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 Sound and musician elementary treatise o 
 
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SOUND AND MUSIC. 
 
CamhriSge : 
 
 PBINTED BY J. AND C. il'. CLAY, 
 AT THE UNIVBESITY PRESS. 
 
SOUND AND MUSIC: 
 
 AN ELEMENTAEY TREATISE ON THE 
 PHYSICAL CONSTITUTION OF 
 
 i¥lu£5wal ^outttis antr l^armonp. 
 
 BY 
 
 SEDLEY TAYLOR, M.A. 
 
 FOBMEBLY PELLOW OF TEINITY COLLEGE, CAMBKIDGE. 
 
 THIED EDITION. 
 
 Hon&ott : 
 MACMILLAN AND CO. 
 
 AND NEW YOEK 
 1896 
 
 [All Rights reserved.'] 
 
First Edition, 1873; Second liiy, Third, 1896. 
 
PREFACE TO THE FIEST EDITION. 
 
 The following treatise, portions of which have 
 been delivered in lectures at the South Kensington 
 Museum, the Royal Academy of Music, and else- 
 where, aims at placing before persons unacquainted 
 with Mathematics an intelligible and succinct ac- 
 count of that part of the Theory of Sound which 
 constitutes the physical basis of the Art of Music. 
 No preliminary knowledge, save of Arithmetic and 
 of the musical notation in common use, is assumed 
 to be possessed by the reader. The importance of 
 combining theoretical and experimental modes of 
 treatment has been kept steadily in view through- 
 out. 
 
 The author has incorporated the chief Acoustical 
 discoveries of Professor Helmholtz, but adopted 
 his own course in explaining them and developing 
 their connection with the previously established 
 
 aS 
 
PREFACE. 
 
 portion of the subject. The present volume, there- 
 fore, even where its obhgations to the great German 
 philosopher are the deepest, is not a mere epitome 
 of his work \ but the result of independent study. 
 
 Tbinitt College, Cambbidge, 
 June, 1873. 
 
 Die Lehre von den Tonempfindungen. Dritte Ausgahe. 
 Braunschweig. 1870. Of this profound and exhaustive treatise 
 it is not too much to say that it does for Acoustics what the 
 Principia of Newton did for Astronomy. 
 
PREFACE TO THE SECOND EDITION. 
 
 The present edition is the result of a careful 
 revision applied to its predecessor. 
 
 An unsatisfactory artificial hypothesis as to the 
 motion of particles in the surface of water-waves, 
 which was avowedly introduced into the first edition 
 for the sake of simplicity, is replaced by a mode 
 of treatment based upon observed facts. 
 
 Points on which special difficulty was ex- 
 perienced by readers of the first edition are here 
 somewhat more fully elucidated. 
 
 The author has, of course, consulted the revised 
 edition ' of Helmholtz's Tonempjindungen. 
 
 Trinity College, Cambkidoe, 
 June, 1883. 
 
 ' Vierte umgearbeitete Ausgahe. Braunschweig, \2>T1. 
 
PREFACE TO THE THIRD EDITION. 
 
 The present edition is substantially identical 
 with its predecessor. 
 
 A reference to the chief acoustical discoveries 
 of the late Professor von Helmholtz, which appeared' 
 on the title-page of the previous editions, has been 
 omitted here. It is no longer called for, now that 
 the eminently fruitful contributions to acoustical 
 knowledge made by that great discoverer are de- 
 finitively incorporated in the mass of our scientific 
 heritage. 
 
 Tbinity College, Cambmdge, 
 December, 1895. 
 
CONTENTS. 
 
 CHAPTER L 
 
 ON SOUND IN GENEEAL AND THE MODE OF ITS TRANSMISSION. 
 
 Sensation of Sound, and its cause, § 1 — Connexion of Sound with 
 motion, § 2 — Velocity of Sound, § 3 — Stationary media of 
 Sound, § 4 — Motion of sea-waves, § 5 — Description of a wave, 
 § 6 — Length, amplitude and form of wave, § 7 — Water-waves, 
 § 8 — Waves due to transverse vibrations, § 9 — Periodic vibra- 
 tions, § 10 — Form of wave and mode of vibration, §§11, 12, 13 — 
 Wave on the surface of a field of standing corn, § 14 — Longi- 
 tudinal vibrations, §§ 15, 16 — Condensation and rarefaction, 
 § 17 — Associated wave, § 18 — Law of pendulum- vibration, 
 § 19 — Pressure and density of air; Mariotte's law, § 20 — Trans- 
 mission of sonorous waves along a tube of uniform bore, § 21 — 
 Unconstrained motion of Sound-waves, § 22 — Musical and non- 
 musical sounds, § 23. pp. 1 — 52 
 
 CHAPTER II. 
 
 ON LOUDNESS AND PITCH. 
 
 Three elements of a musical sound, loudness, pitch and quality, 
 § 24 — Loudness and extent of vibration, § 25 — Pitch and rapidity 
 of vibration; the Syren ; continuity of pitch, §§ 26, 27 — Measure 
 of pitch; vibration-numbers, § 28 — Limits of musical sounds, 
 § 29 — Relative pitch; intervals, § 30 — Tonic intervals of the 
 Major scale; concords and discords, § 31 — Additional notes 
 required for the Minor scale, g 32 — Measure of intervals, § 33 — 
 Vibration-fractions, § 34 — Table of vibration-fractions for the 
 tonic intervals of the Major and Minor scales, § 35 — Calculation 
 of the vibration-numbers of all the notes in a scale from the 
 vibration-number of its tonic, § 36. pp. 53 — 73 
 
X CONTENTS. 
 
 CHAPTER III. 
 
 ON RESONANCE. 
 
 Resonance of pianoforte wires and tuning-forks; cause of the 
 phenomenon, §§37, 38 — Resonance of a column of air ; laws of 
 its production, § 39 — Relation between the length of an air- 
 column and the pitch of its note of maximum resonance, § 40 — 
 Resonance-boxes, § 41 — Helmholtz's resonators, § 42. 
 
 pp. 74—86 
 
 CHAPTER IV. 
 
 ON QUALITY. 
 
 Composite nature of musical sounds in general ; series of con- 
 stituent tones, and law which connects them, § 43 — Experimental 
 analysis of musical sounds, §§ 44, 45 — Nomenclature of the sub- 
 ject, § 46 — Helmholtz's theory of musical quality as depending 
 on the number, orders and relative intensities of the partial-tones 
 present in any given clang, § 47. pp. 87 — 99 
 
 CHAPTER V. 
 
 ON THE ESSENTIAL MECHANISM OP THE PRINCIPAL MUSICAL INSTRU- 
 MENTS, CONSIDERED IN REFERENCE TO QUALITY. 
 
 Sounds of tuning-forks, § 48 — Modes of vibration of an elastic 
 tube, § 49 — Meeting of equal and opposite pulses; formation of 
 nodes, § 50 — Number of nodes formed, § 51 — Nature and rate of 
 segmental vibration, §§ 52, 53 — Motion of a sounding string; 
 quality and pitch of its note, §§ 54, 55— The pianoforte, § 56 — 
 Meeting of equal and opposite systems of longitudinal waves, 
 § 57 — Reflexion of Sound at a closed and at an open orifice, 
 § 58 — Modes of segmental vibration in stopped and open pipes, 
 §§ 59, 60 — Deepest note obtainable from a pipe, § 61 — Relation 
 between length of pipe and pitch of note, § 62 — Theory of re- 
 sonance-boxes, § 63 — Flue-pipes and reed-pipes, § 64 — Construc- 
 tion of flue-pipes and quality of their sounds, § 65 — Mechanism 
 of a reed ; quality of an independent reed, and of a reed associated 
 with a pipe, § 66— Orchestral wind-instruments, § 67 — Mechan- 
 ism of the human voice, § 68 — Synthetic confirmation of Helm- 
 holtz's theory of quality, § 69. pp. \QQ \^\ 
 
CONTENTS. 
 
 CHAPTER VI. 
 
 ON THE CONNEXION BETWEEN QUALITY AND MODE OF VIBRATION. 
 
 Composition of vibrations, §§ 70, 71 — Phase of a vibration ; non- 
 dependence of quality on differences of phase among partial-tones, 
 § 72 — Simple and resultant wave-forms; Fourier's theorem, § 73. 
 
 pp. 142—153 
 
 CHAPTER VII. 
 
 ON THE INTEEFERENCE OF SOUND AND ON 'BEATS.' 
 
 Composition of vibrations of equal periods, § 74 — Two sounds 
 producing silence, § 75 — Beats of simple tones, § 76 — Graphic 
 representation of beats, § 77 — Experimental study of beats, § 78. 
 
 pp. 164—166 
 
 CHAPTER VIII. 
 
 ON CONCORD AND DISCORD, 
 
 Helmholtz's discovery of the nature of dissonance ; conditions 
 under which it may arise between two simple tones, § 79 — Mode 
 of determining the whole dissonance produced by two composite 
 sounds, § 80 — Classification of the tonic intervals of the scale 
 according to their freedom from dissonance, §§ 81-86 — Picture 
 of amounts of dissonance for all intervals not wider than one 
 Octave, § 87 — Consonance dependent on quality, § 88 — Apparent 
 objection to Helmholtz's theory of quality, § 89 — Combination- 
 tones, § 90 — Their use in defining certain consonant intervals for 
 simple tones, § 91 — Divergence from the views of musical theorists, 
 § 92 — Dissonance due to combination-tones produced between the 
 partial-tones of clangs forming a given interval, § 93. 
 
 pp. 167—189 
 
xii CONTENTS. 
 
 CHAPTER IX. 
 
 ON CONSONANT TRIADS. 
 
 Rules for the employment of vibration-fractions, §§ 94-96 — Inver- 
 sion of intervals, § 97 — Definition of a consonant triad, § 98 — 
 Determination of all the consonant tonic triads within one Octave, 
 § 99 — Arrangement in two groups,§ 100 — Mutual relations between 
 the members of each group, § 101 — Notation of Thorough Bass, 
 § 102 — Fundamental and inverted positions of common chords, 
 § 103— Effects of Major and Minor chords, § 104. 
 
 pp. 190—201 
 CHAPTER X. 
 
 ON PURE INTONATION AND TEMPERAMENT. 
 
 Successive intervals of the Major scale, § 105 — Requisites for pure 
 intonation in keyed instruments, §§ 106-108 — Tempering and 
 temperament, § 109 — System of equal temperament, § 110 — Its 
 defects, § 111 — Its influence on vocal intonation, § 112 — Cum- 
 brousness and inefficiency of the established pitch-notation for 
 vocal music, §113 — The 'Tonic Sol-fa' pitch-notation, § 114 — Its 
 simple and eflfective character, § 115 — Relation of the physical 
 theory of consonance and dissonance to the sesthetics of Music, 
 § 116 — Importance of extreme discords; conclusion, § 117. 
 
 pp. 202—223 
 
 EEEATUM. 
 Page 16, line 6, for movement read moment. 
 
SOUND AND MUSIC. 
 
 CHAPTER I. 
 
 ON SOUND IN GENERAL AND THE MODE OF ITS TRANS- 
 MISSION. 
 
 1. In listening to a sound, all that we are im- 
 mediately conscious of is a peculiar sensation. This 
 sensation obviously depends on the action of our 
 organs of hearing ; for if we close our ears the sen- 
 sation is greatly weakened, or, if originally but feeble, 
 altogether extinguished. Persons whose auditory 
 apparatus is malformed, or has been destroyed by 
 disease, may be totally unconscious of any sound, 
 even during a thunderstorm or the discharge of 
 artillery. It would be entii-ely in accordance with 
 the mode of action of our other senses if what 
 we feel as Sound is represented, externally to our- 
 selves, by a state of things very different to the 
 sensation we experience. Analogy, then, indicates 
 
 T. 1 
 
 \\ 
 
2 SENSATIONS AND THEIR CAUSES. [I. § 1. 
 
 that some purely mechanical phenomena external to 
 the ear will prove to be turned into the sensation we 
 call Sound by a process carried on within that organ 
 and the brain with which it is in direct communica- 
 tion. This mechanical agency, whatever may be its 
 nature, is usually set going at a distance from the 
 ear, and, to reach it, must traverse the intervening 
 space. In doing so it can pass through solid and 
 liquid as well as gaseous bodies. For instance, if 
 one end of a felled tree is gently scratched with the 
 point of a penknife, the sound is distinctly audible 
 to a listener whose ear is pressed against the other 
 end of the tree. When a couple of pebbles are 
 knocked together under water, the sound of the 
 blow reaches the ear after first passing through the 
 intervening liquid. That Sound travels through the 
 air is a matter of universal experience, and needs no 
 illustration. In every case accessible to common 
 observation where Sound passes from one point of 
 space to another, it necessarily traverses matter, 
 either in a solid, liquid or gaseous form. We n)ay 
 hence conjecture that the presence of a material 
 medium of some kind is indispensable to the trans- 
 mission of Sound. This important point can be 
 readily brought to the test of experiment, as follows. 
 Let a bell kept ringing by clockwork be placed under 
 the receiver of an air-pump, and the air gradually 
 
I. § 2.] CONNMXIOF OF SOUND WITH MOTION. 3 
 
 exhausted. Provided that suitable precautions are 
 t^rken to prevent communication of Sound to the 
 external air through the body of the receiver, the 
 bell will appear to ring more and more feebly as the 
 exhaustion proceeds, until at last it altogether ceases 
 to be heard. While the air is being readmitted, the 
 sound of the bell will gradually recover its original 
 loudness. This experiment shows that Sound can- 
 not travel in vacuo, but requires for its transmission 
 a material medium of some kind. The air of the 
 atmosphere is, in the vast majority of cases, the 
 medium which conveys to the ear the mechanical 
 impulse which that wonderful organ translates, as it 
 were, into the language of Sound. 
 
 2. Having ascertained that a material medium 
 acts in every case as the carrier of Sound, we have 
 next to examine in what manner it performs this 
 function. The roughest observations suffice to put 
 us on the right track in this enquiry by pointing to 
 a connexion between Sound and Motion. The passage 
 through the air of sounds of very great intensity is 
 accompanied by effects which prove the atmosphere 
 to be in a state of violent commotion. The explosion 
 of a powder-magazine is capable of shattering the 
 windows of houses at several miles' distance. In 
 the case of sounds of only ordinary loudness the 
 accompanying air-motion manifests itself in no such 
 
 1—2 
 
4 VELOCITY OF SOUND. [I. § 3. 
 
 unmistakable way : a little attention will, however, 
 usually detect a certain amount of movement on the 
 part of the sound-producing apparatus, which is 
 probably capable of being communicated to the 
 surrounding air. Thus, a sounding pianoforte-string 
 can be both seen and felt to be in motion : the 
 movements of a finger-glass stroked on the rim by a 
 wet finger can be recognised by observing the thriUs 
 which play on the surface of the water it contains : 
 sand strewed on a horizontal drum-head is thrown off 
 when the drum is beaten. These considerations raise 
 a presumption that Sound is invariably associated 
 with agitation of the conveying medium — ^that it is 
 impossible to produce a sound without at the same 
 time setting the medium in motion. If this should 
 prove to be the case, there would be ground for the 
 further conjecture that motion of a material medium 
 constitutes the mechanical impulse which, falUng on 
 the ear, excites within it the sensation we call Sound. 
 Let us try to form an idea of the hind of motion 
 which the conditions of the case require. 
 
 3. We may conveniently begin by determining 
 the rate at which Sound travels. This varies, indeed, 
 with the nature of the conveying medium. It will 
 suffice, however, for our present purpose to ascertain 
 its velocity in air, the medium through which the 
 vast majority of sounds reach our ears. As long as 
 
I. § 3.] VELOCITY OF SOUND. 5 
 
 we confine our attention to sounds originating at 
 but small distances from us, their passage through 
 the intervening space appears instantaneous. If, 
 however, a gun is fired at a considerable distance, 
 the flash is seen before the report is heard — a proof 
 that an appreciable interval of time is occupied by 
 the transmission of the sound. The occurrence of 
 an echo, in a position where we can measure the dis- 
 tance passed over by the sound in travelling from 
 the position where it is produced to that where it 
 rebounds, gives us the means of measuring the velo- 
 city of Sound ; since we can, by direct observation, 
 ascertain how long a time is spent on the out-and- 
 home journey. The following easy experiment gives 
 a near approximation to the actual velocity of Sound 
 — in fact a much closer one than the rough nature of 
 the observation would have led one to expect. In 
 the North cloister of Trinity College, Cambridge, 
 there is an unusually distinct echo from the wall at 
 its eastern extremity. Standing near the opposite 
 end of the cloister, I clapped my hands rhythmically 
 at a rate such that the strokes and echoes were heard 
 alternately at equal intervals of time. A friend at 
 my side, watch in hand, counted the number of 
 strokes and echoes. The result was that there were 
 76 in half a minute, i.e. 38 strokes and 38 echoes. 
 A little consideration will show that the sound 
 
6 VELOCITY OF SOUND. [I. § 4. 
 
 traversed the cloister and returned to the point of 
 its origination regularly once in each interval of time 
 elapsing between the delivery of a stroke and the 
 perception of its echo. Since each such interval was 
 exactly equal to that between the perception of an 
 echo and the delivery of the following stroke, the 
 whole movement of Sound took place in alternate 
 equal intervals, i.e. in half the observed time, or 
 fifteen seconds. Accordingly the sound travelled to 
 and fro in the cloister 38 times in 15 seconds. The 
 length thus traversed, I found by pacing to be 
 about 419 feet. The velocity of Sound per second 
 
 thus comes out equal to — —z — , or 1061 ' feet and 
 
 15 
 
 a fraction. Sound, then, travels through the air at 
 
 the rate of upwards of 1,000 feet in a second, which 
 
 is more than 600 miles an hour, or about 15 times 
 
 the speed of an express train. In solid and liquid 
 
 bodies its velocity is still greater, attaining in the 
 
 case of steel-wire a speed of from 15,000 to 17,000 
 
 feet in a second ^ or, roughly speaking, about 200 
 
 times that of an express train. 
 
 4. Though the Sound-impulse advances with 
 
 ' This is about 50 feet below its exact value under the 
 circumstances of my observation. See Tyndall's Sound, Third 
 edition, p. 23. 
 
 = Tyndall's Sound, p. 38. 
 
I. §5.] MOTION OF SEA-WAVES. 7 
 
 a steady and high velocity, the medium by which 
 it is transmitted clearly does not share such a 
 motion. Solid conductors of Sound remain, on the 
 whole, at rest during its passage, and a slight 
 yielding of their separate parts is all that their 
 constitution generally admits of. In fluids, or in 
 the air, a continuous forward motion is equally out 
 of the question. The movement of the particles 
 composing the Sound-conveying medium will be 
 found to be of a kind examples of which are con- 
 stantly presenting themselves, but without attract- 
 ing an amount of attention at all commensurate 
 with their interest and importance. 
 
 5. An observer who looks down upon the sea 
 from a moderate elevation, on a day when the wind, 
 after blowing strongly, has suddenly dropped, sees 
 long lines of waves advancing towards the shore at 
 a uniform pace and at equal distances from each 
 other. The effect to the eye is that of a vast army 
 marching up in column, or of a ploughed field 
 moving along horizontally in a direction perpendi- 
 cular to the lines of its ridges and hollows. The 
 actual motion of the water is, however, very differ- 
 ent from its apparent motion, as may be ascertained 
 by noticing the behaviour of a cork or other body 
 floating on the surface of the sea, and therefore 
 sharing its movement. The floating body does not 
 
8 MOTION OF PARTICLES IN A SEA-WAVE. [I. § 5. 
 
 advance with the waves, but rides over their crests 
 and sinks into their troughs as though it were a 
 buoy at anchor. Hence, while the waves travel 
 steadily forward horizontally, each of the fluid 
 particles which compose them describes over and 
 over again a fixed orbit of its own. 
 
 Thus, when we say that the waves advance hori- 
 zontally, we mean, not that the masses of water of 
 which they at any given instant consist advance, but 
 that these masses, by virtue of the separate motions 
 of their individual particles, successively arrange 
 themselves in the same relative positions, so that the 
 cui-ved shapes of the surface, which we call waves, 
 are horizontally transmitted without their materials 
 sharing in the progress. The accompanying figure 
 will show how this happens. 
 
 Fig. 1. 
 A.-""^^ A'- '^^^\ -. ^ B B' 
 
 Let the full curved line AB represent a section 
 of a part of the sea-surface at any given instant 
 made by a vertical plane through the direction of 
 wave-motion, and suppose that during, say, the 
 next ensuing second of time, the separate fluid 
 particles rearrange themselves by virtue of theii' 
 
I. §5.] TRANSMISSION OF WAVES. 9 
 
 respective orbital motions in such a manner that, 
 at the end of that second, they constitute a curved 
 line identical in shape and size with AB, and only 
 differing from it in horizontal position. Let the 
 dotted line A'B' represent the curve thus formed. 
 As the two outlines AB and A'B' are exactly alike, 
 the joint effect produced by the separate particle- 
 movements on the eye of a spectator is just what it 
 would have been had we pushed the curve AB along 
 horizontally untU it came to occupy the position A'B'. 
 In order further to illustrate this point, let us 
 suppose that a hundred men are standing in a line 
 and that the first ten are ordered to kneel down : a 
 Spectator who is too far off to distinguish individuals 
 will merely see a broken line like that in the 
 figure below. 
 
 Now, suppose that after one second the eleventh 
 man is ordered to kneel and the first to stand j after 
 two seconds the twelfth man to kneel and the 
 second to stand; and so on. There will then con- 
 tinue to be a row of ten kneeling men, but during 
 each second it will be shifted one place along the 
 line. The distant observer will therefore see a 
 depression steadily advancing along the line. The 
 
10 TRANSFERENCE OF RELATIVE POSITION. [I. § 6. 
 
 state of things presented to his eye after twenty, 
 sixty and ninety seconds, respectively, is shown in 
 Fig. 2. 
 
 Fig. 2. 
 
 There is here no horizontal motion on the part 
 of the men composing the line, but their vertical 
 motions give rise, in the way explained, to the hori- 
 zontal transference of the depression along the line. 
 The reader should observe that for no two con- 
 secutive seconds does the kneeling row consist of 
 exactly the same men, while in such positions as 
 those shown in the figures, which are separated by 
 ten or more seconds of time, the men who form it 
 are totally difierent. 
 
 6. Let us now return to the sea-waves and 
 examine more closely the elements of which they 
 consist. 
 
 Fig. 3 represents a vertical section of one com- 
 plete wave. 
 
I. §6.] THREE ELEMENTS OF A WAVE. 11 
 
 The dotted line is that in which the horizontal 
 plane forming the surface of the sea when at rest 
 
 PVj.3 
 
 cuts the plane of the figure; it is called the level- 
 line. The distance between the two extreme points 
 of the wave, measured along this line, is called the 
 length of the wave. C is the highest point of the 
 crest DCB ; ^ the lowest point of the trough AED. 
 CF and EG are vertical straight lines through C 
 and E ; HCK and LEM are horizontal straight 
 lines through the same two points. The vertical 
 distance between the lines HK and LM is called the 
 amplitude of the wave. Thus AB is the length of 
 the wave, and, if we produce EG and CF to cut the 
 lines HK and LM m N and P respectively, we 
 have, for its amplitude, either of the equal lines EN, 
 PC. Each of these is clearly equal to FC and GE 
 together, that is to say, the amplitude of the wave 
 is equal to the height of the crest above the level- 
 line together with the depth of the trough below 
 it. In addition to the length and amplitude of the 
 wave, we have one more element, its form. The 
 wave in the figure has its crest shorter than its 
 
12 THREE ELEMENTS OF A WAVE. [I. § 7. 
 
 trough and higher than its trough is deep. More- 
 over the part DC of the crest is steeper than the 
 part CB, while in the trough the parts AE and ED 
 are nearly equally steep. Sea-waves have infinitely 
 various shapes dependent on the character of the 
 wind which originated them and on the depth and 
 configuration of the sea bottom over which they pass. 
 Hence, before we can lay down a wave in a figure, 
 we must know the nature of the wave's curve, i.e. 
 its form. 
 
 Since the crests of the waves are raised above 
 the ordinary level of the sea, the troughs must 
 necessarily be depressed below it, just as, in a 
 ploughed field, the earth heaped up to form the 
 ridges must leave empty the furrows from which it 
 was taken. Each crest being thus associated with a 
 trough, it is convenient to regard one crest and one 
 trough as forming together one complete wave. 
 Thus each wave consists of a part raised above, and 
 a part depressed below, the horizontal plane which 
 would be the surface of the sea were it not being 
 traversed by waves. 
 
 7. The length, amplitude and form of a wave 
 completely determine the wave, just as the length, 
 breadth and height of an oblong block of wood, i.e. 
 its three dimensions, fix the size of the block. These 
 three elements of a wave are mutually independent, 
 
I- § r.] 
 
 WATER.WAVES. 
 
 13 
 
 that is to say, we may alter any one of them with- 
 out altering the other two. This will be seen by 
 a glance at the accompanying figures. 
 
 (l) shows variation in length alone; (2) in am- 
 plitude alone ; (3) inform^ alone. 
 
 Fig. 4 (1) 
 
 (2) 
 
 ^ The reader may rightly object that between the three curves 
 in (2) there are what would, according to ordinary phraseology, 
 be described as dififerences oiform. That term, however, as here 
 used, means generic shape in the same way that the terms 
 'triangular,' 'rectangular,' 'elliptic,' &c. have meanings inde- 
 pendent of differences of mere dimension. 
 
14 WATHB-WAVUS. [I- § 8. 
 
 8. We will next examine more closely the 
 separate movements of particles at the surface of 
 water, and their joint result the transmission of 
 waves. The brothers Heinrich and Wilhelm Weber 
 published, in 1825, a series of very careful obser- 
 vations on water-waves passing along troughs with 
 glass sides not unlike very long and very narrow 
 aquariums. By ingeniously planned arrangements, 
 which it would take too long to describe here, 
 they ascertained that each particle on the surface 
 continued, during the passage of a succession of 
 equal waves, to describe a fixed oval path in a 
 vertical plane, the longer diameter of the oval 
 being horizontal. From the fact that the difference 
 between the horizontal and vertical diameters of 
 the oval diminished as the depth of the water in- 
 creased, the brothers Weber inferred that at the 
 surface of very deep water these diameters would 
 be equal, and the orbits of the particles there- 
 fore exact circles. 
 
 Fig. 5 shows how a series of particles, by de- 
 scribing fixed circular paths in a vertical plane, 
 cause progressive waves to be transmitted hori- 
 zontally. At the top of the figure are 17 spots 
 representing as many particles floating at equal 
 distances along a straight line in the undisturbed 
 surface of very deep water indicated by dots in the 
 
I. § 8.] 
 
 WATHE-WAVES. 
 Fig. 5. 
 
 15 
 
 
 0) lO-fl 
 
 ><* 
 
 (D 
 
 ©d) 
 
 ©(i)(D 
 
 (i)CDd> 
 Ocr>0 
 axDO 
 
 3 ij a a ^ S S^ "^ '»» 
 
16 WATEE-WAVMS. fl. § 8. 
 
 figure. The equal vertical circles are the paths 
 which the particles are respectively about to de- 
 scribe in the direction in which the hands of a 
 watch move, as indicated by a curved arrow in the 
 figure, and with equal uniform velocities. 
 
 (0) represents the state of things at a movement 
 when the group of particles plot out two complete 
 equal waves included by the brackets A and B, each 
 consisting of a crest to the right and trough to the 
 left as marked by the sub-brackets a, a' and b, V 
 respectively. The horizontal dotted line represents, 
 as before, the section of the undisturbed surface or 
 level-line. The reader wOl at once observe, on 
 looking along the line of circles firom left to right, 
 that each particle occupies in its own orbit a 
 position one-eighth of a revolution behind that of 
 the particle to the left of it. 
 
 (1) shows the arrangement of the particles 
 when an interval of time equal to one-eighth of the 
 period of a complete cu-cle-revolution has elapsed from 
 the moment at which they were grouped as in (0). 
 
 Similarly (2) represents the state of things after 
 two-eighths of that period, (3) after three-eighths of 
 it, and so on, until in (8), one complete period of 
 revolution having in all elapsed, the particles are 
 necessarily again in the same positions as those 
 shown in group (O). 
 
I. §9.] WAVE-MOTION. 17 
 
 If now the reader allows his eye to be guided by 
 the oblique dotted lines of the figure, he will readily 
 perceive that the wave A has moved horizontally to 
 the right at each of the above stages and, after the 
 lapse of one complete period of particle-revolution 
 occupies, in (8), the position which the wave B held 
 in (0), i.e. has traversed its own wave-length from 
 left to right. We may express this result generally 
 by saying that while an individual particle performs 
 one complete orbital revolution, the wave advances 
 one wave-length. This is the fundamental pro- 
 position of wave-motion, and should be carefully 
 mastered and remembered. In the figure employed to 
 demonstrate it a wave has been only roughly plotted 
 out by a small number of spots, and its movements 
 estimated at but a few arbitrarily chosen instants. 
 By increasing the numbers of these two elements, 
 however, we might make an indefinitely near ap- 
 proach to continuity both of form and of motion. 
 
 9. The characteristic phenomenon of wave- 
 motion, viz. an apparent forward movement unshared 
 by the materials which give rise to it, though most 
 frequently seen on the surface of water, is by no 
 means confined to fiuid bodies. When a carpet is 
 being shaken, bulging forms exactly like water-waves 
 are seen running along it. A flexible string, one 
 end of which is tied to a fixed point and the other 
 
 T. 2 
 
18 TRANSVERSE VIBRATIONS. [I. §,9. 
 
 held in the hand, exhibits the same phenomenon 
 when the loose end is sharply jerked aside. It has 
 accordingly been found convenient to extend the 
 term wave in order to designate as ' wave-motion ' 
 any movement coming under the definition just 
 given. We proceed to an instance in which th& 
 individual particles, instead of describing circular 
 orbits move to-and-fro, i.e. vibrate, in straight lines 
 perpendicular to the direction of wave-propagation. 
 
 Before applying to this case the method followed 
 in Fig. 5 it will be desirable to define what is meant 
 by a ' complete vibration ' as that term is used by 
 Enghsh writers. 
 
 Let a point move from A' to A along the straight 
 
 Tig.G 
 
 A 
 
 JL' 
 
 line A'A and then back again to A'. A French 
 writer would describe this movement as made up 
 of two complete vibrations executed in opposite 
 directions. An English writer would designate it 
 as one complete vibration, consisting of two half- 
 vibrations performed in opposite directions. This 
 latter usage will be adhered to in the sequel. The 
 
I. § 9.] TRANSVERSE VIBRATIONS. 19 
 
 completion of a vibration is thus to be determined 
 by the moving point having traversed its entire 
 path, in both directions, and this is equally true 
 if instead of starting from either extremity it occu- 
 pies initially an intermediate position such as O and 
 moves from to A, from A to A' and from A' to 
 ; or reversely from O to A', from A' to A and from 
 A to O. 
 
 The ' period ' of a vibration accordingly means 
 the time which it takes the moving point to perform 
 one such complete cycle of its motion. 
 
 Let the 17 spots lying equidistantly along the 
 dotted straight line at the head of Fig. 7 represent 
 as many particles of an elastic string stretched 
 between fixed points' of attachment not shown in 
 the figure. These particles are about to vibrate in 
 straight lines perpendicular to the direction of the 
 string. (0) shows the state of things when the 
 particles plot out two complete waves A and B. 
 The distinction between 'crest' and 'trough' has 
 now disappeared. All that can be said is that each 
 wave is formed of two protuberances lying on oppo- 
 site sides of the undisturbed position of the string, 
 which replaces the 'level-line' of the water-waves 
 and is, like it, represented by a dotted line in the 
 figure. 
 
 By looking along from left to right the reader 
 
 2—2 
 
20 
 
 TRANSVERSE VIBRATIONS. 
 Fig. 7 
 
 rr* 
 
 [I. § 9. 
 
 • : • 
 
 • ! • 
 
 • ,*' ,• 
 
 >•■ .*'■ .• 
 
 • i. yy: y 
 
 ^■\>^ 
 
 *' ,•' i •' 
 
 k 
 
 >< 
 
 i • \m J^ 0\ 
 
 i • i^" •' ; I 
 
 .if'' y; ,••'! > 
 
 
 
 •'; y ;,•' 
 
 m 
 
 « 
 
 H 
 
 y ly y • i • 
 
 V^ 
 
 
 ;_*■' ! y ; y _y • i 
 j y iy .^i' • i • 
 ly' y' •! • i • 
 
 Y • I • i • I " 
 
 • ' i y 
 
 • • 
 
 • • 
 
 "a" Is^ "«■ '»" ^S, >2, -2^ S ^ 
 
I. § 9.] TRANSVERSE VIBRATIONS. 21 
 
 will notice that each particle occupies in its own path 
 a position one-eighth of a period of vibration behind 
 that of the particle next to the left of it. (Compare 
 p. 16.) (1), (2), (3). ..show the positions of the par- 
 ticles after one-eighth, two-eighths, three-eighths... 
 of a complete period of particle- vibration. By follow- 
 ing any one of the vertical lines of spots it will be 
 seen that in the instance selected for this figure, 
 each particle moves more rapidly in the neighbour- 
 hood of its undisturbed position than it does hear the 
 extremities of its swing. In (8) the particles have 
 returned to their original positions and the wave A 
 is where B was in (0). The particles have completed 
 one vibrational cycle and the wave has advanced by 
 its own length. This result may be thus generalised : 
 While an individual particle performs one complete 
 vibration the wave advances one wave-length. The 
 proposition proved above (p. 17) for water-waves 
 is, therefore, also true of waves due to transverse 
 vibrations, i.e. such as are executed perpendicularly 
 to the direction of wave-propagation. 
 
 As waves thus produced are of leading importance 
 in the theory of Sound, it is necessary to study them 
 in some detail. 
 
 Let a particle originally at rest at in the initial 
 line (Fig. 7 his) be cooperating in the transmission 
 of a wave. This wave is drawn in the figure in two 
 
22 
 
 SUCCESSIVE VIBBATIONS. 
 
 [I. § 10. 
 
 positions such that the two points of its curve the 
 most distant from the initial Hne, A and A', are 
 
 situated in two straight lines OA and OA' drawn 
 through in opposite directions, each perpendicular 
 to the initial line. It is evident that, at the moments 
 when the wave is in these positions, the particle 
 originally at will be at A and A' respectively, and 
 that these two points mark the limits of its vibration. 
 Hence the line AA' is the extent of the particle's 
 vibration. But by drawing parallels to the initial 
 line through A and A' it wOl be seen, by reference 
 to the definition in § 6, that AA' is also the ampli- 
 tude of the wave. ' Extent of particle- vibration ' and 
 'amplitude of corresponding wave' are, therefore, 
 only different ways of expressing the same thing. 
 
 10. When a series of continuous equal waves 
 are being transmitted, each particle, after completing 
 one vibration, will instantly commence another pre- 
 cisely equal vibration, and go on doing so as long 
 as the transmission of waves is maintained. This 
 
I. § 10.] 
 
 SUCGESSIVE VIBRATIONS. 
 
 23 
 
 kind of motion, in which the same movement is 
 continuously repeated in successive equal intervals 
 of time, is called 'periodic,' and the time which 
 any one of the movements occupies is called its 
 ' period.' Thus, to continuous equal waves corre- 
 spond continuous periodic particle-vibrations. 
 
 We will next compare the periods of the vibrations 
 which produce waves of different lengths advancing 
 at the same speed. 
 
 In Fig. 8, waves of three different lengths are 
 represented. One wave of (l) is as long as two of 
 
 CO 
 
 c^) 
 
 (?) 
 
 (2), and as three of (3). Therefore, in virtue of the 
 fact proved in § 9, a particle makes one complete 
 
24 SUCCESSIVE VIBRATIONS. [I. § 10. 
 
 vibration in (1) while the long wave passes from A 
 to B, two in (2) while the shorter waves there pre- 
 sented pass over the same distance, and three in the 
 case of the shortest waves of (3). But the velocities 
 of these waves being by our supposition equal, the 
 times of describing the distance AB wUl be the same 
 in (l), (2) and (3). Hence a particle in (2) vibrates 
 twice as rapidly, and in (3) three times as rapidly, 
 as in (l); or conversely, vibration in (l) is half as 
 rapid as in (2), and one-third as rapid as in (3). 
 
 The rates of vibration in (l), (2) and (3) (by 
 which we mean the numbers of vibrations performed 
 in any given interval of time) are, therefore, propor- 
 tional to the numbers 1, 2 and 3, which are them- 
 selves inversely proportional to the wave-lengths in 
 the three cases respectively. We may express our 
 result thus ; The rate of particle-vibration is inversely 
 proportional to the corresponding wave-length. Simi- 
 lar reasoning will apply equally weU to any other 
 case ; the proposition, therefore, though deduced 
 from the relations of particular waves, holds for 
 waves in general. 
 
 The converse proposition admits of easy indepen- 
 dent proof as follows. It has been shown (p. 21) 
 that in one period of particle-vibration a wave 
 traverses its own length. This length must there- 
 fore, if the velocity of the wave remain constant, 
 
I. §11.] MODES OF VIBRATION. 25 
 
 be proportional to the period, i.e. inversely propor- 
 tional to the rate of vibration. 
 
 11. We have now connected the extent of the 
 particle-vibration with the amplitude, and its rate 
 with the length, of the corresponding wave. It 
 remains to examine what feature of the vibratory- 
 movement corresponds to the third element, the form 
 of the wave. 
 
 Fig. 9. 
 
 Suppose that two boys start together to run a 
 race from O to -4, from A to B, and from B back to 
 0, and that they reach the goal at the same moment. 
 They may obviously do this in many different wayi?. 
 For instance, they may keep abreast all through, or 
 one may fall behind over the first half of the course 
 and recover the lost ground in the second. Again, 
 one may be in front over OAO, and the other over 
 OBO, or each boy may pass, and be passed by, his 
 competitor repeatedly during the race. We may 
 regard the movement of each boy as constituting 
 one complete vibration, and thus convince ourselves 
 that a vibratory motion of given extent and period 
 may be performed in an indefinitely numerous variety 
 of modes. Let us now compare the positions of a 
 particle at the expiration of successive equal intervals 
 
26 
 
 VIBRATION-MODE AND WAVE-FORM. [I. § 11. 
 
 of time, when cooperating in the transmission of 
 waves of different forms. 
 
 In each of the three cases in Fig. 10 the front 
 of a wave is shown in the positions it respectively 
 occupies at the end of ten equal intervals of time 
 during which its intersection with the level-line 
 moves from O through the equidistant points 1, 2, 
 3, 4, &c. of the initial line. 
 
 A particle whose place of rest is will neces- 
 sarily assume corresponding positions in the ver- 
 tical line OA : thus the points where this line 
 cuts the successive wave-fronts show the positions 
 
 Fy.lO 
 
 01S3 i S 6 1 a 10 
 
 Oia34SeTBS»lO 
 
I. §12.] CONSTRUCTION OF WAVE-FORM. 27 
 
 of the vibrating particle at equal intervals of 
 time. 
 
 On comparing the three cases it wiU be seen that 
 the mode of the particle's vibration is distinct in 
 each. In (1), it moves fastest at 0, and then slackens 
 its pace up to A. In (2), it starts more slowly than 
 in (l), attains its greatest speed near the middle of 
 OA, and again slackens on approaching A. In (3), 
 the pace steadily increases from O to ^. The different 
 wave-fronts shown in the figure have been purposely 
 constructed with the same amplitude and length, in 
 order that only such variations as were due to differ- 
 ences of form might come into consideration. The 
 reader should construct similar figures with other 
 forms, and so convince himself more thoroughly that 
 to every distinct form of wave there corresponds a 
 special mode of particle-vibration. 
 
 12. Conversely each distinct mode of particle- 
 vibration gives rise to a special form of wave. We 
 will show this by actually constructing the form of 
 wave produced by a given mode of particle-vibration 
 when the mode in which a particle moves is given. 
 
 Suppose that each particle makes one complete 
 vibration per second about its position of original 
 rest in the initial line and that the law of vibration 
 is roughly indicated in Fig. 11. 
 
 AB is the path described by a particle ; its 
 
28 
 
 CONSTRUCTION OF WAVE-FORM. [I. § 12. 
 
 position when in the initial line ; 1, 2, 3, 4, 5, 6... 12, 
 13... 16 its positions after 16 equal intervals of time 
 
 B 
 
 e- 
 7- 
 
 2 
 
 -I 
 
 9- 
 
 10 — 
 
 Jl- 
 12— 
 
 •0,16 
 
 -15 
 
 14 
 -13 
 
 A 
 
 each one-sixteenth of a second : 1 6 coincides with O, 
 as the particle has returned to its starting-point. 
 
 Next, select a series of particles originally at rest 
 in equidistant positions along the initial line, and so 
 situated that each commences a vibration identical 
 with that above laid down in Fig. 11 one-sixteenth 
 of a second after the particle to the lefb of it has 
 started on an equal vibration. Fig. 12 shows the 
 rest-positions of the series of particles 
 
 (Xj, ttj, O,^, Ctg...(tjj, 
 
 in the initial line, and their contemporaneous po- 
 sitions 
 
 during a subsequent vibration. 
 
I. §12.] CONSTRUCTION OF WAVE-FORM. 29 
 
 The moment selected for the figure is that in 
 which the first of the series, a^, is on the point of 
 
 Fig. 12 
 
 .^> 
 
 ^ 
 
 &o i% j«a j^s i«5« H K h \ '"a % «u '"iz *«b "H* '"m \ 
 
 k 
 
 U 
 
 moving vertically upwards out of the initial line, 
 i.e. when the front of a wave of the form in question 
 has just reached the point a„. Since the second 
 particle started one-sixteenth of a second after the 
 first, its position in the figure will be below the 
 initial line at a/ making the line o^ al equal to the 
 line 015 in Fig. 11. The next particle, which is 
 two-sixteenths of a second behind «„ in its path, wUl 
 be at al making a^al equal to 014 in the same 
 figure. In this way the positions of all the points 
 a/a/aj', &c., in Fig. 12 are determined from Fig. 11. 
 They give us, at once, a general idea of the form of 
 the resulting wave. By laying down more points 
 along the line AB in Fig. 11, we can get as many 
 more points on the wave as we please, and should 
 
30 EXTENSION OF THE TERM ' WA VE.' [I. § 13 aud 14. 
 
 thus ultimately arrive at a continuous curved line. 
 This is the wave-form resulting from the given 
 vibration-mode with which we started, and, since 
 only one wave-form can be obtained from it, we 
 infer that each different mode of particle-vibration 
 gives rise to a different form of wave. 
 
 13. It has now been demonstrated that when a 
 wave is produced by particles vibrating in a plane 
 passing through its hne of advance, and in paths 
 perpendicular to that line, the amplitude of the wave 
 is equal to the extent of particle-vibration and the 
 length and form of the wave are determined by the 
 rate and moc?e of vibration respectively. These rela- 
 tions were also shown to hold conversely. 
 
 We will next consider a type of oscillatory 
 movement which is important from its similarity to 
 that to which the transmission of Sound is due. 
 
 14. Anyone who has looked down from a slight 
 elevation upon a field of standing corn on a gusty 
 day, must have frequently observed a kind of thrill 
 running along its surface. As each ear of corn is 
 capable of only a slight swaying movement, we have 
 here necessarily an instance of loave-motion, the ear- 
 vibrations corresponding to the particle-vibrations in 
 the cases already examined. There is, however, this 
 important peculiarity in the instance now before us, 
 that the ears' movements are mainly horizontal, 
 
I. § 15.] 
 
 LONGITUDINAL VIBRATIONS. 
 
 31 
 
 i.e. executed in the line of the wave's advance. The 
 wave, therefore, no longer presents itself to the eye 
 as exclusively a condition of alternate protuberance 
 of outline in opposite directions, but mainly as a state 
 of more tightly, or less tightly, packed ears. The 
 
 JFig 13. 
 
 annexed figure give& a rough idea how this takes 
 place. The wind is supposed to be moving from left 
 to right and to have just reached the ear A. Its 
 neighbours to the right are still undisturbed. The 
 stalk of C has just swung back into its erect posi- 
 tion. The ears about B are closer to, and those 
 about G further apart from, each other than is the 
 case with those on which the wind has not yet 
 acted. 
 
 After this illustration, it will be easy to conceive 
 a kind of wave-motion in which there is no longer 
 any movement transverse to the direction in which 
 the wave is advancing. 
 
 15, Let a series of particles, originally at rest 
 in equidistant positions along a straight line, as in 
 that at the head of Fig. 14, be executing equal 
 
32 
 
 LONGITUDINAL VIBRATIONS. 
 
 [I. § 15. 
 
 periodic vibrations in that line, in such a manner 
 that each is the same fixed amount further back in 
 
 Fig. 14 
 
 B 
 
 (0) 
 
 « 
 
 (a) 
 
 (3) 
 
 (4) 
 
 (--) 
 
 (<f) 
 
 (7) 
 
 (») 
 
 ft 
 
 • 
 
 • 
 
 • 
 
 • 
 
 e 
 
 • 
 
 • 
 
 • 
 
 
 ar 
 
 
 
 a \r V fi ^ 
 
 
 p 
 \ 
 
 • 
 • 
 • 
 • 
 • 
 • 
 • 
 
 • 
 \ 
 
 \ 
 
 • 
 
 • 
 • 
 
 • 
 • 
 
 
 \ 
 • 
 
 \ 
 \ 
 
 • 
 • 
 • 
 
 \ 
 
 \ 
 \ 
 
 • 
 • 
 • 
 
 \ 
 
 \ 
 
 • 
 
 • 
 
 • 
 
 \ 
 
 • 
 • 
 
 • 
 • 
 
 • 
 
 \ 
 
 
 • 
 
 • 
 
 • 
 \ 
 
 \ 
 
 • 
 
 • 
 • 
 
 V ' 
 
 \ 
 
 • 
 
 • 
 
 • 
 • 
 • 
 
 \ 
 
 • 
 
 \ 
 
 • 
 
 • 
 
 • 
 • 
 • 
 • 
 
 • 
 
 \ 
 
 • 
 • 
 • 
 • 
 
 e 
 \ 
 \ 
 \ 
 
 • 
 • 
 • 
 • 
 • 
 • 
 
 \ 
 • 
 
 
 • 
 
 • 
 • 
 • 
 • 
 
 • 
 
 • 
 
 
 Xr a' a . 
 
 
 its path than is its neighbour to the left, and 
 therefore exactly as much more forward than is its 
 neighbour to the right. 
 
I. §15.] LONGITUDINAL VIBRATIONS. 33 
 
 (0) shows the condition of the row of particles at 
 the moment when that on the extreme left is 
 beginning its swing from left to right, which, in 
 accordance with the direction of the wave's advance 
 as indicated by an arrow in the figure, we may 
 call its forward swing. The equidistant vertical 
 straight lines fix the extent of vibration for each 
 separate particle. The constant amount of relative 
 backwardness, or ' retardation ' as it is technically 
 called, between successive particles is, as in Figs. 5 
 and 7, one-eighth of the period of a complete vi- 
 bration. Thus, proceeding from left to right along 
 the line (O), we have the first particle beginning a 
 forward swing, the second, third, fourth and fifth 
 particles entering respectively on the fourth, third, 
 second and first quarters of a backward swing, and 
 the sixth, seventh, eighth and ninth particles on the 
 fourth, third, second, and first quarters of a forward 
 swing. 
 
 Since the ninth particle is just beginning a 
 forward swing, its situation is exactly the same as 
 that of the first. Beyond this point, therefore, we 
 have only repetition of the state of things between 
 the first and ninth particles. The row (O) is there- 
 fore made up of a series of groups, or cycles, of the 
 same number of particles arranged in the same 
 manner throughout. Two such cycles, included by 
 
 T. 3 
 
34 LONGITUDINAL VIBRATIONS. [I. § 15. 
 
 the large brackets A and B, are shown in (O). Each 
 cycle is divided by the sub-brackets a, a' and b, h' 
 into two parts. In a and h the distances between 
 successive particles are less than, and in a' and V 
 greater than, the corresponding distances when the 
 particles occupied their undisturbed positions. The 
 cycles correspond to complete waves on the surface 
 of water, the shortened and elongated portions of 
 each cycle answering to the crest and trough of 
 which each • water-wave consists. 
 
 (1) shows the state of the row of particles when 
 an interval of time equal to one-eighth of the period 
 of a complete particle-vibration has elapsed from the 
 moment shown in (0). The wave A has meantime 
 advanced by one-eighth of its own wave-length into 
 the position pointed to by the dotted lines. 
 
 The following rows (2), (3), (4), &c., show the 
 state of things when two-eighths, three-eighths, four- 
 eighths, &c., of a vibration-period has elapsed since 
 (0). In each successive row the wave A is further 
 advanced by one-eighth of its wave-length. 
 
 In (8), a whole vibration-period has elapsed since 
 (O). Accordingly each particle has performed one 
 complete vibration, and returned to the position 
 which it held in (0). The wave A, meanwhile, has 
 travelled constantly forward so as to be, in (8), 
 where B was in (O), i.e. to have advanced one 
 
I. §16.] LONGITUDINAL VIBRATIONS. 35 
 
 whole wave-length. Hence, while any particle per- 
 forms a complete vibration the wave advances 
 one wave-length. Accordingly, for waves due to 
 longitudinal vibrations, i.e. such as are executed in 
 the line of wave-advance, a proposition holds good 
 corresponding to that established, in § 9 p. 21, for 
 waves due to vibrations performed perpendicularly 
 to the direction of wave-motion. 
 
 16. In the waves shown in Fig. 14, the particles 
 in the bracket a are mutually equidistant, as are 
 also those in the bracket h. This is due to the fact 
 that, in the case there represented, the vibrating 
 particles move uniformly, i.e. with equal velocity, 
 throughout their paths. If we take other modes 
 of vibration, we shall find that this equidistance no 
 longer exists. Fig. 15 shows three distinct modes 
 of vibration with the wave resulting from each, on 
 the plan of Fig. 14. The extent of vibration and 
 length of wave are the same in the three cases. 
 
 In (I) the particles move quickest at the middle, 
 and slowest at the ends of their paths ; in (II) 
 fastest at the ends and slowest in the middle; in 
 (III) slowest at the left end, and fastest at the 
 right. 
 
 The shortest distance separating any two particles 
 contained in a is, in (I), that between 7 and 8 ; in 
 (II), that between 8 and 9 ; in (III), that between 
 
 3—2 
 
36 
 
 LONGITUDINAL VIBRATIONS. [I. § 16. 
 
 5 and 6. The corresponding greatest distances are, 
 in (I), between 2 and 3 ; in (II), between 1 and 2 ; 
 in (III), between 4 and 5. The remaining particles 
 likewise exhibit differences of relative distance in 
 the three cases. Thus, the positions of greatest 
 shortening and greatest lengthening occupy dif- 
 ferent situations in the wave, and the interme- 
 diate variations between them proceed according to 
 different laws, when the modes of particle- vibration 
 are different. The more particles we lay down in 
 their proper positions in a and a', the less abrupt 
 will be the changes of distance between neighbouring 
 
 Fin. 13. 
 
 m 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 a. 
 
 • 
 
 • 
 
 • 
 
 i 
 
 s 
 
 3 
 
 a' 
 
 * 
 
 S 
 
 « 
 
 7 
 
 a 
 
 S 
 
 a 
 
 
 -'' 
 
 
 ~"^- 
 
 ~^^^ 
 
 
 
 
 
 
 • 
 
 • 
 
 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 X 
 
 z 
 
 3 
 
 a' 
 
 4 
 
 5 
 
 6 
 
 7 
 
 a 
 
 • 
 
 8 
 
 ' 
 
 • 
 
 • 
 
 • 
 
 • 
 
 « 
 
 • 
 
 • 
 
 » 
 
 1 
 
 & 
 
 a 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 (m) 
 
 particles. By indefinitely increasing the number of 
 vibrating particles, we should ultimately arrive at a 
 state of things in which perfectly continuous changes 
 of shortening and lengthening intervened between 
 the positions of maximum shortening and maximum 
 lengthening in the same wave. 
 
I. §§17 & 18.] CONDENSATION AND RAREFACTION. 37 
 
 17. Let us now replace our row of indefinitely 
 numerous, but mutually unconnected, particles by 
 the slenderest filament of some material whose parts 
 (like those of an elastic string) admit of being 
 compressed or dilated at pleasure. When any 
 portion of the filament is shortened, a larger quan- 
 tity of material is forced into the space which was 
 before occupied by a smaller quantity. The matter 
 within this space is therefofe more tightly packed, 
 more dense, than it was, i.e. a process of condensa- 
 tion has occurred. On the other hand, when a 
 portion of the filament is lengthened, a smaller 
 quantity is made to occupy the space before occupied 
 by a larger quantity. Here the matter is more 
 loosely packed, more rare, than it was, i.e. a process 
 of rarefaction has taken place. 
 
 Let us now suppose the particles of the filament 
 to be thrown into successive vibrations in the man- 
 ner already so fully explained. Alternate states 
 of condensation and rarefaction will then travel 
 along the filament. It will be convenient to call 
 these states ' pulses ' — of condensation or rarefaction 
 as the case may be. A pulse of condensation and 
 a pulse of rarefaction together make up a complete 
 wave. 
 
 18. The degree of condensation, or rarefaction, 
 existing at any given point of a wave has been 
 
38 ASSOCIATED WAVE. [I. § 18. 
 
 shown to depend on the mode in which the particles 
 of the filament vibrate. It is therefore desirable 
 to have some simple method, appealing directly to 
 the eye, of exhibiting the law of any assigned mode 
 of vibration. We may an-ive at such a method by 
 the following considerations. 
 
 When a line of particles vibrate longitudinally, 
 they give rise to alternate pulses of condensation 
 and rarefaction ; when ' transversely,' they produce 
 alternate protuberances on opposite sides of the line 
 of particles in their positions of rest. Nevertheless, 
 if the vibrations in the two cases are identical in 
 all respects save direction alone, the distance 
 which, at any moment, separates an assigned par- 
 ticle from its position of rest will be the same 
 whether the vibrations are longitudinal or transverse. 
 It is therefore only necessary to construct the wave 
 corresponding to any system of transverse vibrations, 
 in the way shown in § 12, in order to obtain the 
 means of fixing the position of an assigned particle 
 at any given moment for the same system of vibra- 
 tion executed longitudinally. 
 
 Let AB and CD, Fig. 16, be lines of particles 
 executing vibrations transverse to AB and along CD 
 respectively. Let a and h be corresponding particles 
 in their positions of rest. Draw the transverse wave 
 for any given instant of time : the particle originally 
 
I. §19.] ASSOCIATED WAVE. 39 
 
 at a will now be at a', and that originally at h will 
 be at h', if hh' be made equal to aa'. 
 
 Fig.lff. 
 
 By performing the same process for different 
 instants, we can find as many corresponding posi- 
 tions of the longitudinally-vibrating particle as we 
 please. It is true that we learn nothing new by this, 
 since we cannot construct the wave-curve without 
 knowing beforehand the mode of the particle's vibra- 
 tion [§ 12]. Still, when we are dealing with 
 longitudinal particle-vibrations, and require to know 
 the law of the variation of condensation and rarefac- 
 tion at different points of a single wave, it is con- 
 venient to have a picture of the mode of vibration 
 by which, as we know [§ 16], that law is determined. 
 Such a picture we have in the form of the wave 
 produced by the same mode of vibration when 
 executed transversely. Let us call the wave so 
 related to a given wave of condensation and rare- 
 faction its associated wave. 
 
 19. Before leaving this portion of the subject, it 
 will be advisable to draw the associated wave for 
 
40 
 
 PENDULUM-VIBRATION. 
 
 [I. § 19. 
 
 that particular mode of longitudinal vibration in 
 which each particle moves as if it were the extremity 
 of a pendulum traversing a path which is very short 
 compared to the pendulum's length. The meaning 
 of this limitation will be easily seen from Fig. 1 7. 
 
 Let be the fixed point of suspension ; OA the 
 pendulum in its vertical position ; AB a portion of 
 a circle with centre and radius OA ; a, h, c, d, 
 points on this circle ; AD a horizontal straight line 
 through A ; aa', hh', cc', ddf verticals though a, b, 
 c, d, respectively. If the pendulum is placed in the 
 position Oa, and left to itself, it will reach an equally 
 inclined position on the other side of OA, i.e. will 
 swing through twice the angle aOA before it turns 
 back again. Similarly, if started at Ob it wUl swing 
 through twice the angle bOA ; if at Oc, through 
 
I. § 19.] PENDULUM-VIBRATION. 41 
 
 twice the angle cOA ; and so on. Now the extre- 
 mity of the pendulum, when at a, is further from the 
 horizontal line, AD, than when it is at h, since aa' 
 is greater than hh' ; and when at h further than 
 when at c. If we make the pendulum vibrate through 
 only a small angle, by starting it, say, in the position 
 Od, its extremity will throughout its motion be very 
 near to the horizontal straight line AD. If we 
 make the angle small enough, or, which is the same 
 thing, take Ad sufficiently small compared with OA, 
 we may without any perceptible error suppose the 
 end of the pendulum to move in a horizontal straight 
 line instead of in a circular arc, i.e. along d'A 
 instead oidA. To take an actual case, let us suppose 
 the pendulum to be 10 ft. long, and its extent of 
 swing 1 inch on either side of its vertical position. 
 An easy geometrical calculation shows that the 
 end of the pendulum will never be as much as 
 
 ^— r th of an inch out of the horizontal straight line 
 
 drawn through it in its lowest position. This is a 
 vanishing quantity compared to the length of the 
 pendulum; we may, therefore, safely regard the 
 vibration as performed along d'A instead of dA. 
 Such a vibration, though executed in a straight 
 line instead of in the arc of a circle, may be pro- 
 perly called a pendulum-vibration, as expressing the 
 
42 LAW OF PENDULUM-VIBRATION. [I. § 20. 
 
 law according to which it takes place. This law 
 admits of simple geometrical illustration as follows. 
 Let a ball, or other small object, be attached to some 
 part of a wheel revolving uniformly about a fixed 
 horizontal axis, so that the ball goes round and round 
 in the same vertical circle with constant velocity. 
 If the sun is in the zenith, i. e. in such a position that 
 the shadows of all objects are thrown vertically, the 
 shadow of the hall on any horizontal plane below it 
 will move exactly as does the bob of a pendulum. 
 
 The form of the associated wave for longitudinal 
 pendulum-vibrations is shown in Fig. 17 bis. 
 
 Fig. 17 bis. 
 
 Retaining the form of the curve, we may make 
 its amplitude and wave-length as large or as small 
 as we please, as in the case of the waves in Fig. 4, 
 (1) and (2), p. 13. 
 
 20. We have examined the transmission of 
 waves due to longitudinal vibrations along a single 
 very slender filament. Suppose that a great number 
 of such filaments are placed side by side in contact 
 
I. §20.] ATMOSPHERIC PRESSURE. 43 
 
 with each other, so as to form a uniform material 
 column. If, now, precisely equal waves are simul- 
 taneously transmitted along all the constituent 
 filaments, successive pulses of condensation and rare- 
 faction will pass along the column. The parts in 
 any assigned transverse section of the column will, 
 obviously, at any given moment of time, all have 
 exactly the same degree of compression or dilatation. 
 When a pulse of condensation is traversing the 
 section, its parts will be more dense, and when a 
 pulse of rarefaction is traversing it, less dense, than 
 they would be were the column transmitting no 
 waves at all, and its separate particles, therefore, 
 absolutely at rest. Let the column with which we 
 have been dealing be the portion of atmospheric air 
 enclosed within a tube of uniform bore. The pheno- 
 mena just described will then be exactly those which 
 accompany the passage of a sound from one end of 
 the tube to the other. It remains to examine the 
 mechanical cause to which these phenomena are 
 due. 
 
 Atmospheric air in its ordinary condition exerts 
 a certain pressure on all objects in contact with it. 
 This pressure is adequate to support a vertical 
 column of mercury about 30 inches high, as we 
 know by the common barometer. Fig. 18 represents 
 a section of a tube closed at one end, with a 
 
44 ATMOSPHERIC PRESSURE. [I. § 20. 
 
 movable piston fitting air-tight into the other. In 
 (0) the air on both sides of the piston is in the 
 ordinary atmospheric condition, so that the pressure 
 on the right face of the piston is counteracted by an 
 exactly equal and opposite pressure on its left face. 
 
 In (1) the piston has been moved inwards, so as 
 to compress the air on the right of it. That on its 
 left, being in free communication with the external 
 air, is not permanently affected by the motion of the 
 piston. In order to retain the piston in its forward 
 position, it is necessary to exert a force upon it in 
 
 Fig. IS. 
 
 (0) 
 
 (1) 
 (2). 
 
 the direction of the arrow. If this force be relaxed, 
 the piston will be driven back. Since the pressure 
 of the air on the left of the piston is just what it 
 was before, that on its right must necessarily have 
 increased. But this increase of pressure is accom- 
 panied by an increase of density due to the com- 
 pression of the air on the right of the piston. Hence 
 
I. §21.] MARIOTTE'S LAW. 45 
 
 increase of pressure accompanies increase of density. 
 If, as in (2), we reverse the process by moving the 
 piston outwards, the extraneous force must be ex- 
 erted in the opposite direction shown by the arrow. 
 The pressure on the right of the piston is therefore 
 less than the normal atmospheric pressure on its left', 
 i. e. diminution of pressure accompanies diminution 
 of density. By experiments such as the above it 
 was shown, by the French philosopher Mariotte, 
 that the pressure of air varies as its density. 
 
 21. Next, let us take a cylindrical tube open at 
 one end and having a movable piston fitting into the 
 other, as in Fig. 19. 
 
 Fig. 19. 
 
 (O) - 
 
 c 
 
 (.) — I 
 
 I 
 
 In (0) the piston is at rest at A, and the air in 
 its ordinary atmospheric condition of density and 
 pressure. In (1) the piston is pushed inwards as far 
 as C. While it is moving up to this position, the 
 
46 TRANSMISSION OF PULSES. [I. § 21. 
 
 air-particles in front of it are thrown into motion. 
 Suppose that, at the moment when the piston reaches 
 C, the particles at D are just beginning to be dis- 
 turbed. The air which, in (0), occupied AB is now 
 crowded into CD, and is therefore denser than that 
 further on in the tube. Next, let the piston be 
 drawn back to E, (2), as much to the left of its 
 original position. A, (0), as in (l) it was to the 
 right of it. The air in CD, (1), will, while this is 
 taking place, expand into EF; for, being denser, it 
 will also be at a greater pressure than the air to the 
 right of it. It will, therefore, act on this air in the 
 same way as the piston did on the air in contact 
 
 Fig. 19 (repeated). 
 
 (2). 
 (3) 
 
 (*)- — E 
 
 P Q B 
 
 with it when moving from A, (O), to C, (1). Hence 
 the air in FG will be condensed, G being the point 
 where the air particles are just beginning to be dis- 
 turbed at the moment whon the piston reaches the 
 
I. § 21.] TRANSMISSION OF PULSES. 47 
 
 position E. Thus the air at D advances to F. 
 Further, in consequence of the backward motion of 
 the piston, the air in the neighbourhood of C, (1), 
 moves up to E, (2). Thus the air originally in 
 AB now occupies EF, which is greater than AB. 
 It is therefore less dense than in (0), i.e. is in a 
 state of rarefaction. Now, let the piston again ad- 
 vance to H, (3). The air in FG, being at a greater 
 pressure than that in its front and rear, will expand 
 in both directions, causing a new condensation, LM, 
 to be formed further on, and itself becoming the 
 rarefaction KL, co-operating at the same time 
 with the advancing piston to produce in its own 
 rear the condensation HK. In (4) the piston is 
 again where it was in (2). HK has expanded 
 into the rarefaction NO, KL contracted into the 
 condensation OP, LM expanded into the rarefac- 
 tion PQ, and a new condensation, QR, been formed 
 in front. 
 
 The figure makes it clear that each forward stroke 
 of the piston produces a pulse of condensation, and 
 each .backward stroke a pulse of rarefaction ; but 
 that, when once formed, these pulses travel onwards 
 independently of any external force. They do so, as 
 we have seen, in virtue of the relation which connects 
 the pressure of the air with its density, that is to 
 say, on the elasticity of the air. 
 
48 FREELY EXPANDING SOUND-WAVE. [I. § 22. 
 
 If we suppose our movable piston withdrawn 
 from the tube, and a vibrating tvming-fork held with 
 the extremity of one prong close to the orifice of the 
 tube, the conditions of the problem wiU not be 
 essentially modified. Each outward swing of the 
 prong will give rise to a condensed, and each inward 
 swing to a rarefied pidse, and thus, during every 
 complete vibration of the fork one sonorous wave, 
 consisting of a pulse of condensation and a pulse of 
 rarefaction, will be started on its journey along the 
 tube. 
 
 22. We have examined the transmission of 
 Sound along a colunm of air contained in a tube of 
 uniform bore. A more important case is that in 
 which a sound originated at an assigned point 
 spreads out from it freely in all directions. Here we 
 must conceive a series of spherical shells, alternately 
 of condensed and of rarefied air, one inside the other, 
 and having the point of origination of the sound as 
 their common centre. All the shells must be supposed 
 to expand uniformly like an elastic globular baUoon 
 constantly inflated with more and more gas. Of 
 course no shell in any two consecutive positions con- 
 sists of exactly the same materials. The condensations 
 and rarefactions spread progressively, the air-particles 
 merely vibrating about their positions of original rest. 
 The great difference between this case and that last 
 
I. § 23,] MUSICAL AND NON-MUSICAL SOUNDS. 49 
 
 considered lies in the fact, that, as the spherical shells 
 of condensation and rarefaction spread, it is necessary, 
 in order to keep up the wave-motion, to throw larger 
 and larger surfaces of air into vibration ; whereas 
 within the tube the transverse section remained the 
 same throughout. Hence, as the same amount of 
 original disturbing force has to set a constantly in- 
 creasing number of air-particles into motion, it can 
 only do so by correspondingly shortening the distances 
 through which the individual particles move, i.e. by 
 diminishing their extent of vibration. Accordingly, 
 when Sound-waves spread out freely in all directions, 
 the furtlier any given air-particle is from the point 
 at which the sound originated, the smaller will be 
 the extent of the vibration into which it will be 
 thrown when the waves reach it. 
 
 23. Sounds are either musical or non-musical. 
 The vast majority of those ordinarily heard — the 
 roaring of the wind, the din of traflSc in a crowded 
 thoroughfare — belong to the second class. Musical 
 sounds are, for the most part, to be heard only from 
 instruments constructed to produce them. The dif- 
 ference between the sensations caused in our ears by 
 these two classes of sounds is extremely well marked, 
 and its nature admits of easy analysis. Let a note 
 be struck and held down on the harmonium, or on 
 any instrument capable of producing a sustained 
 
 T, 4i 
 
50 STEADY AND UNSTEADY SOUNDS. [I. § 23. 
 
 tone. However attentively we may listen, we per- 
 ceive no change or variation in the sound heard. 
 A perfectly continuous and uniform sensation is 
 experienced as long as the note is held down. If 
 instead of the harmonium we employ the pianoforte, 
 where the sound is loudest directly after the moment 
 of percussion, and then gradually dies away, diminu- 
 tion of loudness is the only change which occurs. 
 
 In the case of non-musical sounds variations of a 
 different kind can be easily detected. In the howl- 
 ing of the wind the sound rises to a considerable 
 degree of shrillness, then falls, then rises again, and 
 so on. On parts of the coast where a shingly beach 
 of considerable extent slopes down to the sea, a sound 
 is heard in stormy weather which varies from the 
 deep thundering roar of the great breakers, to the 
 shrill tearing scream of the shingle dragged along by 
 the retreating surf. Similar variations may be no- 
 ticed in sounds of small intensity such as the rustling 
 of leaves, the chirping of insects, and the like. The 
 difference, then, between musical and non-musical 
 sounds seems to lie in this, that the former are con- 
 stant, while the latter are continually varying. The 
 human voice can produce sounds of both classes. In 
 singing a sustained note it remains quite steady, 
 neither rising nor falling. Its conversational tone, 
 on the other hand, is perpetually varying in height 
 
I. §23.] STEADINESS OF MUtSIG A L SOUNDS. 51 
 
 even within a single syllable ; if it ceases so to vary, 
 it assumes a musical character, witness the epithet 
 ' sing-song ' then commonly applied to it. 
 
 We may then define a musical sound as a steady 
 sound, a non-musical sound as an unsteady sound. 
 It is true we may often be puzzled to say whether 
 a particular sound is musical or not : this arises, 
 however, from no defect in our definition, but from 
 the fact that such sounds consist of two elements, 
 musical and non-musical, of which the latter may 
 be the more powerful, and therefore absorb our at- 
 tention until it is specially directed to the former. 
 For instance, a beginner on the violin often produces 
 a sound in which the irregular scratching of the bow 
 predominates over the regular tone of the string. 
 In bad flute-playing an unsteady hissing sound 
 accompanies the naturally sweet tone of the instru- 
 ment, and may easily surpass it in intensity. In the 
 tones of the more imperfect musical instruments, 
 such as drums and cymbals, the non-musical element 
 is very prominent, while in such sounds as the 
 hammering of metals, or the roar of a water-fall, we 
 may be able to recognise only a trace of the musical 
 element, all but extinguished by its boisterous com- 
 panion. 
 
 We have seen that Sound reaches our ears by 
 means of vibrations executed by the particles of the 
 
 4—2 
 
52 STEADINESS OF MUSICAL SOUNDS. [I. § 23. 
 
 atmosphere. It has also been shown that steadiness 
 is the characteristic feature of musical, as distin- 
 guished from non-musical, sounds. We may infer 
 hence that the motion of the air corresponding to a 
 single musical sound will be itself steady, i.e. that 
 equal numbers of equal vibrations tvill he executed in 
 precisely equal times. This conception of the physical 
 conditions under which musical sounds are produced 
 will suffice for the present. We proceed to consider 
 in detail the various ways in which such sounds 
 may differ from each other, and to investigate the 
 mechanical cause to which each such difference is to 
 be referred. In what follows, by the word ' sound ' 
 will always be meant ' musical sound,' unless the 
 contrary be expressly stated. 
 
CHAPTER II. 
 
 ON LOUDNESS AND PITCH. 
 
 24. A musical sound may vary in three different 
 respects. Let a note be played, first by a single 
 violin, then, by two, by three, and so on, until we 
 have all the violins of an orchestra in unison upon 
 it. This is a variation of loudness only. Next let 
 a succession of notes be played on any instrument 
 of uniform power, such as the harmonium without 
 the expression-stop, or on the principal manual of 
 an organ, only one combination of stops being in 
 either case used. Here we have a variation of pitch 
 only. Lastly, let one and the same note be suc- 
 cessively struck on a number of pianofortes of the 
 same size, but by different makers. The sounds 
 heard will all have exactly the same pitch, and about 
 the same degree of loudness ; nevertheless they will 
 exhibit decided differences of character. The tone 
 of one instrument will be rich and full, of another 
 ringing and metallic, that of a third will be described 
 as ' wiry,' of a fourth as ' tinkling,' and so on. 
 
 Sounds thus related to each other are said to 
 vary in quality only. The instances just considered 
 
54 ELEMENTS OF A MUSICAL SOUND. [II. § 25. 
 
 have the advantage of simplicity, since they allow 
 of changes in loudness, pitch and quality being 
 exhibited separately. They are^ however, less strik- 
 ing than other cases where sounds vary in two, or 
 in all three, of these respects at the same time. A 
 practised ear may be requisite to detect the difference 
 between the tones of two pianofortes, but no one 
 is in danger of mistaking, for instance, a flute for a 
 trumpet. There is here, no doubt, considerable differ- 
 ence of loudness as well as of quality, but let the 
 more powerful instrument be placed at such a dis- 
 tance that it sounds no louder than the weaker one, 
 and the distinction between the two kinds of tone 
 will be still quite decisive. 
 
 Two assigned musical sounds may differ from 
 each other in loudness or pitch or quality, and 
 agree in the other two — or they may differ in any 
 two of these, and agree in the third — or they may 
 differ in all three. There is, however, no other i-espect 
 in which they can differ, and accordingly we know 
 all about a musical sound as soon as we know its 
 loudness, its pitch and its quality. These three 
 elements determine the sound, just as the lengths of 
 the three sides of a triangle determine the triangle. 
 
 25. The loudness of a musical sound depends 
 entirely, as we shall easily see, on the extent of 
 vibratory movement performed by individual par- 
 
II. § 25.] LOUDNESS AND EXTENT OF VIBRATION. 55 
 
 tides composing the conveying medium. A sound- 
 producing instrument can be readily observed to be 
 in a state of rapid tremor. The vibrations of a 
 tuning-fork are recognisable by the eye in the fuzzy, 
 half-transparent, rim which surrounds its prongs after 
 it has been struck ; and by the touch, if we place a 
 finger gently against one of the prongs. The harder 
 we hit the fork the louder is its sound, and the 
 larger, estimated by both the above modes of ob- 
 servation, are its vibrations. The experiment may 
 be tried equally well on any pianoforte whose con- 
 struction allows the wires to be uncovered. It is 
 natural to infer that a vibration on the part of a 
 sound-producing instrument communicates to the 
 particles of the air in contact with it a corre- 
 sponding movement. Thus a sound of given loud- 
 ness is conveyed by vibrations of given extent, and, 
 if the sound increases or diminishes in intensity, 
 the extent of the vibrations which carry it must 
 increase or diminish correspondingly. 
 
 We conclude, then, that the loudness of a musical 
 sound depends solely on the extent of excursion of 
 the particles which constitute the conveying medium 
 in the neighbourhood of our ears. This limitation 
 is clearly essential, since a sound grows more and 
 more feeble the greater our distance from the point 
 where it is produced. This diminution of intensity 
 
56 PITCH AND RAPIDITY OF VIBRATION. [II. § 26. 
 
 as the distance from the origin of sound increases 
 is a direct consequence of the connexion between 
 loudness and extent of vibration. We have seen 
 [§ 22] that the further an air-particle is from the 
 point where a sound is produced, the smaller will be 
 the extent of the vibration into which it is thrown by 
 the sonorous wave. Hence, as the sound advances it 
 will necessarily become feebler, provided always that 
 the waves are permitted to spread out in all direc- 
 tions. If they are confined, say, in a tube, the 
 intensity of the sound will for considerable distances 
 remain practically constant. We have here the 
 theory of the message-pipes which are used to enable 
 a conversation to be carried on between different 
 parts of a building. A whisper, inaudible to a 
 person close to the speaker, may by their means be 
 perfectly well heard by a listener at the other end 
 of the tube. 
 
 26. We have next to enquire to what mechani- 
 cal causes differences in the pitch of musical sounds 
 are to be referred. Rough observation at once in- 
 dicates the direction in which we must look. If we 
 draw the point of a pencil along a rough surface, 
 first slowly and then more quickly, the sound heard 
 wiU become shriller when the movement of the 
 pencil becomes more rapid. As its point passes over 
 the minute elevations and depressions which consti- 
 
II. §26.] THE SYREN. 57 
 
 tute the roughness of the surface, a series of irregular 
 vibrations are set up in the materials of the surface 
 and of the pencil, and by them communicated 
 to the air. The more rapid these vibrations, the 
 shriller is the sound produced. The instrument 
 described below, which is called a ' Syren,' gives 
 us the means of following up with accuracy the 
 hint just obtained. 
 
 Fig,20 
 
 AB is a thin circular disc of metal or cardboard 
 which, by means of a multiplying wheel, can be set 
 in rapid uniform rotation in its own plane about a 
 fixed axis through its centre, C A series of holes 
 (eight in the figure) are punched in the disc at equal 
 distances along a circle having its centre at C A 
 small tube, ah, is held with one end close to one of 
 the holes. If, while the disc is rotating, we blow 
 steadily and continviously into the tube at a, a 
 certain quantity of air will pass through the disc 
 whenever a hole passes across the orifice h of the 
 
58 THE SYREN. [II. § 27. 
 
 tube ah. During the intervals of time which 
 elapse between the passage of adjacent holes across 
 h, no air can pass through the disc. Hence, if the 
 disc be rotating uniformly, a series of such discharges 
 will succeed each other at precisely equal intervals 
 of time. The air on the other side of the disc will 
 necessarily be agitated by the process. Every time 
 that air is driven through one of the holes, an in- 
 crease of pressure occurs close to it, and accordingly 
 a pulse of condensation is formed there. The elastic 
 force of the air will give rise to a pulse of rarefaction 
 during each interval between successive discharges. 
 Hence the Syren supplies us with a regular series of 
 the alternate condensations and rarefactions which, 
 as we have seen, constitute waves of Sound. 
 
 27. While air is being blown steadily into the 
 tube, let the disc be made to rotate slowly, and then 
 with gradually increasing rapidity. At first nothing 
 will be audible but a series of faint intermittent 
 throbs, due to the impact of the air driven through 
 the tube against the successive portions of the disc 
 which separate its holes. This sound may be exactly 
 reproduced by moving the forefinger to and fro 
 rapidly before the lips, while blowing through them. 
 It contributes nothing to the proper musical sound 
 of the instrument, and is only audible in its imme- 
 diate neighbourhood. Presently a deep musical 
 
II. § 27.] CONTINUITY OF PITCH. 59 
 
 sound begins to be heard, which, as the velocity of 
 rotation increases, constantly rises in pitch. The 
 acuteness of the sound thus obtainable depends 
 solely on the speed to which we can urge the instru- 
 ment, and is therefore limited only by the driving- 
 power at our command. The rise of pitch in this 
 experiment is perfectly continuous, that is to say, 
 the sound of the Syren, in passing from a graver 
 to a more acute note, goes through every possible 
 intermediate degree of pitch. It is important that 
 we should familiarise ourselves with this conception 
 of the continuity of the scale of pitch, because in 
 the instrument from which our ideas on this subject 
 are usually obtained — the pianoforte — the pitch 
 alters discontinuously, i.e. by a series of jumps of 
 half a Tone each, and we are thus tempted to ignore 
 the intervening degrees of pitch. The more perfect 
 musical instruments, such as the human voice or the 
 violin, are as capable as the Syren of passing through 
 all degrees of pitch from one note to another in the 
 way called 'portamento' or 'slurring.' 
 
 It is clear from the nature of the Syren's con- 
 struction that the only change which can take place 
 during the rise in pitch is the increased number of 
 impulses communicated to, and therefore of vibrations 
 set up in, the external air, during any given interval 
 of time. If, when a note of given pitch has been 
 
60 MEASURE OF PITCH. [II. § 28. 
 
 attained by the Syren, we check any further increase 
 of velocity and cause the disc to rotate uniformly 
 at the rate which it has just reached, no further 
 alteration of pitch will occur, and the note will be 
 steadily held by the instrument so long as the uni- 
 form rotation of its disc is kept up. Hence the num- 
 ber of aerial vibrations executed in a given time de- 
 termines the pitch of the sound heard. 
 
 28. The Syren, besides teaching us this most 
 important fact, gives us the means of determining 
 the number of vibrations corresponding to any given 
 note. If we know the number of rotations which 
 the disc has performed in a given time, we have only 
 to multiply this number by the number of holes on 
 the disc in order to ascertain how many tube-dis- 
 charges have occurred, and therefore how many 
 corresponding aerial vibrations have been performed, 
 in the period in question. The Syren is provided with 
 a counting-apparatus which registers the number 
 of times its disc rotates per second. 
 
 In order, therefore, to obtain the number of vibra- 
 tions in a second which correspond to an assigned 
 note, we have only to proceed as follows. Let the 
 note be steadily sounded by some instrument of 
 sustained power, e.g. organ or harmonium, and then 
 cause the Syren-sound to mount the scale until its 
 pitch coincides with that of the note under examina- 
 
II. § 29.] LIMITS OF MUSICAL SOUNDS. 61 
 
 tion. During coincidence ascertain by the counting- 
 apparatus the number of rotations which are being 
 performed per second. This, multipHed by the 
 number of holes in the disc, gives the number of 
 vibrations per second required. It will be con- 
 venient, for shortness' sake, to call the number of 
 vibrations per second to which any note is due the 
 vibration-number of the note in question. It is 
 clear, from what has gone before, that any assigned 
 degree of pitch can be permanently registered when 
 once its vibration-number has been ascertained. 
 
 29. Experiment with the Syren shows (§ 27) 
 that below a certain rate of vibration no musical 
 sounds are produced. The position of the absolute 
 limit thus placed to the gravity of such sounds 
 cannot be exactly defined, and probably varies some- 
 what for different ears. The lowest note on the 
 largest modern organs has 16^ for its vibration- 
 number, but it is a moot question how far the 
 musical character of this note can be recognised^ 
 
 ^ Messrs Hill and Sons constructed, for the gigantic organ 
 built by them for Sydney, New South Wales, a set of pedal 
 pipes the deepest of which, 64 feet in length, produced 8\ 
 vibrations per second. Some competent observers declared them- 
 selves able, and some unable, to recognise in the sounds of these 
 pipes degrees of pitch lower than that produced by the 32-foot 
 pipe making 16 J vibrations per second. 
 
62 LIMITS OF MUSICAL SOUNDS. [II. § 29. 
 
 In any case we may, for practical purposes, regard 
 the lower limit of musical sounds as situated in the 
 immediate neighbourhood of this degree of pitch. 
 For some distance above that limit the musical 
 character continues very imperfect, and it is not 
 until we reach about 40 vibrations per second that 
 we get a satisfactory musical sound. The limit 
 which bars the scale of pitch in the opposite direc- 
 tion varies greatly in position with different persons ; 
 some hearing with the utmost distinctness a shrill 
 sound which others declare to have for them no 
 existence. A cleverly -planned little instrument called 
 Gcdton's ivhistle, sold by philosophical instrument 
 makers, the note of which can be made to vary at 
 pleasure within certain limits, brings out these 
 differences of individual perception in a striking 
 manner. 
 
 Sounds well within the limit of general audi- 
 bility are, if above a certain degree of acuteness, 
 painful to the ear, and therefore unfit for musical pur- 
 poses. The highest note of the piccolo, the shrillest 
 sound heard in the orchestra, makes 4,752 vibrations 
 per seconds This we may regard as constituting a 
 practical superior limit to the scale of pitch at the 
 disposal of musical art. The extreme range attain- 
 able by exceptional human voices, from the deepest 
 ^ Taking 264 vibrations for middle-C (German pitch). 
 
II. § 29.] LIMITS OF MUSICAL SOUNDS. 63 
 
 note of a bass to the highest of a soprano, hes, 
 roughly speaking, between 50 and 1,500 vibrations 
 per second. Ordinary chorus voices range from 100 
 to 900 or 1,000 vibrations per second. 
 
 The power of discriminating minute differences 
 of pitch varies greatly with the acuteness of percep- 
 tion possessed by individual observers. An experi- 
 ment which a well-known Cambridge tuner, the late 
 Mr James Ling, was kind enough to make in my 
 presence showed that in the neighbourhood of middle- 
 C of the pianoforte he was able to distinguish, by 
 listening to them successively, two sounds forming 
 with each other an interval so small that it would 
 take about 250 such intervals to make up a single 
 octave. The difficulty of discriminating pitch in- 
 creases rapidly with very low or very high sounds ; 
 the above instance of what can be achieved by the 
 most highly-trained ears applies, therefore, only to a 
 few octaves in the middle region of the scale. The 
 continuously shading-otf gradations presented by 
 the scale of pitch afford no antecedent presumption 
 that particular sets of sounds should be singled out 
 to form the materials of Music. Nevertheless it 
 is found that practically the group of different 
 sounds employed in a musical composition is, re- 
 latively to the whole available stock, extraordinarily 
 limited. 
 
64 RELATIVE PITCH. [II. § 30. 
 
 30. When one sound has been arbitrarily selected 
 as the starting-point, there are certain other sounds, 
 bearing fixed relations of pitch to that previously 
 chosen, which are capable of forming, with it and 
 with each other, melodic and harmonic effects es- 
 pecially pleasing to our ears. These are the notes 
 of the ordinary Major and Minor scales, the original 
 sound of reference being the common tonic\ or key- 
 note, of those scales. In saying that these sounds 
 have fixed mutual relations of pitch, we merely state 
 formally an obvious fact. A familiar melody is 
 recognised equally well whether heard in the deep 
 tones of a man's, or in the shrill notes of a child's 
 voice. Whether the singer ' pitches ' it on a low or 
 on a high note of his voice makes no difference in the 
 melody itself. In fact the correctness with which an 
 air is sung no more depends on the exact pitch of 
 the note on which the singer starts it, than the faith- 
 fulness of a plan depends on the precise scale which 
 the draughtsman has adopted. It is sufficient that 
 the constituent notes of the melody should have fixed 
 mutual relations of pitch, just as, in the plan, the 
 several objects represented need only be drawn in 
 proportion to their actual dimensions. 
 
 1 The tonic may be recognised by its being the note on which 
 a tune almost invariably ends, and on which it must end if an 
 impression of entire completeness is to be conveyed to the hearer. 
 
II. § 31.] INTERVALS. 65 
 
 The difference in pitch of any two notes is called 
 the interval between them : it is on accuracy of 
 intervals that Music essentially depends. 
 
 31. The most important interval in the scale is 
 the Octave. It is that which separates the highest 
 note of a peal of eight bells from the lowest. When 
 a bass and a treble voice sing the same melody 
 together, the notes of the latter are usually an 
 Octave above those of the former. This interval 
 has the property, shared by no other, that if, start- 
 ing from any note we choose, we ascend to that an 
 Octave above it, then to that an Octave above the 
 last, and so on, we get a number of notes which 
 sound perfectly smooth and agreeable when heard 
 all together. The same thing holds good if we take 
 the succession of Octaves in descending order from 
 the note fixed on as the starting-point. Hence the 
 whole scale of pitch may be conveniently divided into 
 a series of Octaves taken upwards and downwards 
 from some one sound arbitrarily selected. Narrower 
 intervals situated in any one Octave are repeated in 
 all the others, so that when we have settled those 
 intervals for a single Octave we have settled them 
 for all the rest. Within the limits of each Octave 
 the Major scale presents us with seven notes, or 
 if we include that which forms the starting point 
 of the next cycle, with eight. The fact that the 
 
 T, 5 
 
66 
 
 INTERVALS. 
 
 [II. § 31. 
 
 eighth note is the Octave of the first explains the 
 meaning of the word ' Octave,' i. e. ' eighth ' {Latin : 
 ' octavus '). 
 
 The eight notes are those of an ordinary peal of 
 the same number of bells, or of the white keys of the 
 pianoforte between two adjacent C's. We may for 
 
 convenience of reference number them 1, 2, 3 8, 
 
 beginning with the lowest note which is their tonic. 
 The following nomenclature is used to describe 
 the intervals formed by the several notes with the 
 tonic. 
 
 Notes forming interval. 
 
 Name of interval. 
 
 1 and 2 
 
 Second 
 
 1 3 
 
 Major Third 
 
 1 4 
 
 Fourth 
 
 1 5 
 
 Fifth 
 
 1 6 
 
 Major Sixth 
 
 1 7 
 
 Major Seventh 
 
 1 8 
 
 Eighth or Octave. 
 
 When two notes of the same pitch are sounded 
 together, e.g. by two instruments or by two voices, 
 the notes are said to be in unison. Though there is 
 here no difference of pitch whatever, it is convenient 
 to rank the unison as an interval. With this ex- 
 planation we may add to the above table that 1 and 
 
II. §32.] CONCORDS AND DISCORDS. 67 
 
 1 form the interval of an unison. The reader must 
 carefully avoid giving to the ' Thirds,' ' Fourths,' &c., 
 vphich he meets v?-ith in Music the meanings attached 
 to the same words in the Arithmetic of fractions, 
 with which they have absolutely nothing to do. A 
 ' Fifth,' for example, does not stand for a fifth part 
 of an Octave, nor indeed for b, fifth part of anything, 
 but for the difference in pitch between the first and 
 fifth notes of the scale. 
 
 The several pairs of notes which form the 
 intervals laid down in our table do not, when 
 sounded together, all produce smooth and agreeable 
 impressions. The five pairs 
 
 1—3, 1—4, 1—5, 1—6, 1—8 
 yield smooth effects : the remaining two, 
 
 1 — 2 and 1 — 7, 
 are decidedly harsh-sounding combinations. The 
 intervals in the first line are therefore classed as 
 concords, those in the second as discords. 
 
 32. The Minor scale always has the notes 1, 4, 5 
 in common with the Major scale, and usually, also, 
 the note 2, for which, however, it occasionally sub- 
 stitutes a sound lying between that note and 1, and 
 forming with the latter a discord called the Minor 
 Second. It invariably substitutes for 3 a sound 
 lying between that note and 2, and forming with 1 
 a consonant interval called the Minor Third. Ac- 
 
 5—2 
 
68 
 
 MAJOR AND MINOR INTERVALS. [II. § 32. 
 
 cording to circumstances it may either retain 6, or 
 replace it by a sound lying between that note and 
 
 5, and forming with 1 a concord called the Minor 
 Sixth. Similarly it may employ 7, or, in the room 
 of that note, a fresh sound situated between it and 
 
 6, and forming with 1 a discord called the Minor 
 Seventh. 
 
 Thus, if we include the Octave, the two scales 
 together give us a series of eleven notes, which, 
 severally combined with the tonic, form ten distinct 
 intervals. They are expressed in musical notation 
 as follows: 
 
 i 
 
 Minor Second. Second. Minor Third. Major Third. Fonrth. 
 
 Fifth. 
 
 S^-^^^=^ 
 
 -j-^r- 
 
 ^mr 
 
 i 
 
 Minor Sixth. Major Sixth. Minor Seventh. Major Seventh. 
 
 Octave. 
 
 ZZ3Z 
 
 ±221 
 
 ^ 
 
 -9^ 
 
 Z^Z 
 
 The reader should endeavour to familiarise him- 
 self with these intervals, so that, when he hears the 
 pair of notes which form any one of them successively 
 sounded, he may at once be able to name the interval. 
 
 33. The Syren enables us to obtain simple nu- 
 merical measures of the intervals exhibited on this 
 page. 
 
 Let a second circular row, containing sixteen 
 holes, be punched in its disc, and the instrument be 
 
IT. § 33.] MEASURES OF INTERVALS. 69 
 
 set rotating uniformly. If we now blow alternately 
 against the 8-hole row and the 16-hole row, we shall 
 find that the sound produced at the latter is precisely 
 one Octave higher than the sound produced at the 
 former. If we increase or diminish the velocity of 
 rotation, both sounds will, of course, rise or fall to- 
 gether, but the interval between them will remain 
 unaffected and equal, as before, to an exact Octave. 
 The number of air-discharges corresponding to the 
 more acute sound is, in this case, evidently twice as 
 large in any given time as the number during the 
 same time for the graver sound. Accordingly we 
 have the following result : — 
 
 When two sounds differ by an Octave, the higher 
 sound makes exactly twice as many vibrations in any 
 assigned time as the lower. 
 
 Next let a row of 12 holes be punched in the 
 disc of the Syren. Taking this row with the 8-hole 
 row, and proceeding as in the last instance, we find 
 that the more acute sound forms a Fifth with the 
 graver one. The numbers of discharges in any given 
 time are here as 12 to 8, i.e. as 3 to 2. The result 
 therefore is as follows : — 
 
 When two sounds differ by a Fifth, the higher 
 sound makes exactly three vibrations during the time 
 in tvhich the lower sound makes two. 
 
 If we take the 16-hole and 12 -hole rows together, 
 
70 MEASURES OF INTERVALS. [II. §§ 34 & 35. 
 
 the interval amounts to a Fourth; accordingly, when 
 two sounds differ by a Fourth, the higher sound makes 
 exactly four vibrations during the time in which the 
 lower sound m,akes three. 
 
 34. The results just obtained admit of being 
 somewhat more concisely stated. During one second 
 of time the upper of two sounds differing by an Octave 
 makes a number of vibrations which is to the number 
 made by the lower sound as 2 to 1 . For a Fifth the 
 ratio is 3 to 2. For a Fourth it is 4 to 3. Remem- 
 bering, then, the definition of the vibration-number 
 of an assigned sound [§ 28], we may express our 
 three results as follows : — 
 
 When two sounds form with each other the 
 intervals of an Octave, a Fifth or a Fourth, their 
 vibration-numbers are to each other, in the first case 
 as 2 to 1, in the second as 3 to 2, in the third as 
 4 to 3. 
 
 A ratio is most easily expressed by a fraction. 
 Thus we may regard the fraction f as denoting the 
 interval of a Fifth. It may be taken as an abbre- 
 viated statement of the fact that, when two sounds 
 form a Fifth with each other, the more acute makes 
 3 vibrations whUe the graver makes 2. 
 
 35. By suitable experiments, similar numerical 
 relations to those already established may be ob- 
 tained for all the intervals already considered. A 
 
II. § 35.] 
 
 VIBBA TION-FRA CTIONS. 
 
 71 
 
 fraction can thus be determined for each interval, 
 in the manner exemplified in the case of the Fifth. 
 We will call this fraction the vibration-fraction of 
 the interval in question. The accompanying table 
 gives, in the second column, the vibration-fractions 
 corresponding to the intervals named in the first ; 
 and, in the third, states the consonant or dissonant 
 character of each interval. 
 
 Name of interval. 
 
 Vibration-fraction. 
 
 Character of interval. 
 
 Unison 
 
 1 
 T 
 
 concord 
 
 Minor Second 
 
 16 
 
 discord 
 
 Second 
 
 9 
 
 ¥ 
 
 discord 
 
 Minor Third 
 
 6 
 
 concord 
 
 Major Third 
 
 5 
 T 
 
 concord 
 
 Fourth 
 
 4 
 
 concord 
 
 Fifth 
 
 3 
 
 ■2" 
 
 concord 
 
 Minor Sixth 
 
 8 
 T 
 
 concord 
 
 Major Sixth 
 
 5 
 
 concord 
 
 Minor Seventh 
 
 16 
 
 discord 
 
 Major Seventh 
 
 1 5 
 
 "8" 
 
 discord 
 
 Octave 
 
 2 
 
 T 
 
 concord 
 
 It is noticeable that the vibration-fractions of 
 the dissonant intervals involve higher numbers than 
 those of the consonant intervals do; the latter, with 
 
72 CALCULATION OF VIBRATION-NUMBERS. [II. § 36. 
 
 the solitary exception of the Minor Sixth, having no 
 number exceeding 6, while the former bring in 9, 
 15 and 16. 
 
 36. By the help of this table we can calculate 
 the vibration-numbers of all the notes within a 
 single Octave which belong to the Major or Minor 
 keys as soon as the vibration-number of their tonic is 
 given. For instance, let middle C of the pianoforte 
 (vib.-no. 264) be the tonic. From the fourth line 
 of the table we see that the vibration-number of 
 D must be to 264 in the ratio of 9 to 8. It must 
 
 9 
 therefore be equal to - x 264, or 297. For Et', by 
 
 8 
 
 exactly similar reasoning, we obtain -x264 or 316^; 
 
 
 
 for E, |x264, or 330. 
 
 The student should work out the remaining cases 
 for himself 
 
 The complete results for the Major scale are as 
 follows : — 
 
 i 
 
 BE 
 
 ~0" 
 2B4 297 SSO 3S2 396 440 495 828 
 
 In order to extend the scale an Octave upwards 
 we have only to multiply each vibration-number 
 by 2. A second multiplication by 2 wUl raise it by 
 another Octave, and so on. Conversely, in order to 
 
II. §36.] CALCULATION OF VIBRATION-NUMBERS. 73 
 
 pass to the Octave below we divide each vibration- 
 number by 2. To descend a second Octave we repeat 
 the operation, and so on. 
 
 Thus the pitch of the tonic absolutely fixes the 
 pitch of every note in the scale of which it is the 
 starting-point. 
 
 Before proceeding to investigate the mechanical 
 equivalent of the third element [§ 24] of a musical 
 sound, its quality, we will briefly examine, in the 
 next chapter, a subject possessing an important 
 bearing on that enquiry. 
 
CHAPTER III. 
 
 ON RESONANCE. 
 
 37. When a sounding body causes another body- 
 to emit sound, we have an instance of a very remark- 
 able phenomenon called resonance. The German term 
 for it, ' co-vibration ' [Mitschwingung), possesses the 
 merit of at once indicating its essential meaning, 
 namely the setting up of vibrations in an instrument, 
 not by a blow or other immediate action upon it, but 
 indirectly as the result of the vibrations of another 
 instrument. In order to produce the effect, we have 
 only to press down very gently one of the keys of a 
 pianoforte, so as to raise the damper without making 
 any sound, and then sing the corresponding note 
 loudly into the instrument. When the voice ceases, 
 the instrument will continue to sustain the note, 
 which will then gradually fade away. If the key is 
 allowed to rise again before the sound is extinct, it 
 will abruptly cease. A similar experiment may be 
 tried, as follows, on any pianoforte which allows the 
 wires to be uncovered. Each note is, as is well known, 
 produced by two or by three wires. Having, as in 
 
III. § 37.] RESONANCE. 75 
 
 the previous case, raised one of the dampers without 
 striking the note, twitch one of the corresponding 
 wires sharply with the finger-nail, and then wait a 
 few seconds. The vibrations will, during this interval, 
 have communicated themselves to the other string 
 or strings belonging to the note pressed down : if, 
 therefore, the first wire be now stopped by applying 
 the tip of the finger to the point where it was at first 
 twitched, the note heard will continue to be sus- 
 tained by the remaining wire or wires. 
 
 A more instructive method of studying resonance 
 is to take two unison tuning-forks, strike one of 
 them, and hold it a short distance from the other. 
 The second fork will then commence sounding 
 by resonance, and will continue to produce its note 
 if the first fork be brought to silence. It is essen- 
 tial to the success of this experiment that the two 
 forks should be rigorously in unison. If the pitch 
 of one of them be lowered by causing a small pellet 
 of wax to adhere to the end of one of its prongs, 
 the effect of resonance will no longer be produced, 
 even though the alteration of pitch be too small to 
 be recognised by the ear. Further, the phenomenon 
 generally requires a certain length of time to develope 
 itself; for, if the silent fork be only momentarily 
 exposed to the infiuence of its sounding feUow, 
 hardly any vibration is communicated. The reso- 
 
76 ACCUMULATED IMPULSES. [III. § 37. 
 
 nance is at first extremely feeble, and gradually 
 increases in intensity under the continued action 
 of the originally-excited fork. Some seconds must 
 elapse before the maximum resonance is attained. 
 The conditions of our experiment show that the 
 resonance of the second fork was due to the trans- 
 mission hy the air of the vibrations of the first, the 
 successive air-impulses falling in such a manner on 
 the fork as to produce a cumulative effect. If we 
 bear in mind the disproportionate mass of the body 
 set in motion compared to that of the air acting upon 
 it, — steel being more than six thousand times as 
 heavy as atmospheric air for equal bulks — we cannot 
 fail to regard this as a very surprising fact. 
 
 Let us examine the mechanical causes to which 
 it is due. Suppose a heavy weight to be suspended 
 from a fixed support by a flexible string, so as to 
 form a pendulum of the simplest kind. In order to 
 cause it to perform oscillations of considerable extent 
 by the application of a number of small impulses, we 
 proceed as follows. As soon as, by the first impulse^ 
 the weight has been set vibrating through a small 
 distance, we take care that every succeeding impulse 
 is impressed in the direction in which the weight is 
 moving at the time. Each impulse thus applied will 
 cause the pendulum to oscillate through a larger 
 angle [§ 19] than before, and, the effects of many 
 
III. § 37.J ACCUMULATED IMPULSES. 77 
 
 impulses being in this way added together, an 
 extensive swing of the pendulum is the result. 
 
 When the distance through which the weight 
 travels to and fro, though in itself considerable, is 
 small compared to the length of the supporting string, 
 the time of oscillation is the same for any extent of 
 swing within this limit, and depends only on the 
 length of the string. My readers will find this im- 
 portant principle illustrated in any manual of Elemen- 
 tary Mechanics, and I must ask them to take it for 
 granted here. For the sake of simplicity, let us sup- 
 pose that we are dealing with a pendulum of such a 
 length as to perform one complete oscUlation in each 
 second, and therefore to make a single forward or 
 backward swing in each half second. It will be clear, 
 from what has been said above, that the maximum 
 effect will be produced on the motion of the pendulum 
 by appljring a forward and a backward impulse 
 respectively during each alternate half-second, or 
 which is the same thing, by administering a pair of 
 to-and-fro impulses during each complete oscillation 
 of the pendulum. We have a simple instance of such 
 a proceeding in the way in which two boys set 
 a heavily-laden swing in extensive motion. They 
 stand facing each other, and each boy, when the 
 swing is moving away from him, helps it along 
 with a vigorous push. 
 
78 CAUSE OF RESONANCE. [III. § 38. 
 
 38. The above considerations enable us to ex- 
 plain how a sounding fork can set another fork in 
 unison with itself into vibration by the action of 
 the intervening air. When a continuous musical 
 note is being transmitted, we know that, at any one 
 point we choose to fix upon, the air is undergoing a 
 series of rapid changes, becoming alternately denser 
 and less dense than it would be but for the passage 
 of the sound. The increase of density is accompanied 
 by an increase of pressure ; its diminution by a 
 diminution of pressure [§ 20]. 
 
 Fig. 21. 
 rlae 
 
 IJ 
 
 Let A, Fig. 21, be the sounding fork, B that 
 whose vibrations are to be excited by resonance, and 
 let us consider the effect of the alternations of pres- 
 sure of the air at a on the prong ha. Let the first 
 pulse which arrives at a be one of condensation, and 
 let its increased pressure cause the prong ha to swmg 
 as far as 6c \ The elastic recoil of the fork would 
 alone cause the prong to spring back to an equally 
 inclined position on the other side of ha. Since, 
 
 ' The extents of vibration shown in this figure we of necessity 
 enormously exaggerated. 
 
III. §38.] CAUSE OF RESONANCE. 79 
 
 however, the period of vibration is by supposition 
 the same for the two forks, a pulse of rarefaction, 
 entaihng diminished pressure, will arrive at a just as 
 the prong is passing through its erect position ha. 
 The prong will therefore swing a little further to 
 the left than it would otherwise have done, say to 
 hd, making the angle abd slightly larger than the 
 angle ahc. The prong then starts to the right and 
 is, while passing through the position ha, again helped 
 on by the arrival there of a new pulse of condensa- 
 tion. Thus the alternate condensations and rare- 
 factions of the Sound-waves play towards the fork 
 exactly the parts sustained by the two boys in our 
 illustration of the swing. Accordingly, these air- 
 impulses are applied under precisely the conditions 
 which we saw to be most favourable to the rapid 
 development of vibratory motion. Their large num- 
 ber makes up for the feebleness of each separate 
 impulse. Accordingly resonance is produced more 
 slowly between unison-forks of low than of high 
 pitch. I find, for example, that, with two forks 
 making 256 vibrations per second, about one second 
 is requisite to bring out an audible resonance ; while 
 with another pair, making 1,920 vibrations per second, 
 I can with difficulty damp the first fork sufficiently 
 soon after striking it to prevent the other from 
 making itself heard. 
 
80 
 
 RESONANT AIR-COLUMNS. [III. § 39, 
 
 39. A column of air is easily set in resonant 
 vibration by a note of suitable pitch. The roughest 
 experiment suffices to establish this fact. We have 
 only to roll up a piece of paper, so as to make a 
 cylinder twelve inches long and an inch or two in 
 diameter, with both ends open, strike a common 
 
 C tuning-fork, I M) " ) > and hold it close to 
 
 one of the apertures. As soon as the fork reaches 
 the position (1) Fig. 22', its tone will unmistakably 
 swell out. In order to estimate the increase of 
 intensity produced, it is a good plan to move the 
 fork rapidly to and fro, a few times, between the 
 positions (1) and (2). 
 
 In the first case we have the full effect of reson- 
 ance, in the second only the unassisted tone of the fork, 
 
 ' This figure was drawn for a cylinder only six inches in 
 length, but suffices for the purpose of illustration for which it 
 is here used. 
 
III. §39.] RESONANT AIR-COLUMNS. 81 
 
 and the contrast is very marked. We may shorten 
 or lengthen our cylinder within certain limits and 
 still obtain the phenomenon of resonance, but the 
 greatest reinforcement of tone attainable with the 
 fork selected will be produced by an air-column 
 about twelve inches long. 
 
 If we close one end of the paper cylinder, by 
 placing it, for instance, on a table, and repeat our ex- 
 periment at the ojaen end, only a very weak resonance 
 is produced ; but we obtain a powerful resonance by 
 
 operating with a fork of pitch | > _^ E= making 
 
 half as many vibrations per second as that before 
 employed. In this case, then, a column of air 
 contained in a cylinder of which one end was closed 
 resounded powerfully to a note an Octave below that 
 which elicited its most vigorous resonance when the 
 air-column was contained in a cylinder open at both 
 ends. 
 
 By operating in this fashion, with forks of dif- 
 ferent pitch, on air-columns of diflferent lengths, we 
 arrive at the following laws, which are universally 
 true : 
 
 1. For every single musical note there exists a 
 corresponding air-column of definite length which, 
 when enclosed in a pipe, open at both ends, re- 
 sounds the most powerfully to that note. 
 
 T. 6 
 
82 RESONANT AIR-COLUMNS. [III. § 40. 
 
 2. The maximum resonance of ail' in a pipe, 
 with one end closed is produced by a note one 
 Octave below that to which a pipe of the same 
 length, open at both ends, resounds the most power- 
 fully. 
 
 40. In order to ascertain the precise relation 
 between the pitch of a note and the length of the 
 corresponding air-column, we will examine the way 
 in which resonance is produced in a column of air 
 contained in a pipe closed at one end. 
 
 Let A, Fig. 23, be the open, and B the closed, 
 ends of the pipe, and let us for a moment replace 
 the contained air by an elastic spiral spring fastened 
 at B, and of length equal to AB. 
 
 Fig. S3. 
 
 uvmwwmMMVsi^imJW 
 
 B A 
 
 Suppose the end of the spring suddenly pushed 
 a little way from A towards B. The coils of the 
 spring nearest A will be squeezed together, and this 
 condensed state of the spring will be transmitted 
 to B, where further movement to the left is stopped 
 by the fixed end of the spring. Hence the coils 
 crowded together near B will begin expanding to- 
 wards the right, i.e. the condensed state of the 
 spring will be turned back, or ' reflected,' at B and 
 will then travel to ^. In virtue of the elasticity 
 
III. § 40.] RESONANT AIR-GOLUMNS. 8.3 
 
 of the spring its free end will now be caused to 
 protrude beyond the mouth of the pipe as far as it 
 was at first pushed into it. The coils near A being 
 thus drawn somewhat more apart than was the case 
 when they were in their undisturbed condition, will 
 exert a pull on those near them, which will in their 
 turn be drawn further apart, and thus a rarefied 
 state of the spring will be produced at A and trans- 
 mitted to B. On reaching the fixed end there, the 
 increased tension will act on the coils near B, i.e. 
 the rarefied state of the spring will be reflected at B, 
 just as its condensation was in the case previously 
 considered. When this rarefaction returns to A, 
 the free end of the spring will momentarily resume 
 the position within the mouth of the pipe to which 
 it was originally pushed. The elasticity of the 
 spring thus causes it to lengthen and shorten as a 
 whole in consequence of the single push originally 
 given it, and this motion would for a time continue, 
 its successive periods being four times the space of 
 time occupied by a pulse of condensation or rarefac- 
 tion in traversing the length of the pipe. The free 
 end of the spring may, however, be pushed and 
 pulled alternately so as to reinforce each pulse as it 
 arrives at the mouth of the pipe, and in this manner 
 the maximum of motion will be communicated to 
 the spring. In this case, one outward and one 
 
 6—2 
 
84 RESONANCE-BOXES. [III. § 41. 
 
 inward impulse must be communicated to the free 
 end of the spring during the time which elapses 
 while a pulse traverses four times the length of the 
 pipe. Reverting to the actual conditions of our 
 problem, we have the resonance of the air-column in 
 place of the alternate lengthening and shortening 
 of the spring. The to-and-fro impulses at A are 
 impressed by a vibrating fork. The Sound-pulse 
 traverses four times the length of the pipe while the 
 fork is performing one complete vibration. We 
 know, however [§ 15 p. 32], that during this latter 
 period the Sound-pulse produced by the fork's action 
 traverses precisely one wave-length corresponding 
 to the pitch of the note produced by the fork. 
 Hence, for maximum resonance in the case of a pipe 
 closed at one end, the wave-length corresponding 
 to the note sounded must be four times as great as 
 the length of the air-column, or the length of the 
 column one quarter of the wave-length. 
 
 41. These principles give us the explanation of 
 a valuable appliance for intensifying the sound of a 
 tuning-fork. Such a fork, when held in the hand 
 after being struck, communicates but little of its 
 vibrations to the surrounding air ; when, however, 
 its handle is screwed into one side of a wooden box 
 of suitable dimensions, in the way shown in Fig. 24, 
 the tone becomes much louder. The vibrations of 
 
III. § 42.] RESONATORS. 85 
 
 the fork pass from its handle to the wood of the box, 
 
 £'^.24 
 
 and thence to the air- column within, which is of 
 appropriate length for maximum resonance to the 
 fork's note. This convenient adjunct to a tuning- 
 fork goes by the name of a ' resonance-box.' 
 
 42. When a number of musical sounds are 
 simultaneously sustained it is generally difficult, and 
 often impossible, for the unaided ear to decide whether 
 an individual note is, or is not, present in the mass 
 of sound heard. If, however, we had an instrument 
 which intensified the note of which we were in 
 
 Fig. 25. 
 
 search, without similarly reinforcing others which 
 there was any risk of our mistaking for it, our power 
 
86 RESONATORS. [HI. § 42. 
 
 of recognising the note in question would be propor- 
 tionately increased. Such an instrument has been 
 invented by Helmholtz. It consists of a hollow ball 
 of brass with two apertures at opposite ends of a 
 diameter, as shown in Fig, 25. 
 
 The larger aperture allows the vibrations of the 
 external air to be communicated to that within 
 the ball ; the smaller aperture terminates in a nipple 
 of convenient form for insertion in the ear of the 
 observer. The air contained in the ball resounds 
 very powerfully to a single note of definite pitch, 
 whence the instrument has been named by its in- 
 ventor a resonator. The best way of using it is, first 
 to stop one ear closely, and then to insert the nipple 
 of the instrument into the other. As often as the 
 resonator's own note is sounded in the external air, 
 the instrument will sing it into the ear of the ob- 
 server with extraordinary emphasis, and thus at once 
 enable him to single out that note from among a 
 crowd of others differing from it in pitch. A series 
 of such resonators, tuned to particular previously 
 selected notes, constitutes an invaluable apparatus 
 for analysing a composite sound into the simple 
 tones of which it is made up. 
 
CHAPTER IV. 
 
 ON QUALITY. 
 
 43. The laws of resonance enable us to establish 
 a remarkable, and by most persons utterly un- 
 suspected fact, viz. that the notes of nearly every 
 regular musical instrument with which we are 
 familiar, are not, as they are ordinarily taken to be, 
 single tones of one determinate pitch, but composite 
 sounds containing an assemblage of such tones. 
 These are always members of a regular series, form- 
 ing with each other fixed intervals which may be 
 thus stated : if we number the separate single 
 tones of which any given sound is made up, 1, 2, 
 3, &c., beginning with the lowest, we have 
 
 (1) The deepest, or ' fundamental,' tone, whose 
 pitch is ordinarily regarded as that of the 
 whole sound. 
 
 (2) A tone one Octave above (l). 
 
88 CONSTITUENTS OF COMPOSITE SOUNDS; [IV. § 43. 
 
 (3) A tone a Fifth above (2), i.e. a Twelfth 
 above (1). 
 
 (4) A tone a Fourth above (3), i.e. two Octaves 
 above (l). 
 
 (5) A tone a Major Third above (4), i.e. two 
 Octaves and a Major Third above (l). 
 
 (6) A tone a Minor Third above (5), i.e. two 
 Octaves and a Fifth above (l). 
 
 These are the most important members of the 
 series. Their vibration-numbers are connected by a 
 simple law, which is easUy deduced from the above 
 relations. If the fundamental tone makes 100 
 vibrations per second, (2) will make twice as many, 
 
 i.e. 200 ; (3), being a Fifth above (2), will have for 
 
 3 
 
 its vibration-number - x 200, or 300. For (4), 
 
 which is a Fourth above (3), we get similarly 
 
 4 5 
 
 - X 300, or 400 ; for (5), - x 400, or 500 ; for (6), 
 
 - X 500, or 600. Thus the numbers, come out 100, 
 
 200, 300 and so on ; or generally, whatever be the 
 vibration-number of (1), those of (2), (3), (4), &c., 
 are respectively twice, three times, four times, &c. 
 as large. If C in the bass clef be selected as the 
 fundamental tone, the series, complete up to the 
 tenth tone, is shown in musical notation as follows : — 
 
IV. § 44.] SERIES OF CONSTITUENT TONES. 89 
 
 „ f^' -?- -e- "? 
 
 eE 
 
 i 
 
 12 S46fi789 10 
 
 The asterisk denotes that the pitch of the 7th 
 tone is not precisely that of the note by which it 
 is here represented, but sHghtly flatter. 
 
 The reader must not suppose that, because the 
 tones into which a note of a musical instrument may 
 usually be decomposed are members of a fixed aeries, 
 all those which we have written down are neces- 
 sarily present in every such note. No more is meant 
 to be asserted than that those which are present, 
 be they few or many, must occupy positions de- 
 termined by the law connecting each tone with 
 its fundamental. The sound may contain, say, 
 (1), (3) and (5) only, or (1), (4) and (8) only, 
 and so on, the rest being entirely absent, but in 
 no case can a tone intermediate in pitch between 
 any two consecutive members of the series make its 
 appearance. 
 
 44. Experimental evidence shall now be pro- 
 duced in support of the extremely important pro- 
 position just enunciated. 
 
 We will begin with the sounds of the pianoforte. 
 
 Let the note BEzznz be first silently pressed down, 
 
90 ANALYSIS OF RESONANCE. [IV. § 44. 
 
 and then B^- p— be vigorously struck, and, after 
 
 three or four seconds, allowed to rise again. The 
 lower note is at once extinguished, but we now 
 hear its Octave sounding with considerable force 
 
 -e- 
 
 from the wires of 3-^ . If we permit the damper 
 
 to fall back on these, by releasing the key hitherto 
 held down, the whole sound is immediately cut off. 
 Next, retaining the same fundamental note, 
 
 ^ n , let ^— 5^ r be quietly freed from its damper, 
 
 and the experiment repeated as before. We shall 
 then hear this note sounding on after the extinction 
 
 of ?zEa=: • Similar results may be obtained with 
 
 the three next tones, ^-°^^^'^ = . but they drop 
 
 off very rapidly in intensity. The tones above 
 
 A are so weak as to be practically inaudible. 
 
 The series of tones produced in this succession of 
 cases can only be due to resonance. But, as has 
 been already shown, the vibrations of any instrument 
 are excited by resonance only ivlien vibrations of the 
 same period are already present in the surrounding 
 air. Accordingly, the only sound directly originated 
 in each variation of our experiment, viz. that of the 
 
 note '^^ o — , must have contained all the tones sue- 
 
IV. §45.] ANALYSIS OF RESONANCE. 91 
 
 cessively heard. The reader should apply the method 
 of proof here adopted to notes in various regions of 
 the key-board. He will find considerable differences, 
 even between consecutive notes, in the number and 
 relative intensities of the separate tones into which he 
 is thus able to resolve them. The higher the pitch of 
 the fundamental tone, the fewer will the recognisable 
 associated tones become, until, in the region above 
 
 ■7- — ■ — , the notes are themselves approximately 
 
 single tones. The causes of these differences will be 
 explained in detail in a subsequent chapter ; it is 
 sufficient here to indicate their existence. The result 
 arrived at, thus far, is that the sounds of the piano- 
 forte are in general composite, the number of consti- 
 tuent tones into which they are resolvable being 
 largest in the lower half of the instrument, and dimi- 
 nishing in its upper half, until at last they become 
 practically tones of only one degree of pitch. 
 
 45, The above resolution has been effected by 
 means of the principle of resonance. It can, however, 
 be performed by the ear directly, though only to a 
 small extent and with less ease. In applying the 
 direct analysis to the sounds of a musical instrument, 
 it is best first to produce gently the note correspond- 
 ing in pitch to the tone which it is wished to isolate, 
 and then to develope the compound sound contain- 
 
92 DIRECT ANALYSIS. [IV. § 45. 
 
 ing it as a constituent. In most cases, provided 
 the observer has kept his attention unswervingly- 
 fixed on the pitch of the note for which he is 
 Ustening, he wUl hear it come out clearly from 
 among the group of tones included in the composite 
 
 sound. If the pianoforte note B^ ^_ be thus 
 
 examined, the Octave, ^ , and Twelfth, ggi 
 
 can generally be recognised with considerable ease ; 
 
 the second Octave, ^ ) " , with a little trouble ; 
 
 the next two tones of the series on page 89 with 
 increasing difficulty, and those which succeed them 
 not at all. The reader approaching the phenomenon 
 for the first time must not be disappointed if, on 
 tiying this experiment, he fail to hear the tones 
 he is told to expect. He should vary its conditions 
 by changing the note struck, in such a way that his 
 attention will not be liable to be diverted by the 
 presence of strong tones more acute than that of 
 
 which he is in search. Thus a note near 
 
 i 
 
 may be advantageously chosen to observe the first 
 Octave, ^ " — ; one near ^=i^=^ to observe the 
 
 m 
 
 Twelfth, ^ ; one near to observe the 
 
IV. § 46.] DIFFICULTY OF DIRECT ANALYSIS. 93 
 
 second Octave, H^ . He may however altogether 
 
 fail in performing the analysis with the unassisted ear. 
 This by no means indicates any aural defect, as he 
 may at first be inclined to imagine. It rather shows 
 that the life-long habit of regarding the notes of in- 
 dividual sound-producing instruments as single tones 
 cannot be unlearned all at once. The case is analo- 
 gous to that of single vision with two eyes, where two 
 distinct and different images are so blended together 
 as to appear to all ordinary observation as one. The 
 acoustical observer who is thus situated must at first 
 rely on the analysis by resonance, and on the testi- 
 mony of those who are able to perform the direct 
 analysis. As he pursues the subject further experi- 
 mentally, his analytical faculty will develope itself. 
 
 46. The composite character of musical sounds, 
 which we have recognised in the case of the piano- 
 forte, and shall have ample opportunity of verifying 
 more generally in the sequel, requires the introduc- 
 tion here of certain verbal definitions and limitations. 
 The phraseology hitherto employed, both in the 
 science of Acoustics and in the theory of Music, 
 goes on the supposition that the sounds of individual 
 instruments are single tones, and therefore, of course, 
 contains no term specially denoting compound sounds 
 and their constituents. ' Sound,' ' note,' and ' tone ' 
 
94 THEORY OF QUALITY. [IV. § 47. 
 
 are used as nearly synonymous. It will be conve- 
 nient to restrict the meaning of the latter so that it 
 shall denote a sound which does not admit of resolu- 
 tion into simpler elements. A single sound of deter- 
 minate pitch will accordingly, in what follows, be 
 called a tone, or simple tone. For a compound sound 
 the word clang will be a serviceable term. The series 
 of elementary sounds into which a clang can be re- 
 solved we shall call its partial-tones, sometimes dis- 
 tinguishing among these, the lowest, or fundamental 
 tone, from the others, or overtones, of the clang. This 
 nomenclature is the direct adaptation of the German 
 terms employed by Helmholtz. Its introduction is 
 due to Professor Tyndall. 
 
 47. This long discussion has paved the way for 
 the complete explanation of musical quality which is 
 contained in the following proposition. The quality 
 of a clang depends on the number, orders and rela- 
 tive intensities of the partial-tones into ivhich it can 
 he resolved. We have here three different causes to 
 which variations in the quality of composite sounds 
 are assigned. 
 
 1. A clang may contain only two or three, or it 
 may contain half-a-dozen, or even as many as fifteen 
 or twenty, well developed partial-tones. 
 
 2. The number of partial-tones pi'esent remain- 
 ing the same, the quality of the resulting sound will 
 
IV. §47.] THEORY OF QUALITY. 95 
 
 vary according as they occupy different positions 
 in the partial-tone series, i.e. on their orders. 
 Thus, a clang containing three tones may consist of 
 (1),(2), (3), or of (1), (3), (5), or of (l), (7), (10), and 
 so on, the quality varying in each instance. 
 
 3. The number and orders of the partial-tones 
 present remaining the same, the quality will vary 
 according to the relative degrees of loudness with 
 which those tones speak. Thus, in the simplest case 
 of a clang consisting of only (l) and (2), either tone 
 may alter in intensity while the other remains 
 constant, and so cause variation in the quality of the 
 sound resulting from their combination. 
 
 It is clear that these three classes of variations 
 are entirely independent of each other, that is to say, 
 any two clangs may differ in the number, orders and 
 relative intensities, of their constituent partial-tones. 
 The variety of quality thus provided for is almost in- 
 definitely great. In order to form some idea of its 
 extent, let us see how^ many clangs of different qua- 
 lity, but of the same pitch, can be formed with the 
 first six partial-tones, by variations of number and 
 order only. We will indicate each group by the 
 corresponding figures inclosed in a bracket ; thus e.g. 
 (1, 3, 5) represents a clang consisting of the first, 
 third and fifth partial-tones. 
 
 All the possible groups, each of course contain- 
 
96 THEORY OF QUALITY. [IV. § 47. 
 
 ing the same fundamental tone, are given in the 
 
 following enumeration : — 
 
 Two at a time : 
 
 (1, 2), (1, 3), (1, 4), (1, 5), (1, 6). 
 
 Total 5. 
 
 Three at a time : 
 
 (1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), {1, 3, 4), 
 {1, 3, 5), (1, 3, 6), (1, 4, 5), (1, 4, 6), (1, 5, 6). 
 
 Total 10. 
 
 Four at a time : 
 
 (1, 2, 3, 4), (1, 2, 3, 5), (1, 2, 3, 6), 
 (1, 2, 4, 5), (1, 2, 4, 6), (1, 2, 5, 6), 
 (1, 3, 4, 5), (1, 3, 4, 6), (1, 3, 5, 6), (1, 4, 5, 6). 
 
 Total 10. 
 
 Five at a time : 
 
 (1, 2, 3. 4, 5), (1, 2, 3, 4, 6), (1, 2, 3, 5, 6), 
 (1, 2, 4, 5, 6), (1, 3, 4, 5, 6). 
 
 Total 5. 
 
 Six at a time : (1, 2, 3, 4, 5, 6). 
 
 Total 1. 
 
 The whole number of groups is 31, or if we allow 
 the fundamental -tone (l) to count by itself as a 
 sound of separate quality, 32. Let us next examine 
 how many clangs of different quality can be obtained 
 from a single combination of three fixed partial-tones 
 by variations of inteyisity only, supposing that each 
 
IV. § 47.] THEORY OF QUALITY. 97 
 
 tone is capable of but two degrees of loudness. Re- 
 presenting one of these by / and the other by p, we 
 indicate, e.g., by (/, p, p) a clang in which the funda- 
 mental tone is sounded forte, and the two overtones 
 piano. The different cases which present themselves 
 are the following : 
 
 (///), {f,p,n ipj,/). ip.pj), if,f,p), 
 if'P'P)' {p>f'P)> {p,p,p) 
 
 or seven in all, since {p, p, p) has the same quaHty 
 as if,/,/). The number of cases increases very 
 rapidly as we take more partial-tones together. 
 Thus a clang of four tones will produce 15 sounds of 
 different quality ; one of five tones 3 1 ; one of six 
 tones 63 ; by variations of intensity only. Alto- 
 gether we could form, with six partial-tones, each 
 susceptible of only two different degrees of intensity, 
 upwards ot four hundred clangs of distinct quality, 
 all having the same fundamental tone. The suppo- 
 sition above made utterly understates, however, the 
 possible variety of quality dependent only on changes 
 of relative intensity. A very slight increase or dimi- 
 nution of loudness, on the part of a single constituent 
 tone, is enough to produce a sensible change of qua- 
 lity in the clang. We should be still far below the 
 mark if we allowed each partial-tone four different 
 degrees of intensity, though even this supposition 
 would bring us more than eight thousand separate 
 
 T. 7 
 
98 THEORY OF QUALITY. [IV. § 47. 
 
 cases. Since many more variations of intensity are 
 practically efficacious, and also since the disposable 
 partial-tones need by no means be limited to the 
 first six, the above calculation will probably suffice 
 to convince the reader that the varieties of quality 
 which the theory we are engaged upon is capable 
 of accounting for are almost indefinitely numerous. 
 This is, in fact, no more than we have a right to 
 demand of the theory, when we reflect on the fine 
 shades of quality which the ear is able to distinguish. 
 No two instruments of the same class are exactly 
 alike in this respect. For instance, grand pianofortes 
 by Broadwood and by Erard exhibit unmistakable 
 differences, which we describe as ' Broadwood tone ' 
 and ' Erard tone.' Less marked, but stiU perfectly 
 recognisable, diiferences exist between individual in- 
 struments of the same class and maker, and even 
 between consecutive notes of the same instrument. 
 To these we have to add the variations in quahty 
 due to the manner in which the performer handles 
 his instrument. On the pianoforte the kinds of tone 
 elicited by a dull slamming touch, and by a lively 
 elastic one, are clearly distinguishable. With other 
 instruments such distinctions are much more marked. 
 On the violin we perceive endless gradations of 
 quality, from the rasping scrape of the beginner up 
 to the smooth and superb tone of a Joachim. A 
 
IV. § 47.] THEORY OF QUALITY. 99 
 
 precisely similar remark applies to wind instruments ; 
 the differences, for example, between first-rate and 
 inferior playing on the hautbois, bassoon, horn, or 
 trumpet, being perfectly obvious to every musical 
 ear. 
 
 In the next chapter we will discuss the quality 
 and essential mechanism of the principal musical in- 
 struments, among which the pianoforte will receive 
 an amount of attention proportionate to its popu- 
 larity and general use. We begin with the simple 
 tones of which all composite sounds are made up. 
 
 7—2 
 
CHAPTER V. 
 
 ON THE ESSENTIAL MECHANISM OF THE PEINCIPAL MUSICAL 
 INSTEUMENTS CONSIDEEED IN EEFEEENCE TO QUALITY. 
 
 1. Sounds of tuning-forks. 
 
 48. Wlien a vibrating tuning-fork is held to 
 the ear, we at once perceive, beside the proper note 
 of the fork, a shrill, ringing and usually rather dis- 
 cordant, sound due to overtones which do not follow 
 the series of such tones set out in § 43. If, how- 
 ever, the fork is mounted on its resonance-box, as in 
 Fig. 24, p. 85, its proper note is so much strength- 
 ened that the extraneous sound is by comparison 
 insignificant. Provided the fork be but moderately 
 excited, e.g. by gentle stroking with a resined bow, 
 the sound heard is practically' a simple tone. It is 
 characterised by extreme mildness, without a trace 
 
 ' Except with very gentle bowing, however, or unless the 
 fork's note has been allowed to grow faint, examination with 
 resonators shows traces of overtones belonging to the ordinary 
 
 series. 
 
V. § 49.] SIMPLE TONES. 101 
 
 of anything which could be called harsh or piercing. 
 As compared with a pianoforte note of the same 
 pitch the fork-tone is wanting in richness and viva- 
 city, and produces an impression of greater depth, so 
 that one is at first inclined to think the fork em- 
 ployed must be an Octave too low. 
 
 It is a direct inference from the general theory 
 of the nature of quality that simple tones can differ 
 only in pitch and intensity. Accordingly we find 
 that tuning-forks of the same pitch, mounted on 
 resonance-boxes and set gently vibrating by a resined 
 bow, exhibit, whatever be their forms and sizes, 
 differences of loudness only. When made to sound 
 with equal intensity by suitable bowing, their tones 
 are absolutely undistinguishable from each other. 
 
 2. Sounds of vibrating strings. 
 
 49. Sounding strings vibrate so rapidly that 
 their movements cannot be followed directly by the 
 eye. It will be well, therefore, to examine how the 
 slower and more easily controllable vibrations of 
 non-sounding strings are performed, before treating 
 the proper subject of this section. Take a flexible 
 caoutchouc tube ten or fifteen feet long and fasten 
 its ends to two fixed objects separated from each 
 other by that distance. The tube can be conveniently 
 set in periodic vibration by impressing a swaying 
 
102 
 
 FORMS OF VIBRATION. 
 
 [V. § 49. 
 
 movement upon it with the hand near either ex- 
 tremity, in suitable time. According to the rapidity 
 of the motion thus communicated, the tube wUl take 
 up different forms of vibration. The simplest of these 
 is shown in Fig. 26. A and B being its fixed ex- 
 tremities, the tube vibrates as a whole between the 
 two extreme positions AaB and AbB. 
 
 Fig. 36 
 
 The tube may also vibrate in the form shown in 
 Fig. 27, where AahB and AcdB are its extreme 
 positions. 
 
 Fig.2.1 
 
 In this instance the middle point of the tube, 
 C, remains at rest, the loops on either side of 
 it moving independently, as though the tube were 
 fastened at C as well as at A and B. For this 
 reason the point C is called a node, from the Latin 
 nodus, a knot. The loops AC and CB are termed 
 ventral segments. 
 
 Fig. 28 shows a form of vibration with two nodes, 
 at C and D, dividing the distance AB into three 
 
V. § 50.] DIRECT AND REFLECTED PULSES. 103 
 
 equal ventral segments. We may also obtain forms 
 with three, four, five, &c., nodes, dividing the tube 
 into four, five, six, &c., equal ventral segments, re- 
 
 FigS.8. 
 
 spectively. The stifihess of very short portions of 
 the tube alone imposes a limit on the subdividing 
 process. Let us examine the mechanical causes to 
 which these effects are due. 
 
 50. If we unfasten one end of the tube, and, 
 holding it in the hand as in Fig. 29, raise a hump 
 upon it, by suddenly jerking the hand transversely 
 
 F1^. ZO ■ 
 
 i«a 
 
 to it through a small distance, the hump will run 
 along the tube until it reaches its fixed extremity B ; 
 it will then be reflected and run back to A, where it 
 will undergo a second reflexion, and so on. At each 
 reflexion the hump will have its convexity 7-eversed. 
 Thus, if while travelling from A towards B its form 
 was that of a, Fig. 30, on its way back it will have 
 
104 DIRECT AND REFLECTED PULSES. [V. § 50. 
 
 the form h. After reflexion at A, it wUl resume its 
 first form a, and so on. Now, instead of a single 
 jerk, let the hand holding the free end execute a 
 series of equal continuous trans vei'se vibrations. Each 
 complete vibration will cause a wave, ah Fig. 31, to 
 
 Fig. 31. 
 
 pass along the tube from A to B, where reflexion wUl 
 reverse the protuberances, so that the wave wiU re- 
 turn from B to A stern foremost. Next let the tube 
 be again fastened at both ends, as before, and the 
 vibrations of the hand impressed at some intermedi- 
 ate point, as C, Fig. 32. 
 
 □ i < - — c ^jT-iT ^^m 
 
 Two sets of waves will now start from C in the 
 directions of the arrows. They will be reflected at A 
 and B, and then their effects wiU intermingle. Let us 
 suppose that the tube has been set in steady motion 
 and, on the removal of the hand, continues its vibra- 
 tions without any external force acting on it. Two 
 sets of equal waves are now moving with equal 
 velocities from A towards B and from B towards 
 A, and we have to determine the motion of the 
 tube under their joint action. 
 
V. § 51.] FORMATION OF NODES. 105 
 
 Suppose that a crest' a, Fig. 33, moving from A 
 towards B, meets an equal trough' h, moving from B 
 towards A, at the point c. An undisturbed particle 
 of the tube situated at this point is solicited by equal 
 forces in opposite directions, and therefore remains at 
 
 rest. The two equal and opposite pulses then proceed 
 to cross each other, but, as a moves to the right and 
 6 to the left with equal speed, there is nothing to give 
 either of them at the point c an influence superior to 
 that exerted in the contrary direction by the other. 
 The particle at c therefore remains at rest under 
 their joint influence, i.e. a node is formed at that 
 point. If a trough had been moving from A to- 
 wards B, and an equal crest from B towards A, the 
 result would clearly have been the same : hence 
 
 A node is formed at every point where two equal 
 and opposite pulses, a crest and a trough, meet 
 each other. 
 
 51. The annexed figure represents two series of 
 equal waves advancing in opposite directions with 
 equal velocities. The moment chosen is that at 
 
 ' Provided that confusion with water-waves be explicitly 
 guarded against, there is no objection to retaining this convenient 
 phraseology for distinguishing between opposite protuberances. 
 
106 FORMATION OF NODES. [V. § 51. 
 
 which crest coincides with crest and trough with 
 trough. The joint effect thus produced does not 
 appear in the figure, our object at present being 
 merely to determine the number and positions of 
 the resulting nodes. For the sake of clearness, one 
 set of waves is represented slightly below the other, 
 though in fact the two are strictly coincident. 
 
 Let the waves ahdf...z be moving from left to 
 right, the waves z'ts'q'... a' frora right to left. The 
 crest klm meets the trough pn'm at m. After these 
 have crossed each other, the trough ghk and the crest 
 rq'p will also meet at m, since km and pm are equal 
 distances. Similarly the crest efg and the trough 
 ts'r will meet at m. Accordingly the point in is a 
 node, and, by exactly the same reasoning, so are a, c, 
 e, g, h, p, r, t, &c. The distances between pairs of 
 consecutive nodes are all equal, each being a single 
 pulse-length, i.e. half a wave-length, of either series. 
 
 Two pulse-lengths, as gk and km, give three nodes 
 g, k, and m ; three pulse-lengths four nodes, and so 
 on. There is thus always one node more than the 
 number of pulses. On the other hand, the fixed ends 
 of the tube, which are the origins of the systems of 
 
V. §52.] NATURE OF SEGMENTAL VIBRATION. 107 
 
 reflected waves, occupy two of these nodes. Deduct- 
 ing them we arrive at this result : — 
 
 The number of nodes formed is one less than the 
 number of the pulse-lengths {or half wave-lengths), 
 ivhich together make up the length of the vibrating 
 tube. 
 
 52. We will now ascertain how the portions of 
 the tube between consecutive nodes move while the 
 
 Tig. 35. 
 
 (O 
 
 (0 
 
 (0 
 
 « 
 
 CO 
 
 two systems of waves are simultaneously passing 
 along it. Let A and B, Fig. 35, be the fixed ends, as 
 before, and let us take five nodes at the points 1,2, 
 3, 4, 5. In (1), the systems of waves coincide, ac- 
 cordingly each point of the tube is displaced through 
 
108 RATE OF SEGMENTAL VIBRATION. [V. § 52. 
 
 twice as great a distance as if it had been acted on 
 by only one system. The tube thus takes the form 
 indicated by the strong line in the figure. In (2), 
 one set of waves has moved half a pulse-length to 
 the right, and the other the same distance to the left. 
 The two systems are now in complete antagonism, 
 the displacements being equal in amount and opposite 
 in direction at every point. The tube is therefore 
 momentarily in its undisturbed position. In (3), each 
 system has moved through a pulse-length, and the 
 maximum displacements are again produced on the 
 tube, but in opposite directions to those of (1). In 
 (4), where the systems have moved through a pulse- 
 length-and-a-half, the tube passes again through its 
 undisturbed position, and, in (5), regains that which 
 it occupied in (l), the systems of waves, meanwhile, 
 having each traversed two pulse-lengths, or one wave- 
 length \ Thus the tube executes one complete vibra- 
 tion in the time occupied by a pulse in passing along 
 a length of the tube equal to tivice one of its own 
 ventral segments. In other words, the tube's rate of 
 vibration varies as the number of segments into ivhich 
 it is divided. It moves most slowly in the form 
 shown in Fig. 26 with but a single segment ; twice 
 
 ' The reader will find that Fig. 35 is rendered more readily 
 intelligible by drawing the two systems of waves in difierent 
 odours, and the successive positions of the tube in black. 
 
V. § 53.] RATE OF SEGMENTAL VIBRATION. 109 
 
 as fast in that of Fig. 27, where it is divided into two 
 segments ; three times as fast with three segments, 
 and so on. It is easy to confirm this by direct ex- 
 periment, the swaying movement of the hand on 
 the tube needing to be twice as rapid for a form of 
 vibration with two segments as for a form with one, 
 and so on. 
 
 53. Instead of comparing the difierent rates at 
 which the same tube vibrates when divided into 
 different numbers of ventral segments, we may com- 
 pare the rates of vibration of tubes of different 
 lengths divided into the same number of segments. 
 
 Let us take as an example the two tubes AB, 
 CD, Fig. 36, each divided by three nodes into four 
 
 Ai ! 1 1— .3 
 
 ''' 1 1 i 'o 
 
 ventral segments. By what has been already shown, 
 the time of vibration of either tube will be that 
 which a pulse occupies in traversing two of its ven- 
 tral segments. Therefore the time of vibration of 
 AB will be to that of CD as ^2 is to C2, i.e. as one 
 half of AB is to one half of CD, or as AB is to CD. 
 This reasoning is equally applicable to any other 
 case. Accordingly we have the general result that, 
 when tubes of different lengths are divided into the 
 
110 MOTION OF A SOUNDING STRING. [V. § 54. 
 
 same number of ventral segments, their times of 
 vibration are proportional to the lengths of the 
 tubes, or, which comes to the same thing, their rates 
 of vibration are inversely proportional to their lengths. 
 The reader should observe that it has been through- 
 out this discussion assumed that the material, thick- 
 ness and tension of the tube, or tubes, in question 
 were subject to no variation whatever. Any changes 
 in these would correspondingly affect the rates of 
 vibration produced, but according to less simple laws 
 than variation in length only. 
 
 54. We are now prepared to examine the motion 
 of a sounding string. Its ends are fastened to fixed 
 points of attachment and the string is excited at 
 some intermediate point, by plucking it with the 
 finger, as in the harp and guitar, by striking it with 
 a soft hammer, as in the pianoforte, or by stroking it 
 with a resined bow, as in the violin and other instru- 
 ments of the same class. The impulses thus set up 
 are reflected at the extremities of the string (in the 
 violin at the bridge and at the finger of the per- 
 former) and behave towards each other exactly as 
 in the case of the vibrating tube considered above. 
 The results there obtained are, accordingly, at once 
 applicable to the case before us. The string may 
 vibrate in a single segment as in Fig. 26. This is 
 the form of slowest vibration with a string of given 
 
V. §54.J POINT OF PERCUSSION. ill 
 
 length, material and tension. Accordingly, when 
 thus vibrating, the string produces the deepest note 
 of which, all other conditions remaining the same, 
 it is capable. The string may also vibrate in the 
 forms shown in Figs. 27, 28, 35, 36, or in forms 
 with larger numbers of segments. The rapidity of 
 vibration in any one of these forms is, as we have 
 seen [§ 52], proportional to the number of segments 
 formed, so that, with two segments, the string 
 vibrates twice, with three, thrice, with four, four 
 times, as fast as in the form with one segment. It 
 follows hence [§ 43] that the notes obtained by 
 causing a string to vibrate successively in forms 
 with 1, 2, 3, 4, 5, &c., segments are all partial- 
 tones of one compound sound, the lowest being of 
 course its fundamental-tone. 
 
 The modes of eliciting the sounds of stringed 
 instruments described on the preceding page are 
 not capable of setting up any one of the above forms 
 of vibration by itself, but give rise to a movement 
 which is the resultant of several such vibration- 
 forms compounded together. Each separate vibra- 
 tion-form thus called into existence sings, as it were, 
 its own note, without heeding what is being done 
 by its fellows. Accordingly, a certain number of 
 tones belonging to one family of partial-tones are 
 simultaneously heard. 
 
112 MODE OF PERCUSSION. [V. § 54. 
 
 How many, and which, members of the general 
 series of partial-tones are present, and with what 
 relative intensities, in the sound of a string set 
 vibrating by a blow, depends on the position of 
 the point at which the blow is delivered, on the 
 nature of the striking object, and on the material 
 of the string. It is clear that a node can never 
 be formed at the point of percussion. Therefore no 
 partial-tone requiring for its production a node in 
 that place can exist in the resulting sound. If, 
 for instance, we excite the string exactly at its 
 middle point, the forms of vibration with an even 
 number of ventral segments, all of which have a node 
 at the centre of the string, are excluded, and only the 
 odd partial-tones, i.e. the 1st, 3rd, 5th, and so on, 
 are heard. In this manner we can always prevent 
 the formation of any assigned partial-tone, by choosing 
 a suitable point of percussion. On the other hand, a 
 vibration-form is in the most favourable position for 
 development when the middle 'point of one of its 
 ventral segments coincides with the point of per- 
 cussion. The more nearly it occupies this position 
 the louder will be the corresponding partial-tone ; 
 while the further it recedes from this position towards 
 that in which one of its nodes falls on the point of 
 percussion, the weaker will the partial-tone become. 
 
 The form and material of the hammer, or other 
 
V. §55.] PITCH OF STRING-SOUNDS. 113 
 
 object with which the string is struck, have also a 
 great influence in modifying the quahty of the sound 
 produced. Sharpness of edge and hardness of sub- 
 stance tend to develope high and powerful over- 
 tones, a rounded form and soft elastic substance 
 to strengthen the fundamental-tone. The material 
 of the string itself produces its effect chiefly by 
 limiting the number of partial-tones. Its stiffness 
 resists division into very short segments, and this 
 implies, for every string, a fixed limit beyond which 
 further subdivision ceases and where, therefore, the 
 series of overtones is cut off. Hence very thin mobile 
 strings are favourable, thick weighty strings un- 
 favourable, to the production of a large number of 
 partial-tones. 
 
 55. Having examined what determines the quality 
 of the sound of a vibrating string, we have next to 
 enquire on what its pitch depends. This term is 
 indeed, strictly speaking, inappropriate to a compo- 
 site sound containing a series of different tones each 
 having its own vibration-number and independent 
 position in the musical scale. If, however, we use 
 the phrase ' pitch of a sound' as equivalent to ' pitch 
 of the fundamental-tone of the sound,' we shall avoid 
 any confusion arising from this circumstance. The 
 pitch of a string-sound depends, of course, on the rate 
 at which the string is vibrating. We have seen 
 
 T. 8 
 
114 LENGTHS AND INTERVALS. [V. § 55. 
 
 that when the material, thickness and tension of a 
 string remain the same, its rate of vibration varies 
 inversely as the length of the string. Accordingly, 
 the vibration-number of a string-sound varies inversely 
 as the length of the string. It follows hence that the 
 relations which connect the vibration-numbers of 
 sounds forming given intervals with each other, hold 
 equally for the lengths of the strings by which those 
 sounds are produced. To verify this by experiment 
 we have only to stretch a wire between two fixed 
 points A and B, Fig. 36 bis, and divide it into two 
 segments by applying a bridge to it at some inter- 
 mediate point C. If AC bears to CB any one of 
 the simple numerical ratios exhibited in the table on 
 
 Fig. 36 {bis.) 
 
 I 1 B 
 
 A C 
 
 p. 71, we obtain the corresponding interval there 
 given by alternately exciting the vibrations of the 
 two segments at any pair of points in -4 C and CB 
 respectively. Thus, if CB is twice as long as AC, 
 the sound produced by the former wUl be one Octave 
 lower than that produced by the latter. If A C is to 
 CB in the proportion of 2 to 3, AC's sound will be 
 a Fifth above CB's ; and similarly in other cases. It 
 was by experiments of this kind that the ancient 
 Greek philosopher Pythagoras is said to have made 
 
V. §55.] LENGTHS AND INTERVALS. 115 
 
 out the existence of a connexion between certain 
 musical intervals and the ratios of certain small 
 integers. Thus an Octave- vfas produced by a string 
 divided into two parts in the proportion of 2 to 1 ; 
 a Fifth was obtained by division in the proportion 
 of 3 to 2, and so forth. The relations existing 
 between these lengths and the vibration-numbers of 
 the notes produced by them were, however, entirely 
 unknown to Pythagoras and his contemporaries ; 
 indeed it was not until the seventeenth century that 
 they were discovered, by Galileo. 
 
 In instruments of the violin class, the pitch of 
 the notes varies according to the position of the 
 finger on the vibrating string. The length of string 
 intercepted between the fixed bridge and the finger 
 admits of being altered at pleasure, and thus every 
 shade of pitch can be produced from such instruments. 
 The resined bow maintains the vibration of the string 
 by alternately dragging it out of its position of rest, 
 letting it fly back again, catching it once more, and 
 so on. The hollow cavity of the instrument rein- 
 forces the string-sound by resonance. The quality 
 of instruments of the violin class is vivacious and 
 piercing. The first eight partial-tones are well 
 represented in their sounds. 
 
 8—2 
 
116 THE PIANOFORTE. ["V. § 56. 
 
 The Pianoforte. 
 
 56. In this instrument each wire is stretched 
 between two pegs, which are fixed into a flat plate 
 of wood called the sound-board. The string is fast- 
 ened to one peg, and coiled round the other, which 
 admits of being turned about its own axis by means 
 of a key of suitable construction. This allows the 
 string to be accurately tuned, since by tightening 
 or loosening the wire we raise or lower its pitch 
 at pleasure. In small instruments two, in larger 
 ones three, wires in unison with each other usually 
 correspond to each note of the key-board. While 
 the pianoforte is not in action a series of small 
 pieces of wood covered with list, called ' dampers,' 
 rest upon the wires. These are connected with the 
 key-board in such a manner that, when a note is 
 pressed down, the corresponding damper rises from 
 its place, and the wires it previously covered remain 
 free until the note is allowed to spring up again, 
 when the damper immediately sinks back into its 
 original position. Each note is connected with an 
 elastic hammer, which deals a blow to its own set of 
 wires and then springs back from them. The wires 
 thus set in motion continue to vibrate until either 
 the sound gradually dies away, or is abruptly ex- 
 tinguished by the descent of the damper. The 
 
V. § 56.] ACTION OF PEDALS. 117 
 
 action of the two pedals is as follows : the soft 
 pedal shifts the hammers in such a way that each 
 hammer only acts on one, or on two, of the wires cor- 
 responding to it, instead of on its complete set of two, 
 or of three, wires. The sound produced by striking 
 a note is therefore proportionately weakened. The 
 loud pedal lifts all the dampers off the wires at once. 
 It thus not only allows notes to continue sounding 
 after the fingers of the player have quitted them, but 
 places wires other than those actually struck in a 
 position to sound by resonance. The number of 
 wires thus brought into play by striking a single 
 note of the instrument will be easily seen to be con- 
 siderable. Suppose, first, that a simple tone, e.g. 
 that of a tuning-fork, is sounding near the wires of 
 a pianoforte with the loud pedal down, its pitch being 
 
 that of middle C, ^ ^ ; the wires of the corre- 
 
 sponding note will of course resonate with it, vibrat- 
 ing in the simplest form with only one ventral seg- 
 
 ment. The wires of the note ^zzscrz , one Octave 
 
 below it, are also capable of producing middle C 
 when they vibrate in the form with two segments. 
 
 So are those of ^ , a Twelfth below it, when 
 
 m 
 
 vibrating with three segments ; those of ;::;3;:^:^ , two 
 
118 QUALITY OF PIANOFORTE SOUNDS. [V. § 56. 
 
 Octaves below it, vibrating with four segments, and 
 so on. Proceeding in this way, we determine a series 
 of notes on the key-board of the pianoforte the wires 
 of which are able to produce a simple tone of the 
 pitch of middle C They obviously follow the same 
 law as the harmonic overtones of a compound sound 
 with middle C for its fundamental-tone, except that 
 the successive intervals are reckoned downwards 
 instead of upwards. The wires of all these notes 
 will reinforce the tone of the tuning-fork by reso- 
 nance. If now we remove the fork and strike middle 
 C on the pianoforte itself we obtain of course a 
 sound consisting of a number of simple tones. To 
 each of these latter there corresponds a descending 
 series of notes on the key-board determined in the 
 manner just explained. A full chord struck in the 
 middle region of the instrument will, therefore, 
 command the more or less active services of many 
 more wires than have been set vibrating by direct 
 percussion. The increase of loudness thus secured 
 is not very considerable, the effect being rather a 
 heightened richness, like that of a mass of voices 
 singing pianissimo. The sustaining power of the loud 
 pedal renders care in its employment essential. It 
 should, as a general rule, be held down only so long 
 as notes belonging to one and the same chord are 
 struck. Whenever a change of harmony occurs, the 
 
V. § 56.] ACTION OF SOUND-BOARD. 119 
 
 pedal should be allowed to rise, in order that the 
 descent of the dampers may at once extinguish the 
 preceding chord. If this precaution is neglected, 
 perfectly irreconcilable chords become promiscuously 
 jumbled together, and a series of jarring discords 
 ensue which are nearly as distressing to the ear as 
 the actual striking of wrong notes. The quality of 
 pianoforte notes varies greatly in different parts of the 
 scale. In the lower and middle region it is full and 
 rich, the first six partial-tones being audibly present, 
 though, 4, 5, 6 are much weaker than 1, 2, 3. 
 Towards the upper part of the instrument the higher 
 partial-tones disappear, until in the uppermost Octave 
 the notes are approximately simple-tones, which 
 accounts for their tame and uninteresting character. 
 The pianoforte shares with all instruments of fixed 
 sounds certain serious defects which will be discussed 
 in detail in a subsequent chapter. 
 
 When a vibrating wire is passing through its 
 undisturbed position, its tension is necessarily some- 
 what less than at any other moment, since, in order 
 to assume the curved segmental form, it must be a 
 little elongated, which involves a corresponding in- 
 crease of tension. Hence the two pegs by which the 
 ends of a wire are attached to the sound-board are 
 submitted to an additional strain twice during each 
 complete segmental vibration. The sound-board. 
 
120 SOUNDS OF OBQAN-PIPES. [V. § 57. 
 
 being purposely constructed of the most elastic wood, 
 yields to the rhythmic impulses acting upon it, and 
 is thrown into segmental vibrations coincident in 
 period with those of the wire. 
 
 These vibrations are communicated to the air in 
 contact with the sound-board, and then transmitted 
 further in the ordinary way. The amount of surface 
 which a wire presents to the air is so small that, but 
 for the aid of the sound- board, its vibrations would 
 hardly excite an audible sound. The reader will not 
 fail to notice that the sound-board of the pianoforte 
 plays the same part as the hollow cavity of the viohn, 
 and is in fact a solid resonator. In the harp, the 
 framework of the instrument serves the same purpose. 
 We have, in this combination of a vibration-exciting 
 apparatus with a resonator, the type of construction 
 adopted in nearly all musical instruments. 
 
 3. Sounds of organ-pipes. 
 
 57. It has been shown [§51] that, when two 
 series of equal waves due to transverse vibrations 
 travel along a stretched wire in opposite directions, 
 nodes are formed at equal distances along it, sepa- 
 rated by ventral segments of equal lengths. Let us 
 now suppose that two series of equal waves due to 
 longitudinal vibrations are traversing, in opposite 
 directions, a column of air contained in a tube of 
 
V. §58.] 'STOPPED' AND 'OPEN' ORGAN-PIPES. 121 
 
 uniform bore. Each set of such waves has its own 
 associated wave-form [§ 18, p. 39]. We have only, 
 therefore, to consider the curves drawn in Fig. 35 as 
 associated waves corresponding to longitudinal air- 
 vibrations, in order to make the conclusions of § 52 
 at once applicable to the case in hand. The result 
 is a series of equidistant nodes, or points of per- 
 manent rest, distributed along the air-column. The 
 intervening portions of air vibrate longitudinally 
 at the same rate as the corresponding ventral 
 segments of Fig. 35. We have here, as in the 
 case of the sounding wire, all the conditions for 
 the production of a musical note of pitch correspond- 
 ing to the rapidity of vibration obtained. It only 
 remains to show that, in the case of every organ- 
 pipe, two sets of equal waves traverse in opposite 
 directions the air-column which it contains. 
 
 Organ-pipes are of two kinds, called respectively 
 * stopped ' and ' open,' — terms which, however, apply 
 only to one end of the pipe ; the other is in both 
 kinds open. 
 
 To begin with first variety : 
 
 c ^37 
 
 58. Let AB, Fig. 37; be the closed end of a 
 
122 REFLEXION AT A CLOSED ORIFICE. [V. § 58. 
 
 stopped pipe, and let a series of pulses of condensa- 
 tion and rarefaction be passing along the air within 
 it in the direction shown by the arrow. First, let a 
 pulse of condensation, CABD, have just reached AB. 
 By supposition, the air in CABD is denser, and there- 
 fore at a higher pressure, than that behind it. It 
 win therefore expand. Forward motion being barred 
 by AB, the expansion must take place entirely in 
 the opposite direction. Hence the pulse of conden- 
 sation is reflected at the end of the pipe, and pro- 
 ceeds to describe its previous course in the reverse 
 direction. Next, suppose CABD to be a pulse of 
 rarefaction. The air in it is at a less pressure than 
 that of the air behind it. As there is a fixed 
 obstacle in front, this rarefaction can only be filled 
 up from behind. The condensed air behind will 
 expand more during the process, and become itself 
 more rarefied, than if there had been no obstacle. 
 A rarefied pulse will therefore return along the pipe. 
 Thus a pulse of rarefaction, equally with one of 
 condensation, is reflected at the closed end of the 
 pipe. Neither pulse suffers any change other than 
 reversal of its direction of motion. Since every 
 pulse is thus regularly reflected at AB, and made 
 to travel back unchanged along the pipe, it follows 
 that a system of equal waves advancing in the 
 direction of the arrow are necessarily met by an 
 
V. § 58.] REFLEXION AT AN OPEN ORIFIGE. 123 
 
 exactly equal system proceeding in the opposite 
 direction. For stopped pipes, therefore, the point 
 required to be proved is made out. 
 
 Let AB, Fig. 38, be one end of an open pipe 
 along which condensed and rarefied pulses are being 
 alternately transmitted in the direction of the arrow. 
 First, let CABD be a pulse of condensation which 
 has just reached AB. If the pipe had continued to 
 the left beyond AB, the particles of air near AB 
 
 C FigSZ 
 
 Af 
 
 B' 
 
 D 
 
 would have moved in the direction of the arrow tUl 
 they were stopped by the increasing pressure in 
 their front. This would correspond to the instant 
 of greatest density at AB. They would then begin 
 to move in the direction opposite to that of the 
 arrow, the pressure in front, i.e. on the left in the 
 figure, being greater than that behind. As, how- 
 ever, the pipe is open and the air free to move in 
 all directions, the condensation just outside its 
 mouth is never great : the particles will move further 
 in the direction of the arrow before being stopped 
 than they would do in the case of a continuous tube, 
 and will, therefore, leave behind them a region in 
 which the air is more rarefied than it would have 
 
124 REFLEXION AT AN OPEN ORIFICE. [V. § 58. 
 
 been at the same time in the corresponding part of 
 the continuous tube. This region of rarefied air 
 will originate a pulse of rarefaction, which will be 
 transmitted in the direction opposite to that of the 
 arrow. Conversely, when a pulse of rarefaction 
 reaches the open end of the pipe, the rarefaction 
 just within its mouth cannot become great, since 
 the air is free to enter from all sides. The particles 
 behind, which in the case of a continuous tube would 
 move in the direction of the arrow and diminish the 
 rarefaction, wdl therefore be sooner stopped : the 
 pressure and density will be greater than at the 
 corresponding place and time in the continuous 
 tube : and thus a pulse of condensation will be 
 produced and transmitted back along the pipe. 
 
 To look at the matter from a somewhat different 
 point of view, the motion of the air in the open pipe 
 wUl be practically the same as it would have been 
 if the pipe had formed part of a continuous tube, but 
 air had been removed at the point corresponding to 
 the opening whenever a pulse of condensation arrived 
 there, and air had been injected at the same point 
 whenever a pulse of rarefaction reached it. 
 
 By the above reasoning^ which the student should 
 
 ' I am indebted for this popular explanation of reflexion at 
 an open orifice to the late Mr Coutts Trotter, formerly Senior 
 Fellow of Trinity College, Cambridge. 
 
V. § 59.] YIBEATION-FORMS FOR A STOPPED PIPE. 125 
 
 carefully compare with that of § 21, it is clear that 
 reflexion takes place at an open as well as at a closed 
 end of a pipe : with this important difference, how- 
 ever, that, in the case of the open orifice, condensation 
 is turned into rarefaction and rarefaction into con- 
 densation, so that the wave returns hind part fore- 
 most. 
 
 59. We will next examine what forms of seg- 
 mental vibration the air in a stopped pipe can adopt. 
 Every such form must necessarily have a node coin- 
 cident with the closed end of the pipe, since no longi- 
 tudinal vibrations are possible there. The impulses 
 constituting the series of direct waves are not, as we 
 shall see presently, originated, like those of a piano- 
 forte string, at some intermediate point, but enter 
 the pipe at its open end. This must therefore be 
 a point of maximum vibration. Now a glance at 
 Fig. 35 shows that the maxima of vibration are at 
 the middle points of the ventral segments. Hence 
 the centre of a segment must coincide with the open 
 end of the pipe. 
 
 The above considerations suffice to solve the 
 problem before us. If the closed end of the pipe is 
 placed at A, Fig. 35(1), the open end must be midway 
 between^ and 1, or between 1 and 2, 2 and 3, 3 and 
 4, and so on. No other form of vibration is possible. 
 
 Fig. 39 shows the air in a stopped pipe of 
 
126 VIBRATION-FORMS FOR A STOPPED PIPE. [V. § 59. 
 
 given length vibrating in four such ways. The 
 vertical lines indicate the positions of the nodes. 
 For the sake of greater clearness, the loops of the 
 associated vibration-forms are in each case drawn in 
 dotted lines. 
 
 (a) 
 
 (b) 
 
 (0 
 
 (°) 
 
 
 In [A) we have half a segment ; in {B) a segment 
 and a half; in (C) two segments and a half; in {D) 
 three segments and a half The numbers of seg- 
 ments into which the air-column is divided in these 
 four cases are, therefore, proportional to 
 i, li, 2^, 3i, 
 i.e. to I, f, I, f, 
 or to the whole numbers 1, 3, 5, 7. 
 
 Now by § 53 it appears that the rate of vibration 
 
V. § 60.] VIBRA T ION-FORMS FOR AN OPEN PIPE. 1 27 
 
 in any form varies as the number of segments 
 which it contains. The vibration-numbers of the 
 sounds produced in the present instance are, therefore, 
 proportional to 1, 3, 5, 7, respectively, i.e. we get the 
 first four odd partial-tones of a sound of which {A) 
 gives us the fundamental-tone [§ 43]. The reasoning 
 here adopted evidently applies equally well to cases 
 in which the air-column is subdivided to any assigned 
 extent. It follows, therefore, that the notes obtain- 
 able from a stopped pipe are all odd partial-tones 
 belonging to one and the same clang. 
 
 60. The case of the open pipe shall next be 
 investigated. Here, as in the previous case, the 
 centre of a ventral segment must coincide with the 
 end of the pipe at which the direct pulses enter. 
 The considerations alleged in § 58 indicate that the 
 same thing must also hold good at the opposite 
 orifice'. 
 
 Referring once more to Fig. 35 (1), we obtain all 
 the possible modes of vibration which satisfy both the 
 above conditions by placing one end of the pipe mid- 
 way between A and 1, and the other successively half 
 way between 1 and 2, 2 and 3, 3 and 4, and so on. 
 
 ' I here assume, for the sake of simplicity, what is not 
 rigorously true, viz. that reflexion at an open end of a pipe is as 
 complete as at a stopped end. It is however approximately true 
 for the pipes employed for musical purposes, whose transverse 
 dimension is small compared to their length. 
 
] 28 VIBRATION-FORMS FOR AN OPEN PIPE. [V. § 60. 
 
 The first four of the cases thus obtained, for a tube 
 of constant length, are shown in the next figure, 
 which is drawn on precisely the same plan as Fig. 39. 
 In each case, the two half-segments at the ends 
 of the pipe make up one whole segment. The num- 
 bers of segments into which the air-column is divided 
 are, therefore, in {A), 1; in [B), 2; in (C), 3; in 
 (Z)), 4. The same law would obviously hold for 
 higher subdivisions. Hence, in the case of an open 
 pipe, the rates of all the possible modes of segmental 
 
 (.A) 
 
 Fig.40. 
 
 (P) 
 
 (O 
 
 (^) 
 
 V 
 
 vibration are as the numbers 1, 2, 3, 4, 5, &c. The 
 notes obtainable from such a pipe are, therefore, 
 the complete series of partial-tones belonging to one 
 and the same clang. 
 
V. §§ 61, 62.] PITCH AND LENGTH OF PIPE. 129 
 
 61. If the slowest forms of vibration, shown at 
 {A) in Figs. 39 and 40, are compared with each other, 
 it will be at once seen that the ventral segment in 
 Fig. 39 is exactly twice as long as that in Fig. 40. 
 Hence, the gravest tone obtainable from a stopped 
 pipe is always one Octave lower than the gravest 
 tone producible from an open pipe of the same 
 length. It has been shown in § 39 that this result of 
 theory is borne out by experiment. 
 
 62. In order to complete this investigation, it is 
 necessary to determine the pitch of the lowest note 
 which a pipe of given length is capable of uttering. 
 By § 52 we know that a complete segmental vibra- 
 tion is performed during the time occupied by a 
 pulse in traversing twice the length of a single seg- 
 ment. In [A) Fig. 39, this is equal to four times the 
 length of the tube. The velocity of the pulse is here 
 the velocity of Sound in air, which, under ordinary 
 conditions of temperature, &c., we may put at 1125 
 feet per second'. The vibration-number of a stopped 
 pipe's lowest tone is therefore found by dividing 
 1125 by four times the length of the pipe expressed 
 in feet. Conversely, the length of a stopped pipe 
 which is to have as its deepest tone a note of given 
 pitch, is found by dividing 1125 by four times the 
 
 ' Tyndall's Sovmd, p. 23. 
 
130 PITCH AND LENGTH OF PIPE. [V. §62. 
 
 vibration-number of the note to be produced. The 
 quotient gives the required length in , feet. For 
 example, middle C of the pianoforte makes 264 
 vibrations per second. The required length in this 
 
 case would be expressed by , which is rather 
 
 more than 1 ft. -^in., i.e. roughly speaking, one foot. 
 An open pipe, to produce the same note, would there- 
 fore have to be two feet in length. 
 
 It has been just shown that the vibration-number 
 of the lowest tone producible, either from a stopped 
 or an open pipe, varies inversely as the length of the 
 pipe. Hence the relations established in § 55 for 
 strings hold also for columns of air contained in pipes. 
 The case of the pipe-sounds is, however, somewhat 
 simpler than that of the string-sounds, since the 
 pitch of the latter depends on the tension of the 
 strings as well as on their lengths, whereas, in the 
 former, pitch depends, under given atmospheric con- 
 ditions, on length alone. Hence we may define a 
 note of assigned pitch by merely stating the length 
 of the stopped, or open, pipe whose fundamental 
 tone it is. The open pipe is commonly preferred for 
 this purpose, and accordingly organ builders call 
 middle C ' 2-foot tone ;' the Octave below it ' 4-foot 
 tone,' and so on. The lowest C on modern piano- 
 fortes is ' 1 6 -foot tone ;' that one Octave lower, which 
 
V. § 63.] PITCH AND LENGTH OP PIPE. 131 
 
 is found only on the very largest organs, * 32-foot 
 tone.' The highest note of the pianoforte, usually A, 
 would be about ' 2 -inch tone.' 
 
 63. The reader should observe that, in the course 
 of this discussion, we have incidentally obtained a 
 more complete theory of resonance than could be 
 given in chapter iii. When a tuning-fork is held at 
 the orifice of a tube, the strongest resonance will be 
 produced if the note of the fork coincides with the 
 fundamental tone of the tube. A decided, though 
 less powerful, resonance ought also to ensue if the 
 fork-note coincides with one of the higher tones of 
 the tube, which, as we know, are all overtones of its 
 fundamental. A resonance-box is only a stopped' 
 pipe under another name. We may therefore eraploy 
 it to test the truth of our result, that the only tones 
 obtainable from a stopped pipe are the odd partial- 
 tones of a clang of which the first is the fundamental 
 tone. I possess a series of forks giving the first seven 
 partial -tones of a clang. When I strike 1, 3, 5 or 7, 
 and hold them before the open end of the stopped 
 resonance-box corresponding to 1, a decided rein- 
 forcement of their tones is heard. If I do the 
 same with 2, 4, or 6, hardly any resonance is pro- 
 
 ' For forks of high pitch it is, however, found best to use 
 resonance-boxes open, at both ends, and therefore corresponding 
 to open pipes. 
 
 9—2 
 
132 CONSTRUCTION OF FLUE-PIPES. [V. §§ 64, 65. 
 
 duced. Thus our theoretical result is experimentally- 
 verified. 
 
 64. Organ-pipes are divided into two classes 
 according as their sounds are originated : 
 
 (1) by blowing against a sharp edge, 
 
 (2) by blowing against an elastic tongue. 
 Those of the first class are called _/?Me-pipes ; those 
 
 of the second class reecZ-pipes. We will consider 
 each class by itself. 
 
 65. Flue-pipes. Here the wind is driven through 
 a narrow slit against a sharp edge placed exactly 
 opposite to it, in the manner shown in Fig. 41, which 
 represents a vertical section of a portion of the pipe 
 near the end at which its sound originates. 
 
 FyAl. 
 
 The air is forced by the bellows through the tube 
 
V. § 65.] QUALITY OF SOUNDS OF FLUE-PIPES. 133 
 
 ab into the chamber c, and escapes through the slit 
 d, thus impinging against the edge e, and exciting in 
 the air-column above segmental vibrations whose 
 period is controlled by the length of the pipe. 
 
 The regular musical note thus produced is ac- 
 companied by a hissing sound, which may be imitated 
 by blowing with the mouth against a knife-edge held 
 in front of it. This sound however contributes 
 nothing to the general effect, being, where a pipe 
 is properly constructed, inaudible except in its im- 
 mediate neighbourhood. 
 
 Stopped wooden flue-pipes of large aperture, 
 blown by only a light pressure of wind, produce 
 sounds which are nearly simple tones ; only a trace 
 of partial tone No. 3 being perceptible. Such tones, 
 like the fork-tones with which they are in fact 
 almost identical, sound sweet and mild, but also 
 tame and spiritless. A greater pressure of wind 
 developes 3 distinctly in addition to 1, and, if it 
 becomes excessive, may spoil the quality by giving 
 the overtone too great an intensity compared to that 
 of the fundamental, or may even extinguish the 
 latter altogether, and so cause the whole sound to 
 jump up an Octave and a Fifth. This result may 
 easily be obtained by blowing with the mouth into a 
 small 6-inch stopped pipe, which can readily be pro- 
 cured at any organ factory. 
 
134 'REED'-PIPES. [V. §66. 
 
 Stopped pipes of narrow aperture develope the 
 third and fifth partial-tones with distinctness. 
 
 In the case of an open pipe the fundamental- tone 
 is never produced by itself According to the dimen- 
 sions of the pipe, and the pressure of wind, it is 
 accompanied by from two to five overtones. Open 
 flue-pipes present, therefore, various degrees of quality 
 which are exhibited in different 'stops' of a large 
 organ. 
 
 66. Reed-pipes. The apparatus by which the 
 sounds of pipes of this class are originated is 
 the following. One end of a thin narrow strip of 
 elastic metal, called a 'tongue,' is fastened to a 
 brass plate, while the other end is free. A rect- 
 angular aperture, very slightly larger than the 
 tongue, is cut through the plate, so as to allow the 
 tongue to swing into and out of the aperture, like 
 a door with double hinges, without touching the 
 edges of the aperture as it passes them. The accom- 
 panying figure shows this piece of mechanism, which 
 is called a ' reed,' in its position of rest. 
 
 FisiS 
 
 It is set in motion by a current of air being driven 
 against the free end of the tongue, which is thus 
 
V. § 66.] MECHANISM OF A REED. 135 
 
 made to swing between limiting positions as shown 
 in the annexed sections. 
 
 Tig.4,3 . 
 
 When the tongue occupies any position inter- 
 mediate between that of {A) and its position of equi- 
 librium, air is blown through the aperture in the 
 direction indicated by the arrows in {A). From the 
 moment that the tongue passes through its equili- 
 brium-position towards that shown in {B), the current 
 of air is barred by the accuracy with which the tongue 
 fits into the aperture beneath it. Only when the 
 tongue again emerges can the air resume its passage. 
 
 A series of equal impulses of air are thus produced 
 at equal intervals of time, and the instrument, there- 
 fore, gives rise to a regular musical sound. Its clang 
 is highly composite, containing distinctly recognisable 
 partial-tones up to the 16th or 20th of the series. 
 Thus a reed does not need to be associated 
 with a resonating column in order to produce a 
 musical sound ; in fact the instrument called the 
 harmonium consists of reeds without such adjuncts. 
 The quality of an independent reed is, however, 
 characterised by too great intensity on the part 
 of the higher partial-tones. It is desirable to correct 
 
136 QUALITY OF REED-SOUNDS. [V. § 66. 
 
 this defect by strengthening the fundamental-tone of 
 the clang. This is done by placing the reed in the 
 mouth of an open pipe whose deepest tone coincides 
 with the fundamental-tone of the reed- clang. This 
 tone will then be most powerfully reinforced by 
 resonance. The other partial-tones of the clang wUl 
 also be strengthened by resonance, but to a smaller 
 and smaller extent as their order rises. The force 
 required to throw a column of air into rapid vibration 
 is greater than that which suffices to set up a slow 
 vibration. Since the force necessary to produce seg- 
 mental vibration increases very rapidly as the sub- 
 divisions of the air-column become more numerous, the 
 very high partial-tones of the reed-clang are practi- 
 cally unsupported by the resonance of the associated 
 pipe. We shall see in the sequel how it happens 
 that the quality of the resulting sound is improved 
 by this circumstance. 
 
 It is clear that sounds varying widely in quality 
 may be obtained by associating a reed with pipes of 
 varying lengths and forms. The resulting quality 
 will be quite different according as the lowest tone 
 which the pipe is capable of strengthening by reso- 
 nance coincides with the fundamental-tone, or with 
 one of the overtones, of the reed-sound. The form 
 of the pipe may also be modified so as to be conical 
 or of any other shape, which will bring in other 
 
V. §§ 67, 68.] ORCHESTRAL WIND-INSTRUMENTS. 137 
 
 changes in its resonating properties. In these ways 
 we have provision for the great variety of quaUty 
 among reed-pipes which we find represented in organ 
 stops of that class. 
 
 4. Sounds of orchestral wind-instruments and of 
 the human voice. 
 
 67. The fiute is in principle identical with an 
 open flue-pipe. The lips, and a hole near the end 
 of the tube, play the parts of the narrow slit and 
 opposing edge. The quality of the instrument is 
 sweet, but too nearly simple to be heard during a 
 long solo without becoming wearisome. Its best 
 effects are produced by contrast with the more 
 brilliant quality of its orchestral colleagues. 
 
 The clarionet, hauthois and bassoon have wooden 
 reeds. 
 
 In the horn and trumpet the lips of the per- 
 former supply the place of a reed. 
 
 68. The apparatus of the human voice is essen- 
 tially a reed (the vocal chords) associated with a 
 resonance-cavity (the hollow of the mouth). 
 
 The vocal chords are elastic bands situated at 
 the top of the windpipe, and separated by a narrow 
 slit, which opens and closes again with great exact- 
 ness as air is forced through it from the lungs. The 
 
138 MECHANISM OF THE HUMAN VOICE. [V. § 68. 
 
 form and width of the sht allow of being quickly 
 and extensively modified by the changing tension of 
 the vocal chords, and thus sounds widely differing 
 in pitch may be successively produced with sur- 
 prising rapidity. In this respect, the human 
 ' reed ' far exceeds any that we can artificially 
 construct. 
 
 The size and shape of the cavity of the mouth 
 may be altered by opening or closing the jaws, 
 raising or dropping the tongue and tightening or 
 loosening the lips. We should expect that these 
 movements would not be without effect on the reso- 
 nance of the contained air, and this proves, on 
 experiment, to be the fact. If we hold a vibrating 
 tuning-fork close to the lips, and then modify the 
 resonating cavity in the ways above described, we 
 shall find that it resounds most powerfully to the 
 fork selected when the parts of the mouth are in one 
 definite position. If we try a fork of different pitch, 
 the attitude of the mouth, for the strongest resonance, 
 is no longer the same. 
 
 Hence, when the vocal chords have originated 
 a reed-sound containing numerous well-developed 
 partial-tones, the mouth-cavity, by successively 
 throwing itself into different postures, can favour 
 by its resonance first one partial-tone, then another ; 
 at one moment this group of partial-tones, at another 
 
V. § 69.] OVERTONES OF THE HUMAN VOICE. 139 
 
 that. In this manner endless varieties of quality 
 are rendered possible. Good vocalisation, therefore, 
 requires the resonating cavity to be so placed as 
 to modify in the way most attractive to the ear 
 the quality of the sounds produced by the vocal 
 chords. 
 
 The complete analysis of the sounds of the hu- 
 man voice into their separate partial-tones presents 
 peculiar difficulties to the unassisted ear, and can 
 hardly be effected without the help of resonators 
 such as those described in § 42. By their aid we can 
 detect in the lower notes of a bass voice, when 
 vigorously produced, shrill overtones reaching as far 
 as No. 16, which is four Octaves above the funda- 
 mental-tone. Under certain conditions these high 
 overtones can be readily heard without recourse to 
 resonators. When a body of voices are singing 
 fortissimo without any instrumental accompaniment, 
 a peculiar shrill tremulous sound is heard which is 
 obviously far above the pitch of any note professedly 
 being sung. This sound is, to my ears, so intensely 
 shrill and piercing as to be often quite painful. I 
 have also observed it when listening to the lower 
 notes of a powerful contralto voice. The reason 
 why these acute sounds are tremulous will be given 
 later. 
 
 69. We close this discussion by describing a 
 
140 COALESCENCE OF PARTIAL-TONES. [V. § 69. 
 
 mode of submitting Helmholtz's general theory of 
 musical quality to a further and very severe test. 
 
 The sounds of tuning-forks mounted on their 
 appropriate resonance-boxes are, as we know, very 
 approximately simple tones [§ 48]. If therefore we 
 allow a number of such sounds, coincident in pitch 
 with the fundamental-tone and with the other partial- 
 tones of any given clang, to be simultaneously pro- 
 duced, the effect on the ear ought, if Helmholtz's 
 theory is true, to be that of a single musical sound, 
 not that of a group of separate notes. To try 
 the experiment in its simplest shape, take two 
 mounted forks forming the interval of an Octave, 
 and cause them to utter their respective tones to- 
 gether. For a short time we are able to distinguish 
 the two notes as coming from separate instruments, 
 but soon they blend into one sound, to which we 
 assign the 'pitch of the loioer fork, and a quality 
 more brilliant than that of either. So strong is the 
 illusion, that we can hardly believe the higher fork 
 to be really still contributing its note, until we 
 ascertain that placing a finger on its prongs at once 
 changes the timbre, by reducing it to the dull, un- 
 interesting quality of a simple tone. The character 
 of a clang consisting of only one overtone and the 
 fundamental may be shown to admit of many dif- 
 ferent shades of quality by suitably varying the 
 
V. § 69.] SYNTHETIC APPARATUS. 141 
 
 relative intensities of the two fork-tones in this 
 experiment. If we add a fork a Fifth above the 
 higher of the first two, and therefore yielding the 
 third partial-tone of the clang of which they form 
 the first and second, the three tones blend as per- 
 fectly as the two did before, the only difference 
 perceptible being an increase of brilliancy. The 
 experiment admits of being carried further with 
 the same result. 
 
 If we were able to produce by means of tuning- 
 forks as many simple tones of the series on p. 89 as 
 we pleased, and also to control at will their relative 
 intensities, it would be possible to imitate in this 
 manner the characteristic quality of every musical in- 
 strument. The unmanageable character of very high 
 forks has as yet prevented this being done for sounds 
 containing a large number of powerful overtones, 
 but an apparatus on this principle was devised by 
 Professor Helmholtz which imitates successfully 
 sounds not involving more than the first six or 
 eight partial-tones. His theory of quality is thus 
 experimentally demonstrated both analytically and 
 synthetically. We will examine in the next chapter 
 certain theoretical considerations which have an 
 important bearing on that theory. 
 
CHAPTER VI. 
 
 ON THE CONNEXION BETWEEN QUALITY AND MODE OF 
 VIBRATION. 
 
 70. It was stated in § 37 that, when a pendulum 
 performs oscillations whose extent is small compared 
 to the length of the pendulum, the period of a 
 vibration is the same for any extent of swing loithin 
 this limit. We will apply this fact to prove that 
 the prongs of a tuning-fork vibrate in the same 
 mode [§ 1 1] as a pendulum. 
 
 When a sustained simple tone is being trans- 
 mitted by the air, we may regard it as originated 
 by a tuning-fork of appropriate pitch and size. But 
 we know experimentally that, by suitable bowing, 
 we may elicit from such a fork tones of various de- 
 grees of intensity, but having all the same pitch. 
 Here, therefore, though the extent of vibration varies, 
 the period remains constant, which is the pendulum- 
 law. Accordingly, the vibrations of a tuning-fork 
 are identical in mode with those of a pendulum. 
 The same thing will hold good of the aerial vibra- 
 
VI. §70.] COMPOSITION OF VIBRATIONS. 143 
 
 tions to which those of a fork give rise. Hence 
 a simple tone is due to vibrations executed according 
 to the pendulum-law. 
 
 Such vibrations will, therefore, give rise to waves 
 of condensation and rarefaction whose associated 
 wave-form is that drawn in Fig. 17 his. It will 
 be convenient to call the vibrations to which a 
 simple tone is due simple vibrations ; and the associ- 
 ated waves simple waves. We proceed to examine the 
 modes of vibration corresponding to composite sounds. 
 
 Let us first take the case of a clang consisting of 
 but two simple tones, the fundamental and its first 
 overtone. A particle of air engaged in transmitting 
 this sound is simultaneously acted upon by two sets 
 of forces which, if they acted separately, would cause 
 it to perform two simple vibrations having periods 
 bearing to each other the ratio of 1 to 2. We have 
 to investigate what its motion will be under the 
 joint action of these forces. The problem before us 
 is the composition of two vibrations executed in the 
 same straight line. 
 
 Suppose that a series of particles originally at 
 rest in the dotted line through O, Fig. 44, and 
 capable of vibrating in paths perpendicular to it, are 
 under simultaneous forces which, if acting separately, 
 would give rise to the transmission of the parts of 
 waves drawn in the figure. Let a and b be the 
 
144 
 
 COMPOSITION OF VIBRATIONS. [VI. § 71. 
 
 respective positions which a particle originally at O 
 would at a given moment occupy if the forces acted 
 separately. 
 
 71. In determining the corresponding position 
 of this particle under the forces acting simultaneously 
 we employ the following principle : — Tlie resultant 
 displacement is equal to the sum or difference of the 
 component displacements, according as these are pro- 
 duced in the same or in opposite directions. 
 
 (1) 
 
 Fi^.44. 
 
 (.z) 
 
 In the application of this principle four different 
 cases present themselves. In (l), crest falls on 
 crest ; in (2), trough on trough ; and the displace- 
 ment, Oc, of the particle from its position of rest, 0, 
 is in both these instances equal to the sum of the 
 displacements Oa and Oh. In (3) and (4), where 
 the crest of one wave meets the trough of another, 
 Oc is equal to the difference between Oa and Oh ; c 
 being above or below the undisturbed line, accordmg 
 
VI. §72.] DETERMINATION OF JOINT WAVE. 145 
 
 as Oa is greater than Oh, as in (3), or less than Oh, 
 as in (4). Thus, each tributary set of forces produces 
 its full effect in its own direction, or, in other words, 
 the displacement due to one set is superposed on 
 that due to the other. 
 
 In order to determine the form of the resultant 
 wave, we have only to apply this principle to suc- 
 cessive particles. We thus obtain an assemblage of 
 points constituting the wave required. 
 
 The problem of the motion of an air-particle 
 engaged in transmitting simultaneously two partial- 
 tones of a clang is, therefore, to be solved as follows. 
 Let each simple vibration be represented by its 
 associated wave. Ascertain by the process just de- 
 scribed to what joint form the superposition of these 
 two waves leads. The result will be the associated 
 wave-form corresponding to the mode of particle- 
 vibration to which the compound sound is due. 
 
 72. Before, however, we can lay down the two 
 tributary waves, an important point remains to be 
 settled. We wUl, for a moment, suppose that the 
 two simple tones on which we are engaged are 
 sustained by two tuning-forks situated as in the 
 annexed figure, and that we are examining the 
 transmission of their resulting clang along the 
 dotted line with respect to which they are sym- 
 metrically situated. 
 
 T. 10 
 
146 DIFFERENCE OF PHASE. [VI. § 72. 
 
 Let the forks have been set sounding by pre- 
 cisely simultaneous blows. They will then commence 
 swinging out of their positions of rest in the same 
 
 FigAS. 
 
 ft 
 
 V 
 
 direction at the same instant. The points in an asso- 
 ciated wave-form which correspond to the initial 
 position of a vibrating particle are those in which 
 the wave-form cuts the initial line. Hence, in laying 
 down the tributary waves we must, in the present 
 instance, begin them at one and the same point in 
 the initial line, and take care that their convexities at 
 that point are turned the same way, as at 0, Fig. 46. 
 
 In this case the two vibrations are said to start 
 in the same phase. 
 
 If one of the two forks have been struck before 
 the other, the former will not necessarily be passing 
 through its equilibrium position at the moment when 
 the latter is swinging out of that position. Hence, 
 both sets of associated waves will no longer necessa- 
 
VI. § 72.] QUALITY AND PHASE. 147 
 
 rily cut the initial line at the same point. The 
 result is of the kind shown in Fig. 47, where three 
 different cases are represented: 
 
 We have here vibrations starting in different 
 -phases. 
 
 It is clear from the figure that all phase- 
 differences can he properly represented by merely 
 causing the waves engaged to assume different po- 
 sitions upon the initial line with reference to each 
 other. The second and third cases are obtainable 
 from the first by sliding the system of shorter waves 
 bodily along the initial line, while the other system 
 of waves retains its position. 
 
 By the help of the instrument mentioned in 
 § 69, Professor Helmholtz has demonstrated that, 
 when a number of partial-tones are independently 
 produced, the clang into which they coalesce has the 
 
 10—2 
 
148 SIMPLE AND RESULTANT WAVE-FORMS. [VI. §73. 
 
 same quality, whatever differences of phase may exist 
 among the systems of sim,ple vibrations to which the 
 constituent partial-tones are due. Accordingly, we 
 may expect to find that not one single waveform, 
 but many such forms, correspond to a sound of given 
 quality and pitch. 
 
 In Figs. 48, 49, 50, the associated wave-form cor- 
 responding to our clang of two partial-tones (§71) 
 is constructed for three degrees of phase-difference. 
 The simple constituent waves are shown in thin, the 
 result of their composition in full lines. In each 
 case two complete wave-lengths of the latter are 
 exhibited. 
 
 Figs. 51 and 52 present two wave-forms drawn, 
 in the same way, for a clang of constant pitch and 
 quality containing the partial-tones 1, 2 and 3. 
 
 The dissimilarity of form, and therefore of corre- 
 sponding particle -vibration, is in both sets of figures 
 very marked. 
 
 73. It has been shown that, by mere alteration 
 of phase, a very great variety of resultant wave-forms 
 can be obtained from two sets of simple waves of 
 given lengths and amplitudes. Each one of these 
 forms will give rise to a cycle of others, if we allow 
 the relative amplitudes of the constituent systems to 
 be changed, while keeping the difference of phase 
 constant. If, therefore, we have at our disposal the 
 
yi. § 73.] SIMPLE AND RESULTANT WA VE-FORMS. 149 
 
150 VARIETY OF RESULTANT WAVE-FOBMS. [VI. §73. 
 
 systems of simple waves corresponding to an unlimited 
 number of partial-tones, and can assign to them any 
 
 Fig.eL 
 
 degrees of intensity and phase-difference that we 
 choose, it is manifest that we may produce by their 
 combination an endless series of different resultant 
 wave-forms. On the other hand, it is not evident 
 that, even out of this rich abundance of materials, 
 we could build up every form of wave which could 
 possibly he assigned. The French mathematician 
 Fourier has, however, demonstrated that there is 
 no form of wave which (unless itself simple) cannot 
 be compounded out of a number of simple waves 
 
YI. § 73.] FOURIER'S THEOREM. 151 
 
 whose lengths' are inversely as the numbers 1, 2, 3, 
 4, &c. He has further shown that each individual 
 wave-form admits of being thus compounded in only 
 one way, and has provided the means of calculating, 
 in any given case, how many and which members of 
 the series will appear, their relative amplitudes and 
 their differences of phase. 
 
 When translated from the language of Mechanics 
 into that of Acoustics, the theorem of Fourier asserts 
 that every regular musical sound is resolvable into 
 a definite number of simple tones whose relative 
 pitch follows the law of the partial-tone series. It 
 thus supplies a theoretical basis for the analysis and 
 synthesis of composite sounds which have been ex- 
 perimentally effected in chapters iv. and v. 
 
 When we are listening to a sustained clang, the 
 air at any one point within the orifice of the ear can 
 have only one definite mode of particle- vibration at 
 any one moment. How does the ear behave towards 
 any such given vibration? It proceeds as foUows. 
 If the vibration is simple, it leaves it alone. If com- 
 posite, it analyses it into a series of simple vibrations 
 whose rates are once, twice, three times &c. that of 
 the given vibration, in accordance with Fourier's 
 theorem. In the former event the ear perceives only 
 a simple tone. In the latter, it is able to recognise, 
 ' Arranged in diminishing order. 
 
152 ANALYSING POWER OF THE EAR. [VI. § 73. 
 
 by suitably directed and assisted efforts, partial- tones 
 corresponding to the rate of each constituent into 
 which it has analysed the composite vibration origi- 
 nally presented to it. The ear being deaf to differ- 
 ences of phase in partial-tones (§ 72), perceives no 
 difference between sounds due to modes of vibration 
 such as those which give rise to the three resultant 
 associated wave-forms shown in Figs. 48, 49, 50, but 
 merely resolves them into the same single parr of 
 partial-tones. Since, however, only one such resolu- 
 tion of a given vibration-mode is possible, the ear 
 can never vary in the group of partial-tones into 
 which it resolves an assigned clang. 
 
 The power possessed by the ear of thus singling 
 out the constituent tones of a clang and assigning 
 to them their relative intensities is unlike any cor- 
 responding capacity of the eye. Take for instance 
 the two curves shown in Figs. 51 and 52, and try 
 to determine, by the eye alone, what simple waves, 
 present with what amplitudes, must be superposed 
 in order to reproduce those forms. The eye wUl be 
 found absolutely to break down in the attempt. 
 
 We have seen that the loudness of a composite 
 sound depends on extent of vibration, and its pitch 
 on rate of vibration. There remains only one variable 
 element, viz. mode of vibration, to account for the 
 quality of the sound. From this consideration it 
 
VI. § 73.] QUALITY AND MODE OF VIBRATION. 153 
 
 follows that some connexion must exist between the 
 quality of a sound and the mode of aerial vibration 
 to which the sound is due. Up to the time of Helm- 
 holtz no advance had been made in clearing up the 
 nature of this connexion. It was reserved for him 
 to show that, while no two sounds of different quality 
 can correspond to the same mode of vibration, an 
 indefinitely large number of modes of vibration may 
 give rise to a sound of only one degree of quality. 
 In other words, mode of vibration determines quality, 
 but quality does not determine mode of vibration. 
 
CHAPTER VII. 
 
 ON THE INTERFERENCE OF SOUND, AND ON ' BEATS.' 
 
 74. In § 71 we examined the principle on which 
 the general problem of the composition of vibrations 
 is solved. We now approach certain very import- 
 ant particular cases of that problem, which it wiU be 
 worth while to solve both independently and as 
 instances of the method repeatedly applied in § 72. 
 
 Suppose that a particle of air is vibrating be- 
 tween the extreme positions A and B while convey- 
 
 Fig.SS. 
 
 ing a sustained simple tone pi-oduced by a tuning-fork, 
 or stopped flue-pipe. Now let a second instrument 
 of the same kind be caused to emit a tone exactly in 
 unison with the first. We will assume that, when 
 the waves of the second tone reach the pai-ticle, it is 
 just on the point of starting from A towards B. 
 Two extreme cases are now possible, depending on 
 the movement which the particle would have exe- 
 cuted in virtue of the later-impressed vibration alone. 
 
VII. § 74.J INTERFERENCE OF SOUND. 155 
 
 First, suppose that movement to be from A along the 
 line AB, either through a greater or less distance than 
 AB, back again to A, and so on. Here the separate 
 effects of the two sets of vibrations will be added to- 
 gether, the particle will, therefore, perform vibrations 
 of larger extent than it would under either component 
 separately. Next, suppose that, under the second set 
 of vibrations alone, the particle would move from A 
 in the opposite direction to its former course, i.e. along 
 BA produced, shown by a dotted line in the figure. 
 In this case the separate effects are absolutely 
 antagonistic ; accordingly the joint result is that due 
 to the difference of its components. The particle 
 wUl, therefore, execute less extensive vibrations than 
 it would have done under the more powerful of the 
 two components acting alone. 
 
 The most striking result presents itself when the 
 two systems of vibrations, besides being in opposition 
 to each other, are also exactly equal in extent. In 
 this case the air-particle, being solicited by equal 
 forces in opposite directions, remains at rest, the two 
 systems of vibrations completely neutralising each 
 other's effects. In general, however, these systems, 
 even when equal in extent of vibration, are neither 
 in complete opposition nor in complete accordance, but 
 in an intermediate attitude, so as only partially to 
 counteract, or support, each other. These conclusions 
 
156 
 
 INTERFERENCE OF SOUND. [VII. § 74. 
 
 admit of being exhibited in a more complete manner 
 by means of associated waves. We have only to lay 
 down the simple wave-forms corresponding to the 
 constituent vibrations, and superpose them as in 
 § 72. The reader will have noticed that the diffe- 
 rences of relative motion described on the preceding 
 page are merely phase-differences. 
 
 Fiy. 54 . 
 
 In Fig. 54, (1), (2), (3), we have two waves of 
 unequal amplitudes in complete accordance, complete 
 antagonism and an intermediate condition, respec- 
 
VII. § 75.] INTERFERENCE OF SOUND. 157 
 
 tively. In Fig. 55, a case of equal and opposite 
 waves is shown. In (l) Fig. 54, the resultant wave 
 is the sum, and in (2) the difference of the compo- 
 nent waves. In (3), we get a wave of intermediate 
 amplitude. These three resultant waves are neces- 
 sarily simple, as otherwise two simple tones in unison 
 would give rise to a composite sound, which would 
 be absurd. In Fig. 55 the wave-form degenerates 
 into the initial line, i.e. no efi'ect whatever is pro- 
 duced. 
 
 Fi0 .SS. 
 
 75. Thus, when one simple tone is being heard, 
 we by no means necessarily obtain an increase of 
 loudness by exciting a second simple tone of the 
 same pitch. On the contrary, we m,ay thus weaken 
 the original sound, or even extinguish it entirely. 
 When this occurs we have an instance of a phe- 
 nomenon which goes by the name of Interference. 
 That two sounds should produce absolute silence 
 seems, at first sight, as absurd as that two loaves 
 should be equivalent to no bread. This is, however, 
 only because we are accustomed to think of Sound as 
 
158 INTERFERENCE OF SOUND. [VII. § 76. 
 
 something possessing an external substantial exist- 
 ence ; not as consisting merely in a state of motion of 
 certain air-particles, and therefore liable, on the 
 application of suitable forces, to be absolutely anni- 
 hilated. 
 
 A single tuning-fork presents an example of this 
 very important phenomenon. Each prong sets up 
 vibrations corresponding to a simple tone, and the 
 two tones so produced are of the same pitch and 
 intensity. If the fork, after being struck, is held 
 between the finger and thumb, and made to re- 
 volve slowly about its own axis, four positions of 
 the fork with reference to the ear wUl be found 
 where the sound goes completely out. These posi- 
 tions are mid-way between the four in which the 
 plane sides of the prongs are held straight before the 
 ear. As the fork revolves from one of these positions 
 of loud sound to that at right-angles to it, the 
 sound gradually wanes, is extinguished in passing 
 the Interference-position, reappears very feebly im- 
 mediately afterwards, and then continues to gain 
 strength until the quarter of a revolution has been 
 completed. 
 
 76. The case of coexistent unisons has now been 
 adequately examined : we proceed to enquire what 
 happens when two simple tones diffeHng slightly in 
 fitch are simultaneously produced. The problem is, 
 
YII. § 76.] BEATS OF SIMPLE TONES. 159 
 
 in fact, to compound two sets of pendulum-vibrations 
 whose periods are not exactly equal. Let us fix on a 
 moment of time at which the two component vibra- 
 tions are in complete accordance, and suppose that a 
 resultant vibration is just commencing from left to 
 right. It will be convenient to call this an outward, 
 and its opposite an inward swing. Since the periods 
 of the two component vibrations are unequal, one of 
 them will at once begin to gain on the other, and 
 therefore, directly after the start, they will cease to be 
 in complete accordance. It is easy to ascertain what 
 their subsequent bearing towards each other will be, 
 by considering two ordinary pendulums of unequal 
 periods, both beginning an outward swing at the 
 same instant. Let A be the slower, B the quicker 
 pendulum. When A has just finished its outward 
 swing, B will have already turned back and per- 
 formed a portion of its next inward swing. Thus, 
 during each successive swing oi A, B will gain a 
 certain distance upon it. When B has, in this 
 manner, gained one whole swing, i.e. half a complete 
 oscillation, upon A, it will begin an inward swing at 
 the moment when A is commencing an outward swing. 
 The two vibrations are here, for the moment, in C07n- 
 plete opposition. After another interval of equal 
 length, B, having gained another whole swing, will 
 be one complete oscillation ahead of A, and they will 
 
160 BEATS OF SIMPLE TONES. [VII. § 76. 
 
 therefore start on the next outward swing together, 
 i.e. the vibrations will be momentarily in complete 
 accordance. Thus, during the time requisite to 
 enable B to perform one entire oscillation more than 
 A, there occur the following changes. Complete 
 accordance of vibrations, lasting only for a single 
 swing of the more rapid pendulum, followed by 
 partial accordance, in its turn gradually giving way 
 to discordance, which culminates in complete opposi- 
 tion at the middle of the period, and then, during its 
 latter half, gradually yields to returning accordance, 
 which regains completeness just as the period 
 closes. 
 
 It follows from this that, in the case of two 
 simple tones differing slightly in pitch, we must 
 hear a sound going through regularly recurring 
 alternations of loudness in equal successive intervals 
 of time, its greatest intensity exceeding, and its least 
 intensity falling short of, that of the louder of the 
 two tones. Each recurrence of the maximum in- 
 tensity is called a heat, and it is clear that exactly 
 one such beat will be heard in each interval of time 
 during which the acuter of the two simple tones 
 performs one more vibration than the graver tone. 
 Accordingly, the number of beats heard in any 
 assigned time will be equal to the number of com- 
 plete vibrations which the one tone gains on the 
 
VII. § 77.] NUMBER OF BEATS PER SECOND. 161 
 
 otlier in that time. We may express this result 
 more briefly as follows : — 
 
 The number of heats per second due to two simple 
 tones is equal to the difference of their respective 
 vihration-numhers. 
 
 77. By means of associated wave-forms we can 
 obtain a graphic representation of beats, which will 
 probably be more directly intelligible than any 
 verbal description. In Fig. 56, the constituent 
 simple waves are laid down, and their resultant is 
 constructed, for the interval of a semi-Tone. The 
 
 vibration-fraction for that interval is — -; i.e. 16 
 
 15 
 
 vibrations of the higher tone are performed in the 
 same time as 15 of the lower. The figure repre- 
 sents completely the state of things from a maxi- 
 inum of intensity to the adjacent minimutn. The 
 time during which this change occurs is one-half of 
 that above-mentioned : accordingly the figure shows 
 8 and 7-J wave-lengths of the respective systems. 
 Thus half a heat is here pictorially represented, the 
 amplitude of the resultant waves steadUy diminish- 
 ing during this period. "We have only to turn the 
 figure upside down, to get a picture of what occurs 
 in the next following equal period. The amplitudes 
 here again increase until they regain their former 
 dimensions. One whole heat is thus accounted for. 
 
 T. 11 
 
162 GRArmC REPRESENTATION OF BEATS. [VII. §77. 
 
VII. §78.] EXPERIMENTAL STUDY OF BEATS. 163 
 
 In addition to the alternations of intensity 
 whicli characterise beats, they also contain variations 
 of pitch. The existence of such variations is both 
 theoretically demonstrable and experimentally re- 
 cognisable, but they are too minute to require ex- 
 amination here'. 
 
 78. The most direct way of studying the beats 
 of simple tones experimentally is to take two unison 
 tuning-forks and attach a small peUet of wax to the 
 extremity of a prong of one of them. The fork so 
 operated on becomes slightly heavier than before ; 
 its vibrations are therefore slightly retarded, and its 
 pitch lowered. When both forks are struck and 
 held to the ear, beats are heard. These will be 
 most distinct when the forks' tones are exactly 
 equally loud, for in this case the minima of intensity 
 will be equal to zero, and the beats will therefore be 
 separated by intervals of absolute, though but mo- 
 mentary, silence. The increase in rapidity of the 
 beats, as the interval between the beating tones 
 widens, may be shown by gradually loading one of 
 the forks more and more heavily with wax pellets, or 
 by a small coin pressed upon them. If it is desired 
 
 ■ The reader may, if he wishes to pursue this subject, refer to 
 a Paper, by the Author, ou 'Variations of Pitch in Beats,' in the 
 Philosophical Magazine for July, 1872, from which Fig. 56 has 
 been copied. 
 
 11—2 
 
164 EXPERIMENTAL STUDY OF BEATS. [VII. § 78. 
 
 to exhibit these phenomena to several persons at 
 once, the forks should first be mounted on their 
 resonant-boxes, and, after the pellets have been 
 attached, stroked with a resined bow, care being 
 taken to produce tones as nearly as possible equal in 
 intensity. Slow beats may also be obtained from 
 any instrument capable of producing tones whose 
 vibration-numbers differ by a sufficiently small 
 amount. Thus, if the strings corresponding to a 
 single note of the pianoforte are not strictly in 
 unison, such beats are heard on striking the note. 
 If the tuning is perfect, a wax pellet attached to one 
 of the wires will lower its pitch sufficiently to produce 
 the desired effect. Beats not too fast to be readily 
 counted arise between adjacent low notes on the 
 harmonium, or, still more conspicuously, on large 
 organs. They are also frequently to be heard in the 
 sounds of church bells, or in those emitted by the 
 telegraph. wires when vibrating in a strong wind. 
 In order to observe them in the last instance, it is 
 best to press one ear against a telegraph-post and 
 close the other : the beats then come out with re- 
 markable distinctness. It should be noticed that, 
 when we are dealing with two composite sounds, 
 several sets of beats may be heard at the same time, 
 if pairs of partial-tones are in relative positions 
 suited to produce them. Thus, suppose that two 
 
VII. § 78.] BEATS OF COMPOUND SOUNDS. 165 
 
 clangs coexist, each of which contains the first six 
 partial-tones of the series audibly developed. Since 
 the second, third, &c. partial-tones of each clang make 
 twice, three times, &c. as many vibrations per second 
 as their respective fundamental-tones, [§ 43] it follows 
 that the differences between the vibration-numbers 
 of successive pairs of partial-tones belonging to the 
 two clangs will be twice, three times, &c., the differ- 
 ence between the vibration-numbers of the two 
 fundamental-tones. Accordingly, if the fundamental- 
 tones give rise to beats, we may hear, in addition to 
 the series so accounted for, five other sets of beats, 
 respectively twice, three, four, five and six times as 
 rapid as they. In order to determine the number of 
 beats per second for any such set, we need only 
 multiply the number of the fundamental beats by 
 the order of the partial-tones concerned. The beats 
 of two simple tones necessarily become more rapid 
 if the higher tone be sharpened, or the lower 
 flattened ; i.e. if the interval they form with each 
 other be widened. The beats may, however, also be 
 accelerated without altering the interval, by merely 
 placing it in a higher part of the scale. Greater 
 vibration-numbers are thus obtained with a pro- 
 portionately larger difference between them, though 
 their ratio to each other remains what it was 
 before. Thus the rapidity of the beats due to 
 
166 BEATS OF A GIVEN INTERVAL. [VII. § 78. 
 
 an assigned interval depends jointly on two ele- 
 ments, the width of the interval and its position 
 in the musical scale ; in other words, on both the 
 relative and absolute pitch of the tones which 
 form it. 
 
CHAPTER VIII. 
 
 ON CONCOKD AND DISCORD. 
 
 79. A question of fundamental importance now 
 presents itself, viz. What becomes of beats when they 
 are so rapid that they can no longer be separately 
 perceived by the ear? In order to answer it, the 
 best plan is to take two unison-forks of medium 
 pitch, mounted on their resonance-boxes, attach a 
 small pellet of wax to a prong of one of them, and 
 then gradually increase the quantity of wax. At 
 first very slow beats are heard, and as long as their 
 number does not exceed four or five in a second, 
 the ear can follow and count them without difficulty. 
 As they become more rapid the difficulty of counting 
 them augments, until at last they cease to be separ- 
 ately recognisable. Even then, however, the ear 
 retains the conviction that the sound it hears is a 
 series of rapid alternations, and not a continuous 
 tone. Its intermittent character is not lost, although 
 the intermittances themselves pass by too rapidly 
 
168 CAUSE OF DliSSONANCE. [VIII. §79. 
 
 for individual recognition. Exactly the same thing 
 may be observed in the roll of a side drum, which no 
 one is in danger of mistaking for a continuous sound. 
 Rapid beats produce a decidedly harsh and 
 grating effect on the ear ; and this is quite what 
 the analogy of our other senses would lead us to 
 expect. The disagreeable impressions excited in the 
 organs of sight by a flickering unsteady light, and in 
 those of touch by tickling or scratching, are familiar 
 to every one. The effect of rapid beats is, in fact, 
 identical with the sensation to which we commonly 
 attach the name of dissonance. Let us examine, in 
 somewhat greater detail, the conditions necessary for 
 its production between two simple tones. If we 
 take a pair of middle- C forks, and gradually throw 
 them more and more out of unison with each other 
 in the way already described, the roughness due to 
 their beats reaches its maximum when the interval 
 between them is about a half-Tone : for a whole Tone 
 it is decidedly less marked, and when the interval 
 amounts to a Minor Third, scarcely a trace of it 
 remains. Hence, in order that dissonance may arise 
 between two simple tones, they must form with each 
 other a narrower interval than a Minor Third. If 
 we call this interval the heating -distance for two 
 such tones, we may express the above condition thus. 
 Dissonance can arise directly between two simple 
 
VIII. §79. THE 'BEATING DISTANCE.' 169 
 
 tones only when they are within heating-distance of 
 each otherK 
 
 It follows from what was shown at the end of 
 Chapter VII. that the same beating interval will 
 give rise to very different numbers of beats per 
 second according as the tones which form it occupy 
 a low or high position in the scale. Such an interval, 
 e.g. a whole Tone, becomes palpably less dissonant 
 as it is successively raised in pitch. Accordingly 
 the beating-distance, which for tones of medium pitch 
 we have roughly fixed at a Minor Third, must be 
 understood to expand somewhat in low, and cor- 
 respondingly contract in high, regions of the scale. 
 
 The general partial-tone series consists of simple 
 tones which, up to the seventh, are mutually out of 
 beating-distance : above the seventh they close in 
 rapidly upon each other. In the neighbourhood of 
 10, the interval between adjacent partial-tones is 
 about a whole Tone; near 16, a semi-Tone; higher 
 in the series they come to still closer quarters. 
 These high partial-tones are, therefore, so situated 
 as to produce harsh dissonances with each other. 
 Where they are strongly developed in a clang, there 
 
 ' It will be shown in the sequel [§§ 89 — 92] that dissonance 
 may, in the case of simple tones forming intervals wider than a 
 Minor Third, arise from beats other than those here under con- 
 sideration. 
 
170 DISSONANT OVERTONES. [VIII. § 80. 
 
 will therefore be a certain inevitable roughness in its 
 quality. This is the cause of the harsh character of 
 trumpet or trombone notes, and also of the shrill 
 tremulous sounds sometimes observed in the human 
 voice (§68). In fact we may regard the partial- 
 tone series above the seventh tone, when fully repre- 
 sented, as contributing mere noise to the clang. 
 This explains why it is advantageous that the lower 
 partial-tones of organ reeds should be more strength- 
 ened by resonance than the higher ones [see § 66]. 
 
 80. It has been shown that, when two simple 
 tones are simultaneously sustained, beats can arise 
 directly between them only under one condition, 
 viz. that the tones shall differ in pitch by less than 
 a Minor Third, or thereabouts. When, however, the 
 two co-existing sounds are no longer simple tones, 
 but composite clangs each consisting of a series of 
 well-developed partial-tones, the case becomes alto- 
 gether different. Let us examine the state of things 
 which then presents itself 
 
 The sounds of most musical instruments practi- 
 cally contain only the first six partial-tones ; we 
 will, therefore, assume this to be the case with the 
 clangs before us. No beats can then arise between 
 partial-tones of the same clang, since no two of 
 them are written a Minor Third of each other. 
 Dissonance due to beats will, however, be produced 
 
VIII. § 81.] DISSONANCE OF TWO CLANGS. 171 
 
 if a partial-tone belonging to one clang is within 
 that distance of a partial-tone belonging to the 
 other clang. Several pairs of tones may be thus 
 situated, and, if so, each will contribute its share of 
 roughness to the general effect. The intensity of 
 the roughness due to any such pair will depend 
 chiefly on the respective orders to which the beating 
 partial-tones belong, and on the interval between 
 them. The lowest partial-tones, being the loudest, 
 produce the most powerful beats, and half-Tone beats 
 are, in general, harsher than those of a whole Tone. 
 In determining the general effect of a combination 
 of two clangs, we have to ascertain what pairs of 
 partial-tones come within beating-distance, and to 
 estimate the amount of roughness due to each pair. 
 The joint effect of all these roughnesses, if there are 
 several such pairs, or the roughness of a single pair 
 if there is but one, constitutes the dissonance of the 
 combination. If there be no dissonance, the combi- 
 nation is described as a perfect concord. When 
 dissonance is present, its amount will decide whether 
 the combination shall be called an imperfect concord 
 or a discord. The line separating the two must 
 therefore, of necessity, be somewhat arbitrarily drawn. 
 81. Let us examine the principal consonant 
 intervals, in the manner above described, beginning 
 with the Octave. 
 
172 CONSONANCE AND DISSONANCE. [VIIT. § 8L 
 
 '.k 
 
 -P-' 
 
 St 
 
 The minims here represent the fundamental- 
 tones ; the crotchets above them corresponding over- 
 tones. Those belonging to the higher clang are 
 only written down as far as the third, since the 
 fourth, fifth and sixth meet no corresponding tones 
 of the lower clang with which to form beating pairs. 
 As long as the tuning is perfect, each partial-tone of 
 the higher clang coincides exactly with one belonging 
 to the lower. No dissonance, therefore, can occur, 
 and the combination is a perfect concord. But 
 suppose the higher C to be slightly out of tune : each 
 of its partial-tones will be correspondingly too sharp 
 or too flat, and three sets of beats will be heard 
 between the partial-tones 2 — 1, 4 — 2 and 6 — 3. 
 When the higher C is as much as a semi-Tone wrong, 
 the result is 
 
 i^k 
 
 iT^- 
 
 -¥-' 
 
 zipi 
 
 The pair 2 — 1 is of the most importance, on 
 account of the prominence of the partial-tones which 
 
VIII. §81.] OCTAVE. 173 
 
 form it. The higher pairs contribute a weaker dis- 
 sonance. As beats of a semi-Tone correspond to 
 about maximum roughness in the middle region of 
 the scale, we have before us an exceedingly harsh 
 discord. As the pitch of the higher note is gradually 
 corrected, the rapidity of the beats diminishes, but 
 the tuning must be rigorously accurate to make 
 them entirely vanish. If the note makes but one 
 vibration per second too many or too few, which 
 corresponds to a difference in pitch of only about a 
 thirtieth part of a whole Tone, the defect of tuning 
 makes itself felt by three sets of beats, of 1, 
 2 and 3 per second respectively. The tuner must 
 keep slightly altering the pitch until he at length 
 hits on that which completely extinguishes the beats. 
 We saw in an earlier part of this inquiry (§ 34) that, 
 when two sounds form with each other the interval 
 of an Octave, their vibration-numbers must be in 
 the ratio of 2 : 1. Long after it had been experi- 
 mentally ascertained that rigorous compliance with 
 this arithmetical condition was essential for securing 
 a perfectly smooth Octave, the reason for this ne- 
 cessity remained entirely unknown, and nothing but 
 the vaguest and most fanciful suggestions were 
 offered to account for it — such as, for instance, that 
 " the human mind delights in simple numerical rela- 
 tions." This attempt at explanation overlooked the 
 
174 COINCIDENCE OF PARTIAL TONES. [VIII. § 82. 
 
 obvious fact that many people who knew nothing 
 either about vibrations or the dehghts of simple 
 numerical relations, could tell a perfect Octave from 
 an imperfect one a great deal better than most men 
 of science. The true explanation, which was left for 
 Helmholtz to discover, lies in the fact that only by 
 exactly satisfying the assigned numerical relation can 
 the partial-tones of the higher clang he brought into 
 exact coincidence with partial-tones of the lower, and 
 thus all beats and consequent dissonance prevented. 
 
 82. No interval naiTower than an Octave can 
 be found which gives an absolutely perfect concord. 
 The nearest approach to such a concord is the 
 Fifth :— 
 
 
 ^: 
 
 Here we get two pairs of coincidences, 3 — 2 and 
 6 — 4, but a certain roughness is caused by 3 of the 
 higher clang being within beating-distance of both 
 4 and 5 of the lower clang. It is true that, since 
 4 and 5 are generally weak, and the beating intervals 
 whole Tones, this roughness will be but slight : still, 
 the dissonance thus caused prevents our classing the 
 Fifth as an interval quite equally smooth with the 
 
VIII. § 83.] 
 
 FIFTH AND FOURTH. 
 
 175 
 
 Octave. The tuning must be perfectly accurate, 
 the Fifth being closely bounded by harsh discords. 
 The result of an error of a semi-Tone is as follows : — 
 
 W^^ 
 
 31^1^ 
 
 --l?i- 
 
 %J 
 
 For every single vibration per second by which 
 the higher clang is out of tune, there will be two 
 beats per second from the pair 3 — 2, with others of 
 greater rapidity, but less intensity, from the higher 
 
 pairs. 
 83. 
 
 For the Fourth we have 
 
 tl 
 
 ^ 
 
 mz 
 
 The amount of dissonance is greater than in the 
 case of the Fifth, since 3 and 2 are usually hotli 
 powerful tones, and therefore produce louder beats 
 than those of 4 — 3 and 5 — 3. There are, further, 
 the beating pairs 5 — 4, 6 — 4 and 6 — 5. Moreover 
 the first pair of coincident partial-tones, 4—3, are 
 in general weaker than the beating pair below them, 
 3 — 2. The Fourth is bounded only on one side by 
 a harsh discord. If its upper clang is half a note 
 
176 MAJOR THIRD AND MAJOR SIXTH. [VIII. § 84. 
 
 too sharp, we have the interval C — i^8, which is 
 treated in the last figure but one. Slight flattening 
 of the F will set the pair 4 — 3 beating ; the disap- 
 pearance of their beats thus secures the accurate 
 tuning of the interval. On the other hand, lowering 
 F weakens the beats of 3 — 2 by widening the dis- 
 tance between those tones, and therefore tends to 
 lessen the whole amount of roughness. These con- 
 siderations go far to explain the fact that a long 
 dispute runs through the history of Music, as to 
 whether the Fourth ought to be treated as a concord 
 or as a discord. The decision ultimately arrived at 
 in favour of the former of these alternatives was per- 
 haps, as Helraholtz suggests, due more to the fact 
 that the Fourth is the inversion' of the Fifth than to 
 the inherent smoothness of the former interval. 
 
 84. Next come the intervals of the Major Third 
 and Major Sixth, which shall be taken together, as 
 they are very nearly equally consonant. 
 
 ii^i^ 
 
 at 
 
 U2Z 
 
 ±?ai 
 
 m 
 
 The dissonance due to the pair 3 — 2, separated 
 by a Tone, in the Sixth, is perhaps about equal to 
 
 ' See § 97 post. 
 
VIII. § 85.] MINOR THIRD AND MINOR SIXTH. 177 
 
 that of the weaker pair 4 — 3, which are only a semi- 
 Tone apart in the Third. The ' definition ' of these 
 intervals, i.e. the accuracy of tuning requisite for a 
 good effect, will, as it depends in each on the fifth 
 partial-tone of the lower clang, be in general not great. 
 
 85. The remaining intervals narrower than an 
 Octave which rank as concords are the Minor Third 
 and Minor Sixth. 
 
 Each contains strong elements of dissonance ; in 
 fact we are here near the boundary between concords 
 and discords. As regards sharpness of definition, the 
 
 2 
 
 
 OE 
 
 ibf2-J 
 
 tones 6 and 5, on which it depends in the first of the 
 two intervals, are, in the sounds of many instruments, 
 weak or even entirely absent, while for the second 
 interval the series of partial-tones must be extended 
 as far as the 8th of the lower clang in order to reach 
 the first coincident pair. Accordingly the Minor 
 Sixth can hardly be said to be defined at all, for 
 clangs of ordinary quality, by coincidence of partial- 
 tones. Its powerful beating pair 3 — 2, separated by 
 the interval of greatest dissonance, a semi-Tone, 
 makes it the roughest of all the concords. On the 
 T. 12 
 
178 DISSONANT INTERVALS. [VIII. §§ 86, 87. 
 
 pianoforte, and other instruments of fixed sounds, 
 the same notes (CE?) which represent the Minor Third 
 have also to do duty as one of the harshest discords, 
 the Augmented Second (CDS), and, in like manner, 
 CA\> stands for the Minor Third and also for the 
 Augmented Fifth (CGt). The extremely defective 
 consonance of the Minor Third and Minor Sixth 
 could hardly be more conclusively shown than by 
 the fact just mentioned. 
 
 86. As regards the dissonant Tonic intervals 
 of the scale, we have, beside those incidentally 
 examined above, the semi-Tone, Tone and Minor 
 Seventh. The first two need not be examined, since 
 obviously each pair of corresponding overtones are 
 brought within the same beating intervals as the 
 two fundamentals. The dissonance resulting is, of 
 course, harsher for the half than for the whole Tone. 
 The Minor Seventh is constituted thus : — 
 
 
 HI 
 
 ^ 
 
 It is the mildest of the discords, in fact in mere 
 smoothness it decidedly surpasses the Minor Sixth. 
 
 87. In order that the reader may see at a glance 
 the whole result of this somewhat laborious discussion, 
 
VIII. § 87.] PICTURE OF INTERVALS. 
 
 179 
 
 we subjoin a graphical representation of the amount 
 of dissonance contained in the several intervals of the 
 scale. The figure is taken, with some slight altera- 
 tions, from one given in Helmholtz's work. The in- 
 tervals, reckoned from middle G, are denoted by dis- 
 tances measured along the horizontal straight line. 
 The dissonance for each interval is represented by 
 the vertical distance of the curved line from the 
 corresponding point on the horizontal line. The cal- 
 culations on which the curve is based were made by 
 Helmholtz for two constituent clangs of the quality 
 of the violin. 
 
 THg.Sr. 
 
 G At) A Bb B C 
 
 The figure indicates the sharpness of definition 
 of an interval by the steepness with which the curve 
 ascends on both sides of it. If we regarded the out- 
 line as that of a mountain chain, the discords would 
 be represented by peaks, and the concords hj passes. 
 The lowness and narrowness of a particular pass 
 would measure the smoothness and definition of the 
 corresponding musical interval. 
 
 12—2 
 
180 CONSONANCE AND QUALITY. [VIII. § 88. 
 
 88. The theory of musical consonance and dis- 
 sonance, our examination of which is now concluded, 
 shows that the degree of smoothness or roughness 
 possessed by an assigned interval is not fixed and 
 invariable, but dependent on the quality of the sounds 
 hy which the interval is held. The results we have 
 arrived at are generally true for sounds containing 
 the first six partial-tones, but they will not apply, 
 without modification, to clangs differently constituted. 
 Let us take a case or two in illustration of this point. 
 Suppose, for instance, we are dealing with sounds 
 such as those of stopped organ-pipes which contain 
 only odd partial-tones (§ 59). It is at once clear 
 from § 82 that the interval of the Augmented Fourth, 
 C — F$, which owes its dissonant character to the 
 beating pairs 3 — 2, 4 — 3 and 6 — 4, wiU become 
 something quite different when the dissonance due 
 to all these pairs disappears, as it must do, since 
 each of them contains at least one partial-tone of an 
 even order. The Minor Sixth would also gain in 
 smoothness with notes of such quality, by the 
 cessation of dissonance due to the pair 3 — 2. 
 
 Further, if the two notes forming any interval 
 are held by instruments of different quality, the 
 smoothness of the combination may greatly depend 
 on which instrument takes the lower and which the 
 upper note. Thus e.g. for the interval of a Fifth 
 
VIII, § 89.] CASE OF SIMPLE TONES. 181 
 
 reference to § 82 will show that the instrument 
 which has the weaker third partial-tone should for 
 the smoother effect take the upper note, since that 
 arrangement reduces to a minimum the dissonance 
 due to the pairs 4 — 3 and 5 — 3. For the interval 
 of a Fourth the opposite arrangement gives the 
 smoother effect by minimising the dissonance 3 — 2. 
 Experiments confirming theoretical conclusions of 
 this kind may be made with great ease on the stops 
 of an organ possessing more than a single manual. 
 
 The above considerations laid down by Helmholtz 
 open a whole field of results which I believe to 
 be entirely in advance of any hitherto obtained by 
 Musical theorists'. 
 
 89. It is possible to draw from the general 
 theory of consonance and dissonance an inference 
 which seems, at first sight, fatal to the truth of 
 the theory itself "If," it may be said, "the 
 difference between a consonant and a dissonant 
 interval depends entirely on the behaviour towards 
 each other of particular overtones ; then, in the case 
 of sounds like those of large stopped flue-pipes, 
 where there are no overtones at all, the distinction 
 between concords and discords ought entirely to 
 disappear, and the interval of a Seventh, for instance, 
 
 ' " Musical theory " is as misleading a term as could possibly 
 be invented to describe conclusions wholly empirical. 
 
182 COMBINATION-TONES. [VIII. § 90. 
 
 to sound just as smooth as that of an Octave. As 
 this is not the fact, the theory cannot be true." 
 
 In order to meet this objection, it will be necessary- 
 first to acquaint the reader with certain known expe- 
 rimental facts which Helmholtz has dragged out of 
 the obscurity in which they had lain for fully a 
 century, and forced to bear testimony in completion 
 of his theory. 
 
 90. Let two tuning-forks of different pitch 
 mounted on their respective resonance-boxes be 
 thrown into powerful vibration by a resined bow. 
 With adequate attention it is possible to recognise, 
 in addition to the tones of the forks themselves, cer- 
 tain new sounds which usually differ in pitch from both 
 the former ones. These sounds, called from the man- 
 ner of their production combination-tones, fall into 
 two categories, with only one of which, discovered 
 in 1740 by a German organist named Sorge, we 
 need now concern ourselves. It consists of a series 
 classed as combination-tones of the Jirst, second, 
 third, &c., orders, of which the first is of the most 
 importance, as it can be heard without difficulty. 
 Its pitch is determined by the following law. The 
 combination-tone of the first order of two simple 
 primary tones has for its vibration-number the differ- 
 ence between the respective vibration-numbers of the 
 primaries. Thus, e.g., if the two primaries make 200 
 
VIII. § 90.] 
 
 CO MB IN A TION- TONES. 
 
 183 
 
 and 300 vibrations per second, and therefore form a 
 Fifth with each other, the combination-tone will 
 make 100 vibrations per second, and therefore lie 
 exactly one Octave below the graver of the two 
 primary tones. By applying this rule we can deter- 
 mine the pitch of the first-order combination-tone 
 for a pair of simple primaries forming any given in- 
 terval with each other. The following table, copied 
 from Helmholtz's work, shows the results for all the 
 consonant intervals not exceeding one Octave. 
 
 Interval. 
 
 Vibration-ratio. 
 
 Difference. 
 
 Depth of the CombiDation-tone 
 below the graver primary. 
 
 Octave 
 
 1 
 
 2 
 
 
 Unison 
 
 Fifth 
 
 2 
 
 3 
 
 
 Octave 
 
 Fourth 
 
 3 
 
 4 
 
 
 Twelfth 
 
 Major Third 
 
 4 
 
 5 
 
 
 Two Octaves 
 
 Minor Third 
 
 5 
 
 6 
 
 1 
 
 Two Octaves & a Major Third 
 
 Major Sixth 
 
 3 
 
 5 
 
 2 
 
 Fifth 
 
 Minor Sixth 
 
 5 
 
 8 
 
 3 
 
 Major Sixth 
 
 In Musical notation the same thing stands thus, 
 the primaries being denoted by minims and the 
 combination-tones by crotchets. 
 
 lE 
 
 ^- 
 
 ^ 
 
 e^ 
 
 
 ti 
 
184 INTENSITY OF COMBINATION-TONES. [VIII. §90. 
 
 Combination-tones are produced with unusual 
 distinctness by the harmonium. If the primaries 
 shown in the treble stave are played on that instru- 
 ment while the pressure of air in the bellows is 
 vigorously sustained, the corresponding combina- 
 tion-tones of the first order, written in the bass, come 
 out with unmistakable clearness. With pianoforte- 
 sounds, on the contrary, combination-tones can be 
 recognised only when the primaries are struck very 
 forcibly, and they are always extremely faint and 
 rapidly evanescent. If, however, the key corre- 
 sponding to the combination-tone sought be first 
 silently pressed down, its note is sometimes sustained 
 by resonance when the keys of the two primaries 
 have been smartly struck "staccato'^. 
 
 Combination-tones of the second order may be 
 treated as if they were first-order tones produced 
 betM^een one or other of the primaries and the com- 
 bination-tone of the first order. Similarly we may 
 regard each combination-tone of the third order as due 
 to a second-order tone paired either with one of the 
 primaries, with the first-order tone or with its own 
 fellow of the second order. Successive subtraction 
 will therefore enable us to determine the vibration- 
 number of a combination-tone of any order from the 
 vibration-numbers of the two primaries. 
 
 ' i.e. allowed to rise again as quickly as possible after the blow. 
 
VIII. § 91.] BEATS OF COMBINATION-TONES. 185 
 
 Combination-tones grow rapidly feebler as their 
 order becomes higher. Those of the first order 
 are usually distinct enough, and those of the second 
 can be heard with a little trouble. The third order is 
 only recognisable when entire stillness is secured, 
 and the greatest attention paid. It is a moot point 
 whether fourth -order tones can be heard at all. 
 
 91. We can now show that the existence of 
 combination-tones prevents intervals formed by two 
 simple tones from altogether lacking the characteristic 
 differences of consonance and dissonance, though 
 those differences are far less marked than in the 
 case of composite sounds. To begin with the Octave. 
 Let us suppose that we have two simple tones form- 
 ing nearly this interval, but that the higher of them 
 is a little sharp, so that the Octave is not strictly 
 in tune, is what is called slightly impure. Let the 
 lower tone make 100, the higher 201, vibrations per 
 second. They will give rise to a combination-tone 
 making 101 vibrations per second (§ 89), and this 
 with the lower primary will produce one heat per 
 second. If the higher primary had been flat instead 
 of sharp, making say 199 vibrations per second, we 
 should have had 99 as combination-tone, giving rise, 
 with 100, to beats of the same rapidity as before. 
 These beats cannot be got rid of except by making 
 the vibration-ratio exactly 1 : 2, i.e. the Octave 
 
186 INFLUENCE OF COMBINATION-TONES. [VIII. §91. 
 
 perfectly pure. The roughness must increase both 
 when the interval widens and when it contracts, so 
 that the Octave, in simple tones, is a well-defined 
 concord bounded on either side by decided discords. 
 This result may be easily verified experimentally by 
 taking two tuning-forks forming an Octave with each 
 other, and throwing the interval slightly out of tune 
 by causing a pellet of wax to adhere to a prong of 
 one of them. On vigorously exciting the forks the 
 beats will be distinctly heard'. 
 
 The Octave is the only interval which is defined 
 by the beats of a combination-tone of the first order 
 with one of the primary tones. For the next 
 smoothest concord, that of the Fifth, we are obliged 
 to have recourse to the second order. Thus, pro- 
 ceeding by successive subtraction, we have : — 
 
 Prunaries 200 301 
 
 C. T. of 1st order 101 
 
 C. Ts. of 2nd order 99 200 
 
 No. of beats per sec. 2 
 
 ' As, however, this mode of treatment produces very percept- 
 ible overtones [§ 48, note\, the experimenter must be ou his guard 
 against attribviting to combination-tone beats an effect which is 
 really due to beats of overtones. There is special risk of being 
 thus led astray in the case of the Octave, where the first-order 
 combination-tone coincides in pitch with the lower of the two 
 primaries. The result of my own observation is that with 
 mounted forks excited in the gentlest possible way there is but 
 slight dissonance even in an impure Octave. 
 
VIII. §92.] INFLUENCE OF COMBINATION-TONES. 187 
 
 The Fifth is, thus, a fairly well-defined concord, 
 though decidedly less sharply bounded than the 
 Octave, owing to the feebleness of the C. T. of the 
 second order. For the Fourth we have : — 
 Primaries 300 401 
 
 C. T. of 1st order 101 
 
 C. Ts. of 2nd order 199 300 
 
 C. Ts. of 3rd order 101 202 98 
 
 No. of beats per sec. 3 
 
 The third-order tones being excessively weak, the 
 interval of a Fourth can scarcely be said to be de- 
 fined at all. Still less can the remaining consonant 
 intervals of the scale be regarded as defined by beats 
 due to combination-tones of yet higher orders. 
 
 92. With two moderately loud simple tones, 
 then, the case stands thus. The interval of a 
 Second is rendered palpably dissonant by direct 
 beats of the primaries ; that of a Major Seventh 
 slightly so by beats of a first-order combination- 
 tone with one of the primaries. There is a certain 
 amount of dissonance in intervals slightly narrower, 
 or slightly wider, than a Fifth, but of a feebler 
 kind than in the case of the Octave, inasmuch as 
 one of the two Combination-Tones producing it is 
 of the second order. Whatever dissonance may exist 
 near the Fourth is practically imperceptible. All 
 other intervals are free from dissonance. Accordingly 
 
188 IMPORTANCE OF OVERTONES. [VIII. § 92. 
 
 all intervals from the Minor Third nearly up to the 
 Fifth, and from a little above the Fifth up to the 
 Major Seventh, ought to sound equally smooth. This 
 conclusion, however inconsistent with the views of 
 Musical theorists, who are apt to regard concord and 
 discord as entirely independent of quality, is strictly 
 borne out by experiment. The intervals lying be- 
 tween the Minor and Major Thirds, and between the 
 Minor and Major Sixths, though sounding somewhat 
 strange, are entirely free from roughness, and there- 
 fore cannot be described as dissonant. 
 
 Helmholtz advises such of his readers as have 
 access to an organ to try the effect of playing alter- 
 nately the smoothest concords, and the most extreme 
 discords, which the Musical scale contains, on stops 
 yielding only approximately simple tones, such e.g. as 
 the flute or stopped diapason. The vivid contrasts 
 which such a proceeding calls out on instruments of 
 bright quality, like the pianoforte and harmonium, 
 or the more brilliant stops of the organ, such as 
 principal, hautbois, trumpet &c., are here blurred 
 and effaced, and everything sounds dull and inanimate 
 in consequence. Nothing can show more decisively 
 than such an experiment that the presence of over- 
 tones confers on Music its most characteristic charms. 
 
 Thus the remark put into the mouth of a sup- 
 posed objector in § 89 turns out to be no objection 
 
VIII. §93.] COMBINATION-TONES OF OVERTONES. 189 
 
 whatever to Helmholtz's theory of consonance and 
 dissonance, but, so far as it represents actual facts, 
 to be valid against a view commonly acted on by 
 Musical theorists. 
 
 93. A point connected with combination-tones, 
 which might otherwise occur as a difficulty to the 
 reader's mind, shall here be briefly noticed. When 
 two clangs coexist, combination-tones are produced 
 between every pair which can be formed of a partial- 
 tone from one clang with a partial-tone from the 
 other. These intrusive sounds will usually be very 
 numerous, and, for aught that appears, might be 
 thought likely to interfere with those originally 
 present to such an extent as to render useless a 
 theory based on the presence of partial-tones only. 
 Helmholtz has removed any such apprehension by 
 showing generally that dissonance due to combina- 
 tion-tones produced between overtones never exists 
 except where it is already present by virtue of direct 
 action among the overtones themselves. Thus the only 
 effect attributable to this source is a somewhat in- 
 creased roughness in all intervals except absolutely 
 perfect concords. No modifications, therefore, have 
 to be introduced on this score into the conclusions of 
 §§ 81-86. 
 
CHAPTER IX. 
 
 ON CONSONANT TRIADS. 
 
 94. In the ensuing portion of this inquiry we 
 shall have to make more frequent use than hitherto 
 of vibration-fractions. It may, therefore, be well to 
 explain at this point the rules for their employment, 
 in order that the student may acquire the requisite 
 facility in handling them. The vibration-fraction of 
 an assigned interval expresses the ratio of the numbers 
 of vibrations performed in the same time by the two 
 notes which form the interval. The particular length 
 of time chosen is a matter of absolute indifference. 
 The upper note of an Octave, for instance, vibrates 
 twice as often as the lower does in any time we 
 choose to select, be it an hour, a minute, a second 
 or a part of a second. In like manner the vibration- 
 fraction ^ indicates that while the lower of two notes 
 forming a Major Third makes four vibrations the 
 higher of them makes five. Therefore while the 
 lower makes one vibration the higher makes |-ths of 
 a vibration. The same reasoning being equally 
 
IX. § 95.] USE OF VIBRATION-FRACTIONS. 191 
 
 applicable to all other cases, it follows that the 
 vibration-fraction of any interval denotes the number 
 of vibrations and parts of a vibration made by the 
 higher of the two notes which form that interval 
 while the lower of them is making a single vibration. 
 
 We will next investigate rules for determining 
 the vibration-fractions of the sum and difference 
 of any two intervals whose vibration-fractions are 
 known. 
 
 95. Suppose that, starting from a given note, 
 we sound a second note, a Fifth above it, and then 
 a third note, a Major Third above the second. What 
 will be the vibration-fraction of the interval formed 
 by the first and third notes, i.e. of the sum of a Fifth 
 and a Major Third ? We wiU, for shortness, call the 
 three notes (l), (2), (3), in order of ascending pitch. 
 The vibration-fractions being, for (l) — (2), f, and, 
 for (2) — (3), I", we proceed thus : — 
 
 While (2) makes 4 vibrations, (3) makes 5 vibrations. 
 Therefore, while (2) makes 1 vibration, (3) makes J vibrations. 
 Therefore, while (2) makes 3 vibrations, (3) makes 3x4 vibrations. 
 
 But while (2) makes 3 vibrations, (1) makes 2 vibrations. 
 Therefore, while (1) makes 2 vibrations, (3) makes 3 x f vibrations. 
 Therefore, while (1) makes 1 vibration, (3) makes |^ x f vibrations. 
 
 Our result, then, is the two vibration-numbers 
 multiplied together. The reasoning is perfectly 
 general, and gives us the following rule. 
 
192 USE OF VIBRATION-FRACTIONS. [IX § 96. 
 
 To find the vibration-fraction of the sum of two 
 intervals, multiply their separate vibration-fractions 
 together. 
 
 96. Next, take the opposite case. Let (2) be 
 a Major Third above (l), and (3) a Fifth above (l), 
 and let the vibration-fraction for the interval (2) — (3) 
 be required. 
 
 While (1) makes 4 vibrations, (2) makes 5 vibrations. 
 Therefore, while (1) makes 1 vibration, (2) makes ^ vibrations. 
 But, whUe (1) makes 2 vibrations, (3) makes 3 vibrations. 
 Therefore, while (1) makes 1 vibration, (3) makes | vibrations. 
 Hence, while (2) makes { vibrations, (3) makes | vibrations. 
 Therefore, whUe (2) makes J of a vibration, (3) makes J x f of a vibration. 
 Therefore, while (2) makes 1 vibration, (3) makes | x f vibrations. 
 
 The result here is the quotient resulting from, the 
 division of the larger vibration-fraction by the 
 smaller : hence we have this general rule : — 
 
 To find the vibration-fraction of the difi^erence 
 of two intervals, divide the vibration-fraction of the 
 wider by that of the narrower interval. 
 
 Thus multiplication and division of vibration- 
 fractions correspond to addition and subtraction of 
 intervals^. 
 
 ' By simply reducing the numerical results, obtained in ^ 95, 
 96, the student will establish the following propositions : 
 ' A Major Third added to a Fifth gives a Major Seventh.' 
 'A Major Third subtracted from a Fifth leaves a Minor 
 Third.' 
 
IX. §§ 97, 98.] INVERSION OF INTERVALS. 193 
 
 97. One of the simplest cases of our second 
 rule occurs when an interval has to be 'inverted.' 
 By the ' inversion ' of any assigned interval narrower 
 than an Octave is meant the difference between it and 
 an Octave, i.e. the interval which remains after it has 
 been subtracted from an Octave. Thus to find the 
 vibration-fraction for the inversion of the Minor Third 
 we merely have to divide 2 by |-, or in other words 
 invert the vibration-fraction of the interval and multi- 
 ply by 2. This applies to all cases. In the particular 
 example selected the result is |- ; the inversion of 
 the Minor Third is therefore the Major Sixth. The 
 relation between an interval and its inversion is 
 obviously mutual, so that each may be described as 
 the inversion of the other. Accordingly the inversion 
 of the Major Sixth is the Minor Third. 
 
 The following table ^ shows the three pairs of 
 consonant intervals narrower than an Octave which 
 stand to each other in the mutual relation of inver- 
 sions. 
 
 Minor Third (f)— Major Sixth (|) 
 
 Major Third (|)— Minor Sixth (f ) 
 
 Fourth (f)— Fifth (f ) 
 
 98. A combination of musical sounds of different 
 
 ' The last two results of this table will be easily verified by 
 the student. 
 
 T. 13 
 
194 CONSONANT TRIADS. [IX. § 99. 
 
 pitch is called a 'chord.' Hitherto we have con- 
 sidered only chords of two notes, or ' binary ' chords. 
 We now go on to chords of three notes, or, as they 
 are usually called, 'triads.' A triad contains three 
 intervals, one between its extreme notes, and one 
 between the middle note and each of the other two. 
 In order that the chord may be free from dissonance, 
 those intervals must all three be concords. 
 
 99. We may, then, search for consonant triads 
 in the following manner. Having selected the lowest 
 of the three notes at pleasure, choose two others each 
 of which forms with the bottom note a consonant 
 interval. Next, examine whether the interval formed 
 by these two notes tvith each other is also a 
 concord. If so, the triad itself is consonant. In 
 order to determine all the consonant triads within an 
 Octave above the fixed bottom note, Ave must assign 
 to the middle and top notes every possible consonant 
 position with respect to the bottom note, and reject 
 all such positions as give rise to dissonant intervals 
 between those notes themselves. The remaining 
 positions will constitute all the consonant triads 
 which have for then- lowest note that originally 
 selected. The intervals at our disposal are, for the 
 middle note, from the Minor Third to the Minor 
 Sixth, and, for the upper note, fi:om the Major Third 
 to the Major Sixth. 
 
TX. § 99.J 
 
 CONSONANT TRIADS. 
 
 195 
 
 In the annexed table ^ the possible positions of 
 the middle note with respect to the bottom note are 
 shown in the left-hand vertical column, the name of 
 each interval being accompanied by its vibration- 
 
 
 Major Thii-d 
 
 Fourth 
 
 Fifth 
 
 Minor Sixth 
 
 Major Sixth 
 
 
 6 
 
 4 
 ■3^ 
 
 1 
 
 8 
 
 6 
 
 s 
 
 Minor Third 
 
 e 
 T 
 
 
 2S 
 
 
 10 
 
 5 
 
 T 
 
 Major Third 
 
 4 
 
 s 
 Fourth 
 
 26 
 
 
 Major Third 
 
 5 
 T 
 
 
 16 
 
 e 
 T 
 
 Minor Third 
 
 32 
 IT 
 
 4 
 
 Fourth 
 
 Fourth 
 
 
 
 9 
 
 
 6 
 "5 
 
 5 
 
 4 
 
 
 
 ^ 
 
 Minor Third 
 
 Major Third 
 
 Fifth 
 
 3 
 
 
 
 
 
 16 
 15- 
 
 10 
 
 
 Minor Sixth 
 
 8 
 
 T 
 
 
 
 
 
 25 
 24 
 
 
 fraction. The possible positions of the top note are 
 similarly shown in the highest horizontal column. 
 Each space common to a horizontal and a vertical 
 column contains the vibration-fraction of the interval 
 formed between the simultaneous positions of the 
 middle and upper notes named at their extremities. 
 
 ' Copied with slight modifications from Helmholtz's work. 
 The reader should verify by calculation each statement made in 
 the table. 
 
 13—2 
 
196 
 
 CONSONANT TRIADS. [IX. §§ 100, 101. 
 
 When this interval is dissonant its vibration- 
 fraction is enclosed in a square bracket. When it 
 is a concord the name of the interval is, in each 
 case, appended. 
 
 100. The following, then, are all the cases ; 
 
 Middle Note. 
 
 Upper Note. 
 
 Minor Third 
 
 Fifth or Minor Sixth 
 
 Major Third 
 
 Fifth or Major Sixth 
 
 Fourth 
 
 Minor Sixth or Major Sixth 
 
 or, in Musical notation. 
 
 i 
 
 ^1=5 
 
 s 
 
 ;:;3i 
 
 They give us two groups of three triads each, 
 which may be arranged thus : — 
 , > jFifth. „. [Minor Sixth. , . [Major Sixth. 
 
 ^^' JMajor Third. ^ ' \MSnox Third. ^''^ iFourth. 
 , , [Fifth. , -. [Major Sixth. , , [Minor Sixth. 
 
 ^"^ \Minor Third.^^ iMajor Third. ^'^' iFourth. 
 
 101. Instead of defining our six consonant 
 triads, as is here done, by the intervals formed by 
 their middle and top notes with the bottom note, 
 we may define them by the intervals separating the 
 middle from the bottom note, and the top firom the 
 
IX. § 101.] 
 
 CONSONANT TRIADS. 
 
 197 
 
 middle note. In order to make this change we have, 
 in each case, a process of subtraction of intervals to 
 perform. Thus e.g. the difference between a Fifth and 
 a Major Third is f x f , i.e. f, or a Minor Third. 
 
 Proceeding in this way, we find that the top 
 and middle notes are separated by the following 
 intervals : 
 
 (a) 
 
 (6) 
 
 («) 
 
 (a) 
 
 (« 
 
 (y) 
 
 Minor Third 
 
 Fourth 
 
 Major Third 
 
 Major Third 
 
 Fourth 
 
 Minor Third 
 
 Hence we may write our two groups as follows : — 
 , ,.JMinorThird.., /Fourth. ,,. [Major Third. 
 
 ^" ' iMajor Third. ^ ' IMinor Third. ^^ '^ iFourth. 
 , „ [Major Third. - , J Fourth. ... [Minor Third. 
 
 ^" ' iMinor Third.^ 'iMajor Third.^^' iFourth. 
 
 It will now be easy to show that the triads of 
 each group are very closely connected together. 
 Take {a'), and let us form another triad from it by 
 causing its bottom note to ascend one Octave, the 
 other two remaining where they were. The middle 
 will then become the bottom note, the top the middle 
 note, and the Octave of the former bottom note the 
 top note. Hence the lower interval of the new 
 triad will be the upper interval of the old one, i.e. a 
 Minor Third. The upper interval of the new triad 
 
198 INVERSION OF TRIADS. [IX. § 102. 
 
 will necessarily be the inversion of the interval which 
 separated the extreme notes of the old triad. This 
 interval is a Fifth [see (a), § 100], and its inversion, 
 by the table in § 97, is a Fourth. Hence the new 
 
 triad is J ' \ which is identical with (&'). 
 
 \Minor Third./ 
 
 If we modify (6') in the same way, the new 
 
 interval is the inversion of the Minor Sixth, i.e. 
 
 the Major Third, and the resulting triad, viz. 
 
 ^ ■ I is identical with (c'). This triad, when 
 
 Fourth. / ^ ' 
 
 similarly treated, brings us back to [a'), and the cycle 
 of changes is complete. By an extension of the 
 word ' inversion,' it is usual to call the triads (6') and 
 (c') the^rs^ and second inversions of the triad (a'). 
 
 Exactly similar relations hold between the mem- 
 bers of the second group of triads : (/8') and (y') 
 are accordingly called the first and second inver- 
 sions of the triad (a'). The proof is exactly like that 
 just given, and wUl be easily supplied by the reader. 
 102. If we choose C as the bottom note of {a') 
 and (a'), these two groups, which are called respec- 
 tively Major and Minor triads, will be expressed in 
 Musical notation by 
 
 ^ 
 
 S 
 
 and ^BtbB 
 
 («') (»') («') (0-) tf) (7') 
 
IX. §§ 103, 104.] INVERSION OF TRIADS. 199 
 
 They may also be defined in the language of 
 Thorough Bass, which refers every chord to its lowest 
 note in accordance with the mode adopted in (a), (6), 
 (c) ; (a), (;8), (y). Thus the triads {a'\ {V), (c') would 
 be indicated by the figures |, |, | respectively, and 
 so would the triads (a') (/8') (y') ; the differences 
 between Minor and Major Thirds and Sixths being 
 left to be indicated by the key-signature. 
 
 The positions [a') and (a') are regarded as the 
 fundamental ones of each group ; [h'), (c') and (/S'), (y') 
 being treated as derived from them by successive 
 inversion. 
 
 103. The fundamental triads bear the name of 
 their lowest notes, thus {a!) and (a') are called re- 
 spectively the Major and Minor common chords of C. 
 
 The remaining members of each group are not 
 named after their own lowest note, but after that 
 which was lowest in the fundamental position of 
 these chords ; thus (&'), (c') and (j8'), (y') are respec- 
 tively Major and Minor common chords of C in their 
 first and second inversions. 
 
 Common chords of more than three constituent 
 sounds can only be formed by adding to the conso- 
 nant triads notes which are exact Octaves' above or 
 below those of the triads. 
 
 104, The marked distinction existing, for every 
 musical ear, between the bright open character of 
 
200 MAJOR AND MINOR EFFECTS. [IX. § 104. 
 
 Major, and the gloomy veiled effect of Minor chords, 
 is connected by Helmholtz with the different ways 
 in which combination-tones enter in the two cases. 
 The positions of the first-order combination-tones 
 for each of the six consonant triads are shown in 
 crotchets in the appended stave, the primaries being 
 indicated by minims. Each interval gives rise to its 
 
 i 
 
 ^^ 
 
 ^ 
 
 f 
 
 (i 
 
 — ''— St SI- 
 
 ^5E 
 
 — I =1 — — I — 
 
 - j— 1-4- 
 
 own combination-tone, but, in the cases of the funda- 
 mental position and second inversion of the C Major 
 triad, two combination-tones happen to coincide. The 
 reader will at once notice that in the Major group 
 no note extraneous to the harmony is brought in by 
 the combination-tones. In the Minor group this is 
 no longer the case. The fundamental position and 
 the first inversion of the triad both bi'ing in an 
 A\> which is foreign to the harmony, and the second 
 inversion involves an additional extraneous note, Bj. 
 The position of these adventitious sounds is not such 
 as to produce dissonance, for which they are too far 
 apart from each other and from the notes of the 
 triad ; but they cloud the transparency of the har- 
 mony, and so give rise to effects chai-acteristic of the 
 Minor mode. 
 
IX. § 104.] MA JOB AND MINOR EFFECTS. 201 
 
 The imperfect nature of Minor compared with 
 Major triads comes out with pecuhar distinctness on 
 the harmonium ; as indeed, from the powerful combi- 
 nation-tones of that instrument (§ 90) was to be 
 anticipated. 
 
CHAPTER X. 
 
 ON PURE INTONATION AND TEMPERAMENT. 
 
 105. The vibration- fractions of the intervals 
 formed by the notes of the Major scale with the tonic 
 are, including the Octave of the tonic, these : — 
 
 8' 4' 3' 2' 3' 8' ^' 
 
 The intervals between the successive notes of the 
 scale are determined by dividing each of these 
 fractions by that which precedes it (§96). Thus the 
 consecutive intervals of the Major scale come out as 
 follows : — 
 
 CDEFGABC 
 
 a 1 1^ i 1 £. Ig. 
 
 8 7 15 B 7 8 T5 
 
 Only three different intervals are obtained, -f is 
 slightly wider than ^^ ; \^ decidedly narrower than 
 the other two. f and ^-^ are called whole Tones, 
 \^ a half-Tone or semi-Tone, though, strictly speak- 
 ing, two intervals of this width added together 
 somewhat exceed the greater of the two whole 
 Tones ; since If x ^f or Mf ^^ to f in the ratio of 
 2048 to 2025. 
 
 Suppose we had a keyed instrument containing 
 
X. § 105.] REQUISITES FOB PURE INTONATION. 203 
 
 a number of Octaves, eacli divided into seven notes 
 forming the ordinary scale as above, so that any 
 Music could be played on it not involving notes 
 foreign to the key of C Ma.] or. But now, suppose 
 we wanted to be able to play in another Major key 
 as well as in that of C, for instance in G. It would 
 be necessary for this purpose to introduce two new 
 notes in every Octave of the key-board. If G is the 
 new tonic, A wUl not serve as the Second of its scale, 
 because the step between tonic and Second is not ^ 
 but |-. Hence we must have a fresh note lying 
 between A and B. Further, F wiU not do for the 
 seventh of the scale of G, as it is separated from G 
 by f instead of ^. This necessitates a second 
 additional note lying between i^and G. If we take 
 as our original Octave that from middle C upwards, 
 we have the following vibration-numbers : — 
 CDEFGABC 
 264 297 330 352 396 440 495 528 
 The new notes, being respectively -f above and 
 -^ helow G, have for their vibration-numbers -f x 396 
 and ^x 396, i.e. 445^ and 371^. The other notes 
 of the scale of G Major can be supplied from that of 
 C Major. Hence these two scales are closely con- 
 nected with each other. Another key nearly related 
 to the key of C is that of F. Its Fourth is | x 352, 
 or 469^, which falls between A and B. Its Major 
 
204 REQUISITES FOR PURE INTONATION. [X. § 106. 
 
 Sixth is f x352, or 586f, which is clearly not the 
 exact Octave of any note between C and C. The 
 corresponding note in that Octave, found by division 
 by 2, is 293^, which comes between C and D. 
 Thus, two more new notes in the Octave must be 
 introduced to make the key of F major attainable. 
 
 106. In order that the reader may see at a 
 glance the variety of sounds which are requisite 
 to supply complete Major scales for the seven keys 
 of C, D, E, F, G, A and B, the vibration-numbers 
 for aU the notes of these scales are calculated out 
 and exhibited in the following table. 
 
 Reducing within the compass of an Octave 
 those notes which lie beyond it, by dividing their 
 vibration-numbers by 2, and arranging the results 
 
 Tonic 
 
 Second 
 
 Major Third 
 
 Fourth 
 
 Fifth 
 
 Major Sixth 
 
 Major 
 Seventh 
 
 0, 264 
 
 297 
 
 330 
 
 352 
 
 396 
 
 440 
 
 495 
 
 D, 297 
 
 3341 
 
 371J 
 
 396 
 
 445^ 
 
 495 
 
 5561 
 
 E, 330 
 
 371^ 
 
 412^ 
 
 440 
 
 496 
 
 550 
 
 618| 
 
 F, 352 
 
 396 
 
 440 
 
 469i 
 
 528 
 
 586| 
 
 660 
 
 G, 396 
 
 4451 
 
 495 
 
 528 
 
 594 
 
 660 
 
 742| 
 
 A, 440 
 
 495 
 
 550 
 
 586| 
 
 660 
 
 733J 
 
 825 
 
 B, 495 
 
 556^ 
 
 618f 
 
 660 
 
 7421 
 
 825 
 
 928^ 
 
X. § 107.] REQUISITES FOB PURE INTONATION. 205 
 
 in order of magnitude, we have eleven notes foreign 
 to the scale of G Major, the positions of which, 
 with reference to the notes of that scale, are 
 as follows : — 
 
 C, 275, 27^^, 293^, D, 309|, E, 334^, F, 366f, 371^, 
 G, 4121 A, 4451 464tL, 469i, B. 
 107. If it is desired to be able to play in the 
 several Minor modes of each of the seven keys, as 
 well as in the Major, additional notes will be called 
 for. Each scale must contain four notes making 
 with the Tonic the intervals Minor Second, Minor 
 Third, Minor Sixth, and Minor Seventh respectively. 
 The following subsidiary table exhibits the vibration- 
 numbers of the sounds forming these intervals with 
 the successive key-notes. 
 
 Tonie 
 
 Minor 
 Second 
 
 Minor 
 Third 
 
 Minor 
 Sixth 
 
 Minor 
 Seventh 
 
 C, 264 
 
 281f 
 
 3164 
 
 422| 
 
 469| 
 
 D, 297 
 
 316| 
 
 356| 
 
 m\ 
 
 528 
 
 E, 330 
 
 352 
 
 396 
 
 528 
 
 586f 
 
 F, 352 
 
 375,1, 
 
 422| 
 
 563^ 
 
 625| 
 
 G, 396 
 
 422f 
 
 4751 
 
 633| 
 
 704 
 
 A, 440 
 
 m\ 
 
 528 
 
 704 
 
 782f 
 
 B, 495 
 
 528 
 
 594 
 
 792 
 
 880 
 
206 REQ UISITES FOR PURE INTONATION. [X. §§ 108, 109. 
 
 Reducing these to one Octave, as before, we find 
 eight notes not included in the previous list, occupy- 
 ing the following positions : — 
 
 C, 281f, D, 312f, 316f, E,F, 356f, 375,^, 391^, 
 G, 422f, A, 475i, B. 
 
 108. Hence, to play perfectly in tune in both 
 Major and Minor modes of the seven keys C, D, E, 
 F, G, A, B, it is necessary to have a key-board with 
 twenty-six notes in every Octave. This number, 
 large as it is, by no means includes all necessary 
 notes. Modem music is written in sharp and fiat 
 keys, i. e. in such whose tonics are not coincident 
 with any one of the notes CDE...B. Moreover, the 
 sharp and flat key-notes are dijBferent from each 
 other. Thus Gt, being a Major Third above E, is, 
 as the first table shows, 412-|-; while ^1? is seen, by 
 the second table, to be 422|-, which is a somewhat 
 sharper note. As the seven keys which have been 
 already examined require 26 notes in the Octave, 
 we may anticipate that the ten additional sharp and 
 flat keys will bring in a still larger number. 
 
 109. It is, however, needless to institute a de- 
 tailed inquiry into these scales, as the facts already 
 established amply suffice to show how serious are the 
 imperfections of tune which inevitably beset instru- 
 ments of fixed sounds, such as the pianoforte, har- 
 
X. § no.] DEFECTS OF KEYED INSTRUMENTS. 207 
 
 monium and organ, containing only twelve notes in 
 each Octave. Pure intonation in the 'natural' keys 
 alone, i.e. those whose tonics are white notes on the 
 board, demands, as has been seen, more than twice 
 this number of available sounds ; and many more 
 stUl, if the keys with tonics on black notes are 
 to be included. Perfect tuning in all the keys 
 being entirely out of the question, a compromise of 
 some kind is the only possible course. Thus we 
 may tune a single key, say C, perfectly ; in which 
 case most of the other keys will be so out of tune 
 as to be unbearable. Or again, we may distribute 
 the errors over certain often-used keys, and accu- 
 mulate them in others which are of less frequent 
 occurrence. 
 
 Expedients of this kind are described as modes 
 of 'tempering,' and the system adopted in tuning 
 any particular instrument is called its 'temperament.' 
 A large number of different methods of tempering 
 have been proposed and tried from very early times 
 in the history of music. 
 
 110. That which has at last been almost univer- 
 sally adopted is the system of equal temperament. 
 It consists in dividing each Octave into twelve pre- 
 cisely equal intervals. Each of these intervals is 
 called a semi-Tone, and any two of them together a 
 Tone or whole Tone. 
 
208 'TEMPERING' AND 'TEMPERAMENT: [X. § 111. 
 
 The Octave of which C, 264, is the lowest note, 
 will contain, on the equal temperament system, the 
 sounds set out below. The vibration-nimibers are 
 given true to the nearest integer. When a note is 
 slightly sharper than that so indicated, this is shown 
 by the sign + attached to the vibration-number in 
 question ; when slightly flatter, by the sign — . For 
 the sake of comparison, the perfect intervals of the 
 same scale are written below the tempered ones. 
 
 C, Cjt, D, D(, E, P, FJt, G, GJt, A, AJf, B 
 264, 280- 296+, 314-, 333-, 352+, 373+, 395+, 419+, 444-, 470+, 498+ 
 
 C, Db, D, Eb, E, F, G, Ab, A, Bb, B 
 264, 282-. 297, 317-, 330, 352, 396, 422+, 440, 469+, 495 
 
 It is clear that the regions of the tempered scale 
 where the tuning is most imperfect are in the neigh- 
 bourhood of the Thirds and Sixths. E and A are 
 nearly three vibrations per second too sharp. The 
 Fourth and Fifth are wrong by only a fraction of a 
 vibration per second. 
 
 111. The intervals of the tempered scale are so 
 nearly equal to those of the perfect scale that, when 
 the notes of the former are sounded successively, it 
 requires a delicate ear to recognise the defective 
 character of the tuning. When, however, more notes 
 than one are heard at a time, the case becomes quite 
 different. We saw in Chapter viii. how rigorously ac- 
 curate the tuning of a consonant interval must be to 
 
X. §lll.j EQUAL TEMPERAMENT. 209 
 
 secure the greatest smoothness of which it is capable. 
 Such intervals were also shown to be generally very 
 closely bounded by harsh discords. Now since, in 
 the system of equal temperament, no interval except 
 that of the Octave is accurately in tune, it follows 
 that every representative of a concord must be less 
 smooth than it would be were the tuning perfect. 
 One of the greatest charms of Music, and especially 
 of modern Music, lies in the vivid contrast presented 
 by consonant and dissonant chords in close juxta- 
 position. Temperament, by impairing, even though 
 but slightly, the perfection of the concords, necessarily 
 somewhat weakens this contrast, and takes the edge 
 off the musical pleasure which, in the hands of a great 
 composer, it is capable of giving us. A fact already 
 once adverted to (§ 85) may be again adduced here, 
 as illustrating the effect of temperament in blurring 
 distinctions of consonance and dissonance, viz. that 
 on the key-board of the pianoforte the same two 
 notes which represent CE'7, which is a concord, 
 also appear in CDt which is a decided discord, 
 and the same thing holds of CA]> and CGt. 
 
 One of the readiest ways of recognising the de- 
 fective character of equal-temperament tuning is, 
 first to allow a few accurate voices to sing a series 
 of sustained chords in three or four parts, without 
 accompaniment, and then, after noticing the effect, 
 
 T. 14 
 
210 EFFECT OF TEMPERAMENT. [X. § 112. 
 
 to let them repeat the phrase while the parts are 
 at the same time played on the pianoforte. The 
 sour character of the concords of the accompanying 
 instrument will be at once decisively manifested. 
 Voices are able to sing perfect intervals, and their 
 clear transparent concords contrast with the duller 
 substitutes provided by the pianoforte in a way 
 obvious to every moderately sensitive ear. 
 
 112. Since the voice is endowed with the power 
 of producing all possible shades of pitch within its 
 compass, and thus of singing absolutely pure inter- 
 vals, it is clear that we ought to make the most of 
 this great gift, and, especially in the case of those 
 persons who are to be public singers, allow during 
 the season of preparation contact with the purest 
 examples of intonation only. Unfortunately the 
 practice of many singing-masters is the very reverse 
 of this. The pupil is systematically accompanied, 
 during vocal practice, on the pianoforte, and thus 
 accustomed to habitual familiarity with intervals 
 which are never strictly in tune. No one can doubt 
 the tendency of such constant association to impair 
 the sensitiveness to minute differences of pitch on 
 which delicacy of musical perception depends. Evil 
 communications are not less corrupting to good 
 ears than to good manners. I am convinced that we 
 have here the reason why so comparatively few of our 
 
X. §112.] VOCAL INTONATION. 211 
 
 trained vocalists, whether amateurs or professionals, 
 are able to sing perfectly in tune. The untutored 
 singing of a child who has never undergone the ear- 
 spoiling process often gives more pleasure by the 
 natural purity of its intonation than the vocalisation 
 of an opera-singer who cannot keep in tune. The 
 remedy is to practise without accompaniment, or 
 with that of an instrument like the violin^, which is 
 not tied down to a few fixed sounds. Even with the 
 pianoforte something might be done by having it, 
 when intended to be used only in assisting vocal 
 practice, put into perfect tune in one single key, and 
 using that key only. 
 
 The services of such an instrument would, no 
 doubt, be comparatively very restricted, but this 
 might not be without a corresponding advantage, if 
 the vocalist were thereby compelled to rely a little 
 more on his own unaided ear, lay aside his corks and 
 swim out boldly into the ocean of Sound. 
 
 ' That a violinist can play 'pure intervals was established 
 by Professor Helmholtz by the following decisive experiment, 
 performed with the aid of Herr Joachim. A. harmonium was 
 employed which had been tuned so as to give pure intervals with 
 certain stops and keys ; and tempered intervals with others. A 
 string having been tuned in unison with a common tonic of both 
 systems, it was found that the intervals played by the eminent 
 violinist agreed with those of the mxtural, not with those of the 
 tempered scale. 
 
 14—2 
 
212 ESTABLISHED MUSICAL NOTATION. [X. § 113. 
 
 113. The Musical notation in ordinary use evi- 
 dently takes for granted a scale consisting of a 
 limited number of fixed sounds. Moreover, it indi- 
 cates directly absolute pitch, and only indirectly 
 relative pitch. In order to ascertain the interval 
 between any two notes on the stave, we must go 
 through a Uttle calculation involving the clef, the 
 key-signature, and perhaps additional ' accidental' 
 sharps or flats. Now these are complications which, 
 if necessary for pianoforte Music S are perfectly gra- 
 tuitous in the case of vocal Music. The vocalist 
 wants only to be told on what note to begin, and 
 what intervals to sing afterwards, i.e. is essentially 
 concerned with absolute pitch only at the start, and 
 need be troubled with it no further. 
 
 The established notation thus encumbers the vo- 
 calist with information which he does not want, and 
 yet fails to communicate the one special piece of in- 
 formation which he does want, viz. the relation which 
 the note to be sung bears to the tonic or key-note. 
 There is nothing in the established notation to mark 
 clearly and directly what this relation is, in each case, 
 intended to be. Unless the vocaHst, besides his own 
 ' part' is provided with that of the accompaniment, 
 
 ' In a Paper published in Vol. i. of the Proceedings of the 
 Musical Association (London, Chappell and Co.), I have en- 
 deavoured to prove that there is no such necessity. 
 
X. § 113.] ESTABLISHED MUSICAL NOTATION. 213 
 
 and possesses some knowledge of Harmony, he can- 
 not ascertain how the notes set down for him are 
 related to the key-note and to each other. The 
 extreme inconvenience of this must have become 
 painfully evident to any one who has frequently sung 
 in concerted Music from a single part. 
 
 A Bass, we will suppose, after leaving off on Ft, 
 is directed to rest thirteen bars and then come in 
 fortissimo on his high E\>. It is impossible for him 
 to keep the absolute pitch of Fi, in his head during 
 this long interval, which is perhaps occupied by 
 other voices, or instruments, in modulating into some 
 remote key ; and his part vouchsafes no indication 
 in what relation the Ely stands to the notes, or chords, 
 immediately preceding it. There remains, then, no- 
 thing for him to do but to sing, at a venture, some 
 note at the top of his voice, in the hope that it may 
 prove to be Fh, though with considerable dread, in 
 the opposite event, of the conspicuous ignominy 
 entailed by a fortissimo blunder. 
 
 The essential requisite for a system of vocal 
 notation therefore is that, whenever it specifies any 
 sound, it shall indicate, in a direct and simple 
 manner, the relation in which that sound stands to 
 its tonic for the time being. A method by which 
 this condition is very completely satisfied shaU now 
 be briefly described. 
 
214 TONIC SOL-FA NOTATION. [X. § 114. 
 
 114. The old Italian singing-masters denoted 
 the seven notes of the Major scale, reckoned from 
 the key-note upwards, by the syllables 
 do, re, mi, fa, sol, la, si, 
 pronounced, of course, in the continental fashion. 
 So long as a melody moves only in the Major mode, 
 without ' chromatic ' notes or modulation, it clearly 
 admits of being written down, as far as relations of 
 pitch only are concerned, by the use of these syllables. 
 The opening phrase of ' Rule Britannia,' for instance, 
 would stand thus : — 
 sol*, do, do, do, re, mi, fa, sol, do, re, re, mi, fa, mi. 
 
 In order to abridge the notation, we may indi- 
 cate each syllable by its initial letter. The ambiguity 
 which would thus arise between sol and si is got rid 
 of by altering the latter syllable into ti. In order to 
 distinguish a note from those of the same name in 
 the adjacent Octaves above and below it, an accent 
 is added either above or below the corresponding 
 initial. Thus d' is an Octave above d ; d^ an Octave 
 below d. 
 
 Where a modulation, i.e. a change of tonic, 
 occurs, it is shown in the following manner. A note 
 necessarily stands in a two-fold relation to the out- 
 going and to the in-coming tonic. The intei-val it 
 
 * The asterisked sol is to be sung an Octave below that not 
 thus marked. 
 
X. § 114.] 
 
 TO NIG SOL-FA NOTATION. 
 
 215 
 
 forms with the new tonic is different from that which 
 it formed with the old one. Each of these intervals 
 can be denoted by a suitable syllable-initial, and 
 the displacement of one of these initials by the 
 other represents in the aptest manner the super- 
 session of the old by the new tonic. The old initial 
 is written above and to the left of the new one. 
 Thus ^indicates that the note re is to be sung, but 
 its name changed to fa. As this is a somewhat 
 difficult point, a few instances are appended, ex- 
 pressed both in the established notation and in that 
 now under consideration. The modulations selected 
 are, from C to G; from Cto i''; from E to C; from 
 GtoFt. 
 
 ^ 
 
 ^=8= 
 
 3= 
 
 P \> i 
 
 ^te^ji^ a ^^^s 
 
 zst 
 
 Immediately after a modulation the ordinary 
 syllable-initials come into use again, and are employed 
 until a fresh modulation occurs. It will be seen at 
 once that the difficulty of ' remote' keys, which is so 
 formidable in the established notation, here altogether 
 
216 
 
 TONIC SOL-FA NOTATION. 
 
 [X. § 115. 
 
 disappears. For instance, a vocal phrase from Spohr's 
 ' Last Judgment ' which in the established notation 
 stands as follows : — 
 
 4.J?/ 
 
 l^ 
 
 ^ 
 
 iici 
 
 rft 
 
 ?^e 
 
 tpr 
 
 s 
 
 q— »g^k=t<- 
 
 ^^ 
 
 takes, in the notation before us, the perfectly simple 
 form, 
 
 s I t \d' mfs\sf I I I s \f m 
 
 As another example, take the following from the 
 same work : — 
 
 g^g^^ 
 
 ^ 
 
 d^f::^: 
 
 A. 
 
 f 
 
 r f 
 
 115. The system of notation of which a cursory 
 sketch has just been given originated with the 
 late Miss S. A. Glover of Norwich, but received 
 its present form at the hands of the late Mr J. 
 Curwen, to whom it also owes the name of ' Tonic- 
 Sol-fa,' by which it is now so widely known. As it 
 is no part of the plan of the present work to go into 
 technical details, only so much has been said about 
 Mr Curwen's system as was necessary to enable the 
 reader to grasp its essential principle. No mention 
 has been made of the notation for Minor and Chro- 
 matic intervals, nor of that for denoting the relations 
 
X. § 115.] TONIG SOL-FA NOTATION. 217 
 
 of time by measures appealing directly to the eye, 
 instead of by mere symbols. On these and other 
 points connected with his system, Mr Curwen's pub- 
 lished works on Tonic Sol-fa give full and thoroughly 
 lucid explanations, Mr Curwen also created an ex- 
 tensive literature of vocal Music, printed in his own 
 notation, which since his death' has been, and is still 
 being, continually enlarged; gave a most remarkable 
 impulse to choral singing ; and established a sys- 
 tem of graded certificate-examinations guaranteeing 
 the attainment of corresponding stages of Musical 
 cultivation. 
 
 I have enjoyed some opportunities of watching 
 the progress of beginners taught on the old system, 
 and on that of the Tonic Sol-fa, and assert without 
 the slightest hesitation that, as an instrument for 
 imparting the power of sight-singing, the new system 
 is enormously, overwhelmingly, superior to the old. 
 In fact, I am prepared to maintain that the compli- 
 cated repulsiveness of the pitch-notation in the old 
 system must be held mainly responsible for the 
 humiliating fact that, of the large number of musically 
 well-endowed persons of the opulent classes who have 
 
 " Mr Curwen died on May 26, 1880. His son, Mr J. Spencer 
 Curwen, is now at the head of the Tonic Sol-fa Association, the 
 publications of which are on sale at 8 and 9 Warwick Lane, 
 London, E.C. 
 
218 TONIC SOL-FA NOTATION. [X. § 115. 
 
 undergone an elaborate vocal training, comparatively 
 few are able to sing even the very simplest Music at 
 sight. Set a young lady thus instructed to sing 
 a psalm-tune she has never seen before, and we all 
 know what the result is likely to be. Now, there is 
 no greater difficulty in teaching a child with a fairly 
 good ear to sing simple music at sight, than there is 
 in teaching him to read ordinary print at sight. A 
 vocalist who can sing only a few elaborately pre- 
 pared songs ought to be regarded as on a level with 
 a school-boy who should be unable to read except 
 out of his own book. If evidence be wanted to 
 make good this assertion, it is at once to hand in 
 the fact that the youngest children, when well 
 trained on the Tonic Sol-fa system, soon obtain a 
 power of steady and accurate sight-suigiag. 
 
 The reader is requested to observe that the above 
 remarks are strictly limited to the achievements of 
 the Tonic Sol-fa system in vocal Music. I express 
 no opinion as to the apphcability of its notation to 
 instrumental Music, nor do I wish to maintain that 
 even in the vocal branch it has arrived at perfection. 
 On the contrary, I am doubtful whether its time- 
 notation, when applied to very complicated rhythmic 
 divisions, does not become more difficult than the 
 system in ordinary use, and I consider the notation 
 adopted for the Minor mode to be seriously de- 
 
X. § 116.] PHYSICS AND ESTHETICS. 219 
 
 fective\ On the main point, however, viz. the 
 decisive superiority of its pitch-notation over that of 
 the established system, and the vitally important 
 consequences as to purity of intonation which neces- 
 sarily follow from this superiority, I desire to express 
 the most confident and uncompromising opinion^ 
 
 116. In closing the enquiry which occupies the 
 preceding chapters, it will be advisable to examine 
 very concisely the bearing of our principal result, 
 the theory of consonance and dissonance, on the 
 aesthetics of music. Dissonance was shown to arise 
 from rapid beats, and the concords were classed in 
 order according to their more or less complete free- 
 dom from dissonance ; the Octave coming first, fol- 
 lowed by the Fifth, Fourth, Major Third and Sixth, 
 and Minor Third and Sixth. This classification was 
 strictly physical, depending exclusively on smooth- 
 ness of eflfect. On its own ground, therefore, this 
 classification is absolutely unassailable, and whoever 
 says, for instance, that a Major Third is a smoother 
 concord than a Fifth or Octave, asserts what is as 
 
 ^ See a pamphlet by me entitled " The Minor Notation of the 
 Tonic Sol-fa system." London, J. Curwen and Sons, 8 and 9 
 Warwick Lane, E.C. 
 
 ^ These topics will be found handled somewhat more at large 
 in a Lecture delivered by me at Manchester on Jan. 11, 1883, 
 entitled "The Tonic Sol-fa Movement.'' London, J. Ourwen 
 and Sons, 8 and 9 Warwick Lane, E.G. 
 
220 PHYSICS AND ESTHETICS. [X. § 117. 
 
 demonstrably false as that the moon goes round the 
 earth in an exact circle. Nevertheless, it by no 
 means necessarily follows that the smoothest con- 
 cords must be the most gratifying to the ear. There 
 may be some other property of an interval which . 
 gives us greater satisfaction than mere consonance. 
 Assuming, for the moment, that such a property 
 does in fact exist, the ear, if called on to arrange the 
 consonant intervals in the order of their pleasantness, 
 might very well bring out a different arrangement 
 from that adopted by physical science on grounds of 
 smoothness alone. Esthetic considerations come in 
 here with the same right to be heard that mechanical 
 considerations possess within their own domain. 
 
 117. Now unquestionably the ear's order of 
 merit is not the same as the mechanical order. It 
 places Thirds and Sixths first, then the Fourth and 
 Fifth, and the Octave last of all. The constant 
 recurrence of Thirds and Sixths in two-part Music, 
 compared with the infrequent employment of the 
 remaining concords, leaves no doubt on this point. 
 In fact these intervals have a peculiar charm about 
 them, not possessed by the Fourth or Fifth to any- 
 thing like the same extent, and by the Octave not 
 at all. 
 
 The thin effect of the Octave undoubtedly de- 
 pends on the fact that every partial-tone of the 
 
X. § 117.] PHTSICS AND ESTHETICS. 221 
 
 higher of two clangs forming that interval coin- 
 cides exactly with a partial-tone of the lower clang. 
 Thus no new sound is introduced by the higher 
 clang ; the quality of that previously heard is 
 merely modified by alteration of relative intensity 
 among its constituent partial-tones. Major and 
 Minor Thirds bring in a greater variety of pitch in 
 the resulting mass of sound than does the Fifth ; but 
 this can hardly be said of the Major and Minor 
 Sixths compared with the Fourth. On the whole, I 
 am inclined to attribute the predilection of the ear 
 for Thirds and Sixths over the other concords to 
 circumstances connected with its perception of key- 
 relations, though I am not able to give a satisfactory- 
 account of them. The ear enjoys, in alternation 
 with consonant chords, dissonances of so harsh a 
 description as to be barely endurable when sustained 
 by themselves. As passages containing such extreme 
 discords take the following. The chord marked* 
 should in each case be played ./irsf hy itself, and then 
 in the place assigned to it by the composer. The 
 effect of the isolated discord is so intensely harsh, 
 that it is at first difi&cult to understand how any 
 preceding and succeeding concords can make it at aU 
 acceptable ; yet the sequence, in both the phrases 
 cited, is of the rarest beauty. 
 
 Considerations such as those just alleged tend to 
 
222 
 
 CONCLUSION. 
 
 [X. § 117. 
 
 show that, while physical science is absolutely au- 
 thoritative in all that relates to the constitution of 
 
 ^ 
 
 w: 
 
 > * 
 
 J=Jz 
 
 Last Chorus, BctcWa 
 "Passion" {St. Matthew). 
 
 m 
 
 m 
 
 bpz 
 
 --^^■ 
 
 > N \ 
 
 ^ *- 
 
 gfcS 
 
 f^HE 
 
 =4^^ 
 
 > ^ 
 
 ai 
 
 ^^ 
 
 =t5=q^ 
 
 ^:±^i 
 
 ^^=^^r'Tr=fTf 
 
 ais^ 
 
 A j^ , ^ * 
 
 
 i 
 
 tfctrH 
 
 ^ 
 
 musical sounds and the smoothness of their combi- 
 nations, the composer's direct perception of what is 
 musically beautiful must mainly direct him in the em- 
 ployment of his materials. It would be a serious error 
 to force upon him a number of rules constructed, on 
 scientific principles, to obtain the maximum smooth- 
 ness of effect ; since mere smoothness is often a 
 matter of extremely secondary importance compared 
 with grandeur of harmony and masterly movement 
 of parts. The nature of the subject may sometimes 
 call for treatment planned to secure exceptional 
 smoothness. In such a case these rules may become 
 of considerable importance. It is well, therefore. 
 
X. § 117.] CONCLUSION. 223 
 
 that a composer should know and be able to handle 
 them, but he should never allow them to fetter his 
 freedom in wielding the higher and more spiritual 
 weapons of his warfare. 
 
 CAMBRIDGE: PKTNTED BY J. & C. F. CLAY, AT THE UNIVEKSITY PRESS.