 
 the upper part. To illustrate this fact, which of course appKes to 
 other intervals, we wiU take two or three cases of Fifths in difierent 
 parts of the musical scale. 
 
 First, take = d = 256, then s = 384. 
 3rd partial of s = 384 X 3 = 1152 
 4th „ d = 256 X 4 = 1024 
 
 number of beats per second = 128 
 
 Now from the table on page 159, we know dxat 128 beats per second, 
 in the neighbourhood of O* = 1,024, is only just within the beating 
 distance ; consequently we may conclude that fifths above middle 
 having the constitution assumed above, are devoid of all roughness 
 whatever. 
 
 Next take C^ = dj = 64, then s^ = 96. 
 3rd partial of Sj = 96 X 3 = 288 
 4th „ dj = 64 X 4 = 256 
 
 number of beats per second = 32 
 
 From the table of page 159, we see that 32 beats per second in the 
 region of C = 256 form a somewhat harsh dissonance. In fact, 
 when Oj and Q^ are strongly sounded on a harmonium, the harsh 
 effect produced is due to the dissonating partials C and D, and 
 consequently this harsh effect is about the same as that obtained by 
 softly sounding the C and D digitals together, — a matter which can 
 be easily put to the proof. 
 
 The above, therefore, explains the fact, that while an Octave 
 may be played anywhere in the Musical Scale, a Fifth cannot be 
 well used below a certain limit. On the other hand, we see that 
 speaking generally, a Fifth is a perfectly consonant interval, when 
 taken above middle ; we might therefore term this, the limit of a 
 perfectly consonant Fifth on the Harmonium, Pianoforte, and 
 stringed instruments in general. 
 
EABMONIOUSNESS OF CONSONANT INTERVALS. 191 
 
 A glance at fig. 80, shows that the Fourth is not so perfect an 
 interval as the Fifth. Its roughness arises chiefly from the beats 
 generated between the usually powerful 2nd partial of the upper 
 tone, and the almost equally loud 3rd partial of the lower one. To 
 this may be added the much softer semitone and tone dissonances 
 between 4th, 5th, and 6th parijals ; and a still slighter disturbing 
 element may be sometimes present in the Summation Tone midway 
 between the 2nd partials. It may be noted also that the 2nd and 
 3rd pairs of dissonating partials, the 4th and 5th pairs, and also the 
 Siunmation Tone, give rise to precisely the same number of beats. 
 
 As in the case of the Fifth, not only will the roughness of this 
 interval vary with the varying intensities of dissonating partials, 
 but other things being equal, with its position in the Musical Scale. 
 
 For example take 0- = d = 384, then f = 512 
 3rd partial of d = 384 X 3 = 1152 
 2nd „ f = 512 X 2 = 1024 
 
 number of beats per second = 128 
 
 which (see table, page 159) is only just within beating distance. It 
 may be noticed that this number is the same that we obtained in 
 the case of the Fifth { q, showing that in order to obtain a Fourth 
 of approximately equal smoothness with a Fifth, we must take the 
 former a Fifth higher in pitch. Thus using the term in the same 
 sense as before, we might call this the lower limit of a perfectly 
 consonant Fourth. 
 
 Coming now to the Thirds, we find in both Major and Minor, 
 that the Third partial of the Upper Tone dissonates with the 4th 
 partial of the lower one ; but while in the latter they form only a 
 tone dissonance, in the former they dissonate at the much more 
 unpleasant interval of a semitone. On the other hand, while softer 
 4th and 5th partials respectively of the Minor Third beat at a 
 semitone distance, the corresponding partials of the Major Third do 
 not beat at all. Further, the Summation tones when present will 
 add to the roughness ; that of the Minor Third being slightly more 
 detrimental than that of the Major. 
 
 The Thirds, in respect to their harmoniousness vary very greatly 
 according to their position in the Musical Scale. They cannot be 
 used very low in pitch, even when they are formed between Simple 
 Tones : for as we have already seen the Thirds j -^ and j q' 
 between Simple Tones are at the beating distance, that is, 0^ and C| 
 
192 HAND-BOOK OF ACOUSTICS. 
 
 are the limits respectively at, and below which, a Major and a 
 Minor Third between Simple Tones, become dissonant. Thirds at 
 or below these limits, between Compound Tones, contain of course 
 these same elements of roughness, between their fundamentals ; to 
 which however must be added, the further roughnesses due to their 
 beating overtones. Thirds, above these limits, must owe their 
 roughness chiefly to beating overtones. 
 
 From fig. 80, we see that the smoothness of a Major or Minor 
 Third between Compound Tones, above the limit just referred to, 
 depends chiefly upon the loudness of the beats between 3rd, 4th, 
 and 5th partials. Now observation shows that in the case of the 
 Voice, Harmonium, and Piano, these partials generally become 
 weak or even altogether absent above middle C ; consequently, 
 Thirds above this region, on these instruments will be as a rule 
 sufBciently smooth. As we descend, however, from this region. 
 Thirds rapidly deteriorate, for in the first place these partials begin 
 to assert themselves, and secondly, the fundamentals are approxi- 
 mating to the beating distance. 
 
 To take an example : 
 
 Let C| = 128, then B, = 160 
 4th partial of 0, = 128 X 4 = 512 
 3rd „ E| = 160 X 3 = 480 
 
 number of beats per second = 32 
 
 and 32 beats per second, in the region of C = 512 are very harsh 
 if at aU prominent. 
 
 Again, 
 
 Let C| = 128, then Bfe, = 163| 
 4th partial of C, = 128 X 4 = 512 
 3rd „ Et2| = 153|-X 3 = 460^ 
 
 number of beats per second = 51^ 
 
 which is within beating distance in the region of C = 512. To 
 this must be added, first, the roughness due to the 153a — 128^25|, 
 beats per second between the fundamentals, which are just about 
 the beating distance, secondly, that due to the possible Summa- 
 tion Tone, and thirdly, that arising from the dissonant 6th and 4th 
 partials. 
 
EABMONIOUSNESS OF CONSONANT INTERVALS. 193 
 
 It should be observed, that, though such 3rd and 4th partials may 
 be absent or weak in tones which are produced softly, they may 
 become very prominent in those tones when sung or played loudly; 
 consequently a Third which may be perfectly smooth and 
 harmonious when softly played or sung, may beconte rough and 
 unpleasant when more loudly produced : a remark which evidently 
 applies to other intervals also. 
 
 The foregoing explains, why Thirds were not admitted to the rank 
 of consonances, until comparatively recent times. For the compass 
 of men's voices (in respect to which, the music among classical 
 nations was chiefly developed) lies chiefly below middle 0, and as 
 we have just seen, Thirds in the lower parts of that compass are 
 actually dissonant. 
 
 We have, in the above, also, the explanation of the rule in 
 harmony which forbids close intervals between the tenor and the 
 bass, when these parts are low in pitch. 
 
 To sum up the comparative smoothness of the Thirds : we find 
 that these intervals may be almost or quite devoid of roughness 
 when somewhat high in pitch, and may even excel the Pourth in 
 smoothness under these circumstances, but that they rapidly 
 deteriorate, as we descend below middle 0. 
 
 For Compound Tones of such constitution as depicted in fig. 80, 
 the Major Sixth seems decidedly equal, if not slightly superior, to 
 the Fourth. As in the case of the latter interval, the 2nd partial 
 of its upper tone dissonates with the 3rd partial of the lower, at 
 the inteiTal of a tone, but the roughness due to dissonances between 
 the 4th and 5th partials in the latter interval is wanting in the 
 former. As a set ofi to this advantage, however, we see that the 
 Summation Tone in the Major Sixth when present, produces a tone 
 dissonance with the 3rd partial of the lower tone. 
 
 On the other hand, the Minor Sixth is the worst interval we have 
 yeit studied. Its chief roughness is due to the semitone dissonance 
 between the 2nd partial of the upper and the 3rd partial of the 
 lower tone, which are usually pretty loud. A subsidiary roughness 
 is seen above between the 3rd, 4th, 5th and 6th partials. 
 
 As an example of the Major Sixth, 
 
 take d = 384, then 1 = 640 
 2nd partial of 1 = 640 X 2 = 1280 
 3rd „ d = 384 X 3 = 1152 
 
 number of beats per second = 128 
 
194 EAND-BOOK OF A00U8TIG8. 
 
 •which we see from the table on page 159 is on the verge of the 
 beating distance. Q- = 384 may be taken therefore, as the limit 
 above which a Major Sixth is a perfectly smooth interval, and 
 below which it gradually deteriorates. 
 
 On the other hand the Minor Sixth at this pitch has still an 
 element of roughness, for 
 
 let n = 384, then d' = 614|- 
 2nd partial of d' = 614f X 2 = 1228f 
 3rd „ n = 384 X 3 = 1162 
 
 number of beats per second == 7&^ 
 which forms a harsh dissonance in this region. 
 
 On comparing the relative smoothness of the Fourth, Major 
 Sixth, and Major Third, no valid reason appears for the precedence, 
 which is usually granted to the first-named interval over the other 
 two ; and in fact no such precedence can be assigned to the Fourth, 
 if the three intervals be judged by the ear alone, imder similar 
 circumstances. As Helmholtz remarks " the precedence given to 
 the Fourth over the Major Sixth and Third, is due rather to its 
 being the inversion of the FiEth, than to its own inherent 
 harmoniousness." 
 
 In the Diminished Fifth and Augmented Fourth or Tritone the 
 2nd, 3rd, and 4th partials of the upper tone dissonate at the interval 
 of a semitone with the 3rd, 1th, and 6th partials of the lower tone ; 
 after mentioning which elements of roughness, it is scarcely 
 worth while to point out the tone dissonance between the 4th and 
 dth partials. 
 
 Although the above results apply fairly weU to all instruments, 
 the tones of which consist of the 1st six partials, to such 
 instruments for example, as the pianoforte, harmonium, open pipes, 
 of organs, and the human voice ; yet it will not do to apply them, 
 in a hard and fast manner, to any instrument whatever. The low 
 tones of the harm.onium, for example, especially if the instrument 
 be loudly played, contain more than the 1st six partials, while those 
 of open organ pipes, gentiy blown, often consist of fewer. Again, 
 the tones of the human voice vary wonderfully in their constitution, 
 not only in different voices, but also, and chiefly, according to the 
 particular vowel sound produced. The influence which the vowel 
 sounds have in modifying the roughness or smoothness of an 
 interval, can best be realized by making a few experiments with 
 men's voices. Let such intervals as the Major and Minor Thirds be 
 
HABMONIOUSNESS OF CONSONANT INTERVALS. 195 
 
 \ 
 
 di di 
 
 m' n' se' 
 
 m, d 
 
 t') 
 
 (*) 
 
 m, 
 
 
 I Hi 
 'd, 
 
 rii 
 
 Fi8. 81. 
 
196 
 
 HAND-BOOK OF AG0U8TIG8. 
 
 \ 
 / 
 
 \ 
 
 / 
 
 d' d' 
 
 S S B 
 
 d' d' 
 
 \ 
 
 (•) 
 
 f» 
 
 ^t 
 
 n" m' 
 
 R, 
 
 01. 01. 01. 01. 
 
 {J 
 
 I 
 i 
 
 Oh. 01. 01. Ob. 
 FiQ. 82. 
 
 06. 01. 01. Ob. 
 
EABMOmOUSNESS OF CONSONANT INTEBVALS. 197 
 
 \ 
 
 (*) 
 
 ii|( m' n' m' 
 
 (' 
 
 in 
 
 pii 
 
 n, 
 
 in 
 
 n 
 
 d' 
 
 (*) 
 
 PI, 
 
 (rii 
 Ob. 01. CI. Oh. 
 
 Ik 
 
 Ob. 01. 01. Ob. 
 Kio. 82. 
 
 (Hi 
 Ob. CI. CI. Ob. 
 
198 HAND-BOOK OF ACOUSTICS. 
 
 sounded by voices at different pitches below middle 0, first on sucli 
 vowel sounds as "a" in father, "i" in pine, and afterwards on 
 " 00 " in cool. The diminution in roughness in the latter case is 
 very striking. The chief cause of the charm of soft singing, 
 doubtless Kes in the fact, that the upper dissonating partials of 
 Thirds, Sixths, &o., beoonie so faint, as to be practically nonexistent. 
 
 Intervals between Compound Tones consisting of the odd partials 
 only, — such tones, for example, as are produced by the narrow 
 stopped pipes of the organ and by the clarionet — may be more 
 briefly noticed. Fig. 81 shows the ordinary consonant intervals 
 between such tones, fully drawn out on the plan of fig. 80. The 
 first thing that strikes us on looking at the figure, is the improve- 
 ment noticeable in each interval ; most of the dissonances of fig. 80 
 having vanished. The reader will be surprised, doubtless, by the 
 apparent inferiority of the Fifth to most of the other intervals — ^to 
 the Thirds — for example ; but we must point out that it is for the 
 most part only apparent. For we have already seen, that the Fifth 
 is a perfectly smooth interval above middle C; consequently the 
 Thirds can only be superior to the Fifth below that limit : and we 
 have shown above that the Thirds rapidly deteriorate as they sink 
 in pitch from that point, in consequence of their fundamentals 
 approaching the Beating Distance. 
 
 Another interesting case is that of Intervals between Compound 
 Tones, one of which consists of only odd, and the other of the full 
 scale of partials ; such intervals as would be produced, for example, 
 by a Clarinet sounding one tone and a Oboe the other. There will 
 be two cases according as the lower tone is sounded on the former, 
 or the latter instrument. Fig. 82 shows the ordinary consonant 
 intervals, drawn out after the manner of the two preceding figures. 
 Each interval is given twice: in those marked Ob CI the lower 
 tone of the interval is supposed to be sounded by the Oboe, while 
 in those marked CI Ob the Clarinet produces the lower tone. 
 
 It is evident at once, that it is not a matter of indifference, to 
 which instrument the lower tone is assigned. The Fifth and Major 
 Third are decidedly better when the lower tone is given to the 
 the Clarinet; while the Fourth, Major Sixth, and Minor Sixth are 
 smoother, when the Oboe takes the lower tone. 
 
 From the foregoing, it is quite dear that no hard and fast line 
 can be drawn between Consonance and Dissonance ; for as we have 
 seen, every interval, between ordinarily constituted Compound 
 
HABM0NI0U8NES8 OF CONSONANT INTERVALS. 199 
 
 Tones, except the Octave, becomes a dissonance •when taken 
 sufiBciently low in pitch. Furthermore, when the intervals of fig. 
 80 are taken in the same region of the musical scale, there is an 
 uniform gradation of roughness, or diminution of smoothness in 
 passing from the Perfect Fifth on the left, to the diminished Fifth 
 on the right, which is usually looked upon as a dissonance. Again, 
 with similar Compotmd Tones, and in the same region of pitch, 
 some so-called dissonances are not inferior to intervals universally 
 termed consonant. Compare for example. 
 
 (0, 
 
 G 
 
 
 
 & 
 
 0, 
 
 0' 
 
 % 
 
 G 
 
 0, 
 
 E. 
 
 Hitherto, we have only considered Intervals not greater than an 
 Octave, and in musical theory, no great distinction is drawn, 
 between an interval, and its increase by an Octave. In reality, 
 however, the addition of an Octave to an Interval, between 
 Compound Tones, does exercise a great influence on its relative 
 smoothness. Fig. 83 shows, on the same plan as before the 
 Twelfth, Eleventh, Tenths, and Thirteenths. 
 
 The first point to be noted about these intervals is that the 
 addition of an Octave to a Fifth, makes the Interval a perfect one ; 
 the addition of the Compound Tone (s) to the Compound Tone (d|) 
 supplying no new partial tone to the latter. A Twelfth is therefore 
 decidedly superior to a Fifth. On the other hand by comparing 
 the Eleventh } |^ of fig. 83 with the Fourth j * of fig. 80 it wiU be 
 seen that the former is the worse of the two : for though the 
 
200 
 
 EANB-BOOK OF ACOUSTIOS. 
 
 I. I. 
 
 ■' •' 8' 
 
 p 
 
 l' 
 
 m 
 
 ( 
 
 
 1 
 
 \ 
 
 S SB 
 
 \ 
 
 m' m' 
 
 % 
 
 I / 
 
 (•) 
 
 (•) 
 
 d, m, 
 
 (•) differential 
 
 J. ^ 
 
 / 
 
 (•) 
 
 V d' 
 
 (•) 
 
 m, 
 
 s 
 
 Fio. 83. 
 
HABMOmOUSNESS OF CONSONANT INTERVALS. 201 
 
 dissonances are at the same interval ia each ; in the latter, it is the 
 dissonance of a 3rd partial against a 2nd, while in the former it is 
 the dissonance of a 3rd partial against a Fundamental. Similarly 
 the Major and Minor Thirteenths are inferior to the Major and 
 Minor Sixths. 
 
 The Major Tenth however is greatly superior to the Major Third, 
 the 3rd and 4th partial dissonance { ^' of the latter, being absent in 
 the former. With regard to the Minor Tenth and Minor Third, 
 although in the former, the 3rd and 4th partial dissonance j^ 
 of the latter has disappeared, yet the semitone dissonance j °° 
 being now between 6th and 2nd partials, wiU. be much more 
 prominent in the former than in the latter, where it only occurs 
 between 5th and 4th partials. Under these circumstances it is 
 difficult to say which is the better interval of the two. Helmholtz 
 holds that the Minor Third is the superior, and so obtains the 
 following symmetrical rule to meet all cases : — 
 
 " Those intervals, in which the smaller of the two numbers 
 expressing the ratios of the vibration numbers is odd, are made 
 worse by having the upper tone raised an Octave ; " while 
 
 " Those intervals, in which the smaller of the two numbers 
 expressing the ratios of the vibration numbers is even, are 
 improved by having the upper tone raised an Octave." 
 
 We conclude the present Chapter, by giving in fig. 84, Helm- 
 holtz's, graphic representation of the relative harmoniousness of 
 musical intervals. 
 
 IH^™8B 
 
 Fio. 84. 
 
 In this figure, the intervals are represented by the horizontal 
 distances CEl?, CE, &c., measured from the point ; while the 
 
202 EAND-BOOK OF A00U8TI08. 
 
 rouglmess of the intervals is shown by the vertical distances of the 
 curved line from the corresponding points E!z, E, &c., on the 
 horizontal line. For example ; the roughness of the interval { ^ 
 IS represented by the length of the vertical line over the point SI ; 
 the roughness of the interval j q, by the short vertical line over the 
 point E, and so on. Thus, if -we liken the curve to the outline of 
 a mountain chain, the dissonances are represented by peaks, while 
 the consonances correspond to passes. 
 
 According to this figure, the consonances in the order of their 
 relative harmoniousness, are. 
 
 Octave, 
 
 Eifth, 
 rEourth, 
 i Major Sixth, 
 (.Major Third, 
 
 Minor Third, 
 
 Minor Sixth. 
 
 In making use of the figure, however, the student must 
 continually bear in mind, the assiunptions on which it was 
 calculated; viz., that the roughness vanishes when there are no 
 beats ; that it increases from this to a maximum for 33 beats per 
 second ; that it diminishes from this point as the number of beats 
 per second increases; and lastly, that the intensity of the partial 
 tones diminishes inversely as the square of their order. The 
 conclusions expressed in the diagram, are therefore only true in 
 those cases in which these assumptions are true, or approximately 
 true. 
 
 StJMMAKT. 
 
 All the consonant intervals between Simple Tones are equally 
 smooth or harmonious. 
 
 Intervals, whether between Simple or Compound Tones, having 
 the following vibration ratios, 
 
 2 3 4 6 
 
 J. J. J. J. &c., 
 
 are perfect in their smoothness; they have no elements of 
 roughness whatever. 
 
EABM0NI0U8NES8 OF CONSONANT INTERVALS. 203 
 
 The consonant intervals less than an Octave vary in smoothness 
 according to the constitution or quality of their constituent Com- 
 pound Tones, and according to their position in the scale of pitch. 
 Hehnholtz's arrangement for average qualities of tone is, 
 
 (1) Fifth, 
 
 (2) Fourth, Major Third, Major Sixth, 
 
 (3) Minor Third, 
 
 (4) Minor Sixth. 
 
 For other consonant intervals greater than an Octave, Hehnholtz's 
 
 rule applies, viz : — 
 
 Those intervals in which the smaller of the two numbers 
 
 expressing the ratios of the vibration numbers, is eoen, are improved 
 
 by having the upper tone raised an Octave, and vice versa ; thus, 
 
 Fifth 4 ; ( TweUth. 
 
 ,, . rrr, . , J uuproved by becomiug ^ , , . „ „ 
 Major Third J ) { Major Tenth. 
 
 Fourth 4 ^ ( Eleventh. 
 
 Minor Third 4( \ Minor Tenth. 
 
 nr • Q- i.-u ,> made worse by becoming{ . 
 
 Major Sixth -fl J Major Thirteenth. 
 
 Minor ,, 4 J I Minor ,, 
 
204 
 
 CHAPTER XVII. 
 
 Chords. 
 
 We have already seen that the Consonant intervals, within the 
 Octave, are the Minor and Major Thirds, the Fourth, the Fifth, and 
 the Minor and Major Sixths. If any two of these intervals be 
 united, by placing one above the other, the interval thus formed 
 between the two extreme tones, may or may not be consonant. In 
 the former case the combination is termed a Consonant Triad. 
 
 In order to obtain all the Consonant Triads within the compass of 
 an Octave, it is therefore only necessary to combine the above 
 intervals two and two, and select those combinations, whose extreme 
 tones form a consonant interval. The following table shows all the 
 combinations of the above iatervals, taken two at a time, whose 
 extreme tones are at a smaller interval than an Octave. 
 
 (1) Minor Third + Minor Third, | X f = |f 
 
 (2) „ + Major „ |xi=i, Fifth 
 
 (3) „ + Fourth, f X ^ = I , Minor Sixth 
 
 (4) „ + Fifth, I X i = f 
 (6) „ + Minor Sixth, i x | = -J^ 
 
 (6) Major Third + Major Third, i X i = ff 
 
 (7) , „ + Fourth, 7X4=4, Major Sixth 
 
 (8) „ + Fifth, I X i = V 
 
 (9) Fourth + Fourth, 4 x 4 = V° 
 
 The only combinations in the above, the extreme tones of which 
 form a consonant interval, are Nos. 2, 3, and 7. But each of these 
 
CHOBDS. 
 
 205 
 
 is capable of forming two Consonant Triads, according as the 
 smaller of the constituent intervals is below or above. Consequently 
 we find that there are altogether six Consonant Triads : viz., 
 
 From (2) ( Minor Third 
 
 I 
 
 [ Major „ 
 
 TMaior Third 
 
 ] 
 
 (_ Minor „ 
 
 From (3) ("Minor Third 
 
 I 
 
 Fourth 
 
 (Fourth 
 
 Is ^ 
 
 ?Mi 
 
 Minor Third 
 
 From (7) ( Major Third 
 fFoTirth 
 
 (Fourth 
 
 f Major Third 
 
 as 
 
 SE 
 
 If the lowest tone (d) of the 1st Triad be raised an Octave, we 
 (d' 
 obtain the 4th Triad above | s ; while if the highest tone (s) of this 
 
 same Triad be lowered an Octave we get the 5th Triad } d. Hence 
 
 V s, 
 the 4th and 5th Triads are usually considered, to be derived from 
 
 the 1st, and are called, respectively, its First and Second Inversions, 
 
 or more briefly, its "h" and " c" positions. 
 
 Again, if the lowest tone (1|) of the 2nd Triad be raised an Octave 
 
 we obtain the 6th Triad j m; while the lowering of the highest tone 
 
 (pi) through the same interval, produces the 3rd Triad ] 1, . Hence, 
 
 (mi 
 as before, the 6th and 3rd Triads are considered to be derived from 
 
 the 2nd and are called its First and Second Inversions, or its "b" 
 
 and " c " positions respectively. 
 
206 
 
 HAND-BOOK OF ACOUSTICS. 
 
 The 1st Triad i m, which has the Major Third below and the 
 Minor Third above, is called a Major Triad, while the second J d 
 which has the Minor Third below and the Major above, is termed 
 a Minor Triad. The Sis Consonant Triads may therefore be 
 arranged as follows : 
 
 MAJOR TEIADS. 
 First Inversion 
 f Fourth 
 
 ( Minor Third 
 
 Normal 
 [ Minor Third 
 
 Major Third 
 
 Da 
 
 ( Major Third 
 ( Minor Third 
 
 D6 
 
 MINOE TEIADS. 
 I Fourth 
 
 ( Major Third 
 
 Second Inversion 
 ( Major Third 
 
 h 
 
 'Fourth 
 
 s ^^ 1^' ^ 1?, I 
 
 Dc 
 
 Minor Third 
 Fourth 
 
 L6 
 
 We shall first consider the Major Triads. An idea of the relative 
 harmoniousness of these Triads may be obtained by the aid of fig. 85, 
 in which these Triads are fully drawn out, on the same plan as in 
 fig. 80, for Compound Tones containing the first six partials. In 
 each Triad, the first sis partials of the lowest tone, the first five of 
 the middle tone, and the first four of the highest tone are given. 
 The partials, that dissonate at the interval of a tone are connected, 
 as before, by a single bne, those dissonating at a semitone distance, 
 by a double line. Whenever any partial dissonates with two 
 partials of the same pitch, the first partial is connected with that 
 partial of the other two which is of a lower order, that is, with the 
 one which is presumably the louder of the two. 
 
 In order to facilitate the comparison of the Triads, an analysis of 
 fig. 85 is given in the following table, in which the first horizontal 
 line gives the names of the Triads ; the next three lines show the 
 
CH0RD8. 
 
 207 
 
 n' n' 
 
 \ 
 
 ( d' 
 
 PI, 
 
 I 
 
 ds 
 
 n' ~^ H< 
 
 \ .. 
 
 I I 
 
 d2 
 
 ^ 
 
 i 
 
 '\ 
 
 'A 
 "1 
 
 No. 1. 
 
 No. 2. 
 Fio. 85. 
 
 No. 3. 
 
208 
 
 HAND-BOOK OF ACOUSTICS. 
 
 corresponding partials 'whicli beat at a semitone distance ; and the 
 remaining lines, tliose that beat at a tone distance. It has not been 
 thought necessary to discriminate between the 16:15 and the 25 : 24 
 beats, nor between the 9 : 8 and 10 : 9. 
 
 
 Do 
 
 Da 
 
 D& 
 
 Partials which beat at 
 
 the interval of a 
 
 semitone. 
 
 3,4 
 
 3,4 
 4, 5 
 
 2,3 
 3,5 
 4,5 
 
 Partials which beat at 
 
 the interval of a 
 
 tone. 
 
 2,3 
 2,3 
 4,6 
 4,6 
 
 3,4 
 3,4 
 
 2,3 
 3,4 
 
 Comparing, in the first place, Dc with Da : after eliminating the 
 semitone dissonance between the 3rd and 4th partial, which ia 
 common to both, there remains in the latter, a semitone dissonance 
 between a 4th and 5th partial, which is absent in the former. On 
 the other hand, among the tone dissonances, Dc has two, between 
 the 2nd and 3rd partials, as against the two between 3rd and 4th 
 partials in Da ; the former being of course the more prominent. 
 The two dissonances of 4th against 6th partials are so slight, that 
 they may be disregarded. It is difficult, on the whole, to decide 
 between these two Triads ; there is probably not much difference 
 between them. Helmholtz gives the preference to Dc, which we 
 have therefore put first in the table. 
 
 If there be any doubt concerning the relative harmoniousness of 
 Dc and Da, there can be none with respect to D6. Comparing it 
 with Da : after discarding the semitone dissonance of a 4th against 
 a 5th partial, which is common to both, there remains in Di the two 
 semitone dissonances of a 2nd against a 3rd and a 3rd against a 5th 
 partial, while in Da there is only that of a 3rd against a 4th. 
 Again, among the tone dissonances, after throwing out that of a 3rd 
 against a 4th partial from both, there remains in D6 the dissonance 
 of a 2nd against a 3rd, while in Da there is the less prominent 
 dissonance of a 3rd against a 4th. Thus D6 is decidedly the least 
 harmonious of the Major Triads. 
 
 Coming now to the Minor Triads : fig. 86 shows them drawn out 
 on the same plan as in fig. " " '" "" ' " 
 
CHORDS. 
 
 209 
 
 I I I 
 
 r T ns ns ma 
 
 c c de^ 
 
 '\ 
 
 1 1 m' 
 
 m 
 
 . sei — 
 
 rn' n' m' 
 
 \ 
 
 
 \ 
 
 \ 
 
 PI' m' 
 
 \ 
 
 ■'I 
 
 PI 
 d 
 
 
 i 
 
 No. 4. 
 
 No. 5. 
 Fio. 86. 
 
 No. 6. 
 
210 
 
 HAND-BOOK OF ACOUSTICS. 
 
 analysis of the results, arranged in tlie same way as in the case of 
 the Major Triads. 
 
 
 ■La 
 
 L6 
 
 Lc 
 
 Partials which beat at 
 
 the interval of a 
 
 semitone. 
 
 3,4 
 4,5 
 
 3,4 
 4,5 
 5,6 
 
 2,3 
 3,5 
 4,5 
 4,5 
 4,6 
 
 Partials which beat at 
 
 the interval of a 
 
 tone. 
 
 3,4 
 3,4 
 3.5 
 
 2,3 
 2,3 
 4,6 
 
 2,3 
 3,4 
 4, 6 
 5,6 
 
 In comparing La and L6, we may first eliminate the semitone 
 dissonances of a 3rd against a 4th, and 4th against 5th partials, 
 which are common to both ; the slight semitone dissonance of a 6th 
 against a 6th still remaining in the case of the latter. Further the 
 two tone dissonances of 2nd against 3rd partials in L6 will be more 
 prominent than the two between 3rd and 4th partials in Lo. For 
 these reasons La seems slightly more harmonious than L6, and has 
 accordingly been placed first in the Table. It is only right to state, 
 however, that Helmholtz places Li before La. 
 
 About Lc there can be no doubt : it has no fewer than five 
 semitone dissonances, including the prominent one of a 2nd against 
 a 3rd partial; it is decidedly the least harmonious of the six 
 Triads. 
 
 It must be recollected that the above results refer only to the 
 ibolated triads ; in order to test these conclusions, each chord must 
 be struck separately, unconnected with any others, and judged 
 entirely by its own inherent harmoniousness. Furthermore, they 
 must not be taken too low in the scale, or beats between the 
 fundamentals may occur; and lastly, it must be borne in mind, 
 that the intervals are supposed in the above, to be in just 
 intonation. 
 
 On comparing Da with La in the above tables, it will be found 
 that they appear very nearly on an equality with regard to their 
 harmoniousness. The same result is obtained on comparing their 
 constituent intervals; each c-onsistin? of ^ ATninp nr\c\ m. Minnr 
 
CHORDS. 
 
 211 
 
 Third, and a Pifth. Thus -we should expect a Minor Triad to 
 sound as well as a Major Triad. This, however, as every one knows, 
 is not the case. The cause of this must be looked for in the 
 Differential Tones. 
 
 The Differential Tones of these Triads, can be found by 
 ascertaining the Differentials generated by their constituent tones. 
 
 Thus in the Triad J m we find from the table on page 130 
 
 Chap. XII, that j ^ generates a Differential d2 ; | ^ generates A^, 
 and il produces d|. Proceeding in this way, we shall find the 
 Differential Tones of the Major and Minor Triads are as follows : 
 
 DrETEBENTIAl ToNES OF THE MAJOE TbIADS. 
 
 Triads, 
 
 Da. 
 
 ^d«r -*- 
 
 D6. 
 
 He. 
 
 m 
 
 'n 
 
 !d^ 
 
 Differential 
 Tones, 
 
 ,^^ 
 
 ^2 -m- 
 
 d,g^=a= 
 
 ,^ 
 
 ^d3 
 
 -_:^- d, :^ 
 
 DlFFEEENTIAL TONES OE THE MiNOE TeIADS. 
 
 Triads, 
 
 Differential 
 Tones, 
 
 La. 
 
 LJ. 
 
 A. ^' 
 
 Lc. 
 
 
212 HAND-BOOK OF ACOUSTICS. 
 
 We see from the above tables tbat tbe DifEerentials of the Major 
 Triads are not only harmless to their respective chords, but actually 
 improve them, supplying as it were a natural and true bass. On 
 the other hand, in the Minor Triads, we find Differential Tones (f 
 and s above) that are entirely foreign to the chords. They are not 
 indeed close enough to beat, nor are they sufficiently distinct to 
 destroy the harmony, but " they are enough to give a mysterious, 
 obscure eflect to the musical character and meaning of these chords, 
 an eflect for -which the hearer is unable to account, because the 
 ■weak diflerential tones on -which it depends are concealed by other 
 louder tones, and are audible only to a practised ear. Hence minor 
 chords are especially adapted to express mysterious obscurity or 
 harshness." 
 
 It must be remembered, that on tempered instruments, the 
 differentials -will not be exactly of the pitch given above, 
 consequently those of the Major Triads -will not fit in so well to the 
 chords. Hence the superiority of Major to Minor chords, though 
 still perceptible on tempered instruments, is not so marked as when 
 the intervals are justly intoned. 
 
 It was sho-wn in the last Chapter, that either tone of a 
 Consonant Interval may be raised or lowered by an Octave, not 
 indeed -without somewhat altering the degree of its harmoniousness, 
 but without losing its consonant character. By thus raising or 
 lowering one or more of their tones, the Consonant Triads may be 
 obtained in a great variety of distributions. We shall proceed to 
 ascertain theoretically, the more harmonioTis of these distributions, 
 in which the extreme tones of the Triad are within the compass of 
 two Octaves. 
 
 In order to ascertain the more harmomous of these distributions 
 of the six fundamental Triads, we shall in the first place have to 
 bear in mind the rules, concerning the enlargement of an Interval 
 by an Octave, which we obtained in the last Chapter (see page 203) ; 
 and in the second place, we shall have to note the eflect of 
 Differential Tones. These two considerations wUl be sufficient to 
 guide us in this enquiry. 
 
 With regard to the fijst, it wUl be convenient to briefly 
 recapitulate the essential part of the results on page 203, via. — 
 Minor Tenths are inferior to Minor Thirds. 
 Elevenths ,, „ Fourths. 
 
 Thirteenths „ ,, Sixths. 
 
 but the Fifth and Major Third are improved by being enlarged by 
 an Octave. 
 
CH0BD8. 
 
 213 
 
 As to the second consideration just referred to, the Differentials 
 of the Consonant intervals, -within the Octave, have been already 
 given in Chapter XII. It mil be convenient, however, to give them 
 here again, together with those of all the other Consonant intervals, 
 •within the Compass of two Octaves, 
 
 TABLE I. 
 
 Interval 
 
 Octave 
 
 Fifth 
 
 Twelfth 
 
 Fourth 
 
 Major 
 3rd 
 
 \t 
 
 (s (n' 
 
 
 < or< 
 
 [3 
 
 Differential 
 
 d 
 
 dj.or 1| 
 
 di or 1 
 
 d| or I2 
 
 d. 
 
 TABLE n. 
 
 Intervals 
 
 Eleventh 
 
 Minor 
 3rd 
 
 Major 
 10th 
 
 Major 
 6th 
 
 Minor 
 6th 
 
 (di (11 
 ] or] 
 
 (s (di 
 
 (ni 
 
 u 
 
 (1 (ni 
 (A (s 
 
 1: 
 
 Differentials 
 
 n or del 
 
 dj or fj 
 
 a 
 
 f| or d 
 
 S| 
 
 TABLE in. 
 
 Intervals 
 
 Minor 10th 
 
 Major 13th 
 
 Minor 13th 
 
 (si (di 
 
 Ui (ni 
 
 
 DiSerentials 
 (approximately) 
 
 V V 
 
 ta or na 
 
 V V 
 
 na' or ta 
 
 ^e' 
 
214 HAND-BOOK OF AC0U8TI08. 
 
 Considering in the first place the distributions of the Major 
 Triads, it is obvious that no such Triad can be injured by 
 difierential tones if the intervals of 'which it consists occur in 
 Tables I or 11 above. For the Differential of each Interval in 
 Table I only duplicates one or other of the Constituent Tones of such 
 interval ; -while the Differential Tone of each interval in Table II 
 ■will either coincide ■with, or duplicate the tone that must be added 
 to that interval, to make it a Major Triad. On the other hand, 
 those Triads ■which contain either of the intervals in Table HE, 
 must be disturbed more or less by their differentials: for in the 
 first place, the differentials are foreign to the scale, and wiU 
 consequently sound strange and disturbing ; and secondly, they 
 may produce audible beats ■with the third tone of the Triad or one 
 of its overtones, as for example, 
 
 {n ^ 
 
 Differentials 
 
 fe' 
 
 d t^ 
 
 m which the s' 'wiU dissonate against the fe'. 
 
 Both the rules conceming the ■widening of the Consonant 
 Intervals, and the Differentials generated by such Intervals, 
 therefore, teach the same fact, viz.: that in selecting ■the most 
 harmonious distributions of the Major Triads, the following 
 intervals must be avoided. 
 
 The Minor Tenth. 
 
 The Thirteenths. 
 
 On examining all the possible distributions of the Major Triad, 
 ■within a compass of two octaves, and rejecting those that contain 
 either a Minor Tenth or a Thirteenth, the following Triads appear 
 the more harmonious, the differentials being shown below : — 
 
CHORDS. 
 
 215 
 
 The most Peefect DiSTEiBUTioisrs of the Major Teiabs. 
 
 I 
 
 It is interesting to observe how closely the above Triads, taken in 
 conjunction with their Differential Tones, approximate to an 
 ordinary Compound Tone. 
 
 The other Distributions of the Major Triads are those that 
 contain the intervals forbidden in the above. They aU generate 
 unsuitable Differentials, which without making them dissonant, 
 cause them to be slightly rougher than those just considered. The 
 following Table contains these Triads, together with the 
 Differentials they generate. 
 
 The Less Pebfeot Distribtjtions of the Majok Triads. 
 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 1 
 
 
 9~r- 
 
 f^ 
 
 
 n*^ 
 
 f--^ 
 
 
 -i^ 
 
 «> 
 
 > — 
 
 -m- 
 d 
 
 ^) :-. 
 
 Y 
 
 (S| 
 
 t> 
 
 (si 
 in 
 fd 
 
 (si 
 di 
 
 r 
 [n 
 
 (d» 
 Si 
 
 ( n 
 
 1 
 
 V 
 
 ta 
 
 Is, 
 
 hta 
 
 (ta 
 
 (Jei 
 (4 
 
 , >ei 
 {Ha 
 f d 
 
216 
 
 HAND-BOOK OF ACOUSTICS. 
 
 In the last two, the Differentials actually produce beats, causing 
 them to be much the least pleasing ; in fact, they are rougher than 
 the better distributions of the Minor Triad. 
 
 We have now to ascertain the more advantageous positions of the 
 
 Minor Triads. Taking < d' th._S:z: as the type of a Minor 
 
 (i tr 
 
 Triad, and keeping within the compass of two Octaves, the 3rd and 
 root of the Triad must have one or other of the foUowing positions 
 
 n 
 
 m 
 
 rv 
 
 (f^ (f;^ (j^ a'^ 
 
 Minor 3rd. Minor 10th. Major 6th. Major 13th. 
 
 But from Table 11 and HI, page 213, we see that these intervals 
 generate respectively the following Differentials — 
 
 i.m 
 
 m 
 
 IV 
 
 f, m=i^= 
 
 na' 
 
 i 
 
 ±xz 
 
 ^ 
 
 IE 
 
 the second and fourth of which are foreign to the scale, while the 
 first and third do not belong to the Minor Triad in question. It 
 follows therefore, that every Minor Triad, must generate at least 
 one disturbing Differential tone. 
 
 Further, in order that there may be only one such Differential, 
 the intervals which "pi" makes with both the " 1 " and the "d" in 
 the above four intervals, must be selected from those in Table I, 
 page 213 ; for if it form with the "1 " or " d" any of the intervals 
 of Tables II and HE, page 213, other differentials not belonging to 
 the Minor chord will be introduced. On examination it will be 
 found, that the following are the only three distributions of the 
 Minor Triad that answer this test, that is, -which have only one 
 disturbing Differential Tone. 
 
CHORDS. 
 
 217 
 
 The Moee Perfect Distributions of the Mestor Triads. 
 
 1 
 
 1 
 
 n 
 
 1M 
 
 IdiS' 
 
 k 
 
 If ^ 
 
 1, 
 
 [d, 
 
 n' -9: 
 
 na 
 di 
 
 The other distributions which do not sound so -well are — 
 The Less Perfect Distribittions of the Minor Triad. 
 
 8 9 
 
 10 
 
 11 
 
 12 
 
 
 is 
 
 1 
 
 
 W: 
 
 TTS^ 
 
 4 
 
 •■e 
 
 <B 
 
 .a 
 
 pia' 
 's 
 1. 
 
 na 
 
 na 
 ' 1, 
 
 I de 
 ' f . 
 
 de 
 
218 
 
 HAND-BOOK OF ACOUSTICS. 
 
 In testing the results of this theoretical investigation, the student 
 must be again reminded, that aU the above intervals are supposed 
 to be in just intonation. On a. tempered instrument, the 
 Combination Tones will be very different from those given above. 
 
 Consonant Teteads. 
 
 If one of the tones of a consonant Triad be duplicated by adding 
 its Octave, a consonant Tetrad or chord of four tones is obtained. 
 A great variety of chords may be thus formed, and the method of 
 investigating them, which we adopted in the case of the siK original 
 Triads, that is, by setting out the partials of each tone, would be 
 somewhat laborious. Nevertheless this method is occasionally 
 convenient, especially when two or three chords only have to be 
 compared with one another. As an example, we will compare the 
 following two Tetrads, the fundamentals being supposed to be of 
 the same pitch, say D. 
 
 
 ^ 
 
 n 
 
 d' 
 
 
 (n 
 
 n 
 s 
 
 & 
 
 I' 
 
 d 
 
 
 fn, 
 
 n 
 
 i 
 
 s 
 
 ^ 
 
 Kg 87 shows the full partials of each tone, up to the 6th in the 
 Bass, 6th in the Tenor, 4th in the Contralto, and 3rd in the Soprano, 
 tone dissonances being connected by a single, and semitone 
 dissonances by a double line, as before; while below are given 
 the Differentials generated by the four fundamentals of each chord. 
 The following is a summary of the partial dissonances : 
 
 
 
 
 I 
 
 
 n 
 
 
 / 
 
 2 
 
 against 3 
 
 2 
 
 against 3 
 
 Semitone 
 
 \ 
 
 
 
 2 
 4 
 
 „ 5 
 „ 6 
 
 Tone 
 
 ( 
 
 I 
 
 1 
 
 2 
 
 „ 3 
 
 9 
 
 2 
 
 „ 3 
 
 A. 
 
CHORDS. 
 
 219 
 
 / 
 
 n 
 
 
 / 
 
 t,^ 
 
 rii 
 
 ni2 
 
 s ^ s 
 
 m 
 
 d' 
 
 ii' 
 
 dj di S| n s (Differentials) S3 du taj d| S| t| 
 Fio. 87. 
 
220 HAND-BOOK OF ACOUSTICS. 
 
 Thus, as far as the partials go, the balance is largely in favour of 
 Ko 1. The differentials tell the same tale; for four of those in 
 No. 1 are identical with partial tones, and the other rather improves 
 the chord than otherwise. On the other hand the ^ta^ in. No 2 is 
 decidedly detrimental, for not only, is it foreign to the chord, but, 
 being at such a low pitch, it will produce audible beats with d|. 
 
 In order to obtain some general results concerning these four-part 
 chords, it wiU be better to apply the results that were arrived at in 
 the case of the Triads. 
 
 Taking the Major Tetrads first, the following may be selected as 
 a typical chord. 
 
 The rule in the case of the Major Triads was, to avoid Thirteenths 
 and Minor Tenths. In transposing the tones of the above chord, 
 therefore, and keeping within the compass of two Octaves, the "s" 
 must not be more than a Minor Third above the " n," otherwise a 
 Minor Tenth will be formed; nor must the " g " be transposed 
 more than a Sixth below the " n," or a Major Thirteenth wiU. 
 occur ; lastly the " d " must not be more than a Minor Sixth above 
 " n," if we wish to avoid making a Minor Thirteenth, but it may 
 be placed as far below it as we please. These rules may be briefly 
 enunciated in the words of Helmholtz : — " Those Major chords are 
 most harmonious, in which the Eoot or the Fifth does not lie more 
 than a Sixth above the Third, or the Fifth does not lie more than a 
 Sixth below it." 
 
 It follows from this, that the "n" and "s" must not be 
 duplicated by the Double Octave. There is another reason for this 
 rule, however, namely that the differentials thus generated, wiU 
 interfere with the other tones of the chord ; thus — 
 
 ( ni f s' 
 
 Differentials t r' 
 
 In the first case the "t" will beat with the "d"" and the 
 
CHORDS. 
 
 221 
 
 "ri" with both "d'" and "n'"; the first being the more 
 objectionable. 
 
 The folio-wing are the Distributions of the Major Tetrads -within 
 the compass of two Octaves, -which contaia no Thirteenths or Minor 
 Tenths, and -which therefore have no disturbing Differential Tones. 
 
 The Most Pebfect Distkib-ctions of the Major Tetrads 
 "within the compass of two octaves. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 [vi 
 )d' 
 
 fd 
 
 
 
 In' 
 )s 
 )n 
 (d 
 
 (di 
 I d| 
 
 si 
 )ni 
 
 (di 
 )n 
 )d 
 
 [ S| 
 
 [ni 
 
 • 
 
 In 
 
 (Si 
 
 )ni 
 
 di 
 
 Id 
 
 fd: 
 
 (di 
 )s 
 )n 
 ( d 
 
 
 7 
 
 ...m.. 
 
 _£_ 
 
 -_*- 
 
 ~w~ 
 
 -s 
 
 zfti 
 
 =1= 
 
 ^= 
 
 
 f' 
 
 
 
 
 
 
 It 
 
 ) • 
 
 
 ; 5 
 
 
 
 
 TT -•- 
 
 -•- 
 
 -S- 
 
 ""#" 
 
 
 -S- 
 -9- 
 
 
 -•- 
 
 /.-Si 
 
 
 
 
 
 
 
 
 PJ. 
 
 
 
 z:a.z 
 
 
 
 
 
 ::«z 
 
 
 
 5C 
 
 > 
 
 
 
 
 
 
 
 
 , ' 
 
 We see from this, that the tones of a Major chord in its First 
 Inversion or " h" position must lie closely together as in 7; that 
 the tones of a Major chord in the Second Inversion or " " position 
 must not have a greater compass than an Eleventh, as m 5, 6, and 
 11 ; but that to Major chords in their normal position more freedom 
 may be allo-wed. 
 
 With regard to Minor Tetrads, -we have already seen that they 
 must have at least one false differential tone. The only Minor 
 Tetrad -with but one such Differential is No 1 in the Table belo-w, 
 •which has the false differential f and its double octave fj. The 
 remaining Minor Tetrads may contain t-wo, three, or even four disturb- 
 ing diflerentials. The foUo-wing Table contains aU those -within the 
 compass of t-wo Octaves, -which generate only two false differentials; 
 such differentials only being sho-wn. 
 
222 
 
 HAND-BOOK OF ACOUSTICS. 
 
 Best Disteibtjtions of Minoe Teteabs. 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 ( 1' 
 
 [1' 
 
 fn' 
 
 (n' 
 
 [ni 
 
 [ni 
 
 [n' 
 
 (n' 
 
 d' 
 
 
 )pii 
 
 \ni 
 
 \\ 
 
 W' 
 
 W' 
 
 d' 
 
 \n 
 
 U' 
 
 1 
 
 
 d' 
 
 )d' 
 
 in 
 
 )1 
 
 1 
 
 1 
 
 d 
 
 )n 
 
 n 
 
 •S 
 
 fi 
 
 [1, 
 
 fd 
 
 111 
 
 fn 
 
 [d 
 
 (li 
 
 111 
 
 fd 
 
 i 
 
 -n "•" 
 
 -«- 
 
 
 
 
 
 
 
 
 
 r /• 
 
 4 
 
 
 m. 
 
 ^-#— 
 
 g_ 
 
 
 
 
 t 
 
 ^=^ — 
 
 _B_ 
 
 — y~ 
 
 -s- 
 
 -i- 
 
 -»- 
 
 
 — «_ 
 
 — S~ 
 
 
 ^KZ 1 
 
 
 --S^ 
 
 — 
 
 
 
 -s^ 
 
 — •■— 
 
 -$r 
 
 J 
 
 ■m- 
 
 
 '•" 
 
 
 
 
 ■*■ 
 
 
 
 
 
 , V 
 
 
 
 
 / V 
 
 
 • rt 
 
 u 
 
 (t 
 
 ( s 
 
 I na 
 
 isi 
 
 U 
 
 ,s 
 
 I na 
 
 L 8| 
 
 g 
 
 
 S V 
 
 
 ^ 
 
 
 ^' 
 
 
 
 ^ 
 
 ® 
 
 (u 
 
 f PB 
 
 ffi 
 
 M^ 
 
 ri; 
 
 ftV 
 
 'f. 
 
 8, 
 
 'f. 
 
 53 
 
 
 ^ 
 
 
 
 
 
 
 
 
 " 
 
 
 
 
 
 
 
 
 
 
 From this Table it is evident, that a Minor chord in its Second 
 Inversion or " c " position, must have its tones close together as in 
 5 ; that the tones of the First Inversion or " 6 " position must be 
 within a Major Tenth, as in 3, 6, and 9. 
 
 We bring this discussion to a conclusion with an extract from 
 the Chapter on Transposition of chords in Helmholtz' work, from 
 which the present Chapter has been largely taken. 
 
 " In musical theory, as hitherto expounded, very little has been 
 said of the influence of the Transposition of chords on harmonious 
 eflect. It is usual to give as a rule that close intervals must not 
 be used in the bass, and that the intervals should be tolerably 
 evenly distributed between the extreme tones. And even these 
 rules do not appeal- as consequences of the theoretical views and 
 laws usually given, according to whicn a consonant interval 
 remains consonant in whatever part of the scale it is taken, and 
 however it may be transposed or combined with others. They 
 rather appear as practical exceptions from general rules. It was 
 left to the musician himself to obtain some insight into the various 
 effects of the various positions of chords, by mere use and 
 experience. No rule could be given to guide him. 
 
CHORDS. 223 
 
 " The subject has been treated here at such length in order to 
 show that a right view of the cause of consonance and dissonance 
 leads to rules for relations which previous theories of harmony 
 could not contain. The propositions we have enunciated agree, 
 however, with the practice of the best composers, of those, I mean, 
 who studied vocal music principally, before the great development of 
 instrumental music necessitated the general introduction of tempered 
 intonation, as anyone may easily convince himself by examining 
 those compositions which aimed at producing, an impression of 
 perfect harmony. Mozart is certainly the composer who had the 
 surest instinct for the delicacies of his art. Among his vocal 
 compositions the Ave verum corpus is particularly celebrated for its 
 wonderfully pure and smooth harmonies. On examining this little 
 piece as one of the most suitable examples for our purpose we find 
 in its first clause, which has an extremely soft and sweet eHect, 
 none but Major chords, and chords of the dominant Seventh. All 
 these Major chords belong to those which we have noted as 
 having the more perfect positions. Position 2 occurs most 
 frequently, and then 8, 10, 1, and 9. It is not till we come to the 
 final modulation of this first clause that we meet with two minor 
 chords, and a major chord in an unfavourable position. It is very 
 striking, by way of comparison, to find that the second clause of 
 the same piece, which is more veiled, longing, and mystical, and 
 laboriously modulates through bolder transitions and harsher 
 dissonances, has many more minor chords, which, as well as the 
 major chords scattered among them, are for the most part brought 
 into imfavourable positions, until the final chord again restores 
 perfect harmony. 
 
 " Precisely similar observations may be made on those choral 
 pieces of Palestrina, and of his contemporaries and successors, 
 which have simple harmonic construction without any involved 
 polyphony. In transforming the Eoman Church music, which was 
 Palestrina's task, the principal weight was laid on harmonious 
 effect, in contrast to the harsh and unintelligible polyphony of the 
 older Dutch system, and Palestrina and his school have really 
 solved the problem in the most perfect manner. Here also we find 
 an almost uninterrupted flow of consonant chords, with dominant 
 Sevenths, or dissonant passing notes, charily interspersed. Here 
 also the consonant chords wholly, or almost wholly, consist of those 
 major and minor chords which we have noted as being in the more 
 perfect positions. But in the final cadence of a few clauses, on the 
 contrary, in the midst of more powerful and more frequent 
 
224 EAND-BOOK OF ACOUSTICS. 
 
 dissonances, we find a predonunance of the unfavourable positions 
 of the major and minor chords. Thus that expression which 
 modem music endeavours to attain by various discords and an 
 abundant introduction of dominant Sevenths, was obtained in the 
 school of Palestriaa by the much more delicate shading of various 
 transpositions of consonant chords. This explains the deep and 
 tender expressiveness of the harmony of these compositions, which 
 sound like the songs of angels with hearts affected but undarkened 
 by human grief in their heavenly joy. Of course such pieces of 
 music require fine ears both in singer and hearer, to let the delicate 
 gradation of expression receive its due, now that modem music has 
 accustomed us to modes of expression so much more violent and 
 drastic." 
 
 " The great majority of Major Tetrads in Palestrina's ' Stabat 
 Mater' are in the positions 1, 10, 8, 5, 3, 2, 4, 9, and of minor 
 tetrads in the positions 9, 2, 4, 3, 5, 1. For the major chords one 
 might almost think that some theoretical rule led him to avoid the 
 bad intervals of the Minor Tenth and the Thirteenth. But this 
 rule would have been entirely useless for minor chords. Since the 
 existence of combinational tones was not then known, we can only 
 conclude that his fine ear led him to this practice, and that the 
 judgment of his ear exactly agreed with the rules deduced from our 
 theory. 
 
 " These authorities may serve to lead musicians to allow the 
 correctness of my arrangement of consonant chords in the order of 
 their harmoniousness. But anyone can convince himself of their 
 correctness on any justly intoned instrument. The present system 
 of tempered intonation certainly obliterates somewhat of the more 
 delicate distinctions, without, however, entirely destroying them." 
 
 Stjhmabt. 
 Triads in their closest distribution. 
 
 Major Triads are smoother or more harmonious than Minor 
 Triads, because their differential tones form part of the chord, 
 which is not the case with the Minor Triads. 
 
 A triad is not equally smooth in its three positions ; arranged in 
 the order of their smoothness, we have for 
 Major Triads, 
 
 1st (c) position or 2nd inversion, as : Dc, Sc, Fc. 
 2nd (a) ,, normal triad, ,, Da, Sa, Fa. 
 
 3rd (6) ,, 1st inversion, ,, D6, S6, Fi. 
 
CHOBDS. 21c> 
 
 Minor Triads. 
 
 1st (&) position or 1st inversion as L5, Ei, M6. 
 
 2nd (a) ,, nonnal triad, ,, Lo, Ea, Ma. 
 
 3rd (c) „ 2nd inversion, ,, Lc, Re, Mc. 
 
 Triads in various distributions. 
 
 The smootliest distributions of the Major Triads are those in 
 which Thirteenths, and Minor Tenths are absent. 
 
 The smoothest distributions of the Minor Triads are the following, 
 taking L as the type. 
 
 Te(/rada. 
 
 Those Major Tetrads are most harmonious in which the Eoot or 
 Pifth does not lie more than a Sixth above the Third ; or the Fifth 
 does not lie more than a Sixth below it. 
 
 To take the D chord for example, those distributions are 
 smoothest in which the d does not lie more than a Sixth above the 
 n; and in which the s does not lie more than a Sixth above or 
 below the same note ; therefore 
 
 The Tones of Major Tetrads in the I position should lie as 
 closely together as possible : and in the c position the 
 extreme compass of the chord should not exceed an 
 Eleventh. 
 
 The tones of a Minor Tetrad in the c position should be aa 
 closely together as possible; and in the J position, the 
 extreme compass of the chord should not exceed a Major 
 Tenth. 
 
 In this Chapter, a chord has been scientifically studied as a thing 
 hy itself; in the art of music it is generally also considered in 
 relation to what goes before and after ; where the requirements of 
 Art do not accord with those of Science in this respect, the latter 
 must give way to the former. 
 
226 
 
 CHAPTER XVIII. 
 
 Temperament. 
 
 In Chap. V, we saw how, by means of Helmholtz's Syren, it may 
 be proved, that, for two tones to be at the interval of a Fifth, it is 
 requisite that their vibration numbers be in the ratio of 3:2; and 
 that for two tones to form the interval of a Major Third, their 
 vibration numbers must be as 5 : 4. 
 
 In Chap. XV we have seen the -reason for this; the vibration 
 numbers m.ust be exactly in these ratios, in order to avoid beats, 
 between Overtones on the one hand, and Combination Tones on the 
 other. 
 
 The vibration numbers of the tones of each of the Major Triads 
 
 is fr' rd' 
 m , 1 1 , i 1 , being therefore in the ratios 6:5:4; the vibration 
 dl ( s t f 
 numbers of all the tones of the diatonic scale can be readily 
 calculated on any given basis, after the manner shown in Chap. V 
 (which the student is recommended to read again, before proceeding 
 with the present chapter). There, 288 having been chosen as the 
 vibration number of d, the vibration numbers of the other notes 
 were found to be as follows : 
 
TEMPERAMENT. 227 
 
 576 = di 
 540 = t ' 
 
 16 
 15 
 
 324 = r 
 
 288 = d 
 
 Tlie vibration ratios of the intervals between the successive notes 
 ■were then caloulated from the numbers thus obtained, and were 
 found to be those given above. 
 The Fifths which can be formed from these tones are : 
 
 [I [\ c (?' {r [t \i 
 
 If, from the vibration numbers above, the vibration ratio of each 
 of these Fifths be calculated, it will be found that each of the 
 following five intervals has the exact vibration ratio of 3 : 2 ; that 
 is to say, the following are Perfect Fifths : 
 
 (s ft (di fri (n' 
 U [n (f Is (r 
 
 Thus, for example. 
 
 fn 
 
 540 _ 3 
 360 ~ 2" 
 
228 BAND-BOOK OF ACOUSTICS. 
 
 The remaining two are exceptions ; for from the vibration numbers 
 above we have 
 
 f I _ 768 _ 64 ( 1 _ 480 _ 40 
 
 t ~ 540 ~ 45 ^r ~ 324 ~ 27' 
 
 Agami, the Minor Thirds that can be formed from the notes of 
 the diatonic scale are : 
 
 {I (s \f (;' 
 
 If, as above, the vibration ratios of these intervals be calculated, it 
 will be found that the following three are true Minor Thirds, that 
 is, the vibration num.bers of their constituent tones are in the exact 
 ratio of 6:5: 
 
 C (?' ( 
 
 The other interval is not a true Minor Third, for : 
 f 384 32 
 
 ( f _ 384 _ 32 
 fr ~ 324 ~ 27" 
 
 Of these three exceptions, we may at once dismiss j V as being 
 a well recognized interval, the Diminished Fifth, less by a Semitone 
 than the Perfect Fifth, for which it is not likely to be mistaken. 
 But it is otherwise with the other two ; they are very nearly a 
 Perfect Fifth and a true Minor Third, respectively, for as we have 
 just seen 
 
 1 _ 120 3 _ 120 
 
 il- 
 
 ,. 32 96 6 96 
 
 ana{ = — =: — while - = — 
 
 27 81 5 80 
 
 The imperfection of these intervals will be best seen by sketching 
 the Compoimd Tones of i ^ up to the beating partials, taking the 
 vibration numbers given above ; thus, — 
 
TEMPERAMENT. 
 
 229 
 
 972 = V 
 
 648 
 
 324 = r 
 
 -II = 960 
 
 480 
 
 We see from this, that -witli these vibration numbers, the mistuned 
 Fifth j ^ produces 972 — 960 = 12 beats per second, between the 
 2nd and 3rd partials. 
 
 Similarly, with the mistuned Minor Third j * ; taking r = 324, 
 24 beats per second would be heard. 
 
 It is obvious, therefore, that, although one might fail to perceive 
 that the r of the scale is not in tune with f and 1 as long as we are 
 concerned with melody only ; yet as soon as these tones are sounded 
 together, the discordance becomes very conspicuous. For true 
 harmony, therefore, another tone is required in the scale, to form a 
 
 Perfect Fifth with 1. Calling this tone rah (r), we can deduce its 
 vibration number in the scale above from the fact that 
 
 vl = 
 
 For 1 = 480, therefore 
 
 3 x\ = 2 
 
 X 480 
 
 V 2 X 480 
 and thus r = = 320 
 
 This tone not only forms a Perfect Fifth with 1, but also a true 
 Minor Third with f , for 
 
 384 _ 6 
 320 ~ 5 
 
 The relations of r with the adjacent tones, which the student cau 
 readily calculate for himself, are as follows : 
 
230 HAND-BOOK OF ACOUSTICS. 
 
 10 
 9" 
 
 10 
 
 The interval between j vj is termed a comma, its vibration ratio 
 being 114 = |i. 
 
 V . 1 
 
 In harmony, r is reqxiired in the Minor Chord J vf and its in- 
 
 t ; but even in melody, r 
 s 
 sounds bett«r than r after the tones f and 1, and similarly r 
 
 better than r after s and t. 
 
 It mil be seen, therefore, that in order to execute a piece of 
 music which is entirely in one key, say Major, and which has no 
 Chromatics, we should require eight tones to the Octave, viz., the 
 eight tones given in the middle column of fig. 88. If, however, 
 the piece of music in question changes key, we shall require other 
 tones. Suppose in the first place it passes into the First Sharp key. 
 The s of the middle column, that is, the dominant of the original 
 key, then becomes the d, or tonic of the First Sharp key, as shown 
 in the right-hand column of fig. 88 Proceeding upwards from the 
 
 tone d, we find that the r and n of this key will correspond to the 
 1 and t of the original key, but a new tone will be required for r. 
 Going downwards the 1|, S|, and f| of the new key correspond to 
 the PI, r, and d of the old, but a new tone wiU be necessary for t|. 
 Thus if the music passes into the First Sharp key, two more 
 tones will be required. 
 
 For a change into the First Flat key, two new tones will also be 
 wanted. For in this case, the f of the original key, becomes the d 
 of the First Flat key, as shown in the left-hand column of fig. 88. 
 Ascending from this d, the r and pi of the new key will correspond 
 exactly to the s and 1 of the original one, but new tones will be 
 
 necessary for r and f . Descending, the n, r, and d of the original 
 key will serve for the t|i L and si of the new one. 
 
TEMPERAMENT. 231 
 
 t n 
 
 i ^ k 
 
 J. g a 
 
 X . 
 
 I f t, 
 
 d f i 
 
 t| n i, 
 
 .' ^ : : 
 
 s", d f , 
 
 Fia. 88. 
 
 The changes of key in modem musio, are, however, rarely 
 confined to the above, and are often very extensive. A study of 
 fig. 89 will show that in general every change of key of one remove 
 either to the right or left of the central key, requires two new 
 tones. Thus starting in Major without Chromatics, 8 tones are 
 required to the octave ; on passing into Q, two new tones are 
 required ; on further changing to D two more will be wanted ; 
 passing from this into A two more have to be brought forward, 
 while if the music then enters the key of B, all the tones of this 
 
232 SAND-BOOK OF ACOUSTICS. 
 
 key will be of diflerent pitches to those in the original key 0. "We 
 have supposed here, that the musio passes gradually through 
 the keys of G, D and A ; of course, if the change be a sudden one 
 from to E, the case would be somewhat different ; The E of the 
 central column would become the d of key E, the A would become 
 its f , the B its s ; only five new tones being therefore required for 
 this key. It may be also noticed, that the E and B of the centre 
 
 column are not of exactly the same pitch as the B and B of last column 
 on the right, which are derived by transition through the inter- 
 mediate keys ; the latter being one comma higher than the former. 
 Similarly each transition to the left of the central key requires 
 
 two new tones. Further, the Eb, Ajj, and D|7 of the extreme left 
 hand columns, are not of the same pitch as the Eb, Ab, D[j of the 
 central column, but are one comma flatter. 
 
 Thus to perform music, which modulates through the major keys of 
 fig. 89, in the major mode only, requires a very large number of 
 tones to the Octave. If to this, the minor mode be also added, a 
 stiU larger number is necessary. Moreover, there are many more 
 keys than those of fig. 89 used in modem music, so that the student 
 will readily perceive that the number of tones to the Octave, thus 
 required in modem music, is very large indeed. 
 
 All this presents no difficulty in the case of the voice, which is 
 capable of producing tones of every possible gradation of pitch 
 withiu its compass, and which, governed by the eax, readily forms 
 the tones necessary to perfect harmony. Nor does it present any 
 real difficulty in the vioUn class of instrvunents, which also may be 
 made to emit tones of every gradation of pitch within their compass. 
 The real difficulty is met with in such instruments as the Organ, 
 Harmonium, Piano, &c., which have fixed tones, and consequently 
 only possess a certain limited number of notes. On these instru- 
 ments, which have but few notes to the Octave (generally only 
 twelve), it is obviously impossible to execute music written in 
 various keys and modes, in true intonation. The only thing that 
 can be done is so to tune the fixed notes of the instrument, that the 
 imperfections shall be as small as possible. The problem therefore 
 is : — ^how so to tune an instrument, with but twelve tones to the 
 Octave, as to be able to play in various keys and modes, with the 
 smallest amount of imperfection. Any system of timing by which 
 this is brought about, is called a Temperament, and the false 
 intervals thus obtained are termed tempered intervals. 
 
TEMPERAMENT. 233 
 
 Dl?A>EbB!7P0&DA Eb" 
 
 vT — s — d 
 t....n — 1 — r 
 
 tt 
 ^r ....s — d ..- f 
 l....r ^r — 8 — d — f 
 
 t| — n — 1 — r 
 
 t|....n....l| 
 s....d....f 
 
 vr — s — d — f 
 t|.... n....l|.. .r 
 
 4. M 1 
 
 t| — n — 1| — r 
 
 vr — g| — d — f 
 n — l|....r 
 
 ^r ....s,....d. ..f| 
 t| — n — Ir — r 
 
 vF . . . .S|.. ..d .. ..fi t|'-..n[ 
 
 r 
 
 vr....S|....d....f| 
 t| — nj.. ..li — r 
 
 d....f| t|....n|....l|....r| 
 
 vri...S|....d....f| 
 ti....n|....lr....r| 
 
 Fio. 89. 
 
 There are many possible Temperaments, but only two are of any 
 practical importance : Mean-Tone Temperament and Equal 
 Temperament. 
 
234 HAND-BOOK OF AOOUSTIGS: 
 
 Mkan-Tonb Tempebament. 
 
 This was the temperament used in tuning the Organ until about 
 50 years ago. Its principle will be seen from the following con- 
 siderations. 
 
 Starting with 0| and tuning upwards four true Fifths consecu- 
 tively we obtain the following notes : 
 
 but as the Fifth j ^ is a true one, while the j p of the diatonic 
 scale is, as we have already seen, smaller by a comma than a true 
 
 Fifth, the A in the above will be a comma sharper than the A of 
 
 / 
 the perfect scale. Consequently the E' wiU also be sharper by a 
 
 comma than the E' two octaves above the E| which makes a true 
 
 Major Third with 0|. 
 
 Using the vibration ratios, we may put the same thing thus. 
 
 The interval between 0| and E' in the above is the sum of four true 
 Fifths, that is 
 
 3 3 3 3 _ 81 
 
TEMPERAMENT. 235 
 
 The interval between Ci and E' is the sum of two Octaves and a 
 Major Third, that is 
 
 2 2 5 _ 20 _ 8C 
 
 i^T^i^T^ie 
 
 Thus the interval between Ei and E' is the difference between the 
 
 above, viz : 
 
 
 
 
 
 
 81 
 
 80 81 
 
 
 16 81 
 
 
 16 ~ 
 
 ~ 16 ~ r6 
 
 X 
 
 80 ~ 80 
 
 that is, a comma. 
 
 Now, Id the Mean-Tone Temperament, each of the four Fifths 
 above is flattened a quarter of a comma, and consequently the E' 
 thus obtained forms a perfectly true Major Third with the C. 
 Thus, starting with C and tuning upwards two of these flattened 
 Fifths, and a true Octave down, we obtain the notes 0, 0, and D ; 
 then again starting from this D, and tuning up two of these Fifths, 
 and another Octave down, we get the additional notes A and B, all 
 the Fifths being a quarter of a comma flat, but j ^ being a true 
 Major Third. 
 
 Now if we start from E, and repeat this process, that is, tune two 
 of these flattened Fifths up, and an Octave down, and again two 
 flat Fifths up, and an Octave down, we shall have obtained 
 altogether the following notes : 
 
 CI 
 
 B 
 
 A. 
 G 
 
 n 
 
 E 
 D 
 
 
236 HAND-BOOK OF AOOUSTIOS. 
 
 the Gil^ thus necessarily forming a true Major Third -with E. Now 
 in the above { c ^s a true Fifth less a quarter of a comma ; but J ^ 
 is a true Major Third; therefore i^ is 3' ^"^^ Minor Third less a 
 quarter of a comma. Again, j ^ is a true Fifth less a quarter of a 
 comma, but it has just been shown that i ^ is a true Minor Third 
 less a quarter of a comma; therefore f^ is a true Major Third. 
 In a similar manner, it may be shown successively, that all the 
 other Major Thirds in the above are true intervals, and that all the 
 Minor Thirds are flatter by a quarter of a comma, than true Minor 
 Thirds. 
 
 Now, starting from C, and tuning two Fifths, each flattened by a 
 quarter of a comma, downwards, and an Octave up : again two flat 
 Fifths down and an Octave up, the following additional tones, 
 printed in italics below, are obtained : 
 
 CI 
 
 B 
 A 
 
 G 
 
 F 
 E 
 
 D 
 
 c 
 
 the Ab thus necessarily forming a true Major Third with C. 
 
 We should have to obtain several more tones, in this way, to form 
 a complete scale in this temperament, but, as we aie supposing but 
 12 tones to the Octave, we must stop here; in fact, we have already 
 exceeded that number, and must throw out either GJorA?; we will 
 suppose the former. 
 
 Now, in the above j |^' is a true Fifth less a quarter of a comma, 
 and as we have just seen j 2> is a true Major Third, therefore j ^|' 
 
TEMPERAMENT. 237 
 
 is a true Minor Third less a quarter of a comma. Again, j ^ is a 
 true Kfth less a quarter of a comma, but we have just shown that 
 j ^ is a true Minor Third less the same amount, therefore j ^ is a 
 true Major Third. Proceeding upwards in this way it may be 
 shown that all the Maj or Thirds in the above are true intervals except 
 i e" ! qi I ^ if ' ^^* *^^ Minor Thirds are flatter than the corre- 
 sponding true intervals by a quarter of a comma. Purther, since 
 the Octaves and Major Thirds are true intervals, it follows that all 
 the Minor Sixths (except four) must be true also. Again, since the 
 Pifths and Minor Thirds are flatter than the corresponding true 
 intervals by a quarter of a comma, it follows that the Eouxths and 
 Major Sixths must be sharper than the corresponding true intervals 
 by the same amount. 
 
 We have seen that the D in the above is derived from 0, by 
 tuning upwards successively, two true Fifths, each less a quarter 
 of a comma, and then an Octave down. Now the E, A, and B above 
 are derived in exactly the same manner from the D, G, and A, 
 respectively. Consequently the four intervals j q j ]^ { q and j J 
 are precisely similar. Moreover, it is easy to show that the interval 
 j ^ is similar to these four; for since I ^ and j ^ have been shown 
 to be true Major Thirds, they are equal to one another; take 
 away the j^from each, and the remaining intervals j^and j^ 
 must be equal. Thus in Mean-Tone Temperament, there is no dis- 
 tinction between the Greater and Smaller step — between the f 
 and y interval. The Major Third, j ^ for example, is composed 
 of two precisely equal intervals, {|and j™, the r being exactly 
 midway between the two tones n and d. The vibration number of 
 this r would thus be the geometrical mean of the vibration numbers 
 of d and n. It is from this circumstance that the term Mean-Tone 
 Temperament is derived. 
 
 In the above, we have seen that j ^ is a true Minor Sixth and its 
 vibration ratio is consequently f : we have also seen that I ^ is a 
 true Major Third, and its vibration ratio is A; therefore the vibration 
 ratio of j qL is 
 
 8 . 5 _ 8 4 _ 32 
 
 5~4~5 5~25 
 
 But j 9ij, as we have seen, is a true Major Third and its vibration 
 
238 HAND-BOOK OF ACOUSTICS. 
 
 ratio is | . Thus the Ab and QJ above are tones of different pitch, 
 and j^ cannot therefore be a true Major Third. Consequently in 
 Mean-Tone Temperament if the number of tones to the Octave be 
 restricted to 12, the Major Thirds cannot all be true. 
 
 With the scale constructed as above, the Major Thirds in the major 
 keys of 0, Q-, D, F, Bb and Ej; are true, but the more remote keys will 
 have one or more of their Major Thirds false; for example, the 
 
 dominant chord of A would have to be played as j Ab and as we 
 
 have just seen j ^ is not a true Major Third. 
 
 Instruments of 12 tones to the Octave, tuned in Mean-Tone 
 Temperament as above, can thus only be used in C and the more 
 nearly related keys, viz., in &, TSi^, F, G and D Major, and C, G, 
 and D Minor : or if Gfl be retained instead of A;>, in B7, F, C, G, 
 D and A Major and G, D and A Minor. The other keys, 
 which are more or less discordant, used to be termed " Wolves." 
 Of course in these "wolves'" not only will some of the Major 
 Thirds be false, but some of the other intervals will differ from 
 what they are in the better keys : for example, j ^' is a true Fifth 
 less a quarter of a comma, therefore if GJJbe retained in preference 
 to a]?, the Fifth j ^' will no longer be equal to this amount. 
 
 Some old instruments, tuned in Mean-Tone Temperament were 
 furnished with additional tones, such as GrJ, DJf and Dl?, thus 
 extending the number of keys that could be employed. The 
 English Concertina, an instrument which is generally tuned on the 
 Mean Tone system, is furnished with G| and DJ as well as A> and 
 Bt7. 
 
 Equal TEMPERAMEifT. 
 
 In this system of tuning, which is the one now universally 
 adopted for key-board instruments, the Octave is supposed to be 
 divided into twelve exactly equal intervals, each of which is termed 
 an equally tempered semitone. In consequence, however, of the 
 extreme difficulty of thus tuning an instrument, these intervals are 
 never exactly equal, and often very far from being so. 
 
 The vibration ratio of the equally tempered semitone is evidently 
 
 for twelve of these intervals added together form an Octave, that is, 
 twelve of their vibration ratios multiplied together must amount to 
 
TEMPERAMENT. 239 
 
 ^. In order to compare the other equally tempered intervals, with 
 the corresponding true ones, we must go a little deeper into the 
 subject. 
 
 Starting from any given tone, say 0, let the other twelve letters 
 in line I below, represent twelve other tones, obtained from it, by 
 successively ascending twelve true Fifths ; and let the letters after 
 in line II denote seven other tones derived from the same tone 
 by ascending seven Octaves : 
 
 s 
 
 S 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 ? 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 I. G D A E B F# C|t Gfl D# A# EJf B# 
 
 n. ccoocooc 
 
 2 2 2 2^ 2 2 2 
 
 1 111 1 T 1 
 
 Inasmuch as the interval between any two successive tones in I is 
 a just Fifth, the vibration ratio of which is -J, it is obvious that the 
 vibration ratio of the interval between the extreme tones of this 
 line is f multiplied by itself twelve times 
 
 12 
 
 = (!) 
 
 Similarly the vibration ratio of the interval between the extreme 
 tones of line II 
 
 7 
 
 Consequently the vibration ratio of the interval between the BJf on 
 the extreme right of Une I and the on the extreme right of II 
 
 12 7 
 
 = (I) - (!) 
 
 12 7 
 
 = (I) X (i) 
 
 531441 
 
 ~ 524288- 
 
 This interval is called the Comma of Pythagoras, which may there- 
 fore be defined as the difference between twelve true Fifths and 
 seven Octaves. This cumbrous vibration ratio cannot be expressed 
 in any simpler way with perfect exactness, but it very nearly equals 
 
 74 
 
 73- 
 
240 SAND-BOOK OF A00U8TIGS. 
 
 This fraction may be used instead of the one above, for all practical 
 purposes, and we shall so employ it here. Eonghly speaking, the 
 comma of Pythagoras is about i of a semitone, or |f of the comma 
 of Didymus which has been often referred to above, and the 
 vibration ratio of which is |^. 
 
 We have just seen that the BJ in Kne I above is sharper than the 
 C in line n beneath it, by the small interval termed the comma of 
 Pythagoras. If therefore each of the twelve true Fifths in I be 
 diminished by -^ of this interval, that is by -Jj- of the ordinary 
 comma (of Didymus), the B# wiU coincide with the C. Thus the 
 tones of I may be written 
 
 m. G D A E B FJf CJ Gjt D# A| F 
 
 in which each successive Fifth, — Gr, Q — D, &c., is a true Fifth 
 diminished by -^ of the ordinary comma. Eeducing these tones so 
 as to bring them within the limit of an Octave they may be 
 arranged in order of pitch thus : 
 
 IV. C# D Djt E F FJ G Gfl A Ait B C. 
 
 By observing the way in which these tones have been obtained 
 above, it will be seen, that the successive intervals C — Cjf, Ojf — D, 
 &c., are all equal to one another. For CJf (see III above) was 
 obtained from C, by ascending 7 of the flat Fifths just referred to ; 
 now if we ascend 5 of these same Fifths from CJ we reach C, and 
 ascending two more of these Fifths from we reach D ; therefore 
 D is obtained by ascending 7 of these flat Fifths from CJ ; 
 consequently the interval C — CJf must be equal to the interval 
 CJ — D, and so on. 
 
 The tones represented in Hne IV above are therefore the tones of 
 Equal Temperament. Thus the Fifths in this Temperament are -Jj- 
 of a comma, or about the ^ part of a Semitone flatter than true 
 Fifths ; and as the Octaves are perfect, the Fourths must be sharper 
 than true Fourths by the same amount. 
 
 The Major Third (0 — E above, for example) was obtained by 
 ascending four true Fifths each flattened by -J,- of a comma and 
 dropping down two Octaves. Now the Major Third obtained by 
 ascending four true Fifths and descending two Octaves, is, as we 
 have already seen, sharper than a true Major Third, by one comma ; 
 therefore the Major Third of the Equal Temperament is sharper 
 than a true Major Third, by a comma, less four times the amount 
 by which each Fifth is flattened, that is, less -jL of a comma ; in 
 
TEMPERAMENT. 
 
 241 
 
 other -words, the Major TMrd of Equal Temperament is -^ of a 
 comma sharper than a true Major Third. 
 
 Again, the Equal Tempered Fifth is too flat by -J,- of a comma, 
 and the Major Third too sharp by -i- of a comma ; therefore the 
 diSerenoe between them, that is, the Minor Third is -;§;- of a comma 
 too flat. Further, as the Octaves are perfect, the Major Sixth 
 must be -Jj- of a comma too sharp, and the Minor Sixth -/j- of a 
 comma too flat. 
 
 The following Table gives in a form convenient for reference the 
 amount by which the Consonant intervals, both in Mean Tone and 
 Equal Temperament, differ from the corresponding intervals in just 
 Intonation. 
 
 Intebtal. 
 
 Mean Tone. 
 
 Equal. 
 
 Minor Thirds 
 
 ^ comma flat 
 
 -jSj- comma flat 
 
 Major Thirds 
 
 true 
 
 -A- » sliarp 
 
 Fourths 
 
 i comma sharp 
 
 iT »J JJ 
 
 Fifths 
 
 i „ flat 
 
 tV .. flat 
 
 Minor Sixths 
 
 true 
 
 TT II 11 
 
 Major Sixth 
 
 J comma sharp 
 
 fr „ sharp 
 
 For further purposes of comparison, the following table is given, 
 the 2nd coltmin of which shows the vibration number of the, tones 
 of the diatonic scale on the basis = 264; to which is added those 
 of the E!?, a Minor Third above C, the Aj7, a Minor Sixth above 0, 
 the Bb, a fourth above F, and the Fff and OJ, a diatom'c Semitone 
 below G and D respectively : aU in true intonation. The 3rd and 
 4th columns give the vibration numbers of the twelve tones, into 
 
 B 
 
242 
 
 EANB-BOOK 
 
 OF A00U8TH 
 
 18. 
 
 
 yfbioh. the Octave is divided in Mean-Tone and Equal Temperament 
 respectively; tlie numbers being calculated as far as tbe fii'st 
 decimal place : 
 
 TOSE. 
 
 Jdst. ^ 
 
 Mban Tone. 
 
 Equal. 
 
 
 CI 
 
 528 
 
 628 
 
 528 
 
 
 B 
 
 495 
 
 493-5 
 
 498-4 
 
 
 B^ 
 
 469-3 
 
 472-3 
 
 470-4 
 
 
 A 
 
 440 
 
 441-4 
 
 444 
 
 
 A^ 
 
 422-4 
 
 422-4 
 
 419-1 
 
 
 Q 
 
 396 
 
 394-8 
 
 395-5 
 
 
 n 
 
 371-2 
 
 368-9 
 
 373-3 
 
 
 p 
 
 352 
 
 353-1 
 
 352-4 
 
 
 E 
 
 330 
 
 330 
 
 332-6 
 
 
 u? 
 
 316-8 
 
 315-8 
 
 313-9 
 
 
 D 
 
 297 
 
 295-2 
 
 296-3 
 
 
 c# 
 
 278-4 
 
 275-8 
 
 279-7 
 
 
 
 
 264 
 
 264 
 
 264 
 
 
TEMPEBAMBNT. 243 
 
 On comparmg tlie Equal with the Mean Tone Temperament, we 
 see that the former has its Fifth better in tune than the latter, but 
 that it is inferior in all its other intervals, especially in the Major 
 Thirds. On the other hand it must be recollected that in. Mean 
 Tone Temperament, it is only possible to play in a limited ntimber 
 of keys, while in Equal Temperament all keys are equally good or 
 equally bad. If it be desired therefore to play in aU keys, the 
 Equal Temperament is decidedly the better ; in fact, the only one 
 possible under these circumstances. 
 
 As Equal Temperament is the one now universally employed on 
 instruments with fixed keys, it wUl be of advantage to be able to 
 compare its intervals with those of just Intonation, without the 
 necessity of using the somewhat cumbrous vibration ratios. We 
 can do this by employing a method devised by Mr. EUis. Suppose 
 a piano to be accurately tuned in Equal Temperament, the Octave 
 being divided into twelve exactly equal parts. Further suppose 
 each of these twelve equal semitones to be accurately divided into 
 100 equal parts ; each of these minute intervals, Mr. Ellis has 
 termed a Cent, so that there are 1,200 equal Cents in the Octave. 
 Eig. 90 shows the magnitudes of the intervals in Equal and in 
 True Intonation, expressed in these Cents. 
 
 The evils of Equal Temperament arise chiefly, of course, from 
 the fact that overtones, which should be coincident, are not so, but 
 produce audible beats. In addition to this, the Differentials, 
 except in the case of the Octave, do not exactly correspond with 
 any tones of the scale, and may generate beats with some adjacent 
 tone, if this latter be sounding at the time. In the case of the 
 Fifths and Fourths, these beats, being very slow, do not produce 
 any very bad eSects : for example, with the Fifth from = 264 
 we have only one beat per second, thus : 
 792 791 
 
 528 
 
 395-5 
 
 264 
 
 Difierential Tone = 395-5 — 264 = 131-5 C, = 132. 
 With the other intervals the case is different, more rapid beats 
 being generated. The 2nd column of the following table shows the 
 number of beats per second produced between partials which are 
 
244 
 
 HAND-BOOK OF ACOUSTICS. 
 
 BauAL. 
 
 Cents. 
 
 Tbttb, 
 
 - 
 
 Semitone 
 
 100 
 
 
 
 112 
 
 Diatonic Semitone : 
 
 It^ 
 
 
 182 
 
 SmaUer Step : { ™ 
 
 
 Tone 
 
 200 
 
 
 
 
 204 
 
 Greater Step : j ^ 
 
 
 Minor Tliird 
 
 300 
 
 
 
 
 316 
 
 Minor Third 
 
 
 
 386 
 
 Major Third 
 
 
 Major Third 
 
 400 
 
 
 
 
 498 
 
 Fourth 
 
 
 Fourtli 
 
 500 
 
 
 
 
 590 
 
 Tritone: {| 
 
 
 Tritone 
 
 600 
 
 
 
 Fifth 
 
 700 
 
 
 
 
 702 
 
 Fifth 
 
 
 Minor Sixth 
 
 800 
 
 
 
 
 814 
 
 Minor Sixth 
 
 
 
 884 
 
 Major Sixth 
 
 
 Major Sixth 
 
 900 
 
 
 
 
 996 
 
 MiDor Seventh: J |_ 
 
 
 Minor Seventh 
 
 1000 
 
 
 
 
 1088 
 
 Major Seventh 
 
 
 Major SsTenth 
 
 1100 
 
 
 
 Octave 
 
 1200 
 
 Octave 
 
 
TEMPERAMENT. 
 
 245 
 
 comcident in just Intonation wlien the interval given in tlie 1st 
 column is played in Equal Temperament, tlie lower tone of each 
 interval being := 264. 
 
 Interval. 
 
 Beats. 
 
 PHti 
 
 1 
 
 Fourth 
 
 1-2 
 
 Major Third 
 
 10'4 
 
 Major Sixth 
 
 12 
 
 Minor Third 
 
 14'5 
 
 Minor Sixth 
 
 16-5 
 
 The piano is specially favourable to Equal Temperament; in fact, 
 this system of tuning was first applied to the piano and subsequently 
 made its way to other key-board instruments. In the first place, 
 the tones of the piano are loud only at the moment of striking, and 
 die away before the beats due to the imperfect intervals have time 
 to become very prominent : and further, music for the piano 
 abounds in rapid passages, and usually, the chords are so frequently 
 changed, that beats have very little time in which to make them- 
 selves heard. Indeed, Hebnholtz makes the suggestion, that it is 
 the unequal temperament, which has forced on the rapid rate of 
 modem music, not only for the piano, but for the organ also. 
 
 On the Harmonium and Organ, the eflects of Temperament 
 becom.e very apparent in sustained chords. On the latter it is 
 especially so m Mixture Stops, the tempered Fifths and Thirds of 
 which, dissonating against the pure Fifths and Thirds in the over- 
 tones, producing what Hehnholtz terms the "awful din" so often 
 heard, when these stops are drawn. 
 
 Many attempts have been made from time to time to construct 
 Harmoniums, Organs, &o., in such a way, and with such a number 
 of tones to the Octave, that the intervals they yield shall be more 
 or less close approximations to True Intonation. The best known 
 of these iastruments are: 
 
 1. HelmholU' s Harmonivm,. This instrument has two Manuals, 
 the tones of each being such as would be generated by a 
 
246 HAND-BOOE OF A00USTI08. 
 
 succession of True Fifths, but those of the one manual are 
 tuned a comma sharper than those of the other. A full 
 description will be found in Helmholtz's " Sensations of 
 Tone." 
 
 2. Bosanquet's Earmonivm,, which possesses eighty-four tones 
 to the Octave and a specially constructed key-board. A full 
 account may be found in "Proceedings of the Eoyal Society," 
 vol. 23. 
 
 3. General Thompson's Enharmonic Organ, which possesses 3 
 manuals and seventy-two tones to the Octave. For 
 particulars of its construction the reader is referred to 
 General Perronet Thompson's work " On the Principles and 
 Practice of Just Intonation." 
 
 4. Colin Brown's Voice Harmonium., the finger-board of which 
 differs entirely from the ordinary one. The principles of its 
 construction are given in " Music in Common Things," parts 
 I&II. 
 
 Stjmmabt. 
 
 To perform music with true Intonation in one key only, without 
 using Ohromatios and in the Major Mode, eight tones to the Octave 
 are required. 
 
 In general, every transition of one remove, either way, from the 
 original key and stiU. keeping to the Major Mode only, requires two 
 new tones. In changes of three or more removes, the number of 
 new tones required is not quite so large as if the changes were 
 made through the intervening keys. 
 
 To modulate therefore in all keys and in both the Major and 
 Minor Mode in true intonation requires a very large number of tones 
 to the Octave — between 70 and 80, in fact. 
 
 This presents no difficulty in the case of the voice and stringed 
 instruments of the Violin Class, fir such instruments can produce 
 tones of any required gradation of pitch ; the difficulty is only felt 
 in instruments with a limited number of fixed tones ; and for such 
 instruments some system of Temperament is necessary. 
 
 A Temperament is any system of tuning other than true intonation ; 
 Intervals tuned on any such system are termed tempered intervals. 
 The object of temperament is so to tune a certain limited number of 
 
TEMPERAMENT. 247 
 
 fixed tones, as to produce, on the whole, the least possible departure 
 from true intonation. 
 
 The limited number of fixed tones just referred to is almost 
 always twelve to the OctaTS. 
 
 The systems of Temperament, ■which hare been most extensively 
 used in Modem Music are Equal Temperament and Mean Tone 
 Temperament. 
 
 Mean Tone Temperament. 
 Chief features: 
 
 (1) The Major Thirds are true. 
 
 (2) The Fifths are J comma flat. 
 
 (3) There is no distinction between the Greater and Smaller 
 Tone. 
 
 When there are but 12 tones to the Octave, however, (1) and (2) are 
 true in only half-a-dozen keys. 
 
 The great disadvantage of this temperament is, that only music 
 in a limited number of keys can be performed on instruments tuned 
 according to this system. 
 
 Equal Temperament. 
 Chief features : 
 
 (1) The Octave is divided into 12 equal intervals, the vibration 
 
 1 2 
 
 ratio of each of which is /„ = 1"0595 or 1'06 nearly 
 
 (2) The Fifths are -jij- command*. 
 
 (3) The Major Thirds are /y comma sharp. 
 The above facts are true in all heys. 
 
 The chief advantage of this temperament is, that music in all 
 heys can be performed on instruments tuned according to this 
 system ; that is to say, aU keys are equally good or equally bad. 
 
 Though it is impossible to obtain true intonation from instruments 
 with but twelve fixed tones to the Octave, yet in the case of the 
 Voice, Violin, and other instruments which may be made to produce 
 tones of any desired pitch, it seems self-evident that true intonation 
 should be the thing aimed at ; inasmuch as it is just as easy with 
 these instruments to make the intervals true as to make them false, 
 provided the ear of the performer has not been already vitiated by 
 the tempered intervals. 
 
248 
 
 QUESTIONS. 
 
 CHAPTER 1. 
 
 1. What is meant by a periodic motion ? 
 
 2. Describe three methods of obtaining a periodic movement. 
 
 8. What is the physical difference between musical eomids and noises? 
 
 4. How can it be demonstrated, that the air in a sounding flue-pipe is in 
 periodic motion ? 
 
 5. Under what circumstances does a periodic motion not give rise to a 
 musical sound ? 
 
 CHAPTER II. 
 
 6. How can it be proved, that some medium is necessary for the 
 transmission of sound ? 
 
 7. Draw a diagram of a water wave, showing clearly what is meant by 
 its length, amplitude, and form. 
 
 8. Draw three water waves of same length and amplitude, but of different 
 forms ; three of same amplitude and form, but of different lengths ; and 
 three of same length and form but of different amplitudes. 
 
 9. Describe the way in which a vibrating tuning-fork originates a sound 
 wave. 
 
 10. In what direction do the air particles in a sound wave vibrate? 
 
 11. What is meant by the length, amplitude, and /orjn of a sound wave ? 
 
 12. State the connection, in a sound wave, between (1) length of wave 
 and duration of particle vibration, (2) amplitude of TFave and extent of 
 particle vibration, (8) form of wave and manner of particle vibration. 
 
 18. Describe the way in which a sound wave transmits itself through the 
 air. 
 
 14. What is an associated wave ? 
 
 15. Describe any method, by which the velocity of sound in air can be 
 determined. 
 
 16. How is the velocity of sound in air affected by temperature ? 
 
QUESTIONS. 249 
 
 17. What is the velocity of sound in aii' at 0° C. ? What is it at 20° C. ? 
 What at 60° Fah. ? 
 
 18. What is the velocity of sound in water ? How has it been deter- 
 mined? 
 
 19. State what you know of the velocity of sound in solids. 
 
 20. Describe an experiment, which illustrates the fact, that solids are, as 
 a rule, good conductors of sound. 
 
 21. A person observes that ten seconds elapse between a flash of lightning 
 and the succeeding thunder clap. What is the approximate distance of the 
 thunder doud from the observer ? 
 
 22. A vessel at sea is seen to fire one of its guns. Thirty-five seconds 
 afterwards, the report is heard. How fai' oflf is the vessel ? (Temperatm'e 
 25° C.) 
 
 CHAPTER HI. 
 
 23. What is the use of the External ear ? 
 
 24. Describe the relative positions in the ear of the (1) Tympanum, (2; 
 Fenestra Ovalis, (3) Fenestra Rotunda. 
 
 25. How is the vibratory motion of the Tympanum transmitted to the 
 Fenestra Ovalis ? 
 
 26. In what part of the internal ear are the Fibres of Corti situated ? 
 What is supposed to be theii' function ? 
 
 27. What is the Eustachian Tube ? What would be the result of this 
 tube becoming stopped up ? 
 
 28 . The cavity of the Middle Ear is in most persons, completely separated 
 from the external air by the Tympanum; but occasionally there is an 
 apertui-e in this latter. Does this necessaiily affect the hearing ? ■ Give 
 reasons for your answer. 
 
 29. What ia the special function of the labyrinth ? 
 
 CHAPTER IV. 
 
 30. What are the three elements which define a musical sound ? 
 
 31. What is the physical cause of variation in pitch? Describe a simple 
 experiment in support of your answer. 
 
 32. What is meant by the vibration number of a musical sound ? 
 
 33. Mention three of the most accurate methods of experimentally 
 determining the vibration number of a given musical sound. 
 
 84. Describe the Wheel Syi'en. 
 
 35. What are the disadvantages of Cargnard de la Tour's Syren ? 
 
 36. Describe the construction of Savart's Toothed Wheel. 
 
 37. Describe the Sonometer or Monoohord. 
 
 38. Describe Helmholtz's or Dove's Double Syren. 
 
250 EANB-BOOK OF A00USTI08. 
 
 39. Describe the principle of the Graphic method of ascertaining the 
 vibration number of a tuning-fork. 
 
 40. Given the vibration number of a musical sound, how can its wave 
 length be determined ? What are the lengths of the sound waves emitted 
 by 4 forks, which vibrate 128, 256, 512, and 1024 times per second, 
 respectively ? (Take velocity of sound as 1100). 
 
 41. The vibration number of a tuning-fork is 532. What wiU be the 
 length of the sound wave it originates (1) in air at 32° Fah., (2) in air at 
 60° Fah. ? 
 
 42. If the length of a sound wave is 3 feet 6 inches when the velocity of 
 sound is 1100 feet per second, what is the vibration number of the sound ? 
 
 43. Calculate the length of the sound wave emitted by an organ pipe, 
 which produces Cj = 32 
 
 44. Calculate the length of the sound wave produced by a piccolo flute, 
 which is sounding C* = 4096. 
 
 45. What are the approximate vibration numbers of the highest and 
 lowest sounds used in music ? 
 
 46. Give the vibration numbers of (1) the C in Handel's time, (2) the 
 French Diapason normal, (3) a Concert Piano, and organ (approximately). 
 
 47. When a locomotive sounding its whistle is passing rapidly through a 
 station, to a person on the platform, the pitch of the whistle appears sharper 
 while the engine is approaching, than it does after it has passed him. Explain 
 this. 
 
 CHAPTER V. 
 
 48. What is meant by the vibration ratio of an interval ? If the vibration 
 numbers of two sounds are 496 and 465 respectively, what is the vibration 
 ratio of the interval between them ? What is this interval cidled ? 
 
 49. What are the vibration ratios of an Octave, a Fifth, and a Major 
 Third? 
 
 50. What are the vibration ratios of a Major and Minor Sixth, and a 
 Minor Thiid? 
 
 51. What is the best way of experimentally proving that the vibration 
 ratios of an Octave, Fifth and Major Third are exactly -i, i, and A respec- 
 tively ? 
 
 52. Given that the vibration numbers of s, m, d, are as 6 : 5 : 4, and 
 that d = 300 ; calculate from these data, the vibration numbers of the 
 other tones of the Diatonic Scale. 
 
 63. Given d = 320, and that the vibration numbers of the tones of a 
 Major Tiiad, in its normal position, are as 6 : 5 : 4 ; calculate from these 
 data the vibration numbers of the other tones of the diatonic scfde. 
 
 64. Given d = 256, and vibration ratios of a Fifth and Major Third are 
 i and A respectively; calculate /com these data, the vibration numbers of the 
 other tones of the diatonic scale. 
 
QUESTIONS. 251 
 
 65. Given d = 240, r = 270, m = 800, f = 320, s = 360, 1 = 400, 
 t =: 450, d' = 480 ; calculate from these data the vibration ratios of a 
 Diminished Fifth, the Greater Step, the Smaller Step, and the Diatonic 
 Semitone. 
 
 66. With the data of question 55, calculate the vibration numbers of fe 
 and se, and the vibration ratios of the Greater Chromatic j *6, and the 
 Lesser Chromatic i °®. 
 
 67. From the data of question 56, show that the interval I ^ is not a 
 
 perfect Fifth ; and calculate the vibration number of the note r, which 
 would form a perfect Fifth to 1. 
 
 58. By means of the results of question 67, ascertain the vibration ratio 
 of the interval from ray to rah. What is this interval called ? 
 
 69. Show from the numbers given in question 65, that j * is not a perfect 
 Minor Third, but that j *>^ is so. 
 
 60. What musical peculiarity does the Octave possess, which is shared by 
 no smaller interval ? 
 
 61. Given: J| = | and { ®g® = f| ; calculate ratio of J ^^. 
 
 62. Given : j a = 1 and j g® = || ; calculate vibration ratio of 1 ^^. 
 
 63. Given : La "= 1728 ^""^ { le ^^ 15 ' calculate vibration ratio of the 
 interval between ta and le. 
 
 64. Given : i ^ = 1 and { ^ = -^ ; calculate vibration ratio of the 
 interval between fe and ba. 
 
 66. Given: jr = i j^ = ^, j f ^ i|, Js = |, ,,d js^e 
 
 = ^ ; calculate vibration ratio of the interval j ^. 
 
 66. Given : { "t' = yI' | 1 = f ' { ie = fs ! <=^<=ulate vibration ratio 
 of the interval ! „„. 
 
 67. Given: { g' = f and j'-* = ^4; calculate vibration ratio of 
 
 (d' 
 (la- 
 
 68. From the data of No. 55, calculate the vibration ratios of a Major 
 Tenth, a Minor Tenth, and a Twelfth. 
 
 69. How is the vibration ratio of the sum of two intervals calculated, 
 when the vibration ratios of these latter are known ? Give an example. 
 
 70. Given the vibration ratios of two intervals, show how the vibration 
 ratio of the difference between these intervaJs can be ascertained. Give an 
 example. 
 
252 HAND-BOOK OF ACOUSTICS. 
 
 CHAPTER VI. 
 
 71. What is meant by the intensity of a musical sound ? How can it be 
 shown experimentally, that the intensity of a sound depends upon the 
 amplitude of the vibrations that give rise to it ? 
 
 72. State the law of Inverse squares. Why does it not appear to be 
 coiTect, under ordinary circumstances ? 
 
 73. What is the principle of the speaking tube ? 
 
 74. Explain the phenomenon of echoes. 
 
 75. If two seconds elapse between a sound and its echo, what is the 
 distance of the reflecting smface ? 
 
 76. Explain one of the causes of the bad acoustical properties of some 
 buildings, and state any remedy you know of. 
 
 77. Why does one's voice appear louder in an empty unfurnished room, 
 than in the same room furnished 1 
 
 78. Give a theoretical proof of the law of Inverse Squares. 
 
 CHAPTER Vn. 
 
 79. How can the phenomenon of resonance or co- vibration be illustrated 
 with two tuning-forks ? What conditions are necessary to the success of the 
 experiment ? 
 
 80. Describe some experiments with stretched strings to illustrate the 
 phenomenon of resonance. 
 
 81. Explain in detail the cause of resonance or co-vibration in the case of 
 tuning-forks or stretched strings. 
 
 82. Explain in detail the cause of resonance in open tubes, showing 
 clearly why the tube must be of a certain definite length, if it is to resound 
 to a note of given pitch. 
 
 83. Explain in detail the cause of resonance in stopped tubes, showing 
 clearly why the tube must be of a certain definite length, if it is to resound 
 to a note of given pitch. 
 
 84. How would you construct a resonator to resound to G' ? Calculate 
 approximate dimensions. 
 
 85. To illustrate the phenomenon of resonance with two tuning-forks, 
 they must be in the most perfect unison ; whereas an approximate unison is 
 suificient in the case of two strings stretched on the same sound-board. 
 Explain this. 
 
 86. I have a tube 1 inch in diameter, open at both ends, which resounds 
 powerfuUy to GJ. What length is it ? (C = 512). 
 
 87. Calculate the length of a stopped tube about 1 inch in diameter, 
 resounding to E|;. 
 
 88. A tube open at both ends, and about an inch in diameter, is lOJ 
 inches long. Calculate approximately the note it resounds to. 
 
QUESTIONS. 253 
 
 89. A tube closed at one end is 14 inches long, and about IJ inches 
 diametei. Calculate approximately the note to which it resounds. 
 
 90. What are resonance boxes ? What are they used for ? What are 
 resonators 1 What ai'e they used for ? Explain the best method of using 
 them. 
 
 91. Describe an experiment to show how the resonance of air-chambers 
 can be optically demonstrated. 
 
 92. Why does the sound of a vibrating tuning-fork die away more quickly 
 when attached to a resonance box, than when held in the hand ? 
 
 93. While singing the other day, I happened to sound D loudly. 
 Immediately a gas globe in the room was heard to give out a tone of the 
 same pitch. I found that this occurred whenever D was sounded in its 
 vicinity, but a tone of any other pitch produced no effect. Explain why 
 the globe emitted this particular note and no other. 
 
 94. Why does a vibrating tuning fork give forth a louder sound, when 
 its handle is applied to a table, than when merely held in the hand ? 
 
 CHAPTER Vm. 
 
 95. Define the terms : Simple Tone, Clang or Compound Tone, Partial, 
 Overtone, Fundamental Tone. 
 
 96. Write down in vertical columns the Partials, that may be heard when 
 any low note is loudly sounded on the pianoforte or harmonium, or by a bass 
 voice, calling the fundamental, d,, r,, m„ f,, s,, 1„ t|, successively. 
 
 97. Write down the vibration numbers of the paiiials which may be 
 heard on a harmonium, calling the fundamental 100. 
 
 98. Write down in vertical columns, the partials that may be heard, when 
 the following tones are struck on a piano : — CSj, Dj, Ej, Pj, Gjjj, Bl?^- 
 
 A 
 
 Q is sounded on a harmonium, or smartly struck on a 
 
 a piano. Write down in 4 columns the various pai-tial tones that may be 
 heard, keeping sounds of the same pitch on the same horizontal line. 
 
 100. From what instruments can simple tones be obtained ? 
 
 101. Write down in a column the relative vibration numbers of the first 
 20 partial tones, naming all those which are constituent tones of the musical 
 scale. 
 
 102. Many persons find great diflSculty in hearing overtones. Explain 
 any methods you know of, which will assist such persons in hearing them. 
 
 103. State what you know concerning the relative intensities of partials 
 on various instruments. 
 
 104. Upon what does the quality of a Compound Tone depend ? Explain 
 fully the meaning of yom- answer. 
 
 99. The chord 
 
254 HAND-BOOK OF ACOUSTICS. 
 
 105. Explain how a Tuning Fork can be kept in a state of continued 
 vibration by an electro-magnet. 
 
 106. Describe the apparatus, which is used for the purpose of keeping a 
 number of forks in continued vibration, by means of electro-magnets and a 
 single current. What relation must exist between the vibration numbers 
 of these forks ? 
 
 107. How may the relative intensities of the sounds of the forks in the 
 above, be modified ? 
 
 108. Describe the apparatus used by Helmholtz, in his experiments on 
 the synthesis of Compound Tones. 
 
 109. What is a pendular vibration ? Describe a method of obtaining a 
 graphic representation of one. 
 
 110. Show by a diagram how to compound two simple associated waves. 
 
 111. Given the quality of a musical tone : is it possible to determine the 
 corresponding wave form ? If not, why not ? 
 
 CHAPTER IX. 
 
 112. Describe an experiment illustrating the use of the sound-board In 
 stringed instruments. 
 
 113. What acts as the sound-board in the harp, and in the violin ? 
 
 114. State the laws of stretched strings, relating to Pitch ; and illustrate 
 them by reference to musical instruments. 
 
 115. Give the experimental proofs of the above. 
 
 116. Take a stretched string and set it vibrating aa a whole. Stop it 
 at half its length and set one of the halves vibrating. Do the same with 
 i- —' ii -I — *n<i —' of its length. What notes will be heard in each case, 
 
 3 4 5 6 7 8 ° 
 
 cilling the first one d, ? What will be the relative rates of vibration f 
 
 117. Define the terms node, and ventral segment. 
 
 118. Describe an experiment to show that a stretched string can vibrate in 
 several segments : say, four. 
 
 119. Explain the cause of the occurrence of partials in the tones which 
 are given by stretched strings. 
 
 120. How can it be proved that any particular partial — say, the 5th — ^is 
 produced by the string vibrating in 6 segments with 4 nodes ? 
 
 121. How could you ensure the absence of the 7th partial in the tone 
 produced by a violin string ? 
 
 122. How is the quality of tone from a stringed instrument affected by 
 the weight and flexibility of the string ? 
 
 123. How could you ensure the absence of any given partial in the tone 
 Horn a stretched string ? 
 
QUESTIONS. 266 
 
 124. What is the best method of proceeding, if we wish to ensure the 
 presence of a particular pai'tial in the tone from a stretched string ? 
 
 125. Explain clearly how nodes ai'e formed in a stretched string. Give 
 an experiment in illustration. 
 
 126. What are the principal circumstances, whioli determine the presence 
 and relative intensities of paiiials, in the tones of stringed instruments ? 
 
 127. Explain fully the effect of pressing down the loud pedal in pianoforte 
 playing. 
 
 128. I press down the loud pedal of a pianoforte and strike E|j in the 
 Bass Clef sharply ; name the strings that will be set in vibration. 
 
 129. How can a Tuning Fork be used, to produce vibrations in a stretched 
 string ? 
 
 130. Explain how the matei'ial of the hammer and the kind of blow, 
 affect the quality of tone in the pianoforte ? 
 
 CHAPTER X. 
 
 131. Describe the phenomenon of the reflection of a sound wave at the 
 end of an open pipe. 
 
 132. Describe the phenomenon of the reflection of sound at the end of a 
 stopped pipe. 
 
 133. Define the terms node and ventral segment as applied to organ pipes. 
 
 134. Explain clearly how a node is formed in an open organ pipe. 
 
 135. Explain how nodes are formed in a stopped organ pipe. 
 
 136. Show by diagrams the positions that the nodes may take in an open 
 organ pipe, and state the pitch of the tone produced in each case relatively 
 to the fundamental tone of the pipe. 
 
 137. Do the same with regard to a stopped organ pipe, 
 
 138. Explain why only the odd series of partial tones occurs in the tones 
 from stopped organ pipes, while open pipes give the complete series. 
 
 139. Wliat is the difference between Harmonics on the one hand, and 
 Overtones or PartiaJs on the other? 
 
 140. Explain the principle of mixture stops on the organ — the 
 " Sesquialtera " for example. 
 
 141. Describe an experimental method of proving the existence of nodes 
 in an organ pipe. 
 
 142. I have two organ pipes each 4ft. 4in. long, one stopped and 
 the other open. Calculate the approximate pitch of the fundamental tone 
 in each. 
 
 143. What will be the pitch (approximate) of an open organ pipe 
 6ft. 6in. long? What would be the approximate pitch if it were stopped ? 
 
256 HAND-BOOK OF ACOUSTICS. 
 
 144. To what length (approxiinate) muot an open organ pipe be cut 
 to produce the F' on the top of the Treble staff? To what length, if 
 stopped ? 
 
 145. What is the general effect of rise of temperature upon organ pipes, 
 and why has it this effect ? 
 
 146. I have two organ pipes, one of metal and the other of wood. They 
 are exactly in unison on a cold day in winter, while on a hot summer's day 
 they are out of tune with one another. Explain the reason of this. Which 
 win be the sharper of the two in wai-m weather ? 
 
 147. How is the pitch of a pipe affected by the wind pressure ? 
 
 148. DetaU the various circumstances that affect the pitch of an organ 
 pipe. 
 
 149. Why cannot the intensity of the sound from an organ pipe be 
 increased by augmenting the wind pressure ? 
 
 150. Describe the two kinds of reed used in musical instruments, and 
 explain their action. 
 
 151. Explain why the tones of a harmonium are so rich in partials. 
 
 152. What is the function of the pipe in an organ reed-pipe ? 
 
 153. How are reeds sharpened and flattened ? 
 
 164. What is the effect of a rise of temperature on the pitch of a reed ? 
 
 155. What is the cause of the peculiar quality of the tones of the 
 Clai-ionet p 
 
 156. Descjibe the principle of the French Horn, showing the origin of its 
 Tones, and how variation in pitch is effected. 
 
 157. To what class of instruments would you assign the Human Voice, 
 ani why ? 
 
 158. What circumstances affect the production of partials, and therefore 
 the quality of the tones of the Human Voice ? 
 
 159. I sing successively the vocal sounds " a" and " oo ; '' state briefly the 
 changes that take place in the shape and size of the mouth, in passing from 
 the former to the latter ; and show how these changes cause the difference 
 in the above sounds. 
 
 160. By increasing the wind pressure in blowing the Flute, the tone rises 
 an octave ; in the Clarionet, it rises a Twelfth. Explain this discrepancy. 
 
 CHAPTER XI. 
 
 161. State the laws connecting the vibration number with the length ot 
 a rod, vibrating longitudinally — 
 
 1st. When the rod is fixed at both ends. 
 
 2nd. When it is fixed at one end only. 
 
 3rd. When it is free at both ends. 
 1616. What partials may occur in each of the above cases? 
 
QUESTIONS. 257 
 
 162. State the laws connecting the vibration number with the length of a 
 rod, vibrating transversely — 
 
 Ist. When the rod is fixed at both ends. 
 2nd. When it is fixed at one end only. 
 3rd. When it is free at both ends. 
 
 163. What partials may occur in each of the above oases ? 
 
 164. State what you know of the partials which may be produced by a 
 tuning fork. 
 
 165. Give sketches showing the various ways in which a tuning fork may 
 vibrate. 
 
 166. When the handle of a tuning fork is applied to a table, the fork's 
 vibrations ai'e communicated to the latter. Explain how this is effected. 
 Account also for the louder sound thus produced. 
 
 167. How does change of temperature affect the pitch of a tuning fork ? 
 
 168. How can the velocity of sound in aii- be approximately ascertainsd, 
 with no other apparatus than an open organ pipe giving C=518 f 
 
 169. Given the velocity of sound in air, how can the velocity of sound in 
 other gases be ascertained ? 
 
 170. Explain the methods by which the velocity of sound in solid bodies 
 is ascertained. 
 
 171. A silver wire is stretched between two fixed points, and caused 
 to vibrate longitudinally. Its length is varied till the tone it produces is in 
 unison with a CI fork (C'=512). It then measures 8ft. 4in. Calculate the 
 velocity of sound in silver. 
 
 172. A copper rod 2ft. lOin. long is fixed in a vice at one end and rubbed 
 longitudinally with a resined leather. The tone it emits is in unison with a 
 C tuning fork (=1024). Find the velocity of sound in copper. 
 
 173. If a lath of metal 4ft. long, fixed at one end, vibrates laterally once 
 a second ; how many vibrations per second wUl it perform, when its length 
 is reduced to 4in. ? 
 
 174. How can the nodal lines of a vibrating plate or membrane be 
 discovered ? Explain the principle of the method. 
 
 175. Describe by means of sketches two or three ways in which a square 
 and a roimd plate may vibrate. 
 
 CHAPTER XII. 
 
 176. What are Combination Tones ? How many kinds of Combination 
 Tones are there ? What are they respectively termed ? 
 
 177. How is the vibration number of a differential tone related to the 
 vibration numbers of its generators ? 
 
 178. Describe some method of producing Differential Tones. 
 
 179. Calculate the Differentials produced by an Octave, Fifth, and Major 
 and Minor Thirds. 
 
258 HAND-BOOK OF ACOUSTICS. 
 
 180. Calculate the Differentials produced by a Fourth, Major and Minor 
 Sixth, a Tone, and a Semitone. 
 
 181. When the following is played on a Harmonium, what Differential 
 Tones may occur ? 
 
 :s 
 
 r :— |r :— d 
 
 d :-lt 
 
 d :-| 
 
 di :— It :1 Is :— If :n 
 n :— Is :f |n :— |r :d 
 
 182. Upon what instruments are Differential Tones very prominent? 
 
 183. When is the Differential intermediate in pitch between its two 
 generators ? 
 
 184. What is meant by a Differential Tone of the let order? Show by 
 an example how Differential Tones of the 2nd and 3rd order are produced. 
 
 185. How is the vibration number of a Summation Tone related to the 
 vibration numbers of its generators ? 
 
 CHAPTER Xm. 
 
 186. Explain by means of diagrams or otherwise, what is meant by 
 Interference of sound waves. 
 
 187. What is meant when it is said that two waves are in the same or 
 opposite ^A(Me .5 
 
 188. Describe and sketch any apparatus that may be used to demonstrate 
 the fact, that two sounds may be so combined as to produce silence. 
 
 189. Describe by help of a diagram the sound waves that emanate from a 
 vibrating fork, clearly showing their alternate phases. 
 
 190. If a tuning-fork be revolved before the ear, alternations of intensity 
 are observed ; clearly explain the whole phenomenon and its cause. 
 
 191. How may Chladni's plates be used to illustrate the phenomenon of 
 interference ? 
 
 192. How are the intensities of sounds related to the amplitudes of their 
 corresponding sound-waves ? 
 
 193. What is a beat ? What is the law connecting the vibration numbers 
 of two tones and the number of beats they generate per second ? 
 
 194. Carefully explain how a beat is produced. 
 
 195. How would you proceed to prove experimentally that the rapidity 
 of the beats increases as the interval between the two generating tones 
 increases ? 
 
 196. Explain the principle of the Tonometer. 
 
 197 . A Tonometer consists of 50 forks, each fork is 4 vibrations per second 
 sharper than the preceding, and the extreme forks form an exact Octave. 
 What is the pitch of the lowest fork ? 
 
 198. Two harmonium reeds when sounded together produce 4 beats per 
 second. The pitch of one of them is then slightly lowered, and on again 
 sounding it with the other, two beats per second are heard. It is then 
 further flattened and again two beats per second are heard. Was it originally 
 sharper or flatter than the unaltered one ? Answer fully. 
 
QUESTIONS. 259 
 
 199. What is the use of the Tonometer? Describe the method of using 
 it. 
 
 200. Why is the Eeed Tonometer iaferior in acomaoy to the tuning-fork 
 Tonometer ? 
 
 CHAPTER XIV. 
 
 201. What use is made of slow beats in music ? 
 
 202. What is the physical cause of dissonance? How would you experi- 
 mentally prove your answer to be correct ? 
 
 203. What is meant by the Beating Distance ? State generally how it 
 varies in different parts of the scale. 
 
 204. How can the phenomenon of beats be imitated by the use of one 
 sound only ? 
 
 205. Why are beats unpleasant to the ear? 
 
 206. Explain a method of detecting, very faint musical sounds by means 
 of beats. 
 
 207. Why do beats cease to be impleasant when they are sufiSciently 
 rapid? 
 
 208. Upon what does the harshness of a dissonance depend. ? Illustrate 
 by examples. 
 
 209. Show by examples that the harsimess of a dissonance does not 
 depend entirely on the rapidity of beats. 
 
 210. What are the Beating Distances, in the regions of 02,0,, C, C, and 
 C? 
 
 211. " The sensation of a musical tone in the region of Cj = 64 persists 
 for J- of a second after the vibrations that give rise to it have ceased." 
 What evidence is there for this statement ? 
 
 212. What is the cause of Dissonance between Simple Tones that are 
 beyond Beating Distance ? 
 
 213. Two forks, the vibrational numbers of which are 100 and 210, 
 dissonate when sounded together. Explain why. How many beats per 
 second may be counted ? 
 
 214. Two forks, the vibration numbers of which are 200 and 296, generate 
 slow beats when sounded together. Explain the cause of this. How many 
 beats per second will be heard ? 
 
 215. Show that a compound tone may contain dissonant elements in 
 itself. 
 
 216. Show clearly, how it is that | ^ is a harsher discord than | ^ and 
 
 217. The interval { ^ sounds less harsh when played by two clarionets, 
 than when played on a harmonium. Explain the reason. 
 
 218. Show by sketches the beating elements present in a Major and 
 Minor Third when played on a piano. 
 
260 HAND-BOOK OF A00U8TI08. 
 
 219. Two harmonium reeds, the vibration numbers of which are 199 and 
 251, produce slow beats when sounded together. Explain the reason. How 
 many beats per second wUl be heard ? 
 
 220. Two forks, the vibration numbers of which are 256 and 168, produce 
 faint beats when sounded together. How many beats per second will be 
 heard? 
 
 221. How many audible beats per second will two harmonium reeds 
 generate, which vibrate 149 and 301 times per second respectively ? 
 
 CHAPTER XV. 
 
 222. How is the interval of an Octave between Simple Tones defined? 
 
 223. How would you proceed to tune two Simple Tones to the interval of 
 a perfect Fifth ; 1st, if you had no other tones to assist you, and 2nd, if you 
 already possessed the Octave of one of the tones ? 
 
 224. How ai'e the intervals of a Fifth and Fourth between Single Toner 
 defined ? 
 
 225. How are the intervals of an Octave and a Fifth between Compound 
 Tones defined ? 
 
 226. Explain the principle involved in tuning two violin strings at the 
 interval of a perfect Fifth. 
 
 227. In the interval of a Fourth, why is it necessary that the vibration 
 numbers of the two tones should be in the exact ratio of 4 . 3 ? 
 
 228. If two harmonium reeds vibrate 501 and 399 times per second 
 respectively, how many beats per second will be heard when they are sounded 
 together ? 
 
 229. The vibration numbers of the C and E|j in an equal-tempered 
 harmonium are 264 and 314 respectively ; how many beats per second will be 
 heard when they are sounded together ? 
 
 230. On an equal-tempered harmonium, C = 264 and A :^ 444. When 
 they are sounded together, how many beats per second will be heard ? 
 
 231. Two harmonium vibrators an exact Octave apart are sounded 
 together. Explain fuUy the result of again sounding them together after 
 one of them has been flattened by one vibration per second. 
 
 232. How is the Octave defined, m the case of stopped organ pipes, the 
 tones of which consist of the 1st and 3rd partials only ? 
 
 233. Which is the easier interval to tune, and why : — the Fifth or Major 
 Third? 
 
 234. Two tones are sounded on a harmonium, the vibration numbers of 
 which are 300 and 401 respectively. How many beats per second may be 
 counted ? 
 
QUESTIONS. 261 
 
 CHAPTER XVI. 
 
 235. What musical intervals are perfectly harmoniouB under all circum- 
 stances ? Why are they so ? 
 
 236. Why does a Fifth sound rough when very low in pitch ? 
 
 237. How is it that singing in Octaves is usually styled singing in unison? 
 
 238. Compare theoretically the following Fifths as played 1st on a har- 
 
 monium, and 2nd on a stopped diapason : — Jf j i^j 
 
 i 
 
 239. Compare the relative smoothness of a Fourth and Major Sixth under 
 similar conditions of quality and pitch. 
 
 240. Compare the relative smoothness of Major and Minor Tliirds between 
 tones of ordinary quality. 
 
 241. Why can the Major Third be used at a lower pitch than the Minor 
 Tliird? 
 
 242. Account for the inferiority of the Minor Sixth to its inversion, the 
 Major Third. 
 
 243. Mention any facts that serve to explain why Thirds were not 
 admitted to the rank of Consonances until comparatively recent times. 
 
 244. Why do such intervals as Thirds, Sixths, &c., sound smoother on the 
 Stopped than on Open pipes ? 
 
 245. In a duet for Oboe and Clarinet, the Fifth sounds smoother when 
 the latter instrument takes the lower tone, but the Fourth is smoother when 
 the Oboe takes the lower tone. Explain this. 
 
 246. Compare in smoothness the intervals of a Fifth and a Twelfth. 
 
 247. Compare the Thirds and Tenths in smoothness. 
 
 248. Show which are the better. Thirteenths or Sixths. 
 
 249. Compare the Fom'th with the Eleventh. 
 
 250. Give general rules referring to the relative smoothness of an Interval, 
 and its increase by an Octave. 
 
 CHAPTER XVII. 
 
 251. Name, and give the vibration ratios of the Consonant intervals 
 smaller than the Octave. 
 
 252. Combine the above, two at a time and calculate the vibration ratios 
 of the intervals thus formed, which are less than an Octave. 
 
 253. Show from the above, that there are only six consonant triads within 
 the Octave. 
 
 254. Name the six consonant triads, and show how they may be considered 
 39 derived from two. 
 
262 HAND-BOOK OF AOOUSTIGS. 
 
 255. Show on physical grounds, that the Ist inversion of the Major triad 
 is inferior to its other two positions. 
 
 256. Prove from acoustical consideiations, that the 2nd inversion of a 
 Minor Triad is inferior to its other two positions. 
 
 257. Account for the fact that /f -g is more harmonious than 
 
 258. Why are Major Triads as a rule more harmonious than Minor Triads ? 
 
 259. In selecting the most harmonious distributions of the Major Ti-iad. 
 what intervals should he avoided, and why ? 
 
 260. Give examples of the more perfect and less perfect distributions of 
 the Major Triads. 
 
 261. What considerations guide us in selecting the more harmonious 
 distributions of the Minor Triad ? 
 
 262. What three distributions of the Minor Triad have only one dis- 
 turbing differential ? 
 
 263. How are consonant Tetrads formed from consonant Triads ? 
 
 264. Compare the relative harmoniousness of the following two Tetrads 
 
 in id 
 
 after the manner of Fig. 87 : — key F. ^ " and key D|j. ^ " 
 
 fd fn 
 
 265. What considerations serve as a guide in selecting the best positions 
 and distributions of a Major Tetrad? Give Helmholtz's rule. 
 
 266. Why may neither the 3rd nor 5th of a Major Triad be duplicated 
 by the double Octave ? 
 
 267. Give examples of good positions and distributions of the Major 
 Tetrad. 
 
 268. State any rule you know of, concerning the distribution of the Ist 
 inversion of the Major Tetrad. 
 
 269. Within what limits should the 2nd inversion of a Major Tetrad lie ? 
 
 270. Write down a Minor Tetrad which has only one disturbing 
 differential, 
 
 271. State any rule you know of, concerning the distribution of the 2nd 
 'nversion of the Minor Tetrad. 
 
 272. Within what limits should the Ist inversion of a Minor Tetrad lie ? 
 
 CHAPTER XVm. 
 
 273. Given that the vibration ratios of the tones of a Major Triad are as 
 6:6:4, show that j J is not a perfect Fifth. 
 
 274. With the data of the last question show that j ' is not a true Minor 
 Third. ' 
 
 275. Calculate, from the data of 273, the vibration ratio of j v' . 
 
 276. When is rah. requii'ed, (1) in harmony (2) in melody P 
 
 V 
 
 277. What are the vibration ratios of jv^aud j 5? 
 
QUESTIONS. 263 
 
 278. How many new tones are requked to form the Major Diatonic Scale 
 of the new Key, when a transition of one remove to the right occurs ? 
 What are they ? Show that your answer is eorrect. 
 
 279. When a transition of one remove to the left occurs in a piece of 
 music, show that two new tones will be required to form the Major Diatonic 
 Scale of the new key. 
 
 280. Show what new tones wiU be requii'ed to form the Major Diatonic 
 Scale, when a piece of music changes suddenly from C to E. 
 
 281. Show what new tones will be required, when a piece of music passes 
 suddenly from the Key of C Major into that of A|j Major. 
 
 282. If music in C Major passes through the Keys of F and B|j to Efj, 
 show thit the C of this last Key is not of the same pitch as the original 0. 
 
 283. "What is meant by temperament ? 
 
 284. Smarting with d, if four true fifths be tuned upwards, and then two 
 octaves downwards, show that the note thus obtained is one comma sharper 
 than the t;ue m. 
 
 285. Shew how the twelve notes of the Octave, in mean-tone tempera- 
 ment, ai'e cetermined. 
 
 286. Why is mean-tone temperament so called ? 
 
 287. Cortpai'e the Consonant intervals in mean-tone temperament with 
 the same in true intonation. 
 
 288. What is the principle of Equal Temperament ? 
 
 289. Wha'. is the Comma of Pythagoras ? How is it obtained ? 
 
 290. Com)are the consonant intervals in equal temperament with the 
 same in true 'utonation. 
 
 291. Compre the consonant intervals in equal, with the same in mean- 
 tone temperanent. 
 
 292. What are the advantages and disadvantages of (1) mean-tone 
 temperament, (2) equal temperament ? 
 
 293. Iq eqial temperament, if C=522, what should be the vibration 
 number of CJ? 
 
 294. Takin; A=435, calculate the vibration numbers of the other eleven 
 tones of the stale in equal temperament. 
 
 295. Takiig C=522-, calculate the vibration numbers of the notes of 
 the diatonic sole in true intonation. 
 
 296. Takinj C=522, calculate the vibration numbers of the notes of the 
 diatonic scale n equal temperament. 
 
 297. Takin; C=522, calculate the vibration numbers of the notes of the 
 diatonic scale n mean-tone temperament. 
 
 298. Asoertiin the vibration numbers of C4, E|7, G|;, F^, and B]? in 
 question 295. 
 
 299. Ascerfcin the vibration numbers of Cl, E|7, G4, F|;, and B7 in 
 question 296. 
 
 300. Ascertan the vibration numbers of Cft, EJ?, Gj, F|;, and B|?, in 
 question 297. 
 
IM 
 
 TRINITY COLLEGE, LONDON, 
 
 HIGHER EXAMINATIONS-JULY, 188+. 
 
 Section B., Div. u.— ELEMENTARY ACOUSTICS 
 
 1. What is the velocity of sound in air, in hydrogen, in oxygm, in water, 
 and in iron wire? How is the velocity in air affected by its density, 
 elasticity, and temperature f 
 
 Am. — For velocity in air and water, see p. 20; in iron, see p. 21; velocity 
 in hydrogen 4164:, and in oxygen 1040 feet per sec. at 0° C. ; or influence 
 of temperature, see pp. 20 & 100. Supposing the temperatu'e to remain 
 constant, if the density of the air increases or diminishes, iti elasticity is 
 increased or diminished in the same proportion, ajid therefore, is the velocity 
 of sound in air varies directly as the square root of its elasticity, and inversely 
 as the square root of its density, no change in velocity will ootur. 
 
 2. What is the physical distinction between musical tonesand noises P 
 Ans. — Musical tones are the result of periodic motions (sei p. 2) ; noises 
 
 of non-periodic motions. 
 
 3. What is meant by nodal points ? 
 Ans.— See pp. 91, 96, & 102. 
 
 4. Two tones differing from each other by, say a semitone are painfully 
 dissonant : will the disagreeable sensation of discord to the eai and brain be 
 either increased or lessened by reducing the difference in pitol between the 
 two tones f State the physical grounds of your opinion. 
 
 Ans. — See pp. 157, 158, et seq. 
 
 5. Upon what physical conditions is the distinctive timbre or quality, of 
 various musical sounds held to depend P Give an outline of theexperimental 
 proofs of the theory commonly adopted. 
 
 ^ns.— See pp. 74, 76, &: 77. 
 
EXAMINATION PAPERS. 265 
 
 II.— JANUAKT, 1885. 
 
 1. Distinguish between consonance and dissonance. Give Helmholtz's 
 theory. 
 
 Arts. — Dissonance is caused by rapid audible beats, between fundamentals, 
 partials, differentials, &o. ; see Chap. XIV. Intervals which give no audible 
 beats are termed Consonances ; the only perfect consonances, under all 
 circumstances , are those having the vibration ratios^, S, J, &c. ; see Chap. 
 XVI. There is no strict line of demarcation between consonance and 
 dissonance ; see pp. 198, 199. 
 
 2. What are harmonics or overtones ? Describe the .ffiolian Harp. To 
 what is its action due p 
 
 Ans. — See pp. 71, 81. The .ffiohan Harp consists of an oblong sound-box 
 of thin wood, about 5 inches deep and 4 feet long, with one or two apertiures 
 cut in the top. Along the upper side of this box, from 7 to 10 gut strings 
 are stretched between bridges at either end, all the strings being tuned in 
 unison. If the Harp thus constructed be placed in a current of air, the 
 strings vibrate in ever varying numbers of segments, thus producing the 
 various harmonic tones of the note to which they are tuned. 
 
 3. How many revolutions of the syi'en per second are requisite to 
 produce C = 256 (a) from a disc of 8 holes? (b) of 12 holes? (c) of IS 
 holes ? 
 
 Am.— (a) A|6 = 32 ; (b) %ff = 21J ; (c) s^s = 16. 
 
 4. To what class of instruments does the human voice belong'? What 
 is the actual voice-producing organ ? 
 
 4ns.— See pp. 114, 115, 117. 
 
 5. Do you consider the human voice a simple tone? Explain your 
 answer. 
 
 4ms.— See pp. 115 & 117. 
 
 III.— JULY, 1885 
 
 1. — ^What is meant by the terms "Absolute" and "Relative" pitch? 
 Give illustrations. 
 
 Ans. — The •• Absolute " pitch of a note defines its position in the realm 
 of pitch, independently of, that is without reference to, a note of any other 
 pitch ; for example, we may say, the Absolute pitch of C = 512, meaning 
 that 512 vibrations per second will produce this note. The " Eelative " 
 pitch of a note defines its position in relation to some other note ; thus we 
 may state the relative pitch of G', by saying that it is a Fifth above C, or 
 of (m) by saying that it is a Major Third above (d). 
 
 2. Explain the difference between a simple and a compound sound. 
 
 Am. — See summaiy, pp. 83 & 84. 
 
266 HAND-BOOK OF ACOUSTICS. 
 
 3 What is meant by a tempered scale f Which is the lower uote Dj^ 
 or C ? If there is any difference, by what iraction is it represented ? 
 
 ^TM.— See pp. 232, 246; C is the lower; j^ = |and {c''=i| 
 
 therefore jg^ = |-^l| = fxif= j^f, consequently also j ^ = 
 
 i^^ thprefnrfi jD* — ifi-l-ilS—iSv 128 _ 2 048 
 128' therelore j (, _ ^^ ______ x jgg — 2025- 
 
 4. Does the staff notation afford any gi'eat advantages oyer any other 
 notation p If so, mention any important features presented by it. 
 
 Ans. — Yes ; it marks absolute pitch better than any other. 
 
 IV JANUAKY, 1886. 
 
 1. How many vibrations per second occur in the lowest and highest tones 
 cognizable by the ordinarily-endowed human ear ? and how many octaves 
 does the audition-power of that ear include P 
 
 Ans.—'&ee pp. 40 & 44 ; tones of from 20,000 to 30,000 vibrations per 
 second can be perceived by most persons, but are too shrill to be used in 
 music ; this would give about eleven octaves as the audition power of the 
 ear. 
 
 2. What opinion do you hold, and on what grounds, concerning the 
 influence of the rise of temperature, occurring in a large crowded hall in the 
 course of a concert, on the pitch of the organ and the orchestral instru- 
 ments P 
 
 Ana. — For organ and other wind instruments, see pp. 100, 101, & 116 ■ 
 for strings, see pp. 90 & 96. 
 
 3. State the rates of transmission per second of sound through any five 
 simple fluids or saline solutions ; how is the velocity influenced, as a general 
 rule, by temperature and density p 
 
 Ans. — Gases are fluids, so that either of the following seta wiU do : 
 
 Air 1090 Solution of common salt - 6132. 
 
 Oxygen 1040 
 
 Hydrogen - 4164 
 
 Carbonic acid 858 
 
 Nitrous oxide 859 
 
 Sulphate of soda 5194. 
 
 Carbonate „ 5230. 
 
 Nitrate ., 5477. 
 
 Chloride of Calcium - 6493, 
 
 increased by rise of temperature and diminished by increase of density. 
 
 4. What is meant by resultant tones, jind how axe they artificially 
 producible ? 
 
 4ns.— See pp. 128, 129, 135. 
 
EXAMINATION PAPERS. 267 
 
 V.-HIGHER ARTS, JULY, 1885. 
 
 ACOUSTICS. 
 
 1. In what sense is the word ipave used in regard to sound P Explain 
 the fact that distance from a sonorous hody lessens the intensity of sound. 
 
 Arts. — See pp. 17, 18 ; see pp. 53 & 54. 
 
 2. Assuming that sound travels at the rate of 1,120 feet per second, 
 calculate the wave-length of a note of 140 vibrations per second. 
 
 . 1120 66 o r 1 
 
 Ans. = i^ = 8 feet. 
 
 140 7 
 
 8. Two notes, one sounded on a comet and the other on a flute, are of 
 the same pitch and intensity, and yet are readily distinguishahle. How is 
 thisP 
 
 Ans. — They differ in quality ; that from the flute is nearly simpZe, while 
 that from the comet contains numerous overtones. 
 
 4. What are Harmonics ? By what simple apparatus would it be possible 
 to prove the existence of any particular haimonic in a given compound 
 tone? 
 
 Ans. — ^Harmonics ai'e termed partials or overtones in this book, see p. 71 ; 
 by means of a resonator, see pp. 63, 65, 66, 67. 
 
 5. What are beats ? and how may they be made optioally evident in the 
 case of two tuning-forks p 
 
 .4ns.— See pp. 144, 145, 152 ; see pp. 148 & 149. 
 
 6. A certain note is obtained from a wire stretched by a weight of 161bs. 
 What weight must be used to obtain the Major Fifth? 
 
 Ans. — ^Vibration numbers are proportional to square roots of stretching 
 weights ; see pp. 87, 89 : therefore if n denotes the required weight in lbs., 
 y/» : y/l& : : 3 : 2, that is, ^n = y^l6 x 3-4.2 = 4x3-;-2 
 = 6, therefore n = 36. 
 
 7. Explain by means of a diagram the manner in which the air vibrates 
 in a stopped organ-pipe. What is the effect of forcing the air-blast in this 
 case? 
 
 Am. — See p. 106 ; on gradually increasing the force of the air blast, the 
 vibration form A and the corresponding tone flrst disappears, then B vanishes. 
 and so on. 
 
 8. Describe any one method by which the positions of nodes and loops in 
 an open pipe may be determined. 
 
 Ans.— See pp. 103, 104. 
 
 9. How may rods he made to vibrate — (a) transversely; (6)longitudinallyr 
 Trace an analogy between rods and organ-pipes. 
 
 Ans.— [a) see p. 121 ; (*) see pp. 118, 119, 120 ; see pp. 119, 120. 
 
268 HAND-BOOK OF ACOUSTICS. 
 
 VI. 
 
 CAMBRIDGE. 
 
 EXAMINATION FOR THE DEGREE OF BACHELOR OF MUSIC 
 
 AND SPECIAL EXAMINATION IS MUSIC FOR THE 
 
 B.A. DEGREE. 
 
 Thuesdat, May 29, 1884. 10 a.m. to 1 p.m. 
 
 1. Describe the nature of a sound-wave in air. What are the " amplitude," 
 the " phase," the " wave-length " ? How does an individual particle of the 
 air move ? 
 
 ^n«.— See pp. 16, 17. See pp. 17, 136, 137, 21, 22. See pp. 18, 21. 
 
 2. Has a sound of given loudness, pitch, and qnahty, a definite wave- 
 fonn p Draw a wave figure for a fundamental note and its second overtone, 
 supposed to be of equal loudness, but in different phases. 
 
 Ans. — No, see p. 85. The figiu-e required is to be constructed on the 
 same principle as fig. 45, p. 83. Draw a curve similar to A, and below it a 
 similai' cm-ve, J the length of A, but of the same amplitude, putting (1) 
 below (2) ; then compovmd the two curves as described on p. 82 ; the scale 
 should be larger than that of fig. 45. 
 
 3. What are vibration-fractions f Why is the fraction corresponding to 
 the sum of two intervals obtained by multiplying together the fractions 
 corresponding to the intervals p 
 
 Ans. — See pp. 45, 50. See p. 49. 
 
 4. Which is the greater, three Major Thirds or an Octave, and by how 
 muchF 
 
 Ans. — The Octave is the greater by the interval corresponding to the 
 vibration ratio i^ ; for, three Major Thirds = 1 v - v - =: -^^. while 
 
 12 5 4 4 4 ~~^ 6 4 
 
 an Octave = ^ = IAS. 
 1 64 
 
 5. State, approximately, the number of vibrations per second which give 
 
 rise to the note ^^ r^ — ; and deduce those of the other notes of the 
 
 1 
 
 diatonic scale of which this note is the Fourth. 
 
 .4m.— Take C = 512 ; then A= 512 x | = 42ej; E' = 640, D'# = 
 600, C'l = 533J, B == 480, A = 426J, G| = 400, F| = 360, E = 320; 
 for method of calculation see pp. 45, 46. 
 
 6. What is meant by the " partials " or " overtones " of a note ? Would 
 you prefer to call the Octave of a note its second partial tone or its first 
 upper partial tone, and for what reason P 
 
EXAMINATION PAPERS. 
 
 269 
 
 Am. — See pp. 71, 84. Its second paiiial tone ; 1st, this term is shorter ; 
 2nd, the latter term is wanting in precision. 
 
 7. Which of the notes of the diatonic scale of C are found among the 
 overtones of C P 
 
 Ans. — See p. 72 ; all are found among very high partials. 
 
 8. What are the nature and cause of the phenomenon of Keaonance ? 
 The dampers of middle C on a pianoforte are raised ; and the Ek next 
 telow is struck sharply. What note or notes wiU be heai'd by resonance ? 
 
 ^ns.— See pp. 57, 68, 69. The G', a Twelfth above middle 0. 
 
 9. What is the physical difference between a note sounded on a tuning- 
 fork and a pianoforte respectively f Why are the hammers of a piano 
 covered with felt ; and what has determined the place on the wii'e at which 
 the hammers are made to strike it ? 
 
 Ans. — The note from the timing-fork is nearly Simple, and what overtones 
 there are do not follow the ordinary partial series; while that from the 
 piano is Compound, and the overtones follow the ordinary law, that is, their 
 vibration numbers are as 2, 3, 4, 6, &o. See p. 94 ; See p. 95. 
 
 10. Two precisely similar tuning-forks aie thrown very slightly out of 
 unison. Describe and explain the phenomenon heard; and state how it 
 changes as the difference between the forks is gi'aduaJly increased. 
 
 Ans. — See pp. 144 to 147 ; see pp. 154, 165. 
 
 11. What is meant by " combination-tones," or " resultant-tones," and 
 their " orders " f Determine their positions in the case of the following two 
 simple primaries : — 
 
 i 
 
 ^ns.— See pp. 128, 129, 
 
 of 2nd order, 
 
 17 
 
 131, 135. 
 
 Differential tone of 1st order, 
 
 IZ2I 
 
 of 3rd order 
 
 4 
 
 iEME 
 
 land 
 
 i& — : of 4th order, ca 
 
 tone, ^t; 
 
 ^ 
 
 m ^"^S 
 
 : nearly. 
 
 and /l\ fe? 
 
 w 
 
 nearly; Summation 
 
 12. What is "equal temperament"? State its advantages and dis- 
 
 ^n».— See pp. 239, &c., and 247. 
 
270 HAND-BOOK OF ACOUSTICS. 
 
 VII. 
 
 Thubsdat, June i, 1885. 9 — 12. 
 ACOUSTICS. 
 
 1 Define wave motion. What are the elements which detennine a 
 wave f If the velocity of a wave be constant how does the length depend 
 upon the time of vibration of its component particles f In the case of wave 
 motion caused by particles vibrating at right angles to a straight line ; if 
 the time of vibration be doubled, the velocity of the wave remaining the 
 same, draw the altered wave. If the time of vibration remain the same but 
 the velocity of each particle be doubled, draw the wave. 
 
 Am. — A wave motion is that kind of motion, in virtue of which, the 
 pai'tides of the medium through which it is passing, successively arrange 
 themselves in the same relative position. Length, amplitude, form. The 
 wave passes through a space equal to its own length, in the time that the 
 component particles take to execute one vibration. If the time of vibration 
 be doubled, the wave must be drawn of twice its former length. If the 
 velocity of each partide be doubled, the time of vibration remaining the 
 same, the wave must be drawn of the original length, but of twice the 
 amplitude. 
 
 2. What is the nature of a sonorous wave ? Upon what does the velocity 
 of sound depend P State Mariotte's law. Why does the velocity of sonnd 
 in air of different densities remain the same if the temperatm'e be unaltered ? 
 
 Am. — See pp. 16, 17. Upon the elasticity and density of the medium. 
 The volume of a gas varies inversely as the pressure, temperature remaining 
 constant. See answer to Trinity College paper, July 1884, No. 1. 
 
 3. Explain the action by which a tuning-fork at rest ia set in vibration 
 by the exdtement of another which is accm-ately in unison with it. 
 
 Am. — See p. 58. 
 
 4. Which overtones of a fundamental tone correspond accurately to 
 notes of its diatonic major scale ? Which approximate most nearly to the 
 remaining notes of the scale P Supposing the fundamental tone to correspond 
 to 264 vibrations per second, determine for each instance the differences 
 alluded to within the compass of the same octave. 
 
 4ns.— See p. 72. See p. 72. Let C = 264; then 7th partial = 1848 ; 
 tUs reduced two Octaves = 462 ; but B|7 = 264 X f X f = 469J, thus 
 difference = 469J — 462 := 7J vibrations per second. Eleventh partial 
 reduced to same Octave =i 863, G = 396, difference = S3. Thirteenth 
 partial reduced = 429, A = 440, difference = 11. Fourteenth partial 
 reduced = 462, B = 495, difference =: 83. Seventeenth partial reduced 
 
 280J, D ::= 297, difference = 16J. Nineteenth pjirtial reduced :^ 313J, 
 
 E = 380, difference = 16J. 
 
EXAMINATION PAPERS. 271 
 
 5. State the laws which govern the \dbratione of stretched strings 
 according as their length, weight, and tension are varied. What muft be 
 the relative thicknesses of the first two strings of a tuned violin in order 
 that their tension may he the same ? 
 
 Ans. — See p. 87 or 96. The vibration numbers are as 3 : 2, therefore the 
 coti'esponding thicknesses must be as 2 : 3. 
 
 6. Give an explanation of the motion of the air in a pipe closed at one 
 end. What overtones are absent, and whyP What is the length of a 
 stopped organ pipe resounding to a note of which the vibration number is 
 100 per second ? 
 
 Ans. — See pp. 105 to 108. Approximately, length = IMl ^ 4 = 2ft. 
 9in. (vibration of sound = 1100). 
 
 7. Two tuning-forks, one a fifth higher in pitch than the other, are set in 
 vibration of equal intensity at the same instant. Draw the " associated 
 wave " corresponding to the composite sound. Upon what does quality of 
 sound depend p How do differences of phase affect the quality of a 
 " clang " ? State Fourier's Theorem. 
 
 Ans. — See fig. 45, p. 83, three waves of B must equal in length two waves 
 of A, the amplitudes being equal. See p. 84. Difi'erences of phase do not 
 affect the quality of a clang. Every form of wave can be resolved into a 
 certain number of simple waves, whose lengths are inversely as the numbers 
 1, 2, 3, 4, &c., and it can be thus resolved in one way only ; see p. 83. 
 
 8. Describe and give an explanation of the effect produced by turning a 
 vibrating tuning-fork round close to the ear. 
 
 Ans. — See p. 141. 
 
 9. What are " beats " ? Explain how they arise and how their number 
 may be calculated. What is meant by the " boating distance " between 
 two notes P Draw the " associated wave " representing a beat caused by two 
 notes differing in pitch by a tone. 
 
 .4ns.— See p. 152. See pp. 144 to 147. See p. 155 or p. 172. Vibration 
 ratio of a tone = f ; therefore, first di-aw 9 equal and simple waves similar 
 to A, fig. 45, then below draw 8 similar waves, making the total lengths of 
 these 8 equal to the total lengths of the 9 ; then compound as described on 
 p. 82 ; see also fig. 76. 
 
 10. Explain what is meant by saying that the Octave is the only perfect 
 concord. 
 
 Ans. — See p. 187 ; the Twelfth is also a perfect concord. 
 
 11. Detennine the vibration fractions representing the intervals between 
 the successive notes of the diatonic scale. What is a Commap What is 
 meant by " equal temperament " ? Point out in what parts of the scale the 
 metholi is most defective. 
 
 ^ns.— See pp. 45 to 47 ; pp. 230, 234, 239 ; pp. 239, 241, 247. 
 
272 HAND-BOOK OF ACOUSTICS. 
 
 VIII. 
 
 LONDON UNIYERSITY. 
 
 INTERMEDIATE EXAMINATION FOB THE DEGREE 01 
 BACHELOR OF MUSIC. 
 
 Monday, December 10th, 1884. 10—1. 
 
 1. Explain dearly what is meant hy (1) Elasticity and (2) Density 
 jyhen it is stated that the Telocity of sound in air is equal to w^§, what 
 
 units are employed in the measurement of the elasticity, density, and 
 velocity ? 
 
 Atis. — (1) Elasticity is that property of matter, in virtue of which a hody 
 tends to recover its form and dimensions, when these have been forcibly 
 changed. (2) The Density of a body is its weight, compared with the 
 weight of an equal bulk of some standard — the standard for gases is usually 
 air, and for liquids, water. The units are immaterial provided they be con- 
 sistent, for example, feet, seconds, and grains may be used, thus : E = 
 pressure of atmosphere in graim per square /ooi, multiplied by the accelera- 
 tion due to gravity, in feet per second; D = weight in grains of a cubic /ooj 
 of air ; then y' 5 will give the velocity of sound, in feet per second. 
 
 2. How does the velocity of propagation of a transverse wave along a 
 wire vary with the density, diameter, and tension of the wire f A wire of 
 given length is tuned to a certain note : how must its tension be changed iu 
 order to raise the pitch of the note a fifth f 
 
 Ans. — ^Velocity varies inversely as square root of density, inversely as 
 diameter, and directly as square root of tension. Let 2n be vibration 
 njimber at tension t, then 3n will be vibration number at required tension T: 
 therefore by the above 
 
 /T ^ 32 tjjat is, 2^ = 2, therefore T=l x t, 
 that is, its tension must be 2J times as great as at first. 
 
 3. A fine string is attached at one end to a prong of a vibrating tuning- 
 fork. What will happen aa the tension of the string is gradually increased 
 from an extremely small amount? What considerations determine the 
 point at which a piano string is struck by the hammer ? 
 
 Ans. — At fii'st the string only flutters irregularly ; then as the tension is 
 increased, it vibrates in a large number of segments, the number decreasing 
 as the tension is increased, till at last only one segment is obtained. See 
 p. 95. 
 
EXAMINATION PAPERS. 273 
 
 4. Briefly describe any method you know for the production of Liasajoua' 
 figures : and show by sbetches the transformations which the figure undergoes 
 when it is produced by two forks vibrating at right angles to one another, 
 the inteiTal between them being nearly, but not exactly, an Octave. 
 
 Am. — The arrangement of the apparatus for the production of Llssajous' 
 figures is the same as that of fig. 79, p. 149, with this exception, that one of 
 the forks should be placed horizontally. If, under these circumstances, the 
 forks be rigorously tuned to the Octave, the figure obtained (in the place of 
 the wavy line in the fig.) is a perfect figure of 8, a parabola, or any distortion 
 of the figure of 8 intermediate between these two ; but whatever figure is 
 obtained, as long as the forks are rigorously in tune, it will not alter in 
 shape. If however the forks are very slightly out of tune, the figui'e will 
 gradually change from the perfect figure of 8 through Its distortions to a 
 parabola, and then gradually back to the perfect 8 again, and so on. 
 
 6. What is the use of placing a tuning-fork upon a resonator ? Will a 
 tuning-fork continue to sound for a longer period when placed upon ? 
 resonator tuned to its own note, or when placed upon a resonator which is 
 incapable of responding to the note of the fork ? State fuUy the reasons for 
 youi' answer. 
 
 Ans. — See p. 64. 
 
 6. How may manometric flames be employed in conjunction with the 
 resonators for the analysis of a compound sound? Describe fully the 
 arrangement of the gas supply, diaphragm, &c., in a manometric flame. 
 
 Ans. — The figure on the left of fig. 55 described on p. 104, shows the 
 ai'rangement of the gas supply, diaphragm, &c., in a manometric flame. 
 When used in conjunction with a resonator, it is best to fix it, in such a way 
 that the diaphragm forms the base of the resonator. With a number of 
 resonators, thus supplied with manometric jets, and tuned to the partials of 
 a compound tone, the presence or absence of any particular overtone may 
 be optically demonstrated. 
 
 IX. 
 
 Afternoon, 3 — 6. 
 
 1. Give an account of the process by which sound is propagated through 
 the air. What is the influence of fog, mist, rain, or snow on the propagation 
 of sound P How do you explain the fact that sounds are often heai'd to a 
 greater distance by night than by day f 
 
 Am. — See pp. 17, 18. Fog, mist, rain, and snow do not obstruct the 
 propagation of sound to any perceptible extent. Because at night the air 
 is generally more homogeneous and therefore favourable to the transmission 
 of sound ; in the day-time, the air is usually less homogeneous — there are 
 warm ascending and cooler descending currents, for example, — and part of 
 tlie sound is reflected in passing from one mass of air to another, the part 
 
274 HAND-BOOK OF ACOUSTICS. 
 
 tranemitted being thus diminished. Furthermore, in the day-time, the 
 temperature generally diminishes as we ascend from the earth's surface, and 
 Professor Reynolds has shown that when this is the case, the sound, which 
 would otherwise proceed horizontally along the earth's sm'face, is tilted 
 upwards to an extent depending on the upward dimiaution of temperature ; 
 now at night, especially when cloudless, the reverse is the case — the air in 
 contact with the earth is colder than that aboye. 
 
 2. Explain how two stretched strings vibrating transversely are tuned to 
 unison — two similar strings are stretched over the same sounding-boai'd : 
 would it be possible for a deaf or blind person to tune them to unison ? 
 
 Ans. — Approximately by ear ; rigorously by beats, see p. 174. A deaf 
 person could tune them, by observing when the one string set the other into 
 most powerful co-vibration, using riders if necessary ; a blind person of 
 course could tune them. 
 
 3. Under what circumstances does interference of sound occur ? 
 Describe an experiment by which it may be shown that two notes of the 
 same character and intensity, but differing from each other in phase by half 
 a wave length, exactly neutralize each other. 
 
 4ns.— See pp. 137 & 138. See p. 139 or 140 or 141. 
 
 4. What are the upper partial tones of a tuning-fork, and how are they 
 produced f Indicate a method by which their vibration-frequency may be 
 determined. 
 
 Ana. — See p. 123. If the stem of a vibrating fork be placed on the 
 string of a monochord and moved to and fro along it, the section of the 
 string will co-vibrate with the fork whenever its length is such as to be in 
 unison with an overtone of the fork. Knowing the vibration number of 
 the whole string, the vibration numbers of the overtones of the fork may 
 be readily calculated, by measuring the co-vibrating lengths of the string 
 
 5. Describe the modes of vibration of a circular metal plate. What is 
 the relation between the pitch of the note produced and the number of 
 diametrical nodal lines P 
 
 Ans. — See p. 125. Vibration number varies as square of number of 
 diametrical nodal lines. 
 
 6. Give an account of Helmholtz's study of the nature of the vowel 
 sounds. 
 
 Ans. — Helmholtz first analysed the various vowel sounds by the aid of his 
 resonators (see pp. 65 & 66), determining the number, order, and intensities 
 of the partials in each vowel sound. He then synthetically constructed the 
 various vowel sounds from simple tones, by means of the apparatus described 
 on pp. 77 to 80. For details, his " Sensations of Tone " must be consulted. 
 
EXAMINATION FAPEBS. 275 
 
 X. 
 
 Tuesday, December 11th. 10 — 1. 
 
 1. State your idea as to the general nature of a musical interval. Explain 
 why in practical music there are limits to the number and magnitude of the 
 intervals used. 
 
 Ans. — ^An interval is the step or distance from one musical sound to 
 another ; its magnitude being measured by the ratio of the vibration numbers 
 of its constituent tones. The essence of melody is motion, and this motion, 
 to produce a pleasurable effect, must be effected in such a manner, that the 
 hearer can readily appreciate its character ; but this is only possible when 
 the intervals are easily measurable and recognizable by the ear ; hence the 
 distances between the successive tones must be definite and limited in 
 number. Furthermore, harmony requires that the notes of the scale should 
 have certain concordant relations to one another, and this is only possible 
 when the number of tones, and therefore the intervals between them, are 
 limited. 
 
 2. Explain fully what is meant by the ratio of a musical interval, and 
 what it expresses. 
 
 Ans. — See pp. 50 or 45. 
 
 3. Give the ratios applicable to all the intervals ordinarily used in modern 
 music. Write the intervals in musical notation and add their names. In 
 the more complicated intervals explain how the ratios are arrived at. 
 
 Ans. — See p. 51. See Chap. V. 
 
 4. Describe the several kinds of semitones, ancient and modern, and give 
 their exact values. 
 
 Ans, — For Diatonic, Greater Chromatic, and Lesser Chromatic Semitones 
 see Chap. V ; the Greek Hemitone is the difference between a Fourth and 
 two Major Thirds, hence its vibration ratio = I X | X | = —^ ; the equal 
 tempered Semitone is the twelfth part of an Octave, hence its vibration 
 
 12 
 
 ratio ^ /2 • the Diatonic Semitone in Mean tone temperament = 
 8 ^^ 
 
 4 
 
 5. Name, and give aocm'ate theoretical definitions of the several intervals 
 which are practically represented on the pianoforte by three semitones, four 
 semitones, and six semitones. 
 
 Am. — Three Semitones on the pianoforte represent the Minor Third and 
 the Augmented Second ; the best way of defining an interval is by stating 
 its vibration ratio ; the vibration ratio of a Minor Third = |^, of an 
 Augmented Second ^ ■^. Four semitones on the pianoforte represent the 
 
 64 IS S9 . 1 
 
 Major Third and the Diminished Fou rth, vibration ratios f ' f4' respectively. 
 
276 
 
 HAND-BOOK OF ACOUSTICS. 
 
 Six pianoforte semitonea represent the Tritone and Diminished Fifth, 
 vibration ratios || & yf respectively. 
 
 6. What is a diatonic scale ? Give the order of the intervals in it, and 
 explain any changes that have been made at different times, in their 
 exact values. 
 
 Am. — The diatonic scale consists of seven tones proceeding by steps from 
 a certain note to its Octave, each tone having as far as possible concordant 
 harmonic relations with all the other tones. See p. 47. The intervals of 
 the ancient Greek diatonic scale were |, |, HI, |, |, |, fff;*e| 
 tone being obtained by taking a Fourth from a Fifth, and the ^% semitone 
 by taking two such tones from a Fourth. In Mean tone temperament the 
 
 intervals were 
 
 v/s v^5 8 
 
 5^5 
 
 V^S v/s ^5 
 
 "2"' 2 '~2 
 
 the 
 
 
 tone being J » true Major Third, and the 
 
 5^5 
 
 5v/5 
 semitone being J the 
 
 difference between 5 such tones and an Octave. In Equal Temperament the 
 
 66 12 66eu 6 
 
 intervals are ^ 2, y' 2, .\/ 2, ^ 2, a/ 2, •y 2, a/ 2, the ■\/ 2 tone being 
 
 1 12 I 
 
 6, and the i/^ semitone the 12 of an Octave. 
 
 7. State what you know of the nature and uses of chromatic notes in 
 modern music. Write examples in musical notation. 
 
 Ans. — Chromatic notes in modern music have two distinct functions, viz: 
 (I) for the pm'poses of modulation, (11) for ornamentation. In (I) their exact 
 positions in the scale are determined by their harmonic relations to the other 
 tones of the scale ; in (II) their positions vary with their melodic relations : 
 for example, in (1) below the F|; must form a true Major Third with D ; 
 
 i 
 
 
 (2) 
 
 &c. 
 
EXAMINATION PAPERS. 
 
 277 
 
 in (2) the Fft as well as the A|j would be taken very near to the G with which 
 they are so closely associated, but there is no logical reason why they should 
 liave any one particular position more than any other ; in (3) a violin-player 
 would probably place the Ftt just midway between the F and Q-, and the 
 same with the C4 and Djt, m order to get equal steps in ihe run. 
 
 XI. 
 
 Aftebnoon 3 — G. 
 
 1. State the vibration numbers for each note in the Octave of the true 
 diatonic major scale of Bjj, taking the second note of this scale at 256 double 
 vibrations per second. 
 
 Ans.—B\y = 227-55, G = 256, D == 2844, E|7 = 303-4, F = 341-3, G = 
 879-2, A = 426-6, B't> = 455-1. 
 
 2. Give also the numbers for the equally tempered scale, keeping the 
 second note the same as before. Explain the mode of calculation. 
 
 Am.—'&\} = 228, C = 256, D = 287 -^-, E|, =304 +, F = 341, G =- 
 383 — , A = 430 — , B|j = 456. 
 
 12 
 
 The vibration ratio of an equal semitone is y/ 2 = 1-06 nearly or 1-06 — , 
 
 6 
 
 „ „ „ „ tone „ y/2 = 1-122 + ; therefore 
 
 B|j = 256 -H 1-122 = 228, 
 C =256, 
 
 D = 256 X 1-122 = 287 + 
 Eb = 287 X 1'06 = 304, & so on. 
 
 3. Write out any melody you can recollect and analyse its rhythm. 
 
 4. Transpose the following melody into the key of A and also one semi- 
 tone higher 
 
 i 
 
 ^ 
 
 s a « 
 
 ^=fc 
 
 ^ 
 
 -r-^ 
 
 ±=t 
 
278 EAND-BOOK OF ACOUSTICS. 
 
 5. Write two bars of melody in each of the yarieties of time ordinarily 
 used in Modern music. 
 
 Ans. — 3, 4 & 5 — Purely musical questions. 
 
 6. State generally how, taking the physical properties of musical sounds 
 as a basis, you would construct the elements of harmony at present in use. 
 
 Ans. — The Octave is the interval between the 1st and 2nd partials of a 
 tone ; the Fifth is the interval between the 2nd and 3rd ; and the Major 
 Third is the interval between the 4th and 6th ; and from these the whole 
 scale, diatonic and chromatic, may be constructed, as shown in Chap. V. 
 Furthermore, the interval between the 3rd and 4th partials gives the Fourth ; 
 that between the 5th and 6th, the Minor Third ; that between the 3rd and 
 5th, the Major Sixth, &c. Again, in order that two tones may not dissonate, 
 their audible partials must not come within beating distance. The Octave 
 satisfies this condition completely, the Fifth to a less extent, and so on ; 
 (see Chap. XVI). ' 
 
 7. Trace any facts in the history of music which iUustrate or have a 
 bearing on such a derivation of the structure of harmony. 
 
 Am. — The Octave is the interval that was earliest used in harmony and it 
 is the very interval that is most prominent when a Compound Tone is 
 heard : (see also p. 187.) The Fifth was the only other interval admitted as a 
 consonance for many years after; and this is the next most prominent 
 interval in a Compound Tone : Similarly with the Fourth. It is only within 
 the last two or three centuries that the Thirds have been ranked as 
 consonances (see p. 193), and these intervals are by no means easily heard in 
 a Compound Tone. 
 
 XII. 
 
 INTERMEDIATE EXAMINATION, 1885. 
 
 MOKNINO, 10 — 1. 
 
 1. state the conditions which determine the fiequoncy of vibration of a 
 stretched string ; and show how each of them is made use of in the con- 
 struction and tuning of the pianoforte. 
 
 .4ns.— See pp. 87, 88, 89. 
 
 2. In drawing water from a tap into a jug, one can tell roughly by the 
 sound of the water, without looking, when the jug is getting full. Explain 
 this. 
 
 Ans. — The water falling into the jug, sets the contained air column 
 vibrating. The pitch of the sound thus produced depends upon the 
 dimensions of this air column. These dimensions become less and less as 
 the water rises in the jug, and the pitch of the sound therefore rises accord- 
 ingly : by this rise in pitch it may be roughly known when the jug is full. 
 
EXAMINATION PAPERS. 279 
 
 3. Describe the Syren, and explain how it is used to determine the pitch 
 of a musical note. 
 
 Ans, — See pp. 31 to 34. 
 
 4. Compare the harmonic tones of a stretched string with those of an 
 open or a stopped flue organ pipe, and account for their agreement or 
 difference. 
 
 Ans. — " Harmonic tones " means " Overtones.'' See pp. 93, 103 to 106. 
 
 5. Explain the use of a resonance-box in connection with a tuning-fork. 
 Is the quality pf the note of the fork in any way modified by the resonance- 
 box? 
 
 Ans.— See pp. 64, 73, 123. 
 
 6. Describe the mode of vibration of the air in a flute, and explain the 
 effect of opening or closing the keys or finger-holes on the pitch of the note 
 sounded. 
 
 Ans. — The mode of vibration of the air in a flute is the same as in an open 
 organ pipe. See pp. 102, 103, 109, 110. 
 
 7. In tuning an organ, it is usual to begin with the " Principal " stop. 
 What physical reason is there for choosing this particular stop in preference 
 to others, say the " Open Diapason " ? 
 
 Ans. — As explained in Chap. XV, intei'vals are best tuned by bringing . 
 some pair of partials into coincidence. In the case of the Fifth for example, 
 the 2nd partial of the higher tone must be brought into unison with the 3rd 
 of the lower; imtil they are in unison, beats are heard. Consequently it is 
 advisable in tuning the organ, to tune first that stop in which the partials 
 are more prominent. Now in the Open Diapason they are less prominent 
 than in the Principal, hence the preference shown to this latter. The 
 Principal having been tuned, the other stops can be tuned from it, by 
 unisons. But another and perhaps the chief reason is that the Principal is 
 an Octave higher than the Open Diapason and therefore the beats are twice 
 as rapid. 
 
 XIII. 
 
 Afteenoon, 3 — 6. 
 
 1. What is the cause of " Interference " ? Explain how interference of 
 sound may be illustrated by a single tuning-fork. Can the sounds emitted 
 by two tuning forks, exactly in unison, interfere with each other P 
 
 ^M.— See pp. 136 to 188. See p. 141. See pp. 138, 139. 
 
 2. Describe the modes of vibration of a bell, and account for the beats 
 which are sometimes heard when a large bell is struck. 
 
 Ans. — Same as circular plates ; see pp. 125, 126. 
 
 3. Enumerate the usual Great Organ stops. Account on physical grounds 
 for the various qualities of tone they produce, and show how they are 
 supplementary to each other. 
 
280 HAND-BOOK OF A00U8TI08. 
 
 Arts. — 1, Stopped diapason ; 2, Open diapason; 3, Dulciaua; 4, Principal; 
 5, Fifteenth ; 6, Mixture. No. 1 is dull in quality, because being a stopped 
 pipe, only the odd partials ai-e present, and usually the 2nd odd pai-tial is 
 faint, so that the tones are nearly simple. Nos. 2 and 3 are somewhat 
 brighter in quality because of the 2nd partial and perhaps the 3rd and 4th. 
 No. 4 which speaks an Octave higher is still brighter, more partials being 
 present because of the smaller scale of the pipes. No. 5 is two Octaves 
 above the Diapasons. In No. 6, two, three, or even more pipes, sounding 
 for example the 15th, 19th, and 22nd above the diapasons, speak for every 
 note. Nos. 5 & 6 are never used alone. When Nos. 1 & 2 or 3 are used 
 together, the ground tone is too much for the faint upper partials ; the tone 
 lacks brightness. To introduce this, Nos. 5 or 6 are brought into play 
 furnishing artificial overtones to Nos. 1, 2, & 3. No. 4 also adds to the 
 brightness of the diapasons by introducing the Octave or 1st overtone. 
 
 4. State and explain the effect of change of temperature on the pitch of 
 a flue organ-pipe. How is it that change of temperature puts the reeds and 
 flue pipes of an organ out of tmie with each other, although the different 
 pipes of the same stop remain in tune among themselves p 
 
 4nj.— See pp. 100 & 101. 
 
 5. What is the cause of dissonance ? Explain why a note and its Minor 
 Third form a less harmonious combination than a note and its Major Third. 
 
 .4ns.— See pp. 154 to 156. See pp. 191, 192, 193. 
 
 6. Explain the general principles of Helmholtz' method of building up 
 a compound musical sound by means of tuning-forks. 
 
 Am.—^ea pp. 76 to 80. 
 
 XIV. 
 
 MoBmNa, 10 — 1. 
 
 1. Explain how and why an arithmetical ratio serves to define the 
 magnitude of a musical interval. 
 
 Ans. — See pp. 44 & 45. 
 
 2. Name, and write in musical notation, the intervals denoted by the 
 foUowing ratios : 
 
 126161006648869 16 26348 
 1' 24' 16' 9' 8' 6' I' 8' 2' 6' 3' i' ¥' l' 2* l' 1* l" 
 
 Ata. — ^Fcr most, see p. 61 ; the others are : - unison ; - = - v ? 
 
 ' ^ ' 1 6 1 '^ 10 — 1 
 
 -^ — ; that is, an Octave less a Minor tone = E \ ~ \ ' ; - = - X - 
 
EXAMINATION PAPERS. 281 
 
 = Major Third + Octave = Major Tenth ; - = - x - = 5th, + 8° = 
 
 12th ; * = J X ^ = Double Octave ;? = ?XjX- = Three Octaves. 
 
 3. Give the ratio for a comma, a mean semitone, and a Greek hemitone. 
 
 . 81' , -„ 2B6 
 Ans.-^; 1-07;^. 
 
 4. Deduce harmonically (not by the use of semitones) the ratios of the 
 tritone, the diminished and augmented fifths, the augmented sixth, and the 
 diminished seventh. 
 
 Am. — Vibration ratios of Major Triad = 6:5:4; from which data, with 
 say C ^ 24, deduce as on p. 45, F ^ 32, B == 45 ; then vibration ratio of 
 the tritone = — ; diminished 5th =: Octave — tritone i^ - x — = — ; 
 
 32' 1 '^ 46 64 
 
 again, ■} mif =i - and ] q = -, therefore augmented 5th ■! qiF = - X j 
 := — ; from above ■] oti = ^ ^.nd ] jj := t> therefore augmented 6th 
 
 I ^i = ^ X 5 = ??5; diminished 5th j "^J = ?^ andl v = " there- 
 
 (. C — 32 4 128 I G| 45, J D' B, 
 
 fore diminished 7th \ r^iiz=— "X. - = — • 
 
 I G J 48 5 T6 
 
 5. State what you know about Bameau, and the system of harmony 
 introduced by him- 
 
 .4ns. — Eameau (1683 — 1764), organist and composer, is chiefly known as a 
 theoretical writer on music. His chief work is the " Traits de I'Harmonie 
 reduite k ses Principes Naturels " published in 1722, in which he maintains 
 that Hai'mony is founded on the natural occurrence of the Octave, Fifth, and 
 Major Third in the compotmd tone itself. The compound tone, containing 
 the tones of a chord among its partials was termed by Eameau, the Funda- 
 mental Base of that chord ; and thus he maintained that all the varieties of 
 position and inversion of the chord were to be classified as one and the same 
 
 harmony. Thus the tones of the Major Tiiad i E ai'e the 6th, 6th, & 4th 
 
 rc (0 
 
 partials of Cj ; and J G ai'e the 8th, 6th, <fc 5th partials of the same tone ; 
 
 (e 
 
 hence these are different positions of one and the same chord on the 
 
 Fundamental Bass Cg . In the Minor Chord, ■< El2, he was obliged to suppose 
 
 JeI2,1 
 
 idd 
 
 two generators E|js and Ca both of which would have G as a partial. 
 
 6. Show how intervals are combined together to form chords ; and give 
 examples of the chords ordinarily used in modem music, naming them, and 
 the intervals composing them. 
 
 .4ns.— See pp. 204 to 206. 
 
 7. Explain and Ulustrate what changes in the intervals occur when the 
 chords are inverted. 
 
 .4n«.— See pp. 204 to 206. 
 
282 
 
 SAND-BOOK OF ACOUSTICS. 
 
 XV. 
 
 Afternoon 3 — 6. 
 
 1. Give the correct Titration numbers of every note of the diatonic 
 major scale of A, on the French pitch (lower A ^ 435). 
 
 Ans.—A = 435, B = 489-375, C'| = 543-75, 'b' = 572-8, D' == 580, 
 E' = 652-5, Fi = 725, G'|; = 815-625, A' = 870. 
 
 2. Also for the notes Aj, B|j, Cfa, Dft, Ffa, and Qjt, deducing them from 
 harmonic relations, with ndtes in the above scale. 
 
 Ans.- 
 
 l^ = 
 
 ^ therefore D'|= 5 y 489-375 = 611-7. 
 
 Aj = ^ y 725 — 453-125. 
 
 c 
 
 5 
 
 X 435 
 
 = 522. 
 
 G< 
 
 _3 
 ~2 
 
 X 522 
 
 = 783. 
 
 F' 
 
 4 
 6 
 
 V 870 
 
 = 696. 
 
 Bt) = 696 X - = 464. 
 
 3. What is the exact interval between the notes B and D, and B and Fl 
 in the above scale ? Explain and account for any peculiarity. 
 
 Ans. — 
 
 f D' 580 i 
 
 t B 489-375 < 
 
 58«X8 116X8 
 
 { 
 
 F*' 725 
 
 B* =" 
 
 725X8 
 3916 
 
 3915 
 40 
 
 &c., see p. 50. 
 
 4. Give the -vibration numbers of all the 12 notes in the above scale on 
 equal temperament. 
 
 ^n«.— A = 435, A| = 461, B =488 +, C = 517 +, C'« = 548, 
 D' = 581 — , D'f = 615 +, E' = 652 — , F' = 691, F'# = 732 + G' = 
 776 — , G'l = 821, A' = 870. ^ 
 
 6. Describe briefly the work of Pythagoras in regard to music, and men- 
 tion anything which still bears his name. 
 
 Ans. — Pythagoras lived from about 570 to 504 b.o. He was the first to 
 observe that when a stretched string was successively sounded as a whole, in 
 halves, in thirds, &c., it produced the Octave, Fifth, &c. ; and he proposed 
 
EXAMINATION PAPBES. 
 
 283 
 
 the problem : " Why is consonance determined by the ratio of small whole 
 nmnbers ?" which has thus taken more than 2,000 years to solve. Pythagoras 
 
 9 9 256 9 9 9 256 
 
 laid down the diatonic scale with the following intervals : -, -, -— , j, :. r, — -• 
 
 8 o 243 8 o o za 
 
 9 9 81 
 
 Thus the Pythagorean Third which bears his name ^ - X - = — and is a 
 
 •' ° 8 8 64 
 
 comma sharper than a true Major Third = - ^ — . The Fifths and Fourths 
 
 9 9 256 4 
 are true in this scale for - X - X ttj = -. For the value of a Pythagorean 
 
 8 8 243 3 
 
 comma see p. 239. 
 
 6. Transpose the following passage into the key of Gjj, writing it in the 
 Bass Clef : 
 
 
 Ai f p f>\ ^ 
 
 *^ 
 
 W^r-w^ 
 
 1 irrr^r 
 
 :t=^ 
 
 7. Write in different kinds of time a few short musical examples to 
 illustrate the meanings of the terms Melody, Harmony, Counterpoint, and 
 Rhythm. 
 
 Am. — 6 & 7 — Purely musiojd questions. 
 
284 
 
 INDEX. 
 
 Page 
 
 Aeroplastic reed 99 
 
 Air, Elasticity of 16 
 
 , Transmission of sound throngh 8 
 
 Amplitude of waves 9, 18, 22 
 
 Antmodes 96, 116 
 
 Appun's tonometer 151 
 
 ABsociated wave 19 
 
 Bars, "Vibrationa of 118 
 
 Basilar membrane 26 
 
 Bassoon 113 
 
 Beating distance 155 
 
 at various pitches 159 
 
 Beats 145 
 
 BeUs 125 
 
 Bosanquet'a Harmonium. 246 
 
 Cagniard de Latonx's Byren 31 
 
 Cents 243 
 
 CMadni's experiments on plates 124 
 
 method of ascertaining vibration 
 
 numbers 121 
 
 Chords 204 
 
 Clang 84 
 
 Clarionet 112 
 
 Cochlea 26 
 
 Colin Brown's Voice Harmonium 246 
 
 Combination Tones '... 128 
 
 Comma 50,230 
 
 of Pythagoraa 239 
 
 Composition of waves 82 
 
 Compound Tones 69, 71, 84 
 
 Condensation 18 
 
 Corti'erods 27, 83 
 
 Co-vibration 57 
 
 Cycloscope 37 
 
 Definition of intervals of Simple 
 
 Tones 173 
 
 Compound Tones 177 
 
 Diapason Normal 42 
 
 Diatonic Semitone 48 
 
 Differential Tones 129 
 
 Dissonance 154 
 
 between Simple Tones from 
 
 Differentials 161 
 
 Dissonances, Classification of 166 
 
 Ear 22 
 
 , Resonator, a 67 
 
 Echoes 54 
 
 Elasticity of Air 16 
 
 Embouchure 99 
 
 Eudolymph 25 
 
 Equal temperament 238 
 
 Eustachian Tube 23 
 
 Paob 
 
 Fenestra Ovalis 24 
 
 Rotunda 24 
 
 Fifth 45,228 
 
 , Diminished 228 
 
 Flageolet 109 
 
 Flue-pipes 98 
 
 Flute 109 
 
 Form of waves 9, 19, 22 
 
 Fourier's theorem 83 
 
 French Horn 114 
 
 French standard of pitch 42 
 
 F iini^fl.TnPTitfl.T Tone 71 
 
 Goltz' theory of the function of the 
 
 Labyrinth 27 
 
 (Graphic method of counting vibra- 
 tions 35 
 
 Greater step 47 
 
 Handel's Fork 42 
 
 Harmonics 102 
 
 Harmonicon 122 
 
 Hautbois 113 
 
 Heat, Effect on length of sound- 
 waves 40 
 
 Pipes 100 
 
 Pitch of tuning-forks 123 
 
 Reeds Ill 
 
 Strings 90 
 
 Velocity of Sound 20,100 
 
 Helmholtz's Resonators 65 
 
 synthetical apparatus for study 
 
 of compound tones 79 
 
 Syren 34 
 
 Theory of the ear 27 
 
 Herman ^coith's Theory of organ 
 
 pipes 99 
 
 Highest tones used in music 40 
 
 Horns 114 
 
 Incus 24 
 
 Intensity 29 
 
 and distance from origin 54 
 
 of partial tones 73 
 
 of sound from pipes 101 
 
 Interference 136 
 
 Komma 60,230 
 
 of Pythagoras 239 
 
 Labyrinth 25 
 
 Lanuna spiralis 26 
 
 Larynx 114 
 
 Lateral vibrations of rods 121 
 
 Law of Inverse Squares 54 
 
285 
 
 Faqe 
 Limits of pitoh. of musical instru- 
 ments 41 
 
 Laws of stretched strings 87 
 
 Length of sound waves 39 
 
 of waves 9,18,21 
 
 Limits of musical pitch 40 
 
 Longitudinal vibrations of rods 118 
 
 Lowest tones 40 
 
 Major Third 45 
 
 Tone 47 
 
 Malleus ._ 24 
 
 Marloye's Harp 120 
 
 Mayei^s ^aphic method of counting 
 
 vibrations 35 
 
 McLeod and Clarke's Cycloscope 37 
 
 Mean-tone Temperament 234 
 
 Meatus of the Ear 23 
 
 Membranes 126 
 
 Membranous Labyrinth 25 
 
 Minor Tone 47 
 
 Modiolus 26 
 
 Mouochord 34 
 
 Musical and unmusical sounds 2 
 
 ^box 121 
 
 Tones, Compound 69 
 
 Nodes in open pipes 102 
 
 in stopped „ 106, &c. 
 
 in stretched strings 91,96 
 
 , To show 92 
 
 Oboe 113 
 
 Octave 44 
 
 Oi^an pipes 99 
 
 Otoliths 26 
 
 Overtones 71 
 
 from open pipes 105 
 
 from stopped ,, 109 
 
 fromreeds Ill 
 
 Partial Tones 71, 84 
 
 , Dif&culty in perceiving ... 74 
 
 • from open pipes 102, &c. 
 
 from stopped ,, 109 
 
 fromreeds Ill 
 
 — , Instruments producing ... 73 
 
 — — ■ ■, Relative intensities of 73 
 
 , Tableof 72 
 
 Particle vibration 15, 18 
 
 Pendular vibrations 80 
 
 Perilymph 25 
 
 Periodic motion 2 
 
 Phase 82, 136 
 
 Piajiof orte, Quahty of tone on 95 
 
 , Effect of pressing down pedals.. 95 
 
 Pipes 98 
 
 , Length and pitch of 100 
 
 Pitch 29 
 
 of strings, affected by heat 90 
 
 , Standards of 42 
 
 Plates, Vibrations of 124 
 
 Pythagoras, Comma of 239 
 
 auality 29, 74 
 
 depends upon partials 74 
 
 of tone affected by material of 
 
 string 94 
 
 point of excitement ... 94 
 
 V Page 
 
 Eah or r 50, 229 
 
 Rarefaction 18 
 
 Reed^ Aeroplastic 99 
 
 mstruments 110 
 
 pipes 112 
 
 Reeds, Overtones from. Ill 
 
 Reflection of soundwaves 64 
 
 Reissner, Membrane of 26 
 
 Resonance 57 
 
 of air columns in open tubes ... 60 
 
 — — closed tubes 61 
 
 of strings 59 
 
 of tuning-forks 58 
 
 -boxes 64 
 
 Resonators 63, 65 
 
 Resonator, Adjustable 66 
 
 , The ear a 67 
 
 Resultant tones 128 
 
 Rods, Vibrations of 118 
 
 Saeculus 25 
 
 Savart's toothed wheel 6, 31 
 
 Scala Media 26 
 
 -Vestibuli 26 
 
 Scale, Vibration num.bers of tones of 46 
 
 Scheibler's tonometer 161 
 
 Semicircular canals 26 
 
 Semitone, Chromatic 49 
 
 , Diatonic 48 
 
 , Equal 247 
 
 Sesquiattera 109 
 
 Simple tone 71,83 
 
 tones produced by pendular 
 
 vibrations 81 
 
 , Instruments producing ... 73 
 
 Singingflame 4 
 
 Smaller step 47 
 
 Sonometer 34 
 
 Sorge 129 
 
 Sound-board 86 
 
 wave 18 
 
 waves, To ascertain the lei^h 
 
 of 39 
 
 Speaking tube 55 
 
 Standards of pitch 42 
 
 Stapedius 24 
 
 Stapes 24 
 
 Strmg, Excited by tuning-fork 92 
 
 , Pitch affected by heat 90 
 
 Strings, Laws of vibrations of 87 
 
 Stringed instruments 86 
 
 Summation tones 129, 134 
 
 Sympathy of tones 57 
 
 Syren, Cagniard de Latour's 31 
 
 , Helmholtz's 34 
 
 ."Wheel 29 
 
 Tartini 129 
 
 Temperament 232,246 
 
 .Equal 238 
 
 , Mean-tone 234 
 
 Temperature, Length of sound wave 
 
 affectedby 40 
 
 .Effect of, on pipes IOC 
 
 , on reeds Ill 
 
 , on strings 90 
 
 , on tuning-forks 123 
 
 , on velocity of sound 20, 100 
 
 Tensor tympani 24 
 
286 
 
 Page 
 Tetrads, Consonant, comparison of... 219 
 
 , distribution of major 221 
 
 minor 222 
 
 Thompaon's Enharmonic organ 246 
 
 Timbre 84 
 
 Tones, mostly compound 69 
 
 Tonic Sol-fa fork 42 
 
 Tonometer 151 
 
 Transmission of sound tbrough air... 8 
 
 Trevelyan's rocker 6 
 
 Triads 204 
 
 , Distribution of major 215 
 
 minor 217 
 
 , Inversions of major ,... 208 
 
 minor 210 
 
 Trombone 114 
 
 Trumpet 114 
 
 Tuning-fork 122 
 
 , EfFectof heat on pitch of 123 
 
 kept vibrating by electro-magnet 77 
 
 Tympanum 23 
 
 Utriculus 25 
 
 Page 
 
 Velocity of sound in air, &c 20 
 
 in gases, how ascertained... 119 
 
 in solids, ,j 119 
 
 Ventral segments of strmgs 91 
 
 inpijffis 102 &c. 
 
 Vibrations of plates 124 
 
 Vibration number 30,42 
 
 numbers of tones of scale 46 
 
 ratio, or vibration fraction 45, 50 
 
 ratios. Composition of 48, 49 
 
 of principal intervals 51 
 
 Violin, Quality of tone on 96 
 
 Vocal ligaments 114 
 
 Vowel souzids 76 
 
 "Water wave 9 
 
 "Wave, Associated 19 
 
 length, To ascertain 43 
 
 , Sound 17 
 
 "Waves, Composition of 82 
 
 on stretched tube or rope 13 
 
 "WerUieim's Velocities of sound in 
 
 solids 21 
 
 "Wheel Syren 29 
 
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THE MUSICAL HERALD: 
 
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