THE SCIENCE OF MUSICAL SOUNDS DAYTON CLARENCE MILLER CORNELL UNIVERSITY LIBRARY dift Of Prof. Guy E. Grantham MUSIC Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924022230654 THE LOWELL LECTURES THE SCIENCE OF MUSICAL SOUNDS O ^^^ o *" THE MACMILLAN COMPANY NEW YORK • BOSTON ■ CHICAGO - DALLAS ATLANTA • SAN FRANCISCO MACMILLAN & CO., Limited LONDON • BOMBAY • CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, Ltd TORONTO THE SCIENCE OFcMUSICAL SOUNDS DAYTON CLARENCE MILLER, D.Sc. PROFESSOR OP PHTSIC8 CASE SCHOOL OF APPLIED SCIENCE NeSd gork THE MACMILLAN COMPANY 1916 All rights reserved Copyright, 1916, By the MACMILLAN COMPANY. Set up and printed. Published March, 1916, sr3 Kotinoati 3Pn«ti J. S. Onshlng Go. — Berwick A Smith do. Norwood, Mass., U.S.A. m PREFACE A SERIES of eight lectures was given at the Lowell Insti- tute in January and February, 1914, under the general title of "Sound Analysis." These lectures have been rewritten- for presentation in book form, and "The Science of Musical Sounds " has been chosen for the title as giving a better idea of the contents. They appear substantially as delivered, though some slight additions have been made and much explanatory detail regarding the experiments and illustrations has been omitted. The important additions relate to the tuning fork in Lecture II and harmonic analysis in Lecture IV, while two quotations are added in the concluding section of Lecture VIII. A course of scientific lectures designed for the general public must necessarily consist in large part of elementary and well known material, selected and arranged to develop the principal line of thought. It is expected that lectures under the aus- pices of the Lowell Institute, however elementary their foun- dation, will present the most recent progress of the science. The explanations of general principles and the accounts of recent researches must be brief and often incomplete ; never- theless it is hoped that the lectures in book form will furnish a useful basis for more extended study, and to further this end they are supplemented by references to sources of additional information. The references are collected in an appendix, citations being made by numbers in the text corresponding to the numbers in the appendix. It is further expected that such lectures will be accompanied by experiments and illustrations to the greatest possible de- gree; the nature and extent of this illustrative material is PREFACE shown as well as may be by the aid of diagrams and pictures, nearly all of which have been especially prepared, and much care has been taken to make them as expressive as possible of the original demonstrations and explanations. The methods and instruments used in sound analysis by the author, and many of the results of such work, were described in the lectures in advance of other publication and it is the intention to supplement the brief accounts here given by more detailed reports in scientific journals. The author is greatly indebted to many friends for the kindly interest shown during the progress of the experimental work here described ; and he is especially under obligation to Professor Frank P. Whitman of Western Reserve University, and to Mr. Eckstein Case and Professor John M. Telleen of Case School of Applied Science, for many helpful suggestions received while the manuscript was in preparation. DAYTON C. MILLER. Cleveland, Ohio, July, 1915. VI CONTENTS LECTUEE I SOUND WAVES, SIMPLE HARMONIC MOTION, NOISE AND TONE Introduction — Sound defined — Simple harmonic motion and curve — Wave motion — The ear — Noise and tone . . . . 1 LECTUEE II CHARACTERISTICS OF TONES Pitch — The tuning fork — Determination of pitch by the method of beats — Optical comparison of pitches — The clock-fork — Pitch limits — Standard pitches — Intensity and loudness — Acoustic properties of auditoriums — Tone quality — Law of tone quality — Analysis by the ear 26 LECTUEE III METHODS OF RECORDING AND PHOTOGRAPHING SOUND WAVES The diaphragm — The phon autograph — The manometric flame — The oscillograph — The phonograph — The phonodeik — The demonstration phonodeik — Determination of pitch with the pho- nodeik — Photographs of compression waves . . . . .70 LECTUEE IV ANALYSIS AND SYNTHESIS OF HARMONIC CURVES Harmonic analysis — Mechanical harmonic analysis — • Amplitude and phase calculator — Axis of a curve — Enlarging the curves — Syn- thesis of harmonic curves — The complete process of harmonic analysis — Example of harmonic analysis — Various types of harmonic analyzers and synthesizers — Arithmetical and graph- ical methods of harmonic analysis — Analysis by inspection — Periodic and non-periodic curves 92 vii CONTENTS LECTURE V INFLUENCE OF HORN AND DIAPHRAGM ON SOUND WAVES, CORRECTING AND INTERPRETING SOUND ANALYSES PAOS Errors in sound records — Ideal response to sound — Actual response to sound — Response of the diaphragm — Chladni's sand figures — Free periods of the diaphragm — Influence of the mounting of the diaphragm — Influence of the vibrator — Influence of the horn — Correcting analyses of sound waves — Graphical presentation of sound analyses — Verification of the method of correction — Quantitative analysis of tone quality 142 LECTURE VI TONE QUALITIES OF MUSICAL INSTRUMENTS Generators and resonators — Resonance — Effects of material on sound waves — Beat-tones — Identification of instrumental tones — The tuning fork — The flute — The violin — The clarinet and the oboe — The horn — The voice — The piano — Sextette and orchestra — The ideal musical tone — Demonstration 175 LECTURE VII PHYSICAL CHARACTERISTICS OF THE VOWELS The vowels — Standard vowel tones and words -^ Photographing, analyzing, and plotting vowel curves — Vowels of various voices and pitches — Definitive investigation of one voice — Classifica- tion of vowels — Translation of vowels with the phonograph — Whispered vowels — Theory of vowel quality .... 215 LECTURE VIII SYNTHETIC VOWELS AND WORDS, RELATIONS OF THE ART AND SCIENCE OF MUSIC Artificial and synthetic vowels — Word formation — Vocal and instru- mental tones — "Opera in English" — Relations of the art and science of music 244 Appendix — Referknces 271 Index 281 THE SCIENCE OF MUSICAL SOUNDS LECTURE I SOUND WAVES, SIMPLE HARMONIC MOTION, NOISE AND TONE Introduction We are beings with several senses through which we come into direct relation with the world outside of ourselves. Through two of these, sight and hearing, we are able to receive impressions from a distance and through these only- do the fine arts appeal to us ; through sight we receive the arts of painting, sculpture, and architecture, and through hearing, the arts of poetry and music. "Undoubtedly, music gives greater pleasure to more people than does any other art, and probably this enjoyment is of a more subtle and pervading nature; every one enjoys music in some degree, and many enjoy it supremely. Sound is also of the greatest practical importance ; we rely upon it continually for the protection of our lives, and through talk- ing, which is but making sounds according to formula, we receive information and entertainment. These facts give ' ample justification for studying the nature of sound, the material out of which music and speech are made. The study of sounds in language is as old as the human race, and the art of music is older than tradition, but the science of music is quite as modern as the other so-called modem sciences. Sound being comparatively a tangible THE SCIENCE OF MUSICAL SOUNDS phenomenon, and so intimately associated with the very existence of every human being, one would expect that if there are any unknown facts relating to it, a large number of investigators would be at work trying to discover them. There has been in the past, as there is now, a small number of enthusiastic workers in the field of acoustics who have ac- comphshed much ; but it is no doubt true that this science has received less attention than it deserves, and especially may this be said of the relation of acoustics to music. Sound Defined Sound may be defined as the sensation resulting from the action of an external stimulus on the sensitive nerve ap- paratus of the ear ; it is a species of reaction to this external stimulus, excitable only through the ear, and distinct from any other sensation. Atmospheric vibration is the normal and usual means of excitement for the ear; this vibration originates in a source called the sounding body, which is itself always in vibration. The source may be constructed especially to produce sound; in a stringed instrument the string is plucked or bowed and its vibration is transferred to the soundboard, and this^in turn impresses the motion upon a larger mass of air'; in the flute and other wind instruments the air is set in motion directly by the breath. The vibration often origi- nates in bodies not designed for producing sounds, as is illustrated by the squeak and rumble of machinery. The physicist uses the word sound to designate the vibra- tions of the sounding body itself, or those which are set up by the sounding body in the air or other medium and which are capable of directly affecting the ear even though there is no ear to hear. 2 THE NATURE OF SOUND There are numerous experiments which demonstrate that a sounding body vibrates vigorously. When a tuning fork, Fig. 1, is struck with a soft felt hammer, it gives forth a continuous sound. The fork vibrates transversely several hundred times a second, though the distance through which it moves is only a few thousandths of an inch. These move- ments, which are too minute and rapid to be ap- preciated by the eye, may be made evident by means of a pith-ball pen- dulum adjusted to rest lightly against the prong; when the fork , is sounding, the ball is violently thrown aside. The powerful longitudinal vi- brations of a metal rod may be exhibited in a like manner with apparatus arranged as shown in Fig. 2. If a piece of rosined leather is drawn along the rod a loud tone is emitted, and the ivory ball resting hghtly against the end is thrown high in the air. At the center of the rod the molecules are at rest, forming a node, while at the ends the vibrations are of relatively large ampUtude. The molecules vibrate hundreds of times per 3 Fig. 1. Tuning fork and pith ball for demonstration of vibration. THE SCIENCE OF MUSICAL SOUNDS second, and the comparatively feeble movements of single molecules, by their cumulative effects in the mass, develop enormous forces, equivalent to a tensile force of several tons. If a glass tube, held at its middle, is rubbed with a piece of wet cloth, the longitudinal vibrations are often so vigorous as to cause the tube to separate into many pieces. A glass bell may be set into transverse vibration by bow- FiG. 2. Longitudinally vibrating rod. ing across the edge. The vibrations cause periodic deforma- tions of the shape, from a circle to an ellipse and back to the circle. The circumference must vibrate in at least four segments,' with the formation of loops and nodes. Balls, suspended from a revolving support, as shown in Fig. 3, may rest against the surface of the bell and may be used to locate the loops and nodes. The vibration may become so violent as to shatter the glass. 4 THE NATURE OF SOUND The vibrations of the source produce various physical effects in the surrounding air, such as displacements, ve- locities, and accelerations, and changes of density, pressure, and temperature ; because of the elasticity of the air, these displacements and other phenomena occur periodically and are transmitted from particle to particle in such a manner Fig. 3. Glass bell. that the effects are propagated outward from the source in radial directions. These disturbances of all kinds, as they exist in the air around a sounding body, constitute sound waves. The velocity of sound is about 1132 feet per second when the temperature of the air is 70° F., a tempera- ture common in auditoriums ; at the freezing temperature, 32° F., it is about 1090 feet per second. Musical sounds of different pitches are all propagated in the open air with the 5 THE SCIENCE OF MUSICAL SOUNDS same velocity. Explosive sounds and sounds confined, as in tubes, are propagated with different velocities.^ Wave disturbances may be transmitted by solid and liquid as well as gaseous matter, but our present study relates mainly to what may be heard, and the explanations are Hmited for the most part to certain features of waves in air, and par- ticularly to the liature of the movements of the air parti- cles when transmitting musical sounds. Simple Harmonic Motion and Curve The simplest possible type of vibration which a particle of elastic matter of any kind may have is called simple harmonic motion; it takes place in a straight line, the mid- dle of which is the position of rest of the particle ; when the particle is displaced from this position, elasticity develops a force tending to restore it, which force is directly proportional to the amount of the displacement; if the displaced particle is now freely released, it will vibrate to and fro with simple harmonic motion. The name originated in the fact that musical sounds in general are produced by complex vibrations which can be resolved into component motions of this type. Other forces than those of elasticity may act in the manner described, as for instance the action of the force of gravity on the bob of a pendulum ; if the bob is considered as swing- ing in a straight line, it has simple harmonic motion, which is also called pendular motion. Simple harmonic motion has several evident features: it takes place in a straight Une ; it is vibratory, moving to and fro ; it is periodic, repeating its movements regularly ; there are instants of rest at the two extremes of the move- ment; starting from rest at one extreme the movement 6 SIMPLE HARMONIC MOTION quickens till it reaches its central point, after which it slackens in reverse order, till it conies to rest at the other extreme. The speed of the particle so moving, the rate at which the speed changes, and other features are very important in a complete study of simple harmonic motion, but for our purpose we need give only a few simple defini- tions. The frequency of a simple harmonic motion is the num- ber of complete vibrations to and fro per second ; the period is the time required for one complete vibration ; the am- plitude is the range on one side or the other from the middle point of thei motion, therefore it is half the ex- treme range of vibration; the phase at any instant is the fraction of a period which has elapsed since the point last passed through its mid- dle position in the direction chosen as positive. Simple harmonic motion is approximated in various me- chanical movements, while a few simple machines reproduce it exactly; 2 this reproduction is always accompUshed by a transformation of uniform motion in a circle into rectilinear motion. The pin-and-slot device has a slotted frame s. Fig. 4, which is movable up and down only ; the pin p of the crank c moves in the slot ; when the crank is turned with uniform angular speed, the frame and all rigidly at- 7 Fig. 4. Simple harmonic motion from mechanical movement. THE SCIENCE OF MUSICAL SOUNDS tached parts, such as the point P, move with simple har- monic motion. The usual starting point for this motion is the middle position of P when it is about to move upward, that is, when the crank is horizontal with the pin at the extreme right and about to turn counterclockwise. One complete vibration is produced when the crank makes one revolution and the point P moves from its mid- position to the extreme upper position, down to the lower extreme, and back to mid-position. The period is the time required for the complete vibration, that is, for one revolution of the crank; the phase at any instant is the fraction of a period which has elapsed since the point last passed through the starting point, and is often ex- pressed by the number of degrees through which the crank has turned in the interval, as is further illustrated in Fig. 6; the amplitude is measured-"by-half the extreme movement, that is, by the length of the crank from center to pin. This device is used in several of the harmonic synthesizers described in Lecture IV. In treatises on mechanics simple harmonic motion is often defined as the projection of uniform motion in a circle upon a diameter of the circle ; this, definition is illustrated by the ',; H' f^ FiQ. 5. Relation of simple harmonic and circular motion. SIMPLE HARMONIC MOTION form of the pin-and-slot apparatus shown in Fig. 5. Turn- ing the crank on the back of the apparatus causes the point P in the diameter to move up and down with a true harmonic /i ^ Phase HPhase 90 %Phase 225° %Phase Jb'iG. U. i-'hases of simple harmonic motion. 375° motion when the point p in the circle revolves with uniform speed; the two points are always in the same horizontal line, or the point in the diameter is always the projection of the one in the circle ; Fig. 6 illustrates the motion in various phases. A crank pin p is pivoted in the center of a rod AB, Fig. 7; the ends of the rod are pivoted to shders which move in two perpendicxilar, straight grooves; when the crank is turned with uniform speed, both of the points A and B move with simple har- monic motion. The dis- placement of either sHder from its central position is always twice the displace- ment of the projection of the point p on the corresponding groove. Fig. 7. Simple harmonic motion from mechanical movement. THE SCIENCE OF MUSICAL SOUNDS A simple harmonic motion can be obtained without the friction of sUders in grooves by employing a pantograph to give the arithmetical mean of two equal and opposite cir- cular motions, as suggested by Everett.^ When the crank c, Fig. 8, is turned, the point P moves up and down in a straight line, so that it is always in the horizontal line con- j Fig. 8. Simple harmonic motion from mechanical movement. Fig. 9. Simple harmonic motion from mechanical movement. necting the points A and B, and therefore, when the wheels rotate with uniform speed, it has simple harmonic motion. A simple harmonic motion is given to any point P on the circumference of a wheel, Fig. 9, when the wheel rolls with uniform speed on the inside of an annulus a, the radius of which is equal to the diameter of the wheel. The point P is always in the horizontal line passing through the point of contact a of the wheel and annulus. 10 SIMPLE HARMONIC MOTION The movement of a sliding block connected to a crank by a pitman rod, as the crosshead of an engine, has a distorted simple harmonic motion, the errors of which may be cor- rected by suitable mechanism ; Fig. 10 shows a device due to Smedley,^ having two crossheads, Ci and Ci, on opposite sides of the crank pin ; during the motion these are oppositely displaced from the true harmonic positions, and the errors are equalized by a system of levers acting on the central Fig. 10. Simple harmonic motion from compensated crosshead movement. sUding block P, which receives simple harmonic motion when the crank revolves uniformly. A simple harmonic motion combined with a uniform mo- tion of translation traces a simple harmonic curve; this condition is illustrated by a pendulum swinging from a fixed point, Fig. 11, and leaving a trace on a sheet of paper mov- ing underneath. The simple harmonic curve, Fig. 12, is perfectly simple, regular, and symmetrical ; in mathematical study it is frequently referred to as a sine curve; a curve of the same form but differing in phase by a quarter period, or 90°, is a cosine curve. As explained in the next section, such 11 THE SCIENCE OF MUSICAL SOUNDS a curve is an instantaneous representation of the condition of motion in a simple wave. Various terms used with regard to simple harmonic motion are also applicable to the curve ; the am- plitude is the height of a crest above the axis, Fig. 12; the period is the time required to trace one wave length consist- ing of a crest and trough ; the fre- quency is the num- ber of periods, or wave lengths traced, per second ; the phase varies along the axis, passing through a complete cycle in one wave length ; waue length Fig. 11. Tracing'a simple harmonic curve. Fig. 12. Sine and cosine curves. the velocity is the rate of translation and is equal to the wave length multiplied by the number of waves per second. Sine curves may differ considerably in appearance, de- 12 WAVE MOTION pending upon the relation of amplitude and wave length (frequency), though all must have the same general prop- erties and be equally regular and simple. All the curves shown in Fig. 13 are simple harmonic, or sine curves, and differ only in amplitude A and frequency n, the relative values of these quantities being shown in the figure. ^=35 n=30 ^=0.1 A=Z5 A=2 'WWV\AAAA/\ =3 A=20 1 = 1 'A^4 A=0.5 n=10 A=4D Fig. 13. Sine curves of various dimensions. Wave Motion The essential characteristic of wave motion is the continu- ous passing onward from point to point in an elastic medium 13 THE SCIENCE OF MUSICAL SOUNDS of a periodic vibration which is maintained at the source. These vibrations, being periodic, produce a series of waves following each other at regular intervals, the speed of propagation depending upon the elastic properties of the medium. There are two distinct motions involved : the vibration of the individual particles about their positions of rest and the progressive outward movement of the wave form. The source of a wave motion may be a disturbance 1 i~ I_ . ' t4 * * * •♦•*♦•♦••• ••*v S tp*"^ IT ■ -:h-,-s-::--:-*f i \ Fig. 14. Machine for illustrating transverse waves. of any type, but for sound waves it consists of vibratory- movements which are either simple harmonic or compounds of such. A simple transverse wave motion is represented by the wave machine shown in Fig. 14; the successive pendulums are given similar periodic transverse vibrations in successive times by a slider s, which is moved from left to right by turn- ing the handle h. The slider produces a wave crest which moves along the row of balls and disappears, being followed periodically by other crests ; the velocity of wave propaga- 14 WAVE MOTION tion is the velocity with which the sUder is moved ; the wave length is the actual distance I from crest to crest of the wave ; amplitude, period, and frequency are illustrated in the vibrations of the pendulums. One of the bars to which is attached one string of each bifilar suspension of a pendulum may be shifted lengthwise and away from the other bar so that the p.endulums can vibrate only in a longitudinal direction ; by moving a slider Fig. 15. Machine for illustrating longitudinal waves. of a second form s, Fig. 15, simple harmonic motions, exactly the same as before except that they are in the direction of propagation, are given to the series of pendulums ; this pro- duces, instead of the crests and troughs of the former wave, condensations and rarefactions in the spacing of the particles which follow each other periodically, and moving forward with the velocity of wave propagation illustrate a longitu- dinal wave motion. In this case the wave length is the dis- tance from one condensation to the next, and the various 15 THE SCIENCE OP MUSICAL SOUNDS other characteristics are substantially the same as for the transverse wave. In fact, one type of wave can be trans- formed into the other and back again by merely shifting one of the suspending bars while the pendulums are vibrat- ing ; this turns the direction of vibration of each pendulum without disturbing the character of the motion. The simple harmonic curve may be considered an instan- taneous representation of a transverse wave; it shows by its shape the nature of the periodic vibration and exhibits the displacements and conditions of motion of a continuous series of particles transmitting the wave. In Fig. 16, A b,.'Aa. F IG. 16. Transverse and longitudinal displacements. represents a row of particles at rest ; if a transverse wave is being transmitted, the particles at some instant will be dis- placed as shown in J5, forming a harmonic curve. If the displacements are of the same amounts but occur in a longi- tudinal direction, upward displacements in B corresponding to forward displacements in C, and vice versa, there results a longitudinal wave of condensation and rarefaction, or of pressure changes; this is the type of sound waves in air. Sound waves usually pass outward from the source in the' form of expanding surfaces of disturbance, and the nature of the pressure changes, as applied to surfaces, may be illus- trated by the spacing of the lines in B, Fig. 17. The rela- 16 WAVE MOTION tions of condensation and rarefaction of the longitudinal wave to crest and trough of the transverse wave are shown in both Figs. 16 and 17. The amounts of displacement, the amplitudes, periods, frequencies, and velocities of propagation are defined in exactly the same way in the two types of waves ; only the directions of displacement differ. It can be shown that both kinds of wave motion are adequately and correctly represented by the harmonic curve. The curve B, Fig. 16, conveys to the eye a much clearer idea of the displacements than does C, though B Fig. 17. Wave of compression. the displacements of the successive particles are of exactly the same amount in both instances. Nearly all of the waves to be studied in these lectures are of the longitudinal type, but they will be represented by curves of transverse displacement. As will be more fully developed in later lectures, several simple harmonic motions of various amplitudes, frequencies, and phases, moving in the same or different directions, often coexist, producing wave motions which are represented by curves of very complex shapes. Sound waves in sohds may be either transverse or longitu- dinal, but the properties of liquids and gases are such that only longitudinal displacements, or pressure changes, can be 17 THE SCIENCE OF MUSICAL SOUNDS transmitted as wave motions. The transmission of a longitudi- nal wave through a solid or liquid body is illustrated by the col- hsion balls, Fig. 18. When the medium is very compressible, such as a gas, the method of propagation is better shown by the apparatus, Fig. 19, which consists essentially of a long, ± flexible spring suspended so as to move horizontally ; if a push of compression is given to one \^-^:J^ end of the spring, it will be transmitted as a wave in a man- ner to be easily followed by the eye. An illustration of two simple harmonic motions at right angles is given by the compound pendulum apparatus shown in Fig. 20, the bob of which is a weight carrying a glass vessel Fig. 18. Collision balls. Fig. 19. Apparatus for illustrating a wave of compression. 18 WAVE MOTION containing sand ; as the pendulum swings the sand flows from a small aperture in the bottom of the vessel and leaves a trace on the paper un- derneatJi. The pendulum may swing to and fro and from side to side; for the first movement its length is Zi, but on account of the arrangement of the two suspending strings, its length for sidewise move- ment is h ; hence the periods of the two movements are different and the bob swings in a peculiar curve compounded of two simple move- FiG. 20. Compound pendulum. 19 Fig. 21. Torsional wave. THE SCIENCE OF MUSICAL SOUNDS ments ; the curve shown in the figure results from periods of the exact ratio of 2 : 3 ; other ratios give characteristic figures ; these curves are known as Lissajous's figures}^ Compound harmonic motion of this kind is made use of in accurate tuning, as described in Lecture II. Various other types of motion besides simple harmonic may generate waves ; a torsional wave, consisting of angular harmonic motion, may be transmitted by a loaded wire, as shown in Fig. 21. The Ear Sound has been defined as the sensation received through the ear, and the definition has been extended to include the external cause of the sensation. All that the ear perceives in the complex music of a grand opera or of a symphony orchestra is contained in the wave motion of the air consist- ing of periodic changes in pressure and completely repre- sented by motion of one dimension, that is, by motion con- fined to a straight line ; or, as Lord Kelvin has expressed it, sound is " a function of one variable." That motion of one dimension is capable of producing these sounds is amply proved by the talking machine ; in the cyhnder type of machine the tracing point moves up and down, and gives a backward-and-forward motion to the diaphragm, each point of which moves in a straight line ; the resulting wave of compression is transmitted by the air to the eardrum. The telephone is another demonstration of the same fact. Some of the disk types of talking machines not only illustrate the movement in one direction, but also demonstrate that a transverse vibration on the record is transformed, through the needle and connecting levers, into an equivalent longitudinal motion at the diaphragm of the sound box. It is marvelous that complex musical re- 20 NOISE AND TONE suits can be produced by such seemingly simple mechanical means. The functioning of the ear, which is a wonderfully com- plex organ, Fig. 22, is but imperfectly understood ; physi- ologists are studying its structure, and psychologists are in- vestigating the manner of the reception and perception of the sensation of sound ; the study of these most interesting questions is quite outside of the province of these lectures, which is confined to the physics of that which may be heard. Fig. 22. Model of the ear, dissected. that is, to sound waves as they exist in the air and to their sources. The ear divides sounds roughly into two classes : noises, which are disagreeable or irritating, and tones, which are received with pleasure or indifference. Noise and Tone Noise and tone are merely terms of contrast, in extreme cases clearly distinct, but in other instances blending; the difference between noise and tone is one of degree. A simple tone is absolutely simple mechanically; a musical tone is more or less complex, but the relations of the com- 21 THE SCIENCE OF MUSICAL SOUNDS ponent tones, and of one musical sound to another, are appreciated by the ear ; noise is a sound of too short dura- tion or too complex in structure to be analyzed or understood by the ear. The distinction sometimes made, that noise is due to a non-periodic vibration while tone is periodic, is not sufficient ; analysis clearly shows that many so-called musical tones are non-periodic in the sense of the definition, and it is equally certain that noises are as periodic as are some tones. In some instances noises are due to a changing period, pro- ducing the effect of non-periodicity ; but by far the greater number of noises which are continuous are merely complex and only apparently irregular, their analysis being more or less difficult. The ear, because of lack of training or from the absence of suitable standards for comparison or perhaps on account of fatigue, often fails to appreciate the character of sounds and, relaxing the attention, classifies them as noises. Small sticks of resonant wood may be prepared. Fig. 23, such that when dropped, the resulting sound is a mixture of noise and simple musical tones. If several of these sticks are dropped together, the sound gives the effect of noise only, while if the sticks are dropped one at a time in proper order, the ear clearly distinguishes a musical melody in spite of the accompanying noise. The drawing of a cork from a bottle expands the contained air ; when the cork is wholly withdrawn, the air, because of its elasticity, vibrates with a frequency dependent upon the size and shape of the bottle. The resulting sound is of short duration and is thought of only as a popping sound, while it is in reality a musical tone. The musical characteristic is made evident by drawing the plugs from several cylindrical bottles. Fig. 22 NOISE AND TONE 23, the tones of which are in the relations of the common chord, do, mi, sol, do. A distinguishable tune can be played on a flute without blowing into it, the air in the tube being set in vibration by snapping the keys sharply against the proper holes to give the tune. A conspicuous instance of the change in classification of a musical composition from noise to music is provided by Wagner's " Tannhauser Overture. " After this overture had been known to the musical public for ten years it was criti- FiG. 23. Sticks and bottles which produce musical noises. cized in the London Times as "at best but a commonplace display of noise and extravagance. ' ' A Frankfort (Germany) critic said in 1853 that " 'Tannhauser,' so far as the public is concerned, may be considered a thing of the past." It was called "shrill noise and broken crockery effects." The eminent musical pedagogue, Moritz Hauptmann (1846), pronounced it "quite atrocious, incredibly awkward in con- struction, long and tedious. It seems to me," he says, "that a man who will not only write such a thing, but 23 THE SCIENCE OF MUSICAL SOUNDS actually have it engraved, has little call for an artistic career." Sidney Lanier, the poet-musician, who better understood this composition, wrote in a letter to his wife,* "Ah, how they have beUed Wagner! I heard Thomas' orchestra play his overture to ' Tannhauser.' The ' Music of the Future ' is surely thy music and my music. Each harmony was a chorus of pure aspirations. The sequences flowed along, one after another, as if all the great and noble deeds of time had formed a procession and marched in review before one's ears, instead of one's eyes. These ' great and noble deeds ' were not deeds of war and statesmanship, but majestic victories of inner struggles of a man. This un- broken march of beautiful-bodied Triumphs irresistibly invites the soul of man to create other processions like it. I would I might lead so magnificent a file of glories into heaven!" As compared with the usual composition of its time " Tannhauser Overture " must be considered as having a com- pUcated construction. There is an accompaniment, quite independent of the main theme, which forms a beautiful background of tone, upon which the noble melody is pro- jected. Many of the early listeners may have given their attention to this accompaniment and so have lost the im- pressiveness of the melody ; to them it was a confused mass of tone producing the effect of noise. ■ The study of noises is essential to the understanding of the qualities of musical instruments, and especially of speech. Words are multiple tones of great complexity, blended and flowing, mixed with essential noises. If with the vowel tone & (mat) we combine a final noise represented by (, the word a+t is produced ; if to this simple combination we add 24 NOISE AND TONE various initial noises, several words are formed, as : 6+at, c+at, /+at, h+at, m+at, p+at, r+at, s+at, t+at, i)+at. However, the study of noises may well be passed until we understand the simpler and more interesting musical tones. Tones are sounds having such continuity and definiteness that their characteristics may be appreciated by the ear, thus rendering them useful for musical purposes; these characteristics are pitch or frequency, loudness or intensity, and quality or tone color. 25 LECTURE II CHARACTERISTICS OF TONES Pitch The pitch of a sound is that tone characteristic of being acute or grave which determines its position in the musical scale; an acute sound is of high pitch, a grave sound is of low pitch. Ex- periment prov^ that pitch depends upon a very simple condition, the num- ber of complete vi- brations per second ; this nimiber is called the frequency of the vibration. One of the sim- plest methods of de- termining pitch is mechanically to create vibrations at a rate which is known and which can be varied as de- sired; the rate is adjusted until the resulting sound is 26 Fig. 24. Serrated disk for demonstrating the de- pendence of pitch upon frequency of vibration. CHARACTERISTICS OF TONES in unison with the one to be measured, then the number of vibrations generated by the machine is the same as that of the sound. If a card is held against the serrated edge of a revolving disk, Fig. 24, the pulsations of the card produce vibrations in the air, and give rise to an unpleasant semi-musical sound, having a recognizable pitch which is measured by the number of taps given to the card per second. The four disks shown in the illustration have numbers of teeth in the ratios of 4 : 5 : 6 : 8, sound- ing the common chord, the pitch of which is dependent upon the rate of ro- tation of the disk. The siren is an in- strument in which the vibrations are produced by inter- rupting a jet of compressed air by means of a revolving disk with holes, as illustrated in Fig. 25 ; the sound is much softer and more musical than that from the ser- rated disk. The siren has been developed into an instru- ment suitable for research, Fig. 26, which enables one in a few minutes of time to determine to one part in a hundred the number of vibrations of common musical sounds. For secTiring greater range or for sounding several tones simul- 27 Fig. 25. The siren. THE SCIENCE OF MUSICAL SOUNDS taneously, the siren is usually provided with two disks, di and di, each having four rows of holes; one or more rows may be used at the same time, each producing its own pitch.^ The disks may be rotated by compressed air on the principle of the turbine, or by an electric motor, as shown in the illustration ; in either case the speed can be controlled, and the number of vibrations is determined with the aid of the revolution counter c between the two disks. J0\ JIB i d, it 1 ' i m - ^ r" Fig. 26. Siren for the determination of pitch. There are various other methods for determining and com- paring the number of vibrations of sounding bodies, which are described in the references.* For the present purpose it will be sufficient to explain those used for the more precise determinations of a fundamental nature, the method of beats, Lissajous's optical method, and the methods of the clock-fork and the phonodeik. 28 CHARACTERISTICS OF TONES The Tuning Fork Perhaps the most important of acoustical instruments is the tuning fork invented in 1711 by John Shore, Handel's trumpeter. The fork reached an almost perfect develop- ment under the exquisite workmanship and painstaking research of Rudolph Koenig of Paris. When properly con- structed and mounted, it gives tones of great purity and constancy of pitch ; it is of very great value in experimental Fig. 27. Tuning forks of various types. work and provides the almost universal method of indicating and preserving standard pitches for all purposes.'^ Fig. 27 shows various forms of tuning forks, while Fig. 42 represents a larger collection, and many special forms are shown in other illustrations. A tuning fork for scientific purposes should be made of one piece of cast steel, not hardened ; the shapes developed by Koenig have not been excelled; the patterns for forks of ordinary musical pitches and those of very high pitches giving loud tones are shown in Fig. 28 ; a fork for As = 435, of the first shape, is 129 millimeters long, not including the 29 THE SCIENCE OF MUSICAL SOUNDS handle; a fork of the second shape, 79 millimeters long, has the pitch 3328. The number of vibrations of a fork is dependent upon the mass of the prongs and the elastic forces due principally to the yoke ; if the prongs are made lighter, by filing on the ends or sides, the pitch is raised ; if the fork is filed near the yoke, the elastic restoring force is diminished and the pitch is lowered. The second shape of fork shown in the figure has a yoke which is very thick in proportion to the prongs, hence it is suitable for high pitches. A standard fork, hav- ing been accurately machined and finished, should be left with the prongs a trifle too long, that is, flat in pitch ; the final tuning should be carried out I very carefully by short- ening both prongs to- I gether till the desired Fig. 28. Shapes of Koenig's tuning forks. frequency is SeCUred. Filing or grinding a fork will heat it, as will also the touch of the fingers ; the heating lowers the pitch of the fork, and if it is tuned while thus heated, it wiU later be found too sharp, that is, the prongs are already too short. Therefore the fiUng should stop while the fork is yet two or three tenths of a vibration flat, and the fork should be allowed to remain at a uniform temperature for a day or two before a comparison is made ; if further tuning is necessary, it must be done with extreme care, and a comparison again made after another interval of rest. Vigorous filing will produce molecular disturbances which subside only after long periods of rest. The methods of comparison are described in the succeeding articles. 30 C CHARACTERISTICS OF TONES Tuning forks are often finished with a bright steel surface, in which case care is required to prevent rust ; smearing with vasehne is a convenient rust preventive. A blued steel finish is excellent ; standard forks are sometimes blued over the entire surface after all machine work has been finished, but before the final tuning; the final adjustment of pitch is made by careful grinding on the ends of the prongs, which are thus made bright, and the surfaces are then very Hghtly etched with a seal. Any further alteration of the fork, or an injury, will disfigure it and will be easily detected. The boxes on which the forks are commonly mounted were fiirst used by Marloye; they are of such dimensions that they form resonance chambers not quite in tune with the fork tone ; if the tuning is perfect, the sound is louder but of short duration, because the energy of the vibration is more rapidly dissipated.^ The box serves a double purpose : it produces a louder sound and it also purifies the tone by reinforcing only the fundamental. When the resonance box is not exactly in tune with the fork, it draws the fork out of its natural frequency by a small amount, a few thousandths of a vibration per second. These effects are considered at greater length under Resonance in Lecture VI. Koenig proved that change of temperature alters the number of vibrations of a fork ; the temperature coefficient was found to be nearly constant for forks of all pitches and to have the value - 0.00011.' The change in the number of vibrations of a fork is found by multiplying its frequency by this coefficient and by the number of degrees of tempera- ture change ; the negative sign means that the frequency is diminished by increased temperature. For instance, a fork giving 435 vibrations per second at 15° C. will have its fre- ' 31 THE SCIENCE OF MUSICAL SOUNDS quency diminished by 0.48 vibration for ten degrees increase in temperature. The pitch of a fork changes slightly with the ampUtude, that is, with the loudness of the tone which the fork is giving ; the greater the amplitude, the less the frequency. For extreme changes in amplitude, the number of vibrations may vary as much as one in three hundred. If a fork is sounded loudly, the pitch will rise slightly as the tone sub- sides ; the true pitch is that corresponding to a small inten- sity." Tuning forks may be excited by bowing across the end of one prong with a violin or a bass bow ; this is perhaps the best method for obtaining the loudest possible response. For usual experimental work the most convenient method is to strike the fork with a soft hammer ; a felt piano hammer head with a flexible spring handle is an excellent tool for the purpose ; a solid rubber ball or a rubber stopper is often used for the hammer head. For sounding the thick high- pitched forks, an ivory hammer is best. Forks should never be struck with metal or other hard substances, for being of soft steel, they are likely to be injured. Forks are often made to sound continuously by means of an electro-magnetic driving arrangement; a fork may be driven by itself, its own vibrations, once started, serving to produce the interrupted current required; such a fork is shown in Fig. 50, page 65. Often a fork is driven by an interrupted, or by an alternating, current produced from some other source ; the ten forks shown in Fig. 179 are all driven by one interrupter fork at the back of the apparatus. In this instance the periods of the forks are exact multiples of that of the interrupter, since a fork wiU respond only to impulses which are in step with its own natural vibrations. 32 CHARACTERISTICS OF TONES When a fork is driven by this method, the prong is inter- mittently urged forward by the magnetic pull. The prong itself is always a very Uttle behind the pull, that is, it lags more or less ; this forcing of the vibration causes the period to be slightly different from that of the same fork vibrating freely." A fork retains its pitch with great constancy; ordinary careless handling causes little change, and even rust, as it slowly proceeds over a period of years, produces but slight effect, rarely exceeding one vibration in two hundred and fifty ; the change usually flattens the pitch, since rust near the yoke affects the fork more than that near the end of the prong. The ordinary wear on a fork is usually greater at the ends which are unprotected, and this causes the pitch to sharpen; rust and wear, then, in some degree produce opposite effects and tend to maintain the original pitch. An account of the tone quahty of the tuning fork is given in Lecture VI, while many illustrations of its usefulness will be found throughout the lectures. Determination of Pitch by the Method of Beats A simple comparison by the ear will enable one who is musically trained to tune certain intervals, such as unisons, octaves, thirds, fourths, and fifths. Two tones nearly in unison produce beats, the number of which per second -is equal to the difference in pitch (see page 183). Beats often occur between the overtones of sounds which are not simple, and under other conditions which need not be considered here. Comparison by ear, based on the method of beats, is the principal means employed in tuning pianos and organs and such stringed instruments as the violin and the guitar. The comparison of a standard tuning fork with an un- D 33 THE SCIENCE OF MUSICAL S0UNl6s known pitch of nearly the same frequency can be made with ease and precision by the method of beats. The imknown sound and that of the standard fork being heard simultaneously, the number of beats per second is determined by counting the number occurring in five or ten seconds ; the required pitch is then that of the standard increased or diminished by the number of beats per second. Usually the ear will decide whether the sound is flatter or sharper than the standard ; in other cases it may be possible to make an easy adjustment to assist in this determination. When the sound is from a tuning fork, one prong may be loaded with a small piece of wax, which will sUghtly lower its pitch ; if there are now more beats per second, the fork is flat, since making it flatter puts it further out of tune, and vice versa. The fork may be adjusted to equaUty with the stand- ard by filing, as already explained, till the beats become fewer and finally cease. When the two sounds approach unison, the interval be- tween beats becomes longer ; when the beats are slow, it is difficult to measure the time between them, for one is not sure of the instant of minimum or maximum sound. It is found that one can count beats with accuracy at the rate of from two to five per second, the count being carried over five or ten seconds, or more ; four beats per second is perhaps the most convenient number. For these reasons an auxihary fork is often used, which is tuned four beats per second sharper than the standard; the fork being tested is then adjusted till it is four beats per second flatter than the auxihary, when it is, of course, exactly in unison with the standard. Sets of forks are made for setting or testing the chromatic scale of equal temperament, as in tuning pianos and organs, 34 CHARACTERISTICS OF TONES in which the comparisons are made by beats. A series of thirteen forks is accurately tuned to the chromatic scale from middle- C to C an octave higher ; an auxihary set of thirteen forks is then tuned so that each is exactly four beats per second sharper than the corresponding fork of the first series; a correctly tuned octave must have its successive tones four beats per second flatter than those of the auxiliary Fig. 29. Sets of forks for testing the accuracy of tuning the chromatic scale. forks. Such forks are shown in Fig. 29 ; for making the tests the auxiliary forks only are actually required, but it is desirable to have the others also. The first and last forks of the scale set, which are an octave apart, give 258.65 and 617.3 vibrations, respectively, for A = 435 ; the auxiliary forks being four vibrations sharp give 262.65 and 521.3 vibra- tions, and are not a true octave apart ; for a true octave the higher fork would be eight vibrations sharp and give 525.3 vibrations ; none of the auxiliary forks gives true musical intervals. 35 THE SCIENCE OF MUSICAL SOUNDS CHARACTERISTICS OF TONES The absolute number of vibrations may be determined by counting the number of beats between the successive forks of a series of fifty or more ranging over one octave, accord- ing to the method devised by Scheibler in 1834.12 Scheibler's tonometer consisted of fifty-six forks having pitches from about 220 to 440, the successive forks differing by four vi- brations per second. This method, which is very laborious, has been used by Ellis and by Koenig. Koenig's masterpiece is perhaps a tonometer consisting of a hundred and fifty forks of exquisite workmanship, and tuned with the greatest care and skill ; it covers the entire range of audible sounds from 16 to 21,845.3 vibrations per second.^' The largest fork is about five feet long, and hasv a cylindrical resonator eight feet in length and twenty inches in diameter. It is possible to find in this series a fork which shall differ from any given musical tone by not more than four beats per second, a comparison with which by the method of beats will determine the pitch of the sound with great ease and -precision. Optical Comparison of Pitches One of the most precise methods for the comparison of frequencies is Lissajous's optical method," which depends upon the geometrical figures traced by two simple harmonic motions at right angles. The motions may be provided by tuning forks which carry mirrors on the prongs, as shown in Fig. 30. A ray of light is reflected from one fork to the other and then to a screen or an observing telescope. When the forks are vibrating, the ray is deflected in two directions, so that the figure on the screen corresponds to the com- pounded motion. The shape of this figure is characteristic of the ratio of the frequencies of the two forks ; for certain 37 THE SCIENCE OF MUSICAL SOUNDS simple ratios, such as 1:2, 1:3, 2:3, etc., the figures are easily recognized by the eye ; and when the ratio is exact, the figure exactly retraces itself, and because of the per- sistence of vision it appears continuous and stationary. If the ratio of frequencies is not exact, the figure changes, because of progressive phase difference, and, passing through a cycle, returns to the original form ; the time for this cyclic change is that required for one fork to gain or lose one com- plete vibration on the exact number corresponding to the indicated ratio. The apphcation of this method is explained in connection with the clock-fork. The Clock-Fork The most precise determinations of absolute pitch are those made by Koenig, who investigated the influence of the resonance box and of temperature on the frequency of a stand- ard fork. He also determined the frequency of the forks used by the Conservatory of Music and the Grand Opera in Paris.^^ By combining the clock-fork of Niaudet with a vibration microscope for observing Lissajous's figures," he developed the beautiful instrument shown in Fig. 31. Fig. -31 is reproduced from an autographed photograph of the original instrument, in the author's possession, while the instrument which was exhibited in the lecture is of more recent construction and is shown in Fig. 32, on page 40. The apparatus is essentially a pendulum clock in which the ordinary pendulum is replaced by a tuning fork; the fork has a frequency of 64, as scientifically defined ; that is, it makes 128 swings per second, counting, both to and fro movements. The clock has the usual hour, minute, and second hands ; but instead of the escapement operating on the second hand to release it once a second, the gearing of 38 CHARACTERISTICS OF TONES the movement is carried one step higher, and a fourth hand IS provided, which goes round once in a second. A very small escapement mechanism is attached to this hand and is so arranged that it is operated by one prong of the tuning fork as it swings to and fro, 128 times a second ; the fork thus releases the wheels regularly, as does an ordinary pen- dulum, and the clock "runs." Moreover, as in the common clock, the escapement not only releases the wheelwork, but it also imparts a small impulse to the fork so as to maintain its vibration as long as the clock runs, that is, for days if desired. Thus we have a tuning fork which will vi- brate continuously, and a clock-work which accurately counts the vibrations. The rate of the fork is ad- justed much as is a pendu- Imn, by moving small weights up or down on threaded sup- ports. If the clock is regu- lated till it keeps correct time, the fork must vibrate exactly 128 times a second, making 1 1,059,200 single vibrations in a day. A change in the rate of the clock of one second per day means a change in the fre- quency of the fork of one part in eighty-six thousand four hundred; that is, when the clock loses one second a day, the fork has a frequency of 63.99926. If it is desired, for instance, to adjust the fork to exactly 63 complete vibra- 39 Fig. 31. Photograph of Koenig's clock- fork bearing his autograph. THE SCIENCE OF MUSICAJ. SOUNDS tions per second, the clock must lose one part in sixty-four, that is, it must lose 22 J minutes a day. By means of the weights the actual fork can be adjusted to have any desired frequency between 62 and 68, this frequency being determined to a ten-thousandth part of a Fig. 32. Clock-fork arranged for verifying another fork. vibration by the rate at which the clock gains or loses. The range of the fork is a musical semi-tone ; by using various multiples of its frequency, it is possible to determine almost any desired musical pitch with precision. The method of using the clock-fork may be illustrated by a concrete example; thus the verification of a standard A = 435 fork reqviires the following procedure. The only 40 CHARACTERISTICS OF TONES integral divisor of 435 which will give a quotient within the limits of frequency of the clock-fork is 7, and the quotient is 62.143; a numerical calculation shows that if the clock loses 41 minutes 46 seconds per day, or 1 minute 44.4 seconds per hour, the tuning-fork pendulum will make 62.143 vi- brations per second ; if the A-fork vibrates exactly 7 times as fast, its frequency must be 435 ; the exact ratio of the frequencies is to be determined by Lissajous's figures with the vibration microscope. The clock-fork carries the objective lens of a microscope, the body of which is attached to the frame. The A-fork is supported so that its Une of vibration is at right angles to that of the lens. Fig. 32, and so that some brightly illuminated point, as a speck of chalk dust on the end of the prong, is visible through the micro- scope ; if the two forks are vibrating, this no. 33. Lissajous's speck is seen to describe the Lissajous ^f^'^ *°'' *'^® '■^*'° curve for the ratio of 1:7, Fig. 33. Sup- pose now the figure goes through its cyclic change once in 5 seconds, then the fork has a frequency of either 434.8, or 435.2. To determine whether the fork is sharp or flat a very small piece of wax is attached to one prong, which will make it vi- brate more slowly. If the cychc change requires a longer time than before, the sUght lowering of pitch has improved the tuning, which condition indicates that the frequency of the fork was 435.2 ; if the change occurs in less time, the lowering of the pitch has made it further from the true value and the fork had a frequency of 434.8. The A-fork may be adjusted by fifing or grinding, near the ends of the prongs to make it sharper or near the yoke to make it flatter, as described on page 30, and the adjustment may be continued tiU any 41 THE SCIENCE OF MUSICAL SOUNDS required accuracy has been obtained; for instance, if the cycUc change occurs in 10 seconds, the error of tuning is ^j vibration per second. The clock-fork is provided with a mirror on the side of one prong so that it may be used to produce Lissajous's figures by the hght-ray method or to record the vibrations directly on a photographic film. Pitch Limits The range of pitch for the human voice in singing is from 60 for a low bass voice to about 1300 for a very high soprano. Fig. 34. Organ pipe over 32 feet long giving 16 vibrations per second. The piano has a range of pitch from 27.2 to 4138.4. The pipe organ usually has 16 for the lowest pitch and 4138 for the highest ; an organ pipe giving 16 vibrations per second, Fig. 34, is nominally 32 feet long, though its actual length is somewhat greater ; there are a few organs in the world having pipes 64 feet long which give only 8 vibrations per second, but such a sound is hardly to be classed as a musical tone ; the frequency 4138 is given by a pipe 1| inches long. ' Neither speech nor music makes direct use of all the sounds which the ear can hear. Helmholtz considered 32 vibrations per second as the lowest limit for a musical sound, that is, one which gives the sensation of a continuous tone ; yet the piano descends to 27 and the organ to 16 or even to 8 vibrations per second. The tuning fork shown in Fig. 35 42 CHARACTERISTICS OF TONES may be made to give from 16 to 32 vibrations per second, according to the position of the weights on the prongs. Experimenters differ widely as to the lower hmit, though nearly all consider Helmholtz's value too high ; perhaps the most trustworthy values are between 12 and 20. vibrations per second, with a general consensus of opinion that the lower limit of audibihty for a musical tone is 16 vi- brations per second. Of coiirse, the ear can hear vibrations when they are fewer in number than 16 per second, but they are heard as separated or discontin-" uous sounds. It is interesting to notice that the frequency of repeti- tion of an impression to pro- duce continuity of sensation for sound is practically the same as for light. The per- sistence of vision is about one tenth of a second, that is, an intermittent visual sensation occurring ten times or more a second produces the effect of a continuous sensation ; for moving pictures the views are usually changed sixteen times a second, and the inter- mittent movement, or vibration, at this rate, gives the im- pression of a continuous motion. If a screen is illuminated with a moving-picture projection apparatus in which there is no picture, the eye perceives a flicker in the general il- 43 Fig. 35.' Large fork giving from 16 to 32 vibrations per second. THE SCIENCE OF MUSICAL SOUNDS lumination when the intermittent shutter of the machine is in operation, unless the number of hght flashes per second exceeds a certain value.^' This value varies from ten to fifty or more per second, according to the intensity of the light. Perhaps Helmholtz's value of 32 for the lower limit of a tone is the flicker limit for the ear. While the upper pitch limit for the musical scale is about 4138, the ear can hear sounds having frequencies of 20,000 Fig. 36. Small organ pipe giving 15,600 vibrations per second. or 30,000, and even more in cases of extreme sensitiveness. Fig. 36 shows what is in form a regular organ pipe, one of the smallest ever made, and much too small to be used in an organ ; the length of the pipe which is effective in pro- ducing the tone is indicated by I in the figure and measures 0.25 inch. This pipe sounds Bg and gives 15,600 complete vibrations per second, a sound which is clearly audible to most listeners. Experiments to determine the upper limit of audibility are often made with a Galton's whistle. Fig. 37, 44 CHARACTERISTICS OF TONES an adjustable whistle or stopped organ pipe of very small dimensions, blown by means of a rubber pressure bulb. The whistle can be set to various lengths, indicated by the graduated scales, giving high-pitched sounds of known frequency. Another experimental method of producing sounds of high pitch is by the longitudinal vibration of short steel bars, Fig. 38. The bars are suspended by silk cords, rh-Hn"'*Hi**^ 1 ' ft ^ ■*■"■■■ Fig. 37. Adjustable whistle for determining the frequency of the highest audible sound. and are struck on the ends with a steel hammer, producing a clear metallic ringing sound, which is the tone desired; the pitch of the sound is determined by the length of the bar, a bar 52.5 miUimeters (2yV inches) long giving 32,768 vibrations per second. Perhaps the most conclusive experiments on audible and inaudible tones of the highest pitch are those of Koenig, extending over a Ufetime of investigation, ^^ in which obser- vations were made with tuning forks, transverse vibrations 45 THE SCIENCE OF MUSICAL SOUNDS of rods, longitudinal vibrations of rods, plates, organ pipes, membranes, and strings. Tuning forks, when properly con- structed and used, proved to be the most suitable source of high tones. Koenig made his first experiments in 1874 when he was forty-one years old and at that time was able to hear tones up to Fs^= 23,000, which he considered the highest directly audible simple tone ; he constructed a set of forks up to Fg = 21,845, which he exhibited at the Centennial Exposi- tion in Philadelphia in 1876. (These forks and much other Fig. 38. Steel bars for testing the highest audible frequency of vibration. interesting acoustic apparatus exhibited by Koenig are now in the laboratory of Toronto University.) In his fifty- seventh year the limit of audibility for Koenig was Eg = 20,480, and in his sixty-seventh year it was D9 f = 18,432. A set of Koenig forks for tones of high pitch is shown in Fig. 39 ; Koenig has made a complete series of such forks extend- ing more than two octaves above the limit of audibiUty to a frequency of 90,000 complete vibrations (180,000 motions to and fro) per second. Sounds which are inaudible are made evident by cork-dust figures in a tube. Fig. 39 ; the stationary air waves produced by the vibration of the fork at the end of the tube cause the cork dust to accumulate in 46 CHARACTERISTICS OF TONES little heaps, one in each half wave length of the sound. The wave length in air for the tone of 90,000 frequency is 1.9 millimeters, or 0.075 inch. Though the pitch of the highest note commonly used in music is 4138, overtones with frequencies of 10,000, or more, probably enter into the composition of some of the sounds of music and speech. The investigation of these tones of very high pitch should not be neglected; however, the Fig. 39. Forks for testing the highest audible frequency of vibration. analytical work discussed is these lectures in Umited to pitches of from about 100 to 5000. While it would be interesting to students of music to con- sider the reasons for the selection of tones of certain pitches to form scales and chords, it would lead us far from our present purpose; it will, however, be useful to notice the location on the musical staff of the octave points of the sounds used in music and to explain the notation which designates a given tone. The- musical staff may be con- sidered as composed of eleven Unes ; to assist in identifj^ng 47 THE SCIENCE OF MUSICAL SOUNDS Table of Equally Tempered Scale, A3 = 435 C-i-Co Co-C, C-Cz C2-C3. C-C. C-Cs Ci-C. C-Gt c 16.17 32.33 64.66 129.33 258.65 517.31 1034.61 2069.22 c« 17.13 34.25 68.51 137.02 274.03 548.07 1096.13 2192.26 D 18.16 36.29 72.58 145.16 290.33 580.66 1161.31 2322.62 D» 19.22 38.45 76.90 153.80 307.59 615.18 1230.37 2460.73 E 20.37 40.74 81.47 162.94 325.88 651.76 1303.53 2607.05 P 21.58 43.16 86.31 172.63 345.26 690.52 1381.04 2762.08 p * 22.86 45.72 91.45 182.89 365.79 731.58 1463.16 2926.32 G 24.22 48.44 96.89 193.77 387.54 775.08 1550.16 3100.33 G» 25.66 51.32 102.65 205.29 410.59 821.17 1642.34 3284.68 A 27.19 54.37 108.75 217.50 436.00 870.00 1740.00 3480.00 a;* 28.80 57.61 115.22 230.43 460.87 921.73 1843.47 3686.93 B 30.52 61.03 122.07 244.14 488.27 976.54 1953.08 3906.17 C 32.33 64.66 129.33 258.65 517.31 1034.61 2069.22 4138.44 the lines, the middle one is omitted except when required for a note, Fig. 40 ; additional lines of short length are used C7 :©:4138 Cei2.2069 = C. -0-1035 = ^ t zcpocbllz w C3-6>-259 :Cpq029l / Ci-e-65 = Co^32 CLi^-16 Fig. 40. Middle C and the several octaves of the musical scale. 48 CHARACTERISTICS OF TONES to extend the compass. The tone called "middle C" is placed on the line between the bass and treble staffs, and is designated by Cs ; in International Pitch this tone has 258.65 vibrations per second ; the musical compass is four octaves upward and downward from middle C, the various octaves bearing subscripts as shown; all the tones of an octave between two C's are designated by the subscript of the lower C ; that is, G 3 is on the second hne of the treble staff, and Gi is on the lowest line of the bass staff, etc. The table opposite gives the pitch numbers for all the tones of the equally tempered musical scale, based on Inter- national Pitch, A3 = 435. Standard Pitches Musical pitch is usually specified by giving the number of vibrations of the note called " VioUn A, " |^ li || , though sometimes it is given by "Middle C," |<^ I II , or by the C an octave higher. The standard of musical pitch has varied greatly, even within the history of modern music, from the classical pitch of the time of Handel and Mozart, when it was A = 422, to the modern American Concert pitch of A = 461.6, a change of more than one and a half semi-tones. Ellis gives a table of two hundred and forty-two pitches, showing values for A ranging from 370 to 567, that is, from F iff to D of the modern musical scale." The condi- tions of use and cause of changes in pitch are described in the references. Especially interesting are the accounts of the changes in Philharmonic Pitch, that of the London Philharmonic Orchestra, which under Sir George Smart, in 1826, was A = 433, and under Sir Michael Costa, in 1845, was raised to A = 455. In America,^" the equivalent of , B 49 THE SCIENCE OP MUSICAL SOUNDS this Philharmonic Pitch is often referred to as Concert Pitch, and it has reached the high Umit of A = 461.6. Not only has the rise in pitch been so great that artists have refused to sing and instrument strings frequently break under the strain, but the lack of uniformity also causes great confusion and trouble. A convention of physicists in Stuttgart in 1834 adopted Scheibler's pitch of A = 440, which has been much used in Germany ; this is perhaps the first standard pitch. As a result of Koenig's researches with the clock-fork, the French "Diapason Normal," A= 435 at the tempera- ture of 20° C, was established in 1859. This was adopted by several of the leading symphony and opera orchestras ; the Boston Symphony Orchestra adopted this pitch upon its organization in 1883. A committee of the Piano Manufacturers' Association of America, of which General Levi K. Fuller was chairman, made an extensive investigation of musical pitch, assisted by Professor Charles R. Cross of Massachusetts Institute of Technology. After consultation with many authorities in this country and Europe, the Committee, in 1891, adopted as the standard the Diapason Normal as determined by Koenig and named it "International Pitch, A= 435," at a temperature of 20° C. (68° F.). This is often called Low Pitch in distinction from Concert or Philharmonic Pitch, which is now referred to as High Pitch. The committee selected as its fundamental standard the type of fork made by Koenig, shown in Fig. 41, which is provided with an adjustable cylindrical resonator and gives a tone of great strength and purity. It has been proposed that A = 438 be made a standard, as a compromise between the Stuttgart A = 440 and the 50 CHARACTERISTICS OF TONES Diapason Normal A = 435 ; for practical purposes there is little difference in the pitches 435, 438, and 440 ; but there should be but one nominal standard, and it seems that the strongest arguments favor the universal adoption of A = 435. The musician should insist that his piano and other instru- ments be tuned to this pitch. Fig. 41. Standard fork. International Pitch, A = 435. Before any standard had been generally estabhshed for musical purposes, Koenig adopted one for his own work, and as tuning forks of his make are widely used in scientific institutions, this pitch, in which middle C = 256, is often referred to as Scientific or Philosophical Pitch. The author urges the use of one pitch only for both scien- tific and musical purposes, viz. A = 435 ; in the tempered 51 THE SCIENCE OF MUSICAL SOUNDS en a 'a 3 S 52 CHARACTERISTICS OF TONES musical scale this gives for middle C 258.65 vibrations per second. This pitch is used exclusively in discussing the results of our sound analysis. In the laboratory of Case School of Apphed Science the scale forks based on C = 256 have been duphcated with new forks based on A = 435 ; Fig. 42 shows the larger part of this collection, there being over two hundred forks in the picture. Intensity and Loudness The loudness of a sound is a comparative statement of the strength of the sensation received through the ear. It is impossible to state simply the factors determining loudness. For the corresponding characteristic of light (illumination) there is a moderately definite standard, commonly called the candle power ; but for sound there is no available unit of loudness, and we are dependent on the subjective com- parison of our sensations.^i Not only are the ears of differ- ent hearers of different sensitiveness, but each individual ear has a varying sensitiveness to sounds of different pitches and, therefore, to sounds of various tone colors. In a, first study of the physical characteristics of sounds we are compelled to consider the intensity not as the loudness perceived by the ear, but as determined by what the physi- cist calls the energy of the vibration. Fortunately, under simple conditions and within the range of pitch of the more common sounds of speech and music, there is a reasonable correspondence between loudness and energy. The energy, or what we will call the intensity of a simple vibratory motion, varies as the square of the amplitude, the frequency remaining constant ; it varies as the square of the frequency, the ampUtude remaining constant ; when both ampUtude and frequency vary, the intensity varies 53 THE SCIENCE OF MUSICAL SOUNDS as the square of the product of amplitude and frequency; or to express it by a formula, representing intensity by I, amplitude by A, and frequency by n, I = n'A^ Since we are to study sounds by means of representative curves or wave lines, we may give attention to the features of the curves which indicate intensity. In Fig. 43 the curve n = l A = \ 1=1 n=l A=l 7=4 n=2 A=l 1=4 n=a3.4 = a3 7= 1 d vAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Fig. 43. Curves representing simple sounds of various degrees of loudness. b has a frequency the same as that of curve a, but its am- plitude is twice as great, hence it represents a sound four times as loud ; the curve c has an amplitude the same as that of a, but its frequency is twice as great, and again its loud- ness is four times that of a ; the curve d has a frequency of 3.3 and an amplitude of 0.3, and it represents a loudness equal to that of a. Then the sounds represented by a and d are of equal loudness ; and those represented by b and c are equal, but are four times as loud as a or d. 54 CHARACTERISTICS OF TONES Caution is necessary when making inferences from simple inspection of photographic records of sound vibrations, since a change of film speed may give an apparent change of frequency when none really exists. When we are studying the records of complex sounds, and n = 10 ^ = 0.3 /= 9 7 = 10 f^ '1 riG.'44. • Curves representing two simple sounds and their combination. practically all sounds are such, a simple measurement of the amphtude of the curve and of the frequency is not sufficient for a determination of the loudness ; it is necessary to analyze the wave into its simple components, to compute the in- tensity due to each component singly, and then to take the sum of these intensities ; Fig. 44 illustrates this condition. Curves a and b have loudnesses represented by 1 and 9, as explained above ; curve c contains both a and b and its true loudness is therefore 10. If it were assumed that the 55 THE SCIENCE OF MUSICAL SOUNDS loudness of a and c are represented by the squares of their measured widths, the value for c would be 1.6 as compared with a, which is only one sixth of its real loudness. A further illustration of the necessity for analysis of a wave before judging of the loudness is shown in Fig. 45, in which a and b are of exactly the same loudness though of different widths. The curve a is composed of two partials, a fundamental and its second overtone, of loudness 1 and 4, respectively ; b is composed of the same partials, and there- FiG. 45. Curves representing the combination of two simple sounds in different phases ; though the curves are of different widths, they represent sounds of the same loudness. fore has the same loudness. The curves differ only in the relation of the phases of the components. Acoustic Properties of Auditoriums The loudness of a sound as perceived by the ear depends not only upon the characteristics of the source, but also upon the pecuharities of the surroundings. Among the features of an auditorium which must be considered are its size and shape, the materials of which it is constructed, its furnishings, including the audience, and the position of the source. 56 CHARACTERISTICS OF TONES The determination of the acoustic properties of audi- toriums is of the very greatest practical importance, and it is also one of the most elusive of problems ; the sounds which most interest us are of short duration and they leave no trace, and the conditions affecting the production, the transmission, and the perception of sound are extremely comphcated. The difficulties of the work are such as to dis- courage any but the most skillful and determined investi- gator. Indeed, the problem has been almost universally considered impossible of solution ; and this opinion has been accepted with so much complacence, and even with satis- faction, that it still persists in spite of the fact that a scientific method of determining the acoustic properties of auditoriums has been developed by Professor Wallace C. Sabine of Harvard University. This method, which is of remarkable practical utility, has been described in archi- tectural and scientific journals.^^ No auditorium, large or small, and no music room, pubHc or private, should be constructed which is not designed in accordance with these principles. Sabine's experiments have shown that the most common defect of auditoriums is due to reverberation, a confusion and diffusion of sound throughout the room which obscures portions of speech. There are other effects, due to echoes, interferences, and reflection in general, all of which have been considered. In many cases these troubles can be remedied, with more or less difficulty, in auditoriums already constructed; this is especially true in regard to reverberation, which is reduced by the proper use of thick absorbing felt placed on the side walls and ceihng. A method for photographing the progress of sound waves in an auditorium is referred to in Lecture III, page 88, which bears indirectly upon the loudness of the sound and 57 THE SCIENCE OF MUSICAL SOUNDS is of great value in designing rooms which shall be free from defects. A soundboard placed behind the speaker may, in some in- stances, distribute the sound in such a way as to remedy certain defects, as has been shown by the elaborate experi- ments of Professor Floyd R. Watson,^^ but the more common faults are not removed by this method. An auditorium has been described by Professor Frank P. Whitman, which was practically unimproved by the use of a soundboard, and was later made altogether satisfactory for pubhc speaking upon the removal of reverberation by Sabine's method.^* It may be added that the stringing of wires or cords across an auditorium can in no degree whatever remove acoustical defects. Tone Quality The third property of tone is much the most compUcated ; it is that characteristic of sounds, produced by some par- ticular instrument or voice, by which they are distingmshed from sounds of the same loudness and pitch, produced by other instruments or voices. This characteristic may be called tone color, tone quality, or simply quality. With comparatively little practice one can acquire the abiUty to recognize with ease any one of a series of musical instruments, when they produce tones of the same loudness and pitch. There is an almost infinite variety of tone quahty ; not only do different instruments have character- istic qualities, but individual instruments of the same fam- ily show delicate shades of tone quality; and even notes of the same pitch can be sounded on a single instrument with qualitative variations. The bowed instruments of the violin family possess this property in a marked degree. No musical instrument equals the human voice in the 58 CHARACTERISTICS OF TONES St. C * SHH! T m ■nmoniBunKE^t ability to produce sounds of varied qualities ; the different vowels are tones, each of a distinct musical quality. The investigation of tone quaUty therefore leads to a study of vocal as well as instrumental sounds. Since pitch depends upon frequency, and loudness upon ampUtude (and frequency), we conclude that quality must depend upon the only other prop- erty of a periodic vibration, the peculiar kind or form of the mo- tion ; or if we represent the vibration by a curve or wave line, quality is dependent upon the peculiarities represented by the shape of the curve. The simplest possible type of vibration, simple harmonic motion, and its representative curve, the sine curve, were described in the preceding lecture. A tuning fork, when properly mounted on a resonance box, gives to the air a single simple harmonic motion, which, being propagated, develops a simple wave. The sensation of such a tone is absolutely simple and pure. The nature of tone quality may be explained with the aid of tuning forks and the wave models ^^ shown in Fig. 46. 59 Fig. 46. Models of three simple waves, having fre- quencies in the ratios of 1 : 2 : 3. THE SCIENCE OF MUSICAL SQUNDS '.4iif^., C X mniHiiir^tm^j Let one of the forks having the pitch C3 be sounded; it will produce a simple wave in the air, which may be repre- sented by the model A ; a second fork, one octave higher, will, when sounding alone, send out twice as many vibrations per second, generating simple waves of just half the wave length, as represented by the model B ; a third fork, vibrating three times as fast as the first, produces waves one third as long, shown by model C. These sim- ple models illus- trate two char- acteristics of tone : pitch, by the frequency or number of waves in a given length, and loudness, by the height or amplitude of the waves. If two forks are sounded at the same time, the two corresponding simple motions must exist simultaneously in the air, and the motion of a single particle at any instant must be the algebraic sum of the motions due to each fork sepa- rately. This condition is shown in Fig. 47, where the wave B has been lowered to rest on the top of A, impressing the form of A upon B, which now exhibits the form of the mo- tion due to the two simple sounds. When the three forks 60 B A 4;)ft'^l»*«tM«W. Fig. 47. Wave form resulting from the composition of two simple waves. CHARACTERISTICS OF TONES ■^ C 3= {■jjiiiiiiniiMitfci X pininDiitiiiiiiaErFi ;;f 'irtritimiiraa x "'Vf,^'' rnrrcEEEnuiMB are sounding, the form of the composite motion is shown by lowering the wave form C upon that of A and B, as shown in Fig. 48. The relative phase of a wave may be shifted by changing the position of one of the forks in relation to the others; this effect is demonstrated by shifting the corresponding wave form side- wise (in the direc- tion of the length of the wave) be- fore the forms are pushed together; the shape of the resulting wave is thus changed while its compo- sition remains the same. This argument may be extended indefinitely to in- clude any num- ber of simple tones of any se- lected frequencies, amplitudes, and phases. There are therefore peculiarities in the motion of a single particle of air which differ for a single tone and for a combination of tones ; and in fact the kind of motion during any one period may be of infinite variety, corresponding to all possible tone qualities. These lectures are concerned almost wholly with the development and the application of this principle. 61 A X Fig. 48. Wave form resulting from the composition of three simple waves, corresponding to a composite sound containing three partials. THE SCIENCE OF MUSICAL SOUNDS '< Law of Tone Quality The law of tone quality was first definitely stated in 1843 by Ohm of Munich, in Ohm's Law of Acoustics, and much of Helmholtz's work of thirty years later was devoted to the elaboration and justification of this law.^^ The law states : all musical tones are periodic ; the human ear perceives pendular vibrations alone as simple tones ; all varieties of tone quality are due to particular combina- tions of a larger or smaller number of simple tones ; every motion of the air which corresponds to a complex musical tone or to a composite mass of musical tones is capable of being analyzed into a sum of simple pendular vibrations, and to each simple vibration corresponds a simple tone which the ear may hear. From this principle it follows that nearly all the sounds which we study are composites. The separate component tones are called partial tones, or simply partials ; the partial having the lowest frequency is the fundamental, while the others are overtones. It sometimes happens that a partial not the lowest in frequency is so predominant that it may be mistaken for the fundamental, as with bells ; and some- times the pitch is characterized by a subjective beat-tone fundamental when no physical tone of this pitch exists. If the overtones have frequencies which are exact multiples of that of the fundamental they are often called harmonics, otherwise they may be designated as inharmonic partials. As the result of elaborate investigation, Helmholtz added the following law: the quality of a musical tone depends solely on the number and relative strength of its partial simple tones, and in no respect on their differences of phase."' Koenig, after experimenting with the wave siren (Fig. 178, 62 CHARACTERISTICS OF TONES page 245), argued that phase relations do affect tone quality in some degree.^* Lindig has used a "telephone-siren" and concludes that the phases of the components influence quaUty of tone only through interference effects.^' Lloyd and Agnew, using special alternating current generators in connection with a telephone receiver, have foimd that the phase differences of the components do not affect the quaUty of tone.'" The question has been extensively in- vestigated by many others, with a consensus of opinion that Helmholtz's statement is justified.'^ In the analysis of sound waves from instruments and voices, described in Lectures VI and VII, the phases of all component tones have been determined. While systematic study of the phases has not yet been made, no evidence has appeared which indicates that the phase relation of the partials has any effect upon the quality of the tone. If tone quality varies with phase relations, the variations certainly are very small in comparison with those due to other influ- ences. The analyses which have been made give abundant evidence that tone quality as perceived by the ear is much influenced by subjective beat-tones. While these tones may be con- sidered as having no physical existence, yet their effects upon the ear are those of real partials, and the laws already stated include them. An explanation of beat-tones is given in Lecture VI, page 183. Fig. 49 shows on the musical staff the relations of a funda- mental tone, Ca = 129, and nineteen of its harmonic over- tones. The numerals in the line below the staff indicate the orders of the several partials. In the next lower Une are given the frequencies of the partials when they are harmonic. Every sound which is represented by a periodic 63 THE SCIENCE OF MUSICAL SOUNDS wave form must have harmonic overtones, as will be more fully explained in Lecture IV; such sounds are generally described as musical. The partial tones of sounds such as the clang of a bell are inharmonic and would not correspond to the scheme shown in the figure. The tones of the musical chromatic scale are determined according to the scheme of equal temperament developed by Bach. The various har- monic overtones of a given sound are not in tune with any notes of the musical scale, except such as are one or more i -rr m y rk I 2 3 4 5 6 7 8 9 10 II 12 13 14 IS 16 17 18 19 20 129 259 388 £17 647 776 905 1035 1164 1233 1423 1552 1681 m 1940 2069 2199 232824572586 C C G C E G B''C D E G''G G*B''B C C*D D*E 129 259 388 517 652 775 922 1035 1161 1304 1463 1550 1642 1843 1953 2069 2192 223 2461 2607 Fig. 49. A fundamental and its harmonic overtones. exact octaves from the fundamental. The notes on the staff in Fig. 49 represent the scale tones which are nearest to the overtones ; the lower Unes in the figure give the desig- nations of the notes and their frequencies in the tempered scale. Overtones can be illustrated by vibrating strings in such a way as to make their nature directly visible. A silk cord may be made to vibrate by a large electrically driven fork, as shown in Fig. 50, with the formation of a single loop due to vibration in the fundamental mode. By changing the ten- sion of the string, it can be made to vibrate in various sub- 64 CHARACTERISTICS OF TONES rrer »jw* a J2 s 65 THE SCIENCE OF MUSICAL SOUNDS Fig. 51. Simple vibrations of a string in various subdivisions, corresponding to harmonic overtones or partials. Fig. 52. Complex vibrations of a string, showing the coexistence of several modes of vibration, representing tones having different qualities. 66 CHARACTERISTICS OF TONES divisions corresponding to its harmonic overtones ; Fig. 51 shows two-loop, three-loop, and five-loop formations, repre- senting the first, second, and fourth overtones. A string vibrating in these forms would emit simple tones only ; if the pitch for the single loop is C2 = 129, the two loops would correspond to the tone C3 = 259, the three loops to G3 = 387, and the five loops to E4 = 645. A string may be made to vibrate in complex modes, with the simultaneous existence ■ of several loop formations. Fig. 52 shows the vibrations with two and four loops, with 44 ^^145 1.1 44444 ^ipiii ■•^ ff»--ti-»' N 4 5 5 ■T- -F -1— ' r^fi 6 5 , m- ■ 6 fh 454 F^ ^. — . — — ■ — m4 •-• ■• 9-9.9. •^•►ff -•tl ► 9- ■ P 9- -9- V p ^ • 0~ Fig. 53. A tune in harmonics. three and six loops, and at the top a much more complex combination ; these forms represent composite sounds, each set of loops corresponding to a partial tone. The multiplicity of tones from one air column, correspond- ing to the several loop formations in a vibrating string, are illustrated by wind instruments, many of which use har- monic tones in their regular scales. The bugle can sound only tones due to the vibration of the air column in various 67 THE SCIENCE OF MUSICAL SOUNDS subdivisions of its fundamental length; it produces the tones of the harmonic series shown in Fig. 49. A flute tube without holes or keys may be made to sound ten or more tones of the harmonic series. A tune can be played on a flute by using the harmonic tones of only three fundamen- tals, requiring two keys which are manipulated by one finger ; the illustration, Fig. 53, shows at the bottom the notes fingered, while those at the top are the harmonic tones sounded ; the small numerals indicate the orders of the partials used for the several tones. Analysis by the Ear Even after the arguments presented, it may seem strange that a single source of sound can emit several distinct tones simultaneously. There is, however, abundant experiinen- tal evidence in support of the statement. By Ustening attentively, one can often distinguish several component tones in the sound from a flute or violin or other instrument. Fig. 54. Helmholtz resonators. Helmholtz, who depended mainly upon the ear for the analysis of composite sounds, developed several methods for assisting the ear in the detection of partial tones.'" He 68 CHARACTERISTICS OF TONES devised the tuned spherical resonator which he used with remarkable success. Fig. 54 shows a series of Helmholtz resonators for the first nineteen overtones of a fundamental having a frequency of 64 vibrations per second; the ten odd-numbered resonators in the series correspond to a fun- damental of 128 vibrations per second and its first nine over- tones. The resonator consists of a spherical shell of metal or glass ; there is a conical protuberance ending in a small aperture, which is to be inserted in the ear ; opposite this aperture is an opening, through which the sound waves influence the air in the resonator. The tuning depends upon the volume of air in the resonator and the size of the opening. If one ear is stopped while a resonator is applied to the other, most of the tones existing in the surrounding air will be damped or, in effect, excluded, while if a component sound exists which is of the same pitch as that of the resonator, this particular simple tone affects the ear powerfully. 69 LECTURE III METHODS OF RECORDING AND PHOTOGRAPHING SOUND WAVES The Diaphragm An adequate investigation of the most interesting char- acteristic of sound, tone quaUty, requires consideration of the form of the sound wave ; for this purpose it is desirable to have visible records of the sounds from various sources which can be quantitatively examined and preserved for comparative study. Nearly all the methods which have been developed for recording sound make use of a diaphragm as the sensitive receiver. A diaphragm is a thin sheet or plate of elastic material, usually circular in shape, and supported more or less firmly at the circumference. The telephone has a diaphragm of sheet iron; in the talking machine sheets of mica are often used, while the soundboard of a piano is a wooden diaphragm; many other materials may serve for special purposes, such as paper, parchment, animal tissue, rubber, gelatin, soap film, metals, and glass. Diaphragms respond with remarkable facility to tones of a wide range of pitch and to a great variety of tone combina- tions. The telephone transmitter, the recording talking machine, and the eardrum illustrate the diaphragm set in vibration by the direct action of air waves; one readily thinks of the diaphragm as being affected by the variations 70 RECORDING AND PHOTOGRAPHING SOUND WAVES in air pressure which constitute the wave, but it is difficult to reahze how the movements can accurately correspond to the composite harmonic motion which represents the par- ticular tone color of a given voice or instrument. However, the reproductions of the telephone and talking machine are convincing evidence that the diaphragm does so respond, at least to the degree of perfection attained by these instru- ments. Not only may sound waves cause a diaphragm to vibrate, but what is even more wonderful, a diaphragm vibrating in any manner may set up sound waves in the air ; this reverse action of the diaphragm is shown in the receiving telephone, magnetism being the exciting cause, and in the machine which talks, the diaphragm of which is mechanically pulled and pushed by the record. The head of a drum is a diaphragm excited by percussion, the soundboard of a piano is caused to vibrate by the action of the strings, and the vocal chords may be considered as a diaphragm set in vibration by a current of air. The usefulness of the diaphragm is limited, and some- times annulled, for both scientific and practical purposes, by certain peculiarities in its action related to what are called its natural periods of vibration ; these effects of the diaphragm are considered in Lecture V. Various instruments employing the diaphragm, which have been useful in research on sound waves, will be de- scribed in the succeeding articles. The Phonautogeaph The Scott-Koenig phonautograph, by which sound waves are directly recorded,'^ was perfected in 1859. The instru- ment consists of a membrane placed at the focus of a para- 71 THE SCIENCE OF MUSICAL SOUNDS bolic receiver or sound reflector, Fig. 55 ; a stylus attached to the membrane makes a trace on smoked paper carried Fig. 55. Koenig's phonautograph for recording sounds. on a rotating cyhnder; a sound produced in front of the receiver causes movements of the membrane which are Fig. 56. Phonautograph records. recorded. A tuning fork with its prongs between the mem- brane and the paper. is mounted on the base of the instru- 72 RECORDING AND PHOTOGRAPHING SOUND WAVES ment ; a stylus attached to one prong of the fork marks a sim- ple wave line by the side of the trace from the membrane. Phonautograph records obtained by Koenig are shown in Fig. 56, the lower one of each pair of traces is that of the sound being studied, combinations of organ pipes in this instance, while the upper trace of each pair is from the tuning fork, enabhng the determination of the frequencies of the recorded tones. These records are not . only small in size, but the essential characteristics , are distorted or obliterated by friction and by the momentum of the stylus. The Manometric Flame In 1862 Koenig devised the manometric capsule in which the flame of a burning gas jet vibrates in response to the variations in pressure in a sound wave.'^ The capsule c, Fig. 57, is divided into two compartments by a partition Fig. 57. Koenig's manometric capsule with revolving mirror. 73 THE SCIENCE OF MUSICAL SOUNDS of thin rubber; the variations of air pressure due to the sound wave are communicated through the speaking tube t to one side of the partition, while the gas supply for the burning jet j is on the other side ; the movements of the diaphragm produce changes in the pressure of the gas which cause the height of the flame to vary accordingly. P '*JAJ*MJA/JJAM/M/JA/XM^kMMM^ re ^•^j UAUmtn t^ mmimmmmmm, jrwrrrfrffffi t er 'll/lljUMm^J-'W\VVvv-AA^612 Vowel curves enlarged from a phonographic record. S Pullay for S/tfa Monemen^ Fig. 64. Scripture's apparatus for tracing talking-machine records. By means of a tracing apparatus, a top view of which is shown in Fig. 64, Scripture has copied talking machine records enlarged 300 times laterally and about 5 times in length.*^ A disk record is rotated very slowly, one turn 77 THE SCIENCE OF MUSICAL SOUNDS in 5 hours, while a tracing point rides smoothly in the groove ; by means of a system of delicate compound levers, the lateral movements of the tracer are registered on a moving strip of smoked paper. The whole apparatus is operated by an electric motor, and when started, may be left to continue the tracing to the end, which operation, with the mag- nification em- ployed, is prac- ticable for a few turns only of the disk. Fig. 65 shows such a tracing from a record of orches- tral music, which as here repro- duced is magni- fied about 150 times laterally and 2^ times in length. These copies are prob- ably the best that have been obtained from phonographic records. Both the process of making the original record in wax and the subsequent enlarging introduce imperfections into the curves ; nevertheless these methods have been of great value in many researches in acoustics. Fig. 65. A tracing, by Scripture, of chestral music. record of or- The Phonodeik For the investigation of certain tone qualities referred to in Lecture VI, the author required records of sound waves 78 RECORDING AND PHOTOGRAPHING SOUND WAVES showing greater detail than had heretofore been obtained. The result of- many experiments was the development of an instrument which photographically records sound waves, and which in a modified form may be used to project such waves on a screen for public demonstration; this instru- ment has been named the "Phonodeik," meaning to show or exhibit sound.*^ The sensitive receiver of the phonodeik is a diaphragm, d, Fig. 66, of thin glass placed at the end of a resonator horn h; behind the diaphragm is a minute steel spindle mounted in jeweled bearings, to which is attached a tiny mirror m ; one part of the spindle is fashioned into a small Principle of the phonodeik. pulley ; a few silk fibers, or a platinum wire 0.0005 inch in diameter, is attached to the center of the diaphragm and being wrapped once around the pulley is fastened to a spring tension piece; light from a pinhole I is focused by a lens and reflected by the mirror to a moving film / in a special camera. If the diaphragm moves under the action of a sound wave, the mirror is rotated by an amount propor- tional to the motion, and the spot of light traces the record of the sound wave on the film, in the manner of the pendulum shown in Fig. 11, page 12. In the instrument made for photography. Fig. 67, the usual displacement of the diaphragm for sounds of ordinary loudness is about half a thousandth of an inch, resulting in 79 THE SCIENCE OF MUSICAL SOUNDS Fig. 67. The phonodeik used for photographing sounds. 80 Recording and photographing sound waves an extreme motion of one thousandth of an inch, which is magnified 2500 times on the photograph by the mirror and Ught ray, giving a record 2| inches wide ; the film commonly employed is 5 inches wide, and the record is sometimes wider than this. The extreme movement of the diaphragm of a thousandth of an inch must include all the small variations of motion corresponding to the fine details of wave form which represent musical quality. Many of the smaller kinks shown in the photographs, such as Figs. 110 and 169, are produced by component motions of the diaphragm of less than one hundred-thousandth of an inch ; the phonodeik must faithfully reproduce not only the larger and slower components, but also these minute vibrations which have a frequency of perhaps several thousand per second. The fulfillment of these requirements necessitates unusual mechanical delicacy; the glass diaphragm is 0.003 inch thick, and is held hghtly between soft rubber rings, which must make an air-tight joint with the sound box ; the steel staff is designed to have a minimum of inertia, its mass is less than 0.002 gram (less than ^V grain) ; the small mirror, about 1 millimeter (0.04 inch) square, is held in the axis of rotation ; the pivots must fit the jeweled bearings more accurately than those of a watch; there must be no lost motion, as this would produce kinks in the wave, which when magnified would be perceptible in the photograph; there must be no friction in the bearings. The phonodeik responds to 10,000 complete vibrations (20,000 movements) per second, though in the analytical work so far undertaken it has not been found necessary to investigate frequencies above 5000. The author wishes to record that the success of the phono- G 81 THE SCIENCE OF MUSICAL SOUNDS deik in meeting these requirements is due to the friendly interest and exceptional skill of Mr. L. N. Cobb, who con- structed the steel staff in accordance with designs which seemed almost impracticable and then mounted it in perfect jeweled bearings. The camera is arranged for moving films of 5 inches in width and of lengths to 100 feet ; there are three separate revolving drums having circumferences of 1, 2, and 5 feet respectively ; there is also a pair of drums, each holding 100 feet of film, arranged for winding the film from one to the other during exposure. The single drums are turned by an electric motor, with film speeds varying from 1 to 50 feet per second. A rheostat for controlling the speed of the motor is placed where it can be reached by the experimenter when he stands near the horn, and there is visible a tachometer which indi- cates the film speed. For general display pictures a speed of 5 feet per second is convenient, while for records to be analyzed 40 feet per second is suitable ; for the latter pur- pose a short record 1 or 2 feet long, made in -^^ or 2V of a second, is sufficient. The camera is provided with several shutters of various types for hand, foot, and automatic electric release, and for any desired time of exposure; and a commutator on the revolving drum may be used to open and close the shutter at desired points in its revolution. Besides the record of the wave there are photographed on the film simultaneously a zero line to give the axis of the curve for analysis, and time signals from a stroboscopic fork, y^^ second apart, to enable the exact determination of pitch from measurements of the film. The axis and time signals are shown in Fig. 96 and in many others; 82 RECORDING AND PHOTOGRAPHING SOUND WAVES when the photograph is intended for display only, these records are sometimes omitted. For visual observations the camera is provided with a horizontal revolving mirror which reflects the vibrating light spot upward on a ground glass in the form of a wave ; an incHned stationary mirror above the ground glass makes the wave visible to the experimenter while the, sound is produced. The speed of the revolving mirror and the dimensions in general are so proportioned that the wave appears on the ground glass in the same size and position as when photographically recorded. The speed of the motor may be adjusted till the wave appears satisfactory and the film speed will be automatically varied to correspond ; the sound is altered in loudness or quality as desired; when a suitable wave appears on the ground glass, the closing of an electric key or the pressure of the foot on a floor trigger makes the photographic exposm-e. The photographs- are all taken under such conditions that the film moves from right to left, giving the time scale in a positive direction, and that a positive ordinate of the curve corresponds to the compression part of the air wave. A sound-recording instrument might best be used out of doors, on the roof of a building for instance, to avoid con- fusion of the records by reflection from the walls ; since it is not convenient to work in such a place, the disturbing factors of the laboratory room are minimized by various precautions, such as padding the walls with thick felt; Fig. 68 shows the room in which the photographs are made ; the phonodeik with the receiving horn stands on a pier, while the Ught and moving-film camera are behind the screen. The tuning fork which flashes the time signal is shown at the right. 83 THE SCIENCE OF MUSICAL SOUNDS 84 RECORDING AND PHOTOGRAPHING SOUND WAVES The Demonstration Phonodeik The vibrator of the phonodeik employed in research is very minute and delicate, and its small mirror reflects too little light to make the waves visible to a large audience. For purposes of demonstration, a phonodeik has been espe- cially constructed, Fig. 69, which will clearly exhibit the principal features of "hving" sound waves.*^ The sound from a voice or an instrument is produced in front of the horn ; the movements of the diaphragm with its vibrating mirror cause a vertical line of light which, falling upon a motor-driven revolving mirror, is thrown to the screen in the form of a long wave ; the movements of the diaphragm are magnified 40,000 times or more, producing a wave which may be 10 feet wide and 40 feet long. With this phonodeik a number of experiments may be made in further explanation of the principles of simple harmonic motion and wave forms. When the revolving mirror is kept stationary, the spot of light on the screen moves in a vertical line as the diaphragm vibrates ; though these movements are superposed, their extreme complexity is shown since the turning points are made evident by bright spots of light. If the mirror is slowly turned by hand, the production of the harmonic curve by the combina- tion of vibratory and translatory motions is demonstrated. With a tuning fork the simplicity of the sine curve is exhib- ited ; with two tuning forks the combination of sine curves is shown; the imperfect tuning of two forks is demon- strated by a slowly changing wave form; the relations of loudness to amplitude and of pitch to wave length may be illustrated. The projection phonodeik is especially suitable for exhibit- 85 THE SCIENCE OF MUSICAL SOUNDS 86 RECORDING AND PHOTOGRAPHING SOUND WAVES ing the characteristics of sounds from various sources ; as seen on the screen the sound waves are constantly in motion, changing shape and size with the sUghtest alteration in frequency, loudness, or quality of the source. (As delivered orally, this Lecture was illustrated with many photographs of sound waves and also by the pro- jection of the sound waves from various sources upon the screen. The greater number of the photographs so used are reproduced in various parts of this book, while the charac- teristics of the sources of sound are described in Lecture VI.) Determination of Pitch with the Phonodeik The photographs obtained with the phonodeik permit a very convenient and accurate determination of pitch ; the time signals are given by a standard tuning fork, record- ing one hundred flashes per second ; it is only necessary to compare the wave length and the time intervals to obtain the frequency. Various photographs, as Fig. 96, show the time signals. A standard clock with a break-circuit attachment may be made to record signals simultaneously with the sound waves ; by counting and measuring, the number of waves per second may be determined with precision. When two sounds are being compared by the method of beats, the exact number (including fractions) of beats per second may be determined by photographing the beats together with the time signals. The phonodeik permits accurate tuning of all the harmonic ratios ; if the spot of light is observed without the revolving mirror, its movements take place in a straight line; two tones sounding simultaneously give a composite wave form, the turning points of which are visible as circles of extra 87 THE SCIENCE OF MUSICyVL SOUNDS brightness on the hne, Uke beads on a string. When the ratio of the component tones is inexact, there is a constant change of wave form which causes the beads to creep along the hne ; when the ratio is exact, the wave form is constant . and the beads are stationary, signifying perfect tuning. Photographs of Compression Waves The methods for recording sound so far described show the movements of a • diaphragm produced by the varying air pressure of the sound wave. Sound waves consist of alternate condensations and rarefactions which are prop- agated through space with a velocity of 1132 feet per second ; for the tone middle C the distance from one com- pression to the next is about four feet. It would be very useful indeed if photographs could be obtained of ordinary sound waves in air, but no practicable means has yet been devised for photographing waves of tliis size. A method due to Toepler ''^ has been successively developed by Mach, Wood, Foley and Souder, and Sabine, by which instantaneous photographs can be obtained of the snapping sound of an electric spark from a Ley den jar. This sound consists of a single wave containing one condensation and one rarefaction, the wave length may be yg inch or less, and the sound is relatively a loud one, that is, the change in density is considerable. If while such a sound wave is passing over a photographic plate in the dark, the wave is instantaneously illuminated by a single distant electric spark, the Ught from the spark will be refracted by the sound wave which will then act as a lens and register itself on the plate. There must be one miniature flash of lightning to make the sound, a sort of minute clap of thunder, and a second distant flash a small fraction of a second later to 88 RECORDING AND PHOTOGRAPHING SOUND WAVES illuminate the thunder wave as it passes outwards. This method is not suitable for recording the peculiarities of ordinary sounds due to various tone qualities, but it is very useful in studying some features of wave propagation. The most beautiful photographs of this kind have been recently obtained by Professor Sabine and appUed by him to the practical problem of auditorium acoustics, to which Fig. 70. Cross-sectional model of a theater, with the photograph of a sound wave entering the auditorium. reference was made in Lecture II. A small cross-sectional model of the auditorium is prepared, as shown in Fig. 70, and the photographic plate is placed behind it ; the sound is produced on the stage, at x, and the resulting wave pro- ceeds on its journey into the auditorium, moving at the rate of 1132 feet per second. The wave length in the experiment is about 2V inch, which is equivalent to a wave length of two feet in the actual audi- 89 THE SCIENCE OF MUSICAL SOUNDS Fig. 71. Position of a sound wave in a theater tSs second after its production on the stage. Fig. 72. Echoes in a theater developed from a single sound impulse in -^ second. 90 RECORDING AND PHOTOGRAPHING SOUND WAVES torium, corresponding to the musical tone one octave above middle C. The wave- in the real auditorium will have reached the position shown in the figure in about yf j second. Various reflected waves or echoes are beginning to appear : Oi is produced by the screen of the orchestra pit, a^ is from the main floor, and as is from the orchestra pit floor. Fig. 71 shows the waves about yfo second later, just before the main wave reaches the balcony, and Fig. 72 shows the waves YoV second after the production of the original sound, when the main wave has reached the back of the gallery. The large number of echo waves which seem to come from many directions are actually generated by the one original impulse. The multiple echoes continue to develop with increasing confusion, until the sound is diffused throughout the auditorium, producing the condition called reverbera- tion. 91 LECTURE IV ANALYSIS AND SYNTHESIS OF HARMONIC CURVES Harmonic Analysis Curves and wave forms such as those obtained with the phonodeik are representative not only of sound, but of many other physical phenomena, and their study is of general importance in science. While inspection and simple measurement will often give some information concerning these curves, as will be explained later, they are in general' too complicated for interpretation in their original forms, and several methods of analysis have been developed which greatly assist in our understanding of them. In the wave method of analysis, often used in optics, the attention is directed to the speed and direction of propaga- tion of the waves in the medium and to their combined effects ; in the harmonic method consideration is given primarily to the vibratory character of the movements of the medium, these vibrations being regarded as compounded of a series of motions, which may be infinite in number, but each of which is of a simple definite type. For the investigation of the complex curves of the sounds of music and speech, the harmonic method of analysis is the most suitable and convenient; it is based upon the im- portant mathematical principle known as Fourier's Theorem, the statement and proof of which was first published in Paris, in 1822, by Baron J. B. J. Fourier.^ For the present 92 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES purpose Fourier's theorem may be stated as follows : If any curve be given, having a wave length I, the same curve can always be reproduced and in one particular way only, by compounding simple harmonic curves of suitable ampli- tudes and phases, in general infinite in number, having the same axis, and having wave lengths of I, \l, \l, and succes- sive aliquot parts of I ; the given curve may have any arbi- trary form whatever, including any number of straight portions, provided that the ordinate of the curve is always finite and that the projection on the axis of a point describ- ing the curve moves always in the same direction. Many of the curves studied by this method can be exactly repro- duced by compounding a limited number of the simple curves; for sound waves the number of components re- quired is often more than ten, and rairely as many as thirty ; in some arbitrary mathematical curves, a finite number of components gives only a more or less approximate repre- sentation, while an exact reproduction requires the infinite series of components. Fourier's theorem may be stated in mathematical form in the Fourier Equation as follows : ^^ ydx- ra r' . z-nx , 1 . 2ttx ra r . ^ttx. i . 4773: I I _. — j_i — . I I ,,^,« -ax sin — ; — [-.... I I rar' Snx ,1 Sttx rsf' ^ttz, ] 4-nx .[tJo y^°^-r''"r T- -^ [tJ, y'°' — d-J-=— +• In this equation y is the ordinate of the original complex curve at any specified point x on the base line, and I is the fundamental wave length. The principal part of this equa- tion is a trigonometric series of sines and cosines and this (or the whole equation) is often referred to as Fourier's Series. 93 THE SCIENCE OF MUSICAL SOUNDS The Fourier equation may be given a sim,pler appear- ance by writing it in a second symbolic form : Itti sin 9 + a2 sin 2 + as sin 3 9 + . . . II &i cos ^ + &2 cos 2 9 + 63 cos 3 ^ + . . . The term Oq is a constant and is equal to the distance between the chosen base Une and the true axis of the curve ; if the base line coincides with the axis, Oo = 0, and this term does not appear in the equation of the curve. Since this term has no relation to the shape of the curve, its value is not required in sound analysis ; the method for evaluating it, however, is described on page 107. The other terms of the equation occur in pairs, as ay sin d, 61 cos 6, etc., and each, whether a sine or cosine term, rep- resents a simple harmonic curve. The successive simple curves of the sine series evidently repeat themselves with frequencies of 1, 2, 3, etc., that is, they have wave lengths in the proportions of 1, \, \, etc., and the same is true of the cosine series. Each of the coefficients ai, hi, a-i, 62, etc., is a number or factor indicating how much of the corresponding simple harmonic curve enters into the composite ; that is, it shows the amplitude, or height, of the simple wave. For the re- production of a given curve it may happen that certain of the simple curves are not required, and the corresponding coefficients then have the value zero and their terms do not appear in the Fourier equation of the curve. A sine and a cosine curve of the same frequency but with independent amplitudes, such as the pairs of curves in the Fourier equation, can be compounded into a single sine (or cosine) curve of like frequency which starts on the 94 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES axis at a point different from that of the component curves, and which has an amplitude dependent upon the ampKtudes of the components. The relation ' of the starting point of the new curve to that of its components is called its phase, as is explained on page 126. This principle may be stated in symbols as follows, a and b being the amphtudes of the given curves, and A that of the resultant, and P the phase of the new curve : a sin 5 +b cos 6 = A sin {d + P), when A = Va^ + b^, and tan P =- . a If the amplitudes a and h are made the base and altitude, respectively, of a right triangle. Fig. 73, then the hypotbe- nuse is the amplitude A of the resultant ^,^ curve and the angle . ^^ which the hypothe- ^-^ nuse makes with the ^^ \ p base is the phase. ^^^ \ If each pair of sine ^ and cosine terms of ^^^ 73 Amplitude and phase relations of compo- the ffeneral Fourier nent and resultant simple harmonic motions. equation is reduced in this manner, and if the origin is on the axis of the curve, the equation may be put into the following equivalent form, consisting of a single series of sines : 2/=Aisin(6i+Pi)+A2sin(2e+P2)+A3sin(36i+P3) + . . .Ill In this equation Ax is the amplitude of the first component (the fundamental tone) and Pi is its phase; while A 2 95 THE SCIENCE OF MUSICAL SOUNDS^ ANALYSIS AND SYNTHESIS OF HARMONIC CURVP:S and Pi determine the second component (first overtone or octave), etc. Form III of the Fourier equation is most suitable for representing the results of the physical analysis of a sound, though the actual numerical analysis is obtained in the first form, I, of the equation. Mechanical Harmonic Analysis The process of analyzing a curve consists of finding the particular numerical values of the coefficients of the Fourier equation so that it will represent the given curve. Fourier showed how this may be done by calculation (see page 133), but as it is a long and tedious process, requiring perhaps several days' work for a single curve, various mechanical devices have been constructed to lessen the labor. The coefficients of the various terms, the quantities in square brackets in equation I, have the following form, n being the order of the term : 2 /•' . 2mrx , J \ y sm — j — ax. These are represented by Oi, bi, etc., in equation II, and are the amplitudes of the component simple harmonic curves. Each definite integral is the area of a certain auxiliary curve on the base I, the nature of which need not be de- scribed here ; *'' this area, divided by I, gives the mean height of the auxiliary curve, which is then multiplied by 2, giving the amplitude of the corresponding component. There are various area-integrating machines, known in their simple forms as planimeters, which can be adapted to the deter- mination of the areas of a given curve under such conditions as to indicate on the dials the numerical values of the H 97 THE SCIENCE OF MUSICAL SOUNDS Fourier coefficients ; in some machines the dial /readings are the coefficients, in others the dial readmgs require further slight reduction. Such machines are called har- monic analyzers. Several types of harmonic analyzers are briefly referred to on page 128. The analyzer devised by Professor Henrici, of London, in 1894, based on the rolling sphere integrator, is perhaps the most precise and convenient yet made.^ An instrument of this type used by the author in the study of sound waves, is shown in Fig. 74, and its operation will be described. The curve to be analyzed, which must be drawn to a specified scale, as is explained later, is placed underneath- the machine; the handles h are grasped with the fingers, and the stylus s is caused to trace the curve, which re- quires movements in two directions. The machine as a whole rests on rollers which permit it to be moved to and from the operator, in the direction of the ampUtude of the curve, and the stylus is attached to a carriage which rolls along a transverse track t in the direction of the length of the curve. The instrument shown has five integrators ; each sphere, made of glass, rests on a roller so that when the curve is traced, the sphere is rotated on a horizontal axis by an amount proportional to the amplitude of the curve; two integrating cyUnders with dial indexes rest against each sphere at points 90° apart. Fig. 75, and, by means of a wire and pulley w are given rotation about a vertical axis pro- portional to the movement along the axis of the curve. While each sphere rolls only in amplitude, the cylinders sliding around the sphere take up components of the amplitude motion which are proportional to the sine and 98 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES cosine of the phase change respectively. The first inte- grator turns once around its sphere while the tracer moves over one wave length of the fundamental curve, that is, while the stylus is being moved the length of the track t, the next integrator turns twice, and the others three, four, and five times in the same interval. In this manner one Fig. 75. The rolling-sphere integrator of the harmonic analyzer. tracing gives the ten coefficients, five sines and five cosines, of the first ten terms of the complete Fourier equation of the curve. In the Henrici analyzer the sizes of the various parts are so proportioned that the effects of the constant factors of the amplitude terms are mechanically incorporated in the dial readings, which are, without reduction (except for the 99 THE SCIENCE OF MUSICAL SOUNDS factor n, mentioned below), the actual amplitudes in milli- meters of the components of the curve traced. When the stylus has been moved over one wave length of the funda- mental, it must have moved over two wave lengths of the second component, three of the third, and so on ; then the integrator for the second component has integrated two waves, and the dial readings are twice the required coefl&- cients ; in general, the readings of the nth integrator are n times too large, they are na„, and n&„. In the study of sound waves the presence of the factor n is a convenience, for the quantities finally desired are the intensities of the Fig. 76. Photograph of the sound from a violin. components, and, as explained on page 167, the loudness of any component is proportional to (nA„)-. By changing the wire to the smaller pulleys v on the integrators, the spheres are turned six, seven, eight, nine, and ten times while tracing the wave, and the dials indicate the sine and cosine coefficients for the components from six to ten. By a reconstruction of the analyzer (in 1910) which it was necessary to carry out in our own instrument shop, the operation of the instrument has been extended from ten to thirty components with precision,' six tracings being required for the larger number.*^ The analysis of the sound wave from the tone B4 = 995, 100 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES played on the E string of a violin, will be considered. This curve, which is shown in Fig. 76, is comparatively simple. When the curve is analyzed with the machine, the opera- tion proceeds in accordance with the method shown in the Fourier equation I, but the mechanical integrators give the result in form II, and the actual equation read from the dials is as follows : y = 151 sin e - 67 cos 9 + 24 sin 2 6i + 55 cos 2 d + 27 sin 3 0+5 cos 3 6. The analyzer, which has five integrators, gives at the same time with the above the coefficients of the terms involving Fig. 77. Curve of a violin tone and its sine and cosine components. 4 d and bd; in this instance the latter coefficients are very small, and for simplicity they are omitted. In other words 101 THE SCIENCE OF MUSICAL SOUNDS the analysis shows this curve to be composed of three com- ponents only, each of which is represented by a pair of sine and cosine terms. In practical work, each pair of sine and cosine terms is at once reduced to a single t^rm, but for the sake of illus- tration the graphic interpretation of the equation in its present form is given in Fig. 77 ; there are six simple cxirves, a sine and a cosine curve for each of the three frequencies, all starting from the same initial line ab; the sine curves are indicated by Si, Sa, and S3, and the cosines by Ci, C2, and C3. These six curves added together, or made into a composite, will accurately reproduce the vioUn curve V. As explained on page 95, each pair of these curves can be reduced to a single equivalent curve, and the six com- ponents thus become three. The reduction for the first pair of terms gives the equation : 151 sin (? - 67 cos e = 165 sin (0 + 336°). y"^. The graphic interpretation of a V/ '"> \ ^^^ reduction is shown in Fig. ■''/ \-\--. 78, in which ri is the resultant / y \ \ '"v of Si and Ci and is the true y" \\ \ 7\ representation of the funda- ''c, \\ "y-J mental of the viohn curve. The \\ / / second and third pairs of curves '\C/ are similarly reduced, giving Fig. 78. The resultant of sine and the CUrveS for the first and cosine curves. i i ^ i second overtones, and the final Fourier equation for the violin curve, in form III, is : y = 165 sin (0 + 336°) + 60 sin (2 0+ 66°) + 27 sin (3 0+ 11°). This is the form of equation usually desired in physical investigations ; its graphical interpretation is shown in 102 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES Fig. 79, which shows the original violin curve at the top, with its three true components, representing partial tones, drawn separately to show the amplitudes and phases (start- ing points) more distinctly. Fig. 79. Curve of a violin tone and its three harmonic components. The complete analysis and synthesis of a more compli- cated curve is described later in this Lecture. Amplitude and Phase Calculator The reduction of the double Fourier series consisting of sines and cosines to the single series of sines with differing 103 THE SCIENCE OF MUSICAL SOUNDS phases is usually carried out by numerical calculation, as has been indicated. The need for a more expeditious method, where a large number of curves are being analyzed, has resulted in the design and construction in our own laboratory of a machine. Fig. 80, which accompUshes the purpose in a satisfactory manner.*^ This ampUtude-and- FiQ. 80. Machine for calculating amplitudes and phases in harmonic analysis. phase calculator is essentially a machine for solving right triangles. The machine has two grooves at right angles to each other, provided with linear graduations; in the grooves are movable shders which carry the graduated hypothe- nuse bar; one end of the hypothenuse is attached to a special angle measurer, while the other end slides through 104 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES a support which also bears an index for reading the length of the bar. The pair of coefficients a and b of the general Fourier equation, which are given by the analyzer, are set off as the base and altitude, respec- tively, of- the triangle, when the length of the hypothenuse is the amphtude A of the re- sultant. The phase of the resultant curve, which is determined by the equation, tan P = - , Fig. 81. Phase angles in four quad- rants. may have any value from 0° to 360°, since a and b may have either the positive or the negative sign. For the same nu- merical values of a and b, and there- fore for the same value of A, there may be four dif- ferent values of the phase angle ; as indicated in Fig. 81, for + a, + b, the angle will have a value between 0° and 90° ; for -a, + b, the angle 105 Fig. 82. Scheme for measuring phases in one quad- rant. THE SCIENCE OF MUSICAL SOUNDS has a value between 90° and 180° ; for -a, — b, it lies between 180° and 270°, and for + a, - b, it is between 270° and 360°. A special angle measurer with four gradua- tions of a quadrant each might be used for the four possible combinations of algebraic signs ; Fig. 82 illustrates a scheme for such graduations. 1 Xj^j \ r" • 1 a- A B ' Fig. 83. Finding the axis of a curve. As a single graduated arc may be provided with two sets of numbers, one on either side, two quadrant graduations are sufficient. To prevent confusion a movable cover is provided for the graduations; this has four apertures so shaped that any one, and one only, of the four sets of num- bers is visible at one time, according to the position of a spring catch attached to the cover. There are four posi- 106 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES tions for this catch marked with the possible combinations of algebraic signs of a and b ; when the cover is set, one reads the true phase angle without any reduction. Axis OF A Curve If the axis of a curve is unknown and is required, it is necessary to determine the first or constant term of the Fig. 84. Finding the true axis of a curve with the planimeter. Fourier equation, forms I and II ; this term consists of the area included between the arbitrarily assumed base line and the curve, divided by the base, and therefore it repre- sents the mean height of the curve from the base. A line drawn through the curve parallel to the base and distant from it by the mean height of the curve, will be the true axis of the curve, since that part of the area between the curve and this axis which is above the axis must be equal to that which is below the axis. 107 THE SCIENCE OF MUSICAL SOUNDS Let it be required to find the axis of the curve shown in Fig. 83. When no base Une is given, any Une parallel to the axis may be used, such as a line touching the crests or troughs of two waves, AB, or any other line through points on two waves which are in the same phase. The area between the assumed base and the curve is measured with a planimeter of any type; this area divided by the wave length is the distance h from the base A 5 to the true axis A'B'. Fig. 84 shows a precision planimeter with a rolling sphere integrator, in position for the axis determi- nation of a curve, that is, for finding the first or constant term of the Fourier equation. The constant term gives no information regarding the nature or shape of the curve, it merely gives its position with regard to the base Une incidentally employed in draw- ing or tracing the curve. Ordinarily this term is not re- quired in sound analysis. Enlarging the Curves For- use with the Henrici analyzer it is necessary that the wave length of the curve which is traced shall be such that when the tracing point moves along its guiding tracks a distance equal to the wave length, the integrator for the first term shall make exactly one revolution aroimd the rolling sphere ; in the instrument illustrated the wave length must be 400 milHmeters, about 16 inches. The photographs of sound waves obtained with the phonodeik have wave lengths varying from 25 to 100 milli- meters ; these waves are enlarged with the apparatus shown in Fig. 85. The photographic film negative of the wave is placed in an adjustable holder /, on an optical-bench pro- jection lantern, the curve being projected on a movable 108 ANAI.YSIS AND SYNTHESIS OF HARMONIC CURVES easel e ; adjustments are made until the projected wave is of the proper size, is well defined, and has its axis horizontal ; the curve is then traced with a pencil on a sheet of paper. The initial point is chosen merely with reference to con- venience in determining the length of one wave, as a, Fig. 96, page 122, where the curve crosses the axis. The time re- quired for the operation of enlarging a curve is less than five minutes. Thus all curves as analyzed are of the same wave length, Fig. 85. Apparatus for enlarging curves by projection. regardless of their original size and frequency, and as they are drawn on a standard sheet of paper, 19 by 24 inches, filing is facilitated. The harmonic synthesizer, described later, draws curves of this same wave length, 400 milli- meters, which permits a direct comparison of the analyzed and synthesized curves. Some of the simpler analyzers mentioned later may be used with a curve of any size such as the original photo- graph, but the results read from the machine require further reduction for each individual curve and component ; as 109 THE SCIENCE OF MUSICAL SOUNDS already stated, Avith the Henrici analyzer, the machine readings are final, requiring no reduction, so far as analysis is concerned. Where many curves are being exhaustively studied by analysis and synthesis, the enlargement to stand- ard wave length is not a disadvantage. Synthesis op Harmonic Curves It is often required to perform the converse of the ana- lytical process which has been described, that is, to recom- bine several simple curves to find their resultant or com- posite curve; this is harmonic synthesis. The synthesis of curves can be accomplished by calculation in some in- stances, and always by graphic methods by adding the measured ordinates of the component curves and plotting the results ; since both of these methods are laborious, machines called harmonic synthesizers have been designed to faciUtate the work. I A harmonic synthesizer is a machine which will generate separate simple harmonic motions of various specified fre- quencies, amphtudes, and phases, and will combine these into one composite motion which is recorded graphically. One of the earliest synthesizers was made. in London about 1876 by Lord Kelvin (see page 129), to be used as a tide-predicting machine ; it is based upon the pin-and-slot device described in Lecture I. A cord fixed at one end, Fig. 86, passes around several pulleys and at the other end is attached to a pencil, which makes a trace on a moving chart. The pulley o is attached to a pin-and-slot device, which moves up and down with a simple harmonic motion ; the cord will transmit this motion to the pencil, doubled in amount ; if the chart moves continuously, the trace is a simple harmonic curve of a frequency depending upon the 110 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES rapidity with which the crank-pin is rotated, and of an ampli- tude depending on the distance of the pin from the center of its crank; the wave length of the cvirve depends upon the speed with which the chart moves. If another pulley b is attached to a second pin-and-slot device rotating twice as fast as the first, it will give the pencil a simple harmonic motion of twice the frequency of the first. It is evident from the manner in which the cord passes i m ^ fa ' '' 9b MBifawlj||MgWMB8a8B8BMBBBBfeP8i!ia!MWI^BMia?»r''y>v W ^^iilHB^ i ■ - ^-H Fig. 86. Apparatus illustrating the method of harmonic synthesis. around the system of pulleys that if the two devices operate simultaneously, the pencil will have a composite motion which is the sum of the two components, and the trace will be the synthetic curve. The scheme may be extended to include any number of simple harmonic motions of any desired frequencies, phases, and amplitudes. Two harmonic synthesizers, especially for the study of sound waves, have been designed and constructed in the laboratory of Case School of Applied Science, one having 111 THE SCIENCE OF MUSICAL SOUNDS 112 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES ten components and the other thirty-two. The ten-com- ponent machine, finished in 1910, was soon found inade- quate for the study of musical sounds and was dismounted in 1914, upon the completion of the thirty-two-component synthesizer shown in Fig. 87. The latter machine is per- haps more convenient for the study of harmonic curves in general than any other which has been constructed; it draws curves with great accuracy and on a very large scale ; the drawing board is 24 by 34 inches in size, but by shifting the paper and pen a curve of almost any size may be drawn. The largest single component curve may be 28 inches wide and have a wave length of 32 inches ; the highest com- ponent may be 4 inches wide. The wave length commonly used is 400 millimeters, about 16 inches, the same as that used with the analyzer, but larger or smaller wave lengths are easily arranged. Thirty of the elements are provided with gears giving the relative frequencies 1, 2, 3 ... 30 ; the other two elements are arranged with change-gears, like a lathe head, which permits their easy setting for higher or lower frequencies or for inharmonic frequencies. The machine can be quickly set to give the frequencies 1, 2, 4, 6, and all even terms to 60 ; or for the series 1, 3, 6, 9, and all multiples of 3 to 90. The mechanical arrangements permit the amplitude and phase of any component to be readily set to any value ; all the graduated circles and scales are on the upper surface of the machine and are of white celluloid. There are special scales showing the phase and amphtude of the synthesized curve. All the motions which affect the separate components as they are being com- pounded and synthetically drawn are provided with ball bearings, to eliminate friction and lost motion ; the motion is so accurately transmitted to the pen that a wave can be I 113 THE SCIENCE OF MUSICAL SOUNDS clearly drawn in which the amplitude is less than 0.2 milli- meter (less than -j^^ inch). This machine is described in detaU, with specimen curves, in the Journal of the Franklin Institute.^' A ten-component curve can be synthesized in about five minutes, while the machine may be set for thirty com- ponents and the curve drawn in twelve minutes. In the study of sound waves the synthesizer is chiefly used to verify the correctness and sufficiency of the analyses. The several unit devices of the machine having been set to reproduce the separate components in exact sizes and phases, the tracer will draw the resultant curve. If this resultant curve is exactly like the original which was analyzed, the analysis is correct and complete, and the fact is recorded by tracing the synthetic curve over the original in a con- trasting color of ink. Fig. 99, page 127, shows the synthetic reproduction of the analysis of an organ-pipe curve (Figs. 96 and 98) drawn on the photograph itself. The synthesizer is also useful for drawing a curve corre- sponding to the average of several photographed curves, and for drawing curves of any assumed composition, as in trial analysis. After a photographed curve has been analyzed and the components have been corrected for in- strumental disturbances, as explained in Lectm-e V, it is often useful to draw the corrected synthetic curve, as is illustrated in Fig. 132, page 173. The synthesizer is useful in preparing illustrations, such as many of those required in these lectures ; it would be very difficult to draw the curves of correct form by any other means. When the synthesizer is set for any curve, if the handle is turned till the phase circle for the first component reads 114 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES 0°, the circles for the other components show, without cal- culation, the relative phases, or, as sometimes called, the epochs, of the several components ; the tracing point will now be at what may be considered the initial point of the wave, though in general this will not be where the curve crosses the axis. These relations are further explained in connection with the analysis of the curve shown in Fig. 96, on page 122. The mathematician finds the haCrmonic synthesizer useful for the investigation of many kinds of curves ; the proper- ties of periodic func- tions and the conver- ^'°- ^^- ^ g«°°'«t™^i f°™- gency of series can be shown graphically. The equation of the wave form made up of two straight lines, as shown in Fig. 88, is represented by the infinite series 2/ = 2 [sin X + I sin 2 a; + f sin 3 x -|- J sin 4 x -f- . . .], the wave length being equal to 2 ir = 6.28^. The manner in which such an angular geometrical figure may be built up from smooth curves is shown by drawing curves representing different numbers of terms of the series. The first term only, y = 2 sin x, is represented in Fig. 89, a; b, c, d, e, and / represent the curves obtained when two, three, four, five, and ten terms, respectively, are used. Fig. 90 is the curve obtained when thirty terms are included. These ciu-ves are graphic illustrations of the convergence of this series ; the more terms employed, the closer the result approximates the given form; an infinite number of terms would be required to reproduce the figure exactly. 116 THE SCIENCE OF MUSICAL SOUNDS a 1 term b 2 terms d 4 term.s 5 terms / 10 terms Fig. 89. Forms obtained by compounding 1, 2, 3, 4, 5, and 10 terms of the series y = 2 [sin a; + J sin 2 a; + J sin 3 a; + . . .]. 116 ANALYSIS AND SYNTHESIS OP HARMONIC CURVES Fig. 91 is a curve made up of the same components as enter into the curve shown in Fig. 90 ; the only difference is 27T Fig. 90. Curve obtained by compounding 30 terms of the series y = 2 [sin x + i sin 2 X + i sin 3 x + .]. that the phase of each component has been changed by 90° ; that is, the sines become cosines. A further interesting variation is obtained by using the Fig. 91. Curve obtained by compounding 30 terms of the series y = 2 [sin (x + 90°) + i sin {2 x + 90°) + i sin (3 x + 90°) + .], which is equivalent to 2/ = 2 [cos a; + I cos 2 x + i cos 3 x + . .]. 117 THE SCIENCE OF MUSICAL SOUNDS odd-numbered terms only of the first series, producing the form shown in Fig. 92. If the phases of the alternate terms of the odd-term series yVv^- — — v^*/v/N/yl 77 \/Vv^/-^-■— -v-i^^/^-^Jyi 27r — ~\/y Fig. 92. Curve obtained by compounding 15 terms of the series y = 2 [sin x + i sin S X + i sin 5 x + . , .]. are changed by 180°, the curved form shown in Fig. 93 is obtained. ^^ The arbitrary nature of the curves that rtiay be studied Fig. 93. Curve obtained by compounding 15 terms of the series 1/ = 2 [sin x + i sin (3x + 180°) + J sin 5 x + * sin (7 a; + 180°) +...], or 3/ = 2 [sin s - i sin 3 X + J sin 5 X — J sin 7 a; + . . .]. by the Fourier method is further illustrated by the analysis and synthesis of a portrait profile. The original portrait is shown in the center of Fig. 94, while a tracing of the 118 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES profile is given at the left, 0. The curve was analyzed to thirty terms, but the coefficients of the terms above the eighteenth were neghgibly small. The equation of the s Fig. 94. Reproduction of a portrait profile by harmonic analysis and synthesis. curve is as follows, the numerical values corresponding to a wave length of 400 : y = 49, + 13, + 4. + + + 0, + 0, + 0, + 0, ,6 sin ,8 sin ,5 sin 7 sin 6 sin 3 sin 7 sin 4 sin 5 sin ( e + 302°) ( 3 ^+195°) {5e+ 80°) ( 7 6+ 34°) ( 9 e + 331°) (11 e+ 89°) (13 e + 103°) (15 e + 169°) (17 e + 207°) +17, + 7, + 0, + 0, + + + 0, .4 sin ,1 sin .6 sin ,6 sin .3 sin 5 sin 3 sin 5 sin 4 sin {2 6 + 298°) ( 4 & + 215°) ( 6 fl + 171°) ( 8^ + 242 J (10 e + 208°) (12 e + 229°) (14^ + 305°) (16^ + 230°) (18 e+ 64°). This equation was set up on tlie synthesizer, and the portrait, as drawn by the machine, is shown at the right, *S, Fig. 94. 119 THE SCIENCE OF MUSICAL SOUNDS If mentality, beauty, and other • characteristics can be considered as represented in a profile portrait, then it may be said that they are also expressed in the equation of the profile. Since the profile is reproducible by compounding a num- ber of simple curves, it is possible to compound the simple tones represented by these curves in such a way that the resulting wave motion of the combined sounds shall be the periodic repetition of the profile. Fig. 95 is a drawing of such a wave. The reproduction of vowel wave forms, shown Fig. 95. Wave form obtained by repeating a portrait profile. in Fig. 181, page 250, is a similar synthetic experiment. In this sense beauty of form may be hkened to beauty of tone color, that is, to the beauty of a certain harmonious blending of sounds. The Complete Process of Harmonic Analysis A curve having been provided, such as the photograph of a sound wave, an electric oscillogram, a diagram of baro- metric pressures, or a chart of temperatures, its complete analysis by the Fourier harmonic method may be con- veniently carried out in accordance with the following scheme : (a) The curve is redrawn to the standard scale required by the Henrici analyzer, so that the wave length is 400 millimeters. Time required : five minutes. (b) The curve is traced with the analyzer, one tracing giving five sine and five cosine coefficients of the complete Fourier equation of the curve, determining five components. 120 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES By changing the wire attached to the tracer from one set of pulleys on the integrators to another set, a second tracing gives five more pairs of coefficients, determining ten com- ponents of the cui;ve. Continued tracings will give fifteen, twenty, twenty-five, and thirty components. Time re- quired : for each tracing, including making the record, five minutes ; for ten components, ten minutes. (c) Each pair of sine and cosine terms as given by the analyzer is reduced by the triangle machine, to determine the true ampUtude of the corresponding component, together with its phase. Time required : for ten pairs of terms, including making the record, five minutes. (d) The correctness and completeness of the analysis are verified by setting the synthesizer for the values of the ampUtudes and phases of the several components and then reproducing the original curve. Time required : for ten components, five minutes ; for thirty components, twelve minutes. (e) The numerical quantities of the analysis are pre- served on cards suitable for filing; the synthetic curve is drawn, superposed on the original curve, forming a per- manent record of the degree of approximation secured. Time required : included in the time given for the operations (&), (c), id). (/) The synthesizer may be used to draw each component separately, in its true amplitude and phase. Time re- quired : for ten components, twenty minutes. (g) The true axis of the curve may be determined with the planimeter. Time required : three minutes. The times mentioned for the several operations are those required when a number of curves are being analyzed in routine ; if a single curve is analyzed by itself, a longer time 121 THE SCIENCE OF MUSICAL SOUNDS will be consumed. The analysis of a curve as ordinarily- understood involves only the operations (6) and (c), re- quiring about fifteen minutes for ten components. Card forms, Fig. 97, 5 by 8 inches in size, have been arranged for preserving the data of analysis and reduction, as required in the study of sound waves. One card con- tains the data for ten components. The cards for the first ten components, n = 1 to n = 10, are white in color ; for a larger number of components, cards of different colors are used, buff for values of n from 11 to 20, and salmon for n = 21 to n = 30 ; blue cards are used for additional information, averages, etc. The data relating to any com- ponent are given in the vertical column under the value of n corresponding to the order of the component. Example of Harmonic Analysis As a further illustration of harmonic analysis, let it be required to analyze the curve of an organ-pipe tone, shown Fig. 96. Photograph of the sound wave from an organ pipe. in Fig. 96. The curve is traced twice with the analyzer, the necessary change in the wire being made between the two tracing* ; then each of the ten pairs of sine and cosine 122 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES terms is reduced with the triangle machine for finding the resultant amplitudes and phases. The actual time required for the complete determination of the ten ampUtudes and ten phases, including the recording of the results on the analysis card, was thirteen minutes. (The actual analysis of this curve was extended to twenty components, but it was found to contain only twelve components of appreciable size.) Fig. 97 is a reproduction of the card containing the com- No.|fe30 CASE SCHOOL OF APPLIED SCIENCE DEPARTMENT OF PHY ^ ANALYSIS OF SOUND-WAVES Source ©,i,»,o^ Vijpt -"R-.A*,^, Otwrt,!' SICS 1-10 Tone Ik C. ^ Abs.N ifeO llPurposp (y-„jj,ll»,i.ij6,. M9I.T ■W3 *.»v II lf.^,ix«*^f.Ml, compoDeDt,n 1 2 3 4 5 6 7 8 9 10 na. + 22,< + 39.6 + 101.1 + 76.1 + 41.5- - 49.1 + -H.8 + M.9 - II.S - 7.1 nb„ + 31.1 - 87.2 - 42.5 - 7S -26.1 - (0.6 - 3.8 -66,7 -36.9 -21.7 nA. 36. J (32.0 /09.4 76.8 SI. 6 SO.^ 44". 111 38.7 ZZA K.i ) nA.k„ |nA.k.l' A. 9C,.S 66,0 36.S 13,? /0.3 S.4 &A 8.9 4,3 23 P. 7(f° 3 19.° 337." SSH." 330.' M?." 35-4.° MO." 2,52* zr2.° A„k. AirpUtude,% Phase 0° /67° (03° 5-0,° 310.° ZSO.' /SZ." 41.' 28a° zir Intensity,^ Remarks Sum Analyzed 'CL'JvA Synthesized Fig. 97. Card form for the record of the analysis of a sound wave. plete records of the analysis of the above curve (for the first ten components). The first two lines, «a„ and n&„, , are the coefficients (each multiplied by n, the order of the component) of the sine and cosine terms of the Fourier equation, form II, as read from the dials of the analyzer. Each pair of numbers is reduced with the amplitude-and- phase calculator, giving nAn and P„; each of the multiple amplitudes, nAn, is divided by the corresponding value of 123 THE SCIENCE OF MUSICAL SOUNDS n, giving A„; A„ and P„ are the coefficients and phases of form III of the Fourier equation. The mathematical equation of the organ-pipe curve (twelve components) is, then, as follows, the wave length being equal to 400 : 2/ = A + 96.5 sin ( 6+ 76°) + 66.0 sin ( 2 6 + 319°) + 36.5 sin ( 3 6 + 337°) + 19.2 sin ( 4:6 + 354°) + 10.3 sin ( 5 6 + 330°) + 8.4 sin ( 6 + 347°) + 6.4 sin ( 7 6 + 354°) + 8.9 sin ( 8 ^ + 290°) + 4.3 sin ( 9 6 + 252°) + 2.3 sin (10 6 + 252°) + 2.2 sin (110 + 230°) + 1.5 sin (12 + 211°) The graphic interpretation of this equation is given in Fig. 98. The equation as a whole is represented by the original curve at the top ; each of the twelve sine terms corresponds to one of the simple curves 1, 2, . . . 12. The numerical values of the several coefficients (96.5, 66.0, etc.) are the actual amplitudes, Ai, A 2, etc., of the component curves, expressed in milUmeters, for a wave length, ab, of 400 millimeters. If the curve is drawn to any other scale, the coefficients must be changed in the same proportion as is the wave length. The phases of the several components (76°, 319°, etc.) express the positions of the curves, length- wise, with respect to the initial Une, ai. The Une photographed as the axis often is not the mathe- matical axis of the curve. The true axis is found, as de- scribed on page 107, with the planimeter. The curve was traced with the planimeter, showing that the area of that part of the ciu:ve which Ues below the horizontal hne exceeds that above by 2704 square millimeters ; this quantity, divided by the wave length, 400, gives 6.75 millimeters as the distance of the true axis below the assumed axis. The true axis is the dotted line a'b' in Fig. 98 ; if the curve is 124 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES Fig. 98. An organ-pipe curve and its harmonic components. 125 THE SCIENCE OF MUSICAL SOUNDS traced with respect to this Une, the areas above and below the hne will be found equal. The distance of the true axis from the assumed line, - 6.75, is the numerical value of the coefficient A(, of the equation of "the curve. The analysis means that the original curve is equivalent to the simultaneous sum, or coniposite, of the several com- ponent curves. If the axes of the twelve components all coincided with the axis a'b', then the algebraic sum of the twelve ordinates of the curves at any point x (along the line yz) would be equal to the ordinate xy of the original curve. The ordinate of the first component for the point a; is + Xiyi ; for the second component it is + X2y2 ; for the third it is zero, x^ ; for the fourth it is - x^yi, etc. ; the sum is positive and equal to xy. For the point u, the sum of the ordinates of the components on the line uw ds zero, that is, the curve crosses the axis at this point. This graphic representation of the analysis of a curve is in accord- ance with the principles illustrated in the models of three waves shown in Lecture II, page 59. If the separate simple sounds from twelve tuning forks (or other source) produce motions in the air represented by the twelve component curves, then the composite tone of all would produce a composite motion represented by the original curve. The meaning of the phases (or epochs) of the several components may be further explained by reference to the figure. The starting point for tracing with the analyzer is arbitrarily selected ; it may be, for instance, the point a, Figs. 96 and 98, where the photographed curve crosses the photographed axis, which may or may not be the true axis. The phases obtained by analysis then give the rela- tions of the several component curves to the assumed initial line ai. In the example here shown, the phase of the first 126 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES component is 76° ; this means that where the curve crosses the initial Une, the first component has already progressed 76° (one wave length equals 360°) from its own zero point. The zero point for the first component is then /eV of the wave length to the left of the initial line, at c in Fig. 98. The phases of the other components have similar interpre- ^1 1 ^M I 1 ■ M ■ ^^RH m Ws^K^ H K ;^i Bl ^^^BBfcfe^.^ ffisA m^ffi H W i ^S PP Jl 1 i 1 Kfiifciii terifrim 1 1 ;-"\";"" -• ' , / ^^^^^^Kf4- \ I^^hI ^^v Jj^H ^^^^1 ^^^H^^^^^ - "'/'I^^Sij^^a^X ^'■■ii^f'^^,.f s 1 , 1^^ 1 1 Fig. 99. Proof of the analysis of a curve by synthesis. tations, each being measured in terms of its own wave length. It is sometimes desirable to consider the beginning of a curve as the point where the phase of the first component is zero ; the initial line would then be at the point d, and the wave length would be de. This point, in general, is not where the curve crosses the axis and there is no way of determining it in advance of analysis. The phases of the several higher components at the 127 THE SCIENCE OF MUSICAL SOUNDS point where that of the first component is zero are con- venient for the comparison of phases ; these are obtained by subtracting from each phase, as obtained by analysis, n times the phase of the first component. These relative phases are determined without calculation with the syn- thesizer as explained on page 114, and they are recorded on the card in the line labelled "Phase" as shown in Fig. 97. The verification of an analysis is made by synthesis. The equation of the curve is set up on the synthesizer, and the curve is drawn by machine, superposed on the enlarged drawing of the curve which was used with the analyzer. For illustration in this instance, the original photograph has been enlarged and the synthetic curve has actually been drawn by machine on the photograph. Fig. 99 is a reproduction of the original and synthetic curves. The likeness is sufficiently close for general purposes; a more exact reproduction would probably require the inclusion of a very large number of higher components, all of which have very small amplitudes. Fig. 100. Kelvin's tidal harmonic analyzer. Various Types of Harmonic Analyzers and Synthe- sizers The general methods of harmonic analysis and synthesis which have been described in detail in the preceding pages 128 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES are applicable to all kinds of investigations requiring such treatment. However, variations of the mechanical devices are sometimes desired to obtain special results. Brief men- tion will be made of several other types of instruments. Probably the first useful application of harmonic analysis was to tidal analyzing and predicting machines by Lord Kelvin, in 1876. The rise and fall of the tides, having been Fig. 101. Kelvin's tide predictor. observed at a given port for a year or more, is represented by a curve which is then analyzed. The tidal components being known, it is possible to synthesize these for future dates, that is, to predict the tides.^" Kelvin's analyzer is shown in Fig. 100, and the predictor in Fig. 101. A tide-predicting machine of remarkable completeness and perfection has recently been constructed by the United K 129 THE SCIENCE OF MUSICAL SOUNDS ANALYSIS AND SYNTHESIS OF HARMONIC CURVES States Coast and Geodetic Survey at Washington; Fig. 102 is a general view of this instrument/^ Professor A. A. Michelson has de- vised a very ingen- ious harmonic syn- thesizer and ana- lyzer, for eighty components, which he has applied most effectively to the study of light waves."^ A Michel- son synthesizer, of recent construction, for twenty compo- nents, is shown in Fig. 103; the ma- chine also serves as an analyzer. The given wave form is cut on the edge of a sheet of card or metal, which is then applied to set the machine ; a curve is drawn by means of which the ampli- tudes of the reqtiired components may be determined in a manner described in the references. Harmonic analyzers are employed in electrical engineer- 131 J ^B c i 3t 1 Fig. 103. Michelson's harmonic analyzer and syn- thesizer for twenty components. THE SCIENCE OF MUSICAL SOUNDS ing for the study of alternating-current waves and other periodic curves. The curves being investigated often have few components, or the interest is centered in a few com- ponents ; in such cases simpler forms of analyzers may be used in which the in- tegrating is performed by a planimeter of the ordinary type. Figures 104, 105, and 106 show instruments of this kind designed by Rowe, Mader, and Chubb, respectively.*' These machines may be used with a wave of any size, such as the original oscillogram ; the curve is traced with the stylus, giving one component ; by changing one or more gear wheels and again tracing, another component is found, and so on. Fig. 104. Rowe's harmonic analyzer. Fig. 105. Mader's harmonic analyzer. Several other types of harmonic analyzers are described in Horsburgh's "Instruments of Calculation" and in Morin's " Les Appariels d' Integration." *^ These books also describe many subsidiary instruments and processes which are helpful in numerical work. 132 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES Fig. 106. Chubb's harmonic analyzer. Arithmetical and Graphical Methods of Harmonic Analysis Mention has already been made of the application of harmonic analysis to the study of acoustics, the tides, elec- tricity, optics, and mathematics. The method is also use- ful in the investigation of other more or less periodic phe- nomena. In meteorology it is appMed to the study of hourly or daily temperature changes, barometric changes, etc.^^ In astronomy the periodicity of sun spots, magnetic storms, variable stars, etc., may be treated by harmonic analysis.^® In mechanical engineering, valve motions and other mechanical movements may be investigated.^' The method is also used in geophysics, in naval architecture, and in the study of statistics. ^^ When the number of curves to be analyzed is small and especially when the number of components is Umited, it may not seem necessary to provide a machine for perform- 133 thp: science of musical sounds ing the analyses. While the general solution of the prob- lem was given by Fourier in his original work " La Th^orie Analytique de la Chaleur " (Paris, 1822),** yet the labor in- volved in the numerical reduction is very great. Many arithmetical and graphical schemes for faciUtating the work have been developed by Wedmore, Chfford, Perry, Kintner, Steinmetz, Rosa, Runge, Grover, S. P. Thompson, and others. In general a set of coordinates of the curve is measured, and these measures are reduced in accordance with the scheme selected to give the amplitudes and phases of the components. Steinmetz gives general formulae systematically arranged for the calculation of any number of components, of odd or even order."' Several numerical examples are given, selected from electrical engineering, while another is the determination of the first seven components of a diagram of mean daily temperatures. Runge's method depends upon a scheme of grouping the terms so as to facilitate the numerical work.^" A number of ordinates of the curve, n, are measured. For odd com- ponents only, the ordinates are evenly distributed over a half wave and give ^ n components ; for odd and even components the ordinates are evenly distributed over the whole wave, and give ^n - I components. Runge gives schemes for 12, 18, and 36 ordinates. Bedell and Pierce give a scheme and an example for determining the odd com- ponents from 18 ordinates.^^ Carse and Urquhart give the scheme with numerical examples for odd and even com- ponents from 24 ordinates.** F. W. Grover gives six schedules according to Runge's method with examples for the calculation of the odd components from 6, 12, and 18 ordinates ; and also schedules for both odd and even com- 134 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES ponents from 6, 12, and 18 ordinates.''^ There is also given a special multiplication table for all the sine and cosine products required. When one of these schedules is appli- cable, the method as described by Grover is probably the most expeditious available for numerical analysis. H. O. Taylor has developed a convenient method for constructing complete schedules adapted to general or special condi- tions.^^ S. P. Thompson has provided several schedules also based on Runge's method, which are very expeditious. More recently he has developed an approximate method of har- monic analysis in which all multiplication by sines and cosines is dispensed with, and only a few additions and sub- tractions of the numerical values of the ordinates is re- quired.^' The method is applicable only to periodic curves in which the components higher than those being calculated are absent ; if higher components are present, their values may be added to those of the lower components in certain cases. Thompson gives schedules for the first three com- ponents, suitable for the analysis of valve motions, a schedule for the first seven components, and one for the odd components to the ninth, and a special schedule suit- able for tidal analysis. A large number of graphical methods for harmonic analysis have been devised. These are suitable for curves having only a few components, but it is doubtful whether they are any more expeditious than the equivalent arithmetical methods, and usually they are not so precise. A convenient graphical method is that devised by Perry.^* Numerous other methods are described in "Modern Instruments of Calculation" and in the volumes of the Electrician.^^ For comparison the curve which was analyzed by machine 135 THE SCIENCE OF MUSICAL SOUNDS in 13 minutes, as described on page 122, was analyzed by Steinmetz's method, requiring about 10 hours to obtain ten components, and by Grover's method the time was about 3 hours for eight components odd and even (the largest number for which a scheme is arranged). (Grover mentions that by his method eight odd components can be determined in less than an hour by one familiar with the process.) The curve shown in Fig. 76, page 100, which was known to have but three components, was analyzed by Thompson's short method, the three ampUtudes and phases being evaluated in fifteen minutes. The analysis of the same curve (for five components) by machine required less than seven minutes. Analysis by Inspection A famiUarity with the effects produced by various har- monics on the shape of a wave will often enable one to judge by inspection what harmonics are present. Fig. 107. Symmetrical wave form of an electric alternating current. If a wave consists of alternate half-waves which are exactly of the same shape but opposite in direction, that is, if the wave is' a symmetrical one with respect to its axis, it can contain only odd-numbered components ; if 136 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES a wave is not symmetrical, it must contain some even- numbered components, and it may contain both odd and even. Sound waves belong to the latter class, no in- stance having been observed of a symmetrical sound wave, except that of a tuning fork which has only one component and is a simple sine curve. Electric alter- nating-current waves are usually of the first kind, con- taining only odd-numbered components ; such a wave is shown in Fig. 107. In some instances a particular high partial may be promi- nent and so impress its effect on the wave as to produce Fig. 108. Photograph of the vowel a in father, intoned upon the pitch n = 159. distinct wavelets or ripples on the main wave form even though this is itself irregular ; the order of such a partial is at once determined by merely counting the number of such wavelets occurring in one fundamental wave length. Fig. 108 shows a wave for the vowel a in father ; this curve is evidently complex, but there are six distinct sub-peaks on one wave, and the sixth partial is prominent. Since the frequency of the fundamental is known to be 159, that of the sixth partial is 954. Analysis shows that the first ten components of this curve have the following amplitudes, corresponding to a wave length of 400 : 137 THE SCIENCE OF MUSICAL SOUNDS 1=4 II = 15 III = 18 IV = 12 V =20 I =60 VII = 21 VIII = 16 IX = 2 X = 3 An instance where analysis by inspection is sufficient is given in Fig. 136, on page 187, which shows the wave from a tuning fork having but one overtone. The partial is prominent and produces very definite sub-peaks; it is found that there are about twenty-five wavelets to four large waves, that is, the partial has a frequency about 6.25 times that of the fundamental and is inharmonic. •■■■'"^ ■■■■■r^ ■■• * ■. i ■ y Fig. 109. Clarinet wave showing beats produced by the higher partials. In some instances the peaks due to the partials are very pronounced in portions of a wave and almost disappear in other parts ; this indicates that there are beats between certain partials. If there is one beat per wave length, it is produced by two adjacent partials ; if there are two beats, then the orders of the partials differ by two. The average distance between sub-peaks is found by measurement and, when compared with the wave length, gives the average order of the prominent partials, from which it is then usually possible to specify their exact orders. While this method is most useful for waves having a few components, such as alternating-current waves, it may be applied to a 1.38 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES wave as complex as that of the tone of a clarinet, Fig. 109. Since there is one beat per wave, there are two prominent adjacent overtones. Actual measurement of these wavelets shows an average length of 3| millimeters ; the fundamental measures 38 millimeters, about 11| times the length of the sub- wave ; the conclusion is that the eleventh and twelfth partials are producing the beat. The correctness of this conclusion is proved by the actual analysis which gives the following values for the first twelve components of the curve, corresponding to a wave length of 400 : I = 29 II = 7 III = 20 ir = 1 V = 2 VI = 6 VII = 6 VIII = 8 IX = 16 X = 9 XI = 30 XII = 35 A photograph of the sound of the explosion of a sky- rocket in a Fourth of July celebration is shown in Fig. 110. Mh\ Fig. 110. Photograph of the sound of the explosion of a skyrocket. The rocket was about a quarter of a mile from the recording apparatus, the sound entering the laboratory through two open windows. While this curve as a whole is not periodic, yet two or more periodicities are clearly shown. The time signals are j^-^ second apart, and comparison shows one frequency of about 130 per second producing the principal feature of the curve ; superposed upon this is a much higher frequency of about 2000 per second. 139 THE SCIENCE OF MUSICAL SOUNDS Periodic and Non-periodic Curves The Fourier analysis is suitable and complete for any curve whatever within the distance called one wave length, even though there is no repetition of this form ; if the por- tion analyzed is successively and exactly repeated, that is, if the curve is periodic, it represents a wave motion and the analysis represents the entire wave. A periodic wave is shown in Fig. HI, which is a photograph of the vowel a in mat. If a curve representing some physical phenomenon is periodic, then each separate term of the Fourier equation !fi!'Wk¥W¥^¥^ Fig. 111. A periodic curve ; a photograph of the vowel a in mat. of the curve may be presumed to correspond to something which has a physical existence ; it is the behef in this- state- ment, amply supported by investigation, which leads one to analyze sound waves by this method ; as explained under Ohm's Law, page 62, each term is presumed to cor- respond to a simple partial tone which actually exists. If the curve representing the physical phenomenon is non-periodic, any portion of the curve may be analyzed, and it will be completely represented as to form by the Fourier equation, within the limits analyzed, but not beyond these limits. In this case, the separate terms of the Fourier series may not correspond to anything having a separate physical existence ; in fact the equation may be presumed 140 ANALYSIS AND SYNTHESIS OF HARMONIC CURVES to be merely an artificial mathematical formula for the short irregular line which has been analyzed. The analysis of the profile portrait, described on page 119, illustrates this application of Fourier analysis ; there is no periodicity of the wave form, and the separate terms of the equation can have no real significance. There is no general method for analyzing non-periodic curves, that is, for curves containing incommensurable (in- harmonic) or variable components ; such a method is very much desired for the study of noises and of sounds from such sources as bells, whispered words, the consonant \Af v mm Fig. 112. A non-periodic curve ; a photograph of the sound from a bell. sounds of speech, and in fact all speech sounds except the simple vowels ; this need is probably the greatest of those unprovided for.^*'^* A non-periodic curve, a photograph of the sound from a bell, is shown in Fig. 112 ; there is no apparent wave length in this curve, and an analysis of any portion of it would probably give an equation containing an infinite number of terms, though the real sound is un- doubtedly compounded from a finite number of partials which are inharmonic and therefore indeterminate. Much information may be obtained from such curves by making skillfully assumed analyses. Ul LECTURE V INFLUENCE OF HORN AND DIAPHRAGM ON SOUND WAVES, CORRECTING AND INTERPRETING SOUND ANALYSES Errors in Sound Records The photographs and analyses of sound waves obtained by the complicated mechanical and numerical processes described in Lectures III and IV, are, unfortunately, not yet in suitable form for determining the tone characteristics of the sounds which they represent. Before these analyses can furnish accurate information they must be corrected for the effects of the horn and the diaphragm of the recording instrument, a correction involving fully as much labor as was expended on the original work of photography and analysis. For the sake of greater emphasis, it may be di- rectly stated that the neglect of the corrections for horn a-nd diaphragm often leads to wholly false conclusions regard- ing the characteristics of sounds, since horns and diaphragms of different types give widely differing curves for precisely the same sound. For research upon complex sound waves, a recording instrument using a diaphragm should possess the following characteristics : (a) the diaphragm as actually mounted should respond to all the frequencies of tone being investi- gated ; (&) it should respond to any combination of simple frequencies ; (c) it should not introduce any fictitious fre- quencies; (d) the recording attachment should faithfully 142 ERRORS IN SOUND RECORDS transmit the movements of the diaphragm; and (e) there must be a determinate, though not necessarily simple, relation between the response to a sound of any pitch and the loudness of that sound. It is well known that the response of a diaphragm to waves of various frequencies is not proportional to the ampUtude of the wave ; the diaphragm has its own natural periods of vibration, and its response to impressed waves of frequencies near its own is exaggerated in degrees de- pending upon the damping. The resonating horn also greatly modifies waves passing through it. Therefore, it follows that the resultant motion of the diaphragm is quite different from that of the original sound wave in the open air. The theory of these disturbances for simple cases is complete, but what actually happens in a given practical apparatus is made indeterminate by conditions which are comphcated and frequently unknown. There being no available solution of this problem, it was necessary to make an experimental study of these effects as they occur in the phonodeik.*^ It has been proved that the phonodeik possesses several of the characteristics mentioned ; (a) it has been shown by actual trial that it responds to all frequencies to 12,400 ; (6) various combinations of simple tones up to ten in num- ber have been actually produced with tuning forks, and the photographic records have been analyzed; (c) in each case the analysis shows the presence of all tones used, and no others. We have then only to determine the accuracy with which the response represents the original tones, the quaUfications (d) and (e) mentioned above; this requires the investigation of all the factors of resonance, interfer- ence, and damping. 143 THE SCIENCE OF MUSICAL SOUNDS Ideal Response to Sound In Lecture II it has been explained that the intensity or loudness of a simple sound is proportional to the square of the amplitude multiplied by the square of the frequency, that is, to {nAy- A recording apparatus having ideal response must fulfill the following condition : let all the ~ C2 D EF G A BC3 D EF C A BC, D EF C A BC5 D EF C A Bq D EF G A BC, 12 . loo n Fig. 113. Curve showing the amphtudes for a sound of varying pitch but of constant loudness. tones of the musical scale (simple tones) from the lowest to the highest, and all exactly of the same loudness, be sounded one after the other and be separately recorded ; let the amplitudes of the various responses be measured, and each amplitude be multiplied by the frequency of the tone pro- ducing it ; then the squares of the products of amplitude and frequency must be constant throughout the entire series. A curve of ideal amplitudes is given in Fig. 113, the verti- cal scale of which is one of linear measure, centimeters for 144 ERRORS IN SOUND RECORDS instance, and the horizontal scale is a logarithmic scale of frequencies. The divisions represented by the light verti- cal lines correspond to the successive tones of the musical scale, as is explained later in this lecture. The properties of this curve are as follows : if a simple sound having the pitch Cs = 259 produces a record which has an amplitude represented by the ordinate a of the curve, then a sound of exactly the same loudness, but one octave lower in pitch, is correctly represented by a record the amplitude of which is the ordinate h; further, the sound A4, having 870 vibra- tions per second, and of the same loudness as either of the others, should produce an amplitude measured by the much shorter ordinate, c ; and similarly for any note of the scale. Actual Response to Sound The determination of the actual response of a recording apparatus requires a set of standards of tone intensity for the entire scale of frequencies under investigation. The practical fulfillment of this requireijaent for a time seemed an impossibility. A manufacturer of organ pipes who became interested in the problem provided two complete sets of pipes, an open diapason of metal and a stopped dia- pason of wood, especially voiced and regulated to uniform loudness throughout, according to his skilled judgment. The stopped diapason pipes, Fig. 114, sixty-one in number, range in pitch from C2 = 129 to C? = 4138 ; the scale is extended by nineteen specially voiced metal pipes to a pitch 12,400. The adjustment of these pipes for uniform loud- ness has been improved and verified by two experimental methods. While this scale is arbitrary and of moderate precision, it is the only available method by which progress has been possible, and its use has led to most interesting L 145 THE SCIENCE OF MUSICAL SOUNDS results in studying the responses and correction factors of the phonodeik under various conditions. The sounds from the several pipes of the two sets have been photographed and analyzed. The analyses show that the open diapason pipes have a strong octave accompanying the fundamental, while the stopped pipes give practically simple tones ; the latter are used exclusively in obtaining the correction factors as explained later. These pipes are sounded in front of the phonodeik, one Fig. 114. A set of organ pipes of uniform loudness. at a time, and the resulting amphtudes of vibration of the diaphragm are recorded photographically. The film is stationary ; the first pipe, Cj, is sounded steadily ; the shutter is released, giving an exposure of about ^-^ second ; the spot of light which is vibrating back and forth in a straight line falls on the film, making several excursions within the time of exposure, and records the amplitude of the vibration ; the record for the first pipe is C in Fig. 115. The film is moved lengthwise about a quarter of an inch, and while the second pipe, Ciij[, is sounding, the result- ing amplitude is photographed. The process is continued 146 ERRORS IN SOUND RECORDS until the amplitudes produced by the sixty-one pipes are recorded. The vertical scale of such a chart represents linear amplitude, while the horizontal scale is a logarithmic scale of frequencies which is described on page 168. A curve may be drawn through the upper ends of these amplitude records, showing the " responsivity " of the apparatus under the conditions of the experiment. Fig. 115 shows the responses used in correcting the analysis of the organ-pipe curve shown on page 122, while the inter- pretation of the responses is given on page 163. Fig. 115. Photographic record of the ampUtudes of vibration for the organ pipes of uniform loudness. The irregular curve of Fig. 116 is the response obtained with one of the earhest forms of phonodeik ; it shows an almost startling departure from the ideal response repre- sented by the smooth curve. What produces the range of mountains, with sharp peaks and valleys ? Why is there no response for the frequency 1460 ; why is it excessive for frequencies from 2000 to 3000? There were five suspected causes : (1) unequal loudness of the pipes ; (2) the dia- phragm effects ; (3) the mounting and housing of the dia- phragm; (4) the vibrator attached to the diaphragm; (5) the horn. 147 THE SCIENCE OF MUSICAL SOUNDS The investigation of these peculiarities was most tantaliz- ing ; the peaks acted like imps, jumping about from place to place with every attempt to catch them, and chasing and pushing one another in a very exasperating manner. Perhaps two months' continuous search was required to find the causes of "1460" and "2190" alone. The investi- gations led to many improvements in the phonodeik and to Cj D EF G A BC3 D EF G A BC, D EF C A BC5 D EF C A BC^ D EP G A BC, i)gi;i^i=i=i.iiisis|g^i;isiiiiiii8iifefei;i)siiisiii8^^ 128 ».=ri=s_ 259 517 1035 20e9 4138 Fig. 116. A response curve obtained with an early form of phonodeik. a practical method of correction for the departures from' ideal response. Response of the Diaphragm ' Experiments have been made with diaphragms of several sizes and thicknesses, and of various materials, such as iron, copper, glass, mica, paper, andalbumen. The experi- ments described in this section concern circular glass dia- phragms having a t-hickness of 0.08 miUimeter, and held around the circumference, either firmly clamped between hard cardboard gaskets and steel rings, or loosely clamped 148 INFLUENCE OF DIAPHRAGM ON SOUND RECORDS between soft rubber gaskets ; the diaphragm is entirely free, there being no horn or housing of any kind. The silk fiber of the phonodeik vibrator is attached to the diaphragm a ill D,= :4i( EF G A ■ I I.I. I, BC, D -Mr ■ EF G A BC. D EP EF G A BC. B EP G A BC, I .1 .1 1°. I .1 I .1.1 . I r mmmn mmtmmm mmmmmn iiiiig|j|jiiij|j|jiiiji ig|giiiiiiiiiii§iiiiiii 259 517 1035 2069 Fig. 117. Resonance peaks for diaphragms of different diameters. 4138 to record its movements. The pipes already described were sounded in succession in front of the diaphragm, and observationsXwere made of the response under various 149 THE SCIENCE OF MUSICAL SOUNDS conditions. The responses of three glass diaphragms of 22, 31, and 50 milUmeters diameter, respectively, are shown in a, b, and c. Fig. 117. When the pitch of the pipe being sounded is near the natural frequency of the diaphragm, the latter moves easily and responds vigorously ; the diaphragm is in sym- 176 iM C, D EF G A BCj D EF G A BC, D EF A BC, D EF G A BO. D EF G A EC 128 > S58 517 1035 2069 4138 Fig. 118. Effects of clamping and distance on the response of a diaphragm. pathy or in resonance with the tone ; the response curve shows a peak for such a resonance condition. The natural period of the largest diaphragm had a frequency of 366, corresponding to which there was a large response, as shown in the lower curve of the figure. Two other peaks repre- sent the natural overtones of the diaphragm; these over- tones have frequencies 3.28 and 6.72 times that of the funda- mental, and are inharmonic. The other curves show that 150 INFLUENCE OF DIAPHRAGM ON SOUND RECORDS the natural period of the diaphragm rises in pitch as the diameter decreases, and that the actual response becomes less. The lower curve, c. Fig. 118, is the response of a glass diaphragm 31 millimeters in diameter, held lightly in the clamping rings. When the clamping is tightened, the response is as shown in the middle curve ; the natural period is increased from 640 to 916, while the amplitude is reduced from 228 to 160. The upper curve shows the response for the same diaphragm as for the middle curve, but with the pipes at a greater distance ; the curve is of the same general shape, while the response is of diminished amplitude. Chladni's Sand Figures Chladni's method of sand figures has been employed in studying the conditions of vibration of the diaphragm." A plate or diaphragm, clamped at the edge or at an interior point, may be made to vibrate in many different modes. When sand is strewn on the plate it is observed that por- tions are moving up and down, throwing the sand into the air. There are certain Unes toward which the sand gathers, indicating that these parts are relatively at rest ; the lines on which the sand accumulates are called nodal lines, and form patterns or figures which are always the same for the same note, but differ, for each change of pitch or quality. It is thus shown that a diaphragm vibrates in various subdivisions. A diaphragm of glass held in circular rings was placed horizontally, the vibrator being attached to. the under side ; sand was then sprinkled over the diaphragm which was made to respond in succession to each one of the eighty pipes of frequencies from 129 to 12,400. The characteristic nodal 151 THE SCIENCE OF MUSICAI. SOUNDS lines produced in each instance were either sketched or photographed. When the test sound has a pitch equal to the natural frequency of the diaphragm, 366, the diaphragm vibrates Fig. 119. Different modes of vibration of a diaphragm, shown by sand figures. as a whole vigorously ; there are no nodal lines, except the circumference, and even this is probably in motion when the clamping is Ught. There are no nodal Unes for tones -within the octave lower and the octave higher than the natural frequency. 152 INFLUENCE OF DIAPHRAGM ON SOUND RECORDS As the pitch of the test sound rises, the area of the plate which can respond seems to be less than the whole, and this part moves, with the formation of a nodal boundary line of more or less irregular shape ; Fig. 119, a, is the photograph of the pattern for the frequency 977. Since parts of the plate are now at rest, no part can vibrate through a large amplitude, and the response is greatly diminished, as shown in the response curve. Fig. 117. A second maximum re- sponse is obtained from a sound corresponding to the first overtone of the diaphragm having a frequency of 1200, about 3.28 times that of the fundamental; this is represented by the second, smaller peak of the curve. The nodal figure on the diaphragm is a circle of medium size. As the pitch of the test sound rises, the figures again become irregular and of smaller area, and two concentric nodal circles appear, b, Fig. 119, corresponding to the second overtone, of a frequency of 2460, 6.72 times that of the fundamental. As the pitch rises stiU higher, the areas become smaller, with the formation of three, four, and five concentric circles, and other designs. The photographs c and d show the nodal lines for frequencies of 2600 and 10,400. Free Pebiods of the Diaphragm Besides the two methods already described, one by direct measure of the response, the second by means of the Chladni sand figures, a third method of determining the diaphragm characteristics has been used, that of photographing the free-period effects. ' The diaphragm is given a single dis- placement, and upon release is allowed to vibrate freely. This displacement may be produced by the noise of a hand- clap, or by attaching a fine thread to the diaphragm which 153 THE SCIENCE OF MUSICAL SOUNDS is gently pulled aside and the thread then burned. The frequency is given by comparison with the time signals. The curve a, Fig. 120, was obtained with a free, uncov- ered diaphragm ; two distinct frequencies are shown at the beginning of the motion, 1000 and 3100 ; the latter persists foi" ^irr second, while the former lasts about -^^ second. Curve b was obtained when the diaphragm was inclosed in a housing forming a front and back cavity, but with no horn; the air cushions damp the vibrations, there being Fig. 120. Free-period vibrations of a diaphragm. nine vibrations now, while before there were twenty-two ; the frequency of the diaphragm has been reduced to 400, and there is a higher frequency of 2190 due probably to the natural period of the air in the chambers. When a horn is added, the curve c is obtained ; the frequency of vibration of the air in the horn is 264, and it continues to vibrate for about a tenth of a second ; the frequency of the diaphragm is now 530, and that of the back cavity is 2190 ; these vi- brations die out in about xl^ second, as before. 154 INFLUENCE OF DIAPHRAGM ON SOUND RECORDS Influence op the Mounting of the Diaphragm In the early experiments it was thought desirable, in order to protect the diaphragm from indirect sounds, to inclose it in a housing; various shapes and sizes of front and back coverings, shown in Fig. 121, were tried. The diaphragm is in effect between two cavities, and it was found that each produces its own complete and independent resonance effects, and that these influence each other through the diaphragm. When the frequencies of these cavities are in certain ratios, the re- sponse of the diaphragm is annulled by in- terference ef- fects ; at other times these cavities pro- duce exagger- ated responses. Experiment indicated that the back should be uncovered, since the effect of a sound produced in front of the horn is ordinarily of no influence on the back of the diaphragm. The front must, of course, be covered and the connections between the cover and the horn, and the cover and the dia- phragm, must be air-tight. The best results were obtained by using a shallow cup-shaped front cover with an opening for the horn, which may have a diameter about one fourth of that of the diaphragm. If the front cover is close to the diaphragm, the damping effect of the air cushion may be too great. 155 Fig. 121. Shapes of front and back coverings for a diaphragm. THE SCIENCE OF MUSICAL SOUNDS Influence of the Vibbator Elaborate studies have been made of the influence of the vibrator on the response of the apparatus; among the factors investigated are the mass and shape of the vibra- tor, size and shape of the mirror, material and length of the fiber, material and length of the tension piece and its hysteresis and damping effects, amount of tension, moduli of elasticity, and temperature ; computations also have been made of the inertias, accelerations, forces, and natural periods, of the various parts, and their resonances and inter- ferences for frequencies up to 10,000 ; the differential equa- tions of motion of the actual system have been formed and solved. It is quite out of place to explain the details of this work here ; the final practical result is the demonstration that for frequencies less than 5000 the vibrator produces no appreciable effect on the record. This conclusion is verified by the results of a further study with the Chladni sand figures ; besides the set of figures described previously, with the vibrator attached to the diaphragm, a second complete set of figures was ob- tained without the vibrator; the two sets are practically identical except for a shifting of the nodal fines for high frequencies. Influence of the Horn A horn as used with instrimients for recording and repro- ducing sound is usually a conical or pyramidal tube, the smaller end of which is attached to the soundbox contain- ing the diaphragm, while the larger end opens to the free air. The effect of the horn is to reinforce the vibrations which enter it due to the resonance properties of the body 156 INFLUENCE OF THE HORN ON SOUND RECORDS of air inclosed by the horn. The quantity and quality of resonance depends mainly upon the volume of the inclosed air and somewhat upon its shape. If the walls of the horn are smooth and rigid, they produce no appreciable effect upon the tone. But if the walls are rough or flexible, they may absorb or rapidly dissipate the energy of vibrations of Fig. 122. Experimental horns of various materials, sizes, and shapes. the air of certain frequencies and thus by subtraction have an influence upon tone quality. The horn of itself cannot originate any component tone, and hence cannot add any- thing to the composition of the sound. The horn is an air resonator and not a soundboard ; any vibrations which the walls of the horn may have are relatively feeble and are re- ceived from the air which is already in vibration, while in the case of a soundboard, the air receives its vibration from 157 TJIE SCIENCE OP MUSICAL SOUNDS the soundboard as a source. Because the horn operates through the inclosed air, it is a very sensitive resonator, and hence its usefulness when its action is understood and properly appHed. A horn used in connection with the diaphragm very greatly increases the response, but it also adds its own nat- ural-period effects, which are quite complex. A variety of horns, shown in Fig. 122, were used in the experiments ; these are of various materials, sheet zinc, copper, thick and thin wood, and artificial stone ; one horn was made with double walls of thin metal, and the space between was filled with water. Probably the most rigid material, such as stone or thick metal, gives the best results. For con- venience, however, sheet zinc is used with the phonodeik, and so long as the horn is supported under constant conditions, which are involved in the "correction curve", described later, this material is satisfactory. A horn such as is used in these experiments has its own natural tones, which can be brought out by blowing with a mouthpiece as in a bugle ; these tones are a fundamental with its complete series of overtones. The fundamental pitch can be heard by tapping the small opening with the palm of the hand. When a horn is added to the diaphragm, the response is greatly altered; Fig. 123' shows the response curves for three horns of different lengths. In each curve the peaks corresponding to the fundamental of the horn, the octave, and the other overtones up to the seventh, are distinct. These peaks are indicated in the figure by Hi, H2, etc., while the diaphragm peak is marked D. In the upper curve, the peak due to the diaphragm comes between the peaks for the fundamental and the octave of 158 INFLUENCE OF THE HORN ON SOUND RECORDS the horn ; the latter peaks are pushed apart, one being lowered in pitch and the other raised, so that the interval between them is two semitones more than an octave ; the peak for the third partial is in its proper position. The horn reacts upon the diaphragm, causing it to have a period different from that which it had before the horn was applied. 129 259 517 1035 2099 4138 Fig. 123. Resonance peaks for horns of various lengths. The middle curve shows the response with a longer horn ; the diaphragm peak now comes between the peaks of the second and third partials, and both these and that of the fourth are displaced. The lower curve represents the re- sponse for a still longer horn. A long horn seems to respond nearly as well to high tones as does a short one, while the response to low tones is much greater; the response below the fundamental of the horn is very feeble. The horn selected should be of such a 159 THE SCIENCE OF MUSICAL SOUNDS length that its fundamental is lower than the lowest tone under investigation. For the study of vowel sounds, the horn employed has a length of about 48 inches, giving a fundamental frequency of about 125. It is important that there are no holes, open joints, or leaks of any kind in the walls of the horn, because tones with a node in the position of a hole will be absent. A hole 129 259 517 1085 2069 4138 Fig. 124. Resonance peaks for horns of various flares. one milhmeter in diameter is sufficient to alter the response. The flare of the horn has a great influence upon the re- sponse ; Fig. 124 shows the responses obtained with three horns of the same length, but of different flares. The upper curve is the response for a narrow conical horn, the large end of which has a diameter equal to one fifth of the length ; the middle curve is for a wide cone, the diameter of the open end being one half of the length ; the lower curve 160 INFLUENCE OF THE HORN ON SOUND RECORDS is for a horn of flaring, bell shape. Widening the mouth increases the effect in a general way; the bell flare makes the natural periods indefinite, and heaps up the response near the fundamental, diminishing that for the higher tones. The shape selected for use with the phonodeik is a cone 258 517 1035 Fig. 125. Resonance effect of the horn. of medium flare ; this gives a good distribution of response, and the resonances are definite, but not too sharp to allow of correction. The curve a, Fig. 125, is an actual response curve con- taining both horn and diaphragm effects, b is the diaphragm response, while c and d are the" ideal curves previously- explained. The effect of the natural period of the diaphragm M 161 THE SCIENCE OF MUSICAL SOUNDS is represented by the sharp peak D. If the diaphragm responded ideally, the curve 6 would coincide with c through- out its length, and a would then indicate the effects due to the horn alone. The ordinates of the curve a have been multiplied by such factors as will reduce the diaphragm curve to the ideal, and the results are plotted in e. The difference between the curve e and the ideal d is the effect due to the horn, corrected for the peculiarities of the dia- phragm, in the manner explained later in this lecture. The line / indicates the location of the natural harmonic overtones of the horn ; the curve e shows by its peaks that the horn strongly reinforces the tones near its own funda- mental, and, in a diminishing degree, those near all of its harmonics. The resonance of the horn increases the effects of all tones corresponding to the complete series of har- monics which the horn itself would give if it were blown as a bugle or hunting horn. The diaphragm peak D comes between the peaks for the third and fourth partials of the horn, and it in effect divides the partials into two groups which are pushed apart in pitch; the amount of this displacement is shown by the gap in the Une / near the fourth point. COERECTING ANALYSES OF SoUND WaVES The investigation of the effects of the horn and diaphragm of a sound-recording apparatus, a few details of which have been described, involved an unexpected amount of labor ; it is estimated that the time required was equivalent to that of one investigator working eight hours for every working day in three and a half years. It has been shown that the horn and diaphragm introduce many distortions into the curves obtained with their aid, and that the dis- 162 CORRECTING ANALYSES OF SOUND WAVES tortions vary greatly with the conditions of the instrument. Unless these errors are eUminated and the true curves found, the records of the instrument will be without value because the incorrect and false curves can lead to no rational conclusions whatever. One may wonder, if the horn pro- duces such disturbances, why it is not dispensed with in scientific research ; the horn has been retained because the 128 _=:=■:=. 258 517 1035 2069 413B Fig. 126. Curves used in correcting analyses of sound waves. sensitiveness of the recording apparatus is increased several thousand-fold by its use. Whenever records are made for analytical purposes, the condition of the phonodeik is usually determined, as pre- viously explained, by photographing its response to each of a set of sixty-one organ pipes of standard intensity, covering a range of frequencies from 129 to 4138. The actual response of the phonodeik in the form for research is shown by the irregular curve a, Fig. 126, while h is the 163 THE SCIENCE OF MUSICAL SOUNDS desired or ideal response. The shape of the curve will vary with every change in the size or condition of the horn or diaphragm, with temperature changes, with the tight- ness of clamping of the diaphragm, with the nature of the room in which the apparatus is used, and with other condi- tions. The sharp resonance peak D is due to the free period of the diaphragm. The ordinates of this curve give the amplitudes of the phonodeik records for sounds of various pitches, aU being of the same intensity. The ciurve shows that the tone of any particular pitch, whether a single simple tone or a single component of a complex sound, in general produces a response either too large or too small as compared with the response due to other pitches. When any sound has been photographed with the phonodeik in the conditions here represented, and the amplitudes of all the components of the complex tone have been determined by analysis, it is necessary to correct each individual amplitude by multi- plying it by a factor corresponding to its particular pitch. The factor for a tone of any pitch is the number by which the ordinate of the actual response curve for the given pitch must be multipUed to give the ordinate of the ideal curve. These correction factors are obtained from a correction curve, c. Fig. 126, determined in the following manner. The sixty-one ordinates of the ideal curve b corresponding to semitones of the scale are divided by the corresponding ordinates of the response curve a; the quotients are the correction factors for these particular pitches. These fac- tors are then plotted on the chart, and a correction curve is drawn through the points as shown at c. The correction curve is an inverse of the response curve, where one has a peak the other has a corresponding valley. 164 CORRECTING ANALYSES OF SOUND WAVES It is convenient to make a correction curve showing the factors for all pitches, since a number of photographs of sounds are often made with the phonodeik in the same con- dition, and the analyses of all are corrected from the same response curve. " The analysis of a curve having been recorded on a card, as explained on page 123, the correction factors, k, are measured from the chart of the correction curve, and are No. lead CASE SCHOOL OF APPUKD SCIENCE DEPARTMENT OF PHY ^ANALYSIS OF SOUND-WAVES Source ©,.QO.Y\T lis P.— '•9.eeA\is.i Ov,oe" SICS Dateat>»i\ 2.3, I-IO Tone ' c , ' Abs. N Z 6 '^ISK component, n 1 2 3 4 5 6 7 8 9 10 na„ + 2Z£ + 39.4 + I0(./ + 7C.I + 4-».i- -49.1 + 44.a + ?19 - //.i- - 7,1 nb. -f M.| -87,2 - 12.5 - 75- -26.1 - /o,fc - 3.6 -66.7 -36.9 - 21.7 nA. 94.4 m.o /09.1 7i.6 S/.C, S0.3 4i:o 7 /.I 3B.7 22.9 k.,(l70s) o.b 0.7 IS Z.l 0,8 l.o 0.6 0.4 OS o.« nA.k. S7.9 9Z.^ lil.l ICI.3 4 5 6 7 8 9 to 15 Fig. 130. Distribution of energy in sounds from various sources. of energy from the source. The usefulness of such curves in connection with the study of vowel tones is more fully explained on pages 220 and 228. These diagrams and curves showing the distribution of the energy in a sound, are not unlike the spectrum charts 171 THE SCIENCE OF MUSICAL SOUNDS and emission curves obtained in the study of light soiu-ces. Corresponding to a monochromatic light we have what may be called the "mono-pitched" sounds of the tuning fork. This soui;id is simple, containing but a single com- ponent, and its diagram consists of one "strong Une, " as shown in Fig. 130. Other sources emit complex sounds, the energy being variously distributed among the several partials, as shown, for particular instances, in the figure. The horn gives the most uniformly distributed emission of sound energy, and its tone may be said to correspond acous- tically to white hght. The characteristics of instrumental and vocal tones are more fully discussed in the succeeding lectures. Verification of the Method of Correction As a test of the sufficiency of the method which has been developed for correcting analyses, one hundred and thirty photographs of tones from nine different instruments were made with four distinctly different combinations of horn and diaphragm, giving for each tone four sets of curves which are wholly unUke. A long horn was used with a large and a small diaphragm, and also a short horn with each diaphragm ; the responses were such that the peaks in one instance corresponded in pitch with the valleys in another. Response and correction curves were made for each combi- nation. After correction the various analyses of any one tone were identical. Fig. 131 shows the photographs of the tone from an organ pipe made with three horn-and-diaphragm combinations ; these would hardly be taken for records of the same sound. After analysis, the components for each curve were cor- rected for horn and diaphragm effects and then recom- pounded with the synthesizer, the three corrected curves 172 CORRECTING ANALYSES OF SOUND WAVES Fig. 131. Three curves for the same tone, made under different conditions. being shown in Fig. 132. These curves show how success- ful the method is in reducing unUke curves for the same sound to practical identity. Fig. 132. The three curves of Fig. 131 corrected for instrumental effects. 173 THE SCIENCE OF MUSICAL SOUNDS Quantitative Analysis of Tone Quality In Lectures III, IV, and V there has been described a quantitative method for the analytical study of tone quaUty. The method includes the arrangement of the working appara- tus, and schemes for computing, reducing, presenting, and filing the results. The results so obtained are expressed in terms of the relative loudness of the various partial tones of an instrument. A determination of the relative loud- ness of the sounds of one instrument as compared with another is no doubt of much interest, but this is not included in the present discussion. In the definitive study of a musical instrument or voice it is desirable that a large number of tones be photographed, perhaps four per octave, three semitones apart, through- out the whole compass ; the tones should be sounded in three different intensities as p, mf, and /, or five intensities may be studied, by adding p-p and ff; to eliminate errors, two or more combinations of horn and diaphragm may be Used; the source of sound may be placed at various dis- tances from the horn ; response and correction curves must be taken before and after each change in the recording apparatus ; such a study of one instrument may require a year's time for its completion. Any scheme less com- prehensive than this will not give an adequate idea of the tone quahty of a musical instrument. 174 LECTURE VI TONE QUALITIES OF MUSICAL INSTRUMENTS Generators and Resonators Previous lectures have demonstrated that, in general, the sounds from musical instruments are composite, that is, all those which can be said to have characteristic quality are made up of a larger or smaller number of partial tones of various degrees of loudness. A scientific definition of the quality of a musical tone requires a statement of what particular partial tones enter into its composition and of the intensities and phase relations of these partials. In order to understand a musical instrument, we need to know how its tones are generated and controlled by the performer. The sound producing parts of a musical instrument, in general, perform two distinct functions. Certain parts are designed for the production of musical vibrations. The vibrations in their original form may be almost inaudible, though vigorous, because they do not set up waves in the air, as is illustrated by the vibrations of the string of a viohn without the body of the instrument ; or the vibra- tions may produce a very undesirable tone quality because they are not properly controlled, as in the case of the reed of a clarinet without the body tube. Other parts of the instrument receive these vibrations, and by operation on a 175 THE SCIENCE OF MUSICAL SOUNDS larger quantity of air and by selective control, cause the instrument to send out into the air the sounds which we ordinarily hear. These parts, which may be referred to as generator and resonator, are illustrated by the following combinations : a tuning-fork generator and its box resona- tor ; the strings and soundboard of a piano ; the reed and body tube of a clarinet ; the mouth and body tube of an organ pipe ; the vocal cords and mouth cavities of the voice. In the piano the soundboard acts as a universal resonator for all the tones emitted by the instrument ; in the organ each pipe constitutes its own separate resonator; in the flute the body tube is adjusted to various different conditions by means of holes and keys, each condition serv- ing for several tones. The resonator cannot give out any tones except those received from the generator, and it may not give out aU of these. The generator must therefore be capable of pro- ducing the components which we wish to hear, and these must in turn be emitted in the desired proportion by the resonator. If the generator produces partial tones which are undesirable, the resonator should be designed so that it will not reproduce them ; if the generator produces tones which are of musical value but which the resonator does not reproduce, we do not hear them, and it is as though they were not produced at all. It follows that we can hear from a given instrument nothing except what is produced by the generator, and further we can hear nothing except what is also reproduced by the resonator ; hence it may be that the most important part of an instrument is its resonator. The quality of any tone depends largely upon the kind and degree of sympathy, or resonance, which exists between the genera- tor and the resonator. 176 TONE QUALITIES OF MUSICAL INSTRUMENTS Resonance Every vibrating body has one or more natural periods in which it vibrates easily ; to tune a sounding body is to adjust its natural period to a specified frequency. If a body capa- ble of vibration is excited by any means whatever, and the exciting cause is removed, the body will usually vibrate freely in its natural frequency, or with its free period. If the exciting cause operates in this same frequency, the two are in resonance, that is, they are in tune ; under these condi- tions the response of the body receiving the vibration is a maximum. If the exciting cause differs in frequency but slightly from that natural to the other body, there will still be response but in a lesser degree, that is, the resonance is not so sharp.^^ When the two bodies are quite out of tune, there will be very little resonance, and while the second body may still be made to vibrate, the response will be small. These conditions are well illustrated by the response curves described in the previous lecture. When the resonator is out of tune with the generator, it is often made to vibrate with the generator, and it is then said to have forced vibration. In forced vibration, the two bodies have different natural frequencies, and the resulting forced frequency is in general not that natural to either body, each drawing the other more or less to a common intermediate frequency. In musical instruments usually the generator is much less influenced than is the resonator ; for instance, a tuning fork in connection with a resonance box not exactly in tune, draws the air in the box to its own frequency much more easily than the air draws the fork. Koenig found that for a fork of 256 vibrations per second the maximum alteration of its frequency due to the draw- N 177 THE SCIENCE OP MUSICAL SOUNDS ing effect of the resonance box is produced when the box has a natural frequency of either 248 or 264. The fork in causing forced vibration of the air in the box draws the air from a frequency of 248 to 256.036 in the first case, and from 264 to 255.964 in the second; that is, the box being out of tune by 8 vibrations, the fork is forced out of its natural frequency by 0.036 vibration. When the box is out of tune by a musical semitone, the effect on the fork is less, being about 0.025 vibration.^' When a body is exactly in tune with the generator, that is, when it is in resonance, it may take up the vibrations with great ease and vigor ; such a response is often called sympathetic vibration. This is often disagreeably illustrated by the rattle of bric-a-brac in a music room, or by the buzz of some part of the action of a piano or of a machine. Sym- pathetic vibration is demonstrated by means of two forks which are exactly in tune ; if one fork is sounded loudly for a few seconds, the other fork is set in audible vibration, the only medium of communication being the air. There are two distinct kinds of resonators. One kind having no definite vibration frequency of its own, responds to tones of any frequency and to combinations of these; it can reproduce all gradations of tone quahty. A plate, such as the soundboard of a piano, is representative of this kind of resonator. The second type of resonator possesses a more or less definite natiiral frequency and, because of selective control, it reproduces sounds of particular quality only. Such a resonator will respond not only to tones corresponding to its fundamental, but also to tones in uni- son with its overtones. The second kind of resonator is typified by the cylindrical brass box of the standard tun- ing fork described on page 51. This box has a very definite 178 TONE QUALITIES OF MUSICAL INSTRUMENTS fundamental frequency and overtones which are high in pitch and not in tune with any overtone of the fork ; there- fore only the fundamental of the fork is reinforced, and the result is a pure simple tone. A resonator does not create any sound ; it can only take up the energy of vibration of the generator and give it out in a different loudness. It follows that for a given blow to a fork or a string, the more perfect the tuning of the resona- tor, the louder will be the sound and the shorter will be its duration. If the strings of two different pianos are struck with the same force of blow, that piano which gives the loudest sound will probably have the shortest duration of tone, while the one which begins the sound with moderate loudness will continue to sound longer or will "sing" better. The loudness and the duration of the sound from an instru- ment are dependent upon the damping or absorption of the vibration in the instrument and its surroundings. The energy of the waves which travel outward from a sound- ing body is derived from the vibration of the body ; usually not all of the energy of vibration is transferred, some being absorbed and transformed into heat through friction and the viscosity of the body. When the loss of energy is rapid, the amplitude of vibration decreases rapidly, and the vibra- tions are said to be damped.™ These effects must be con- sidered with resonance and consonance in the complete study of musical instruments. Effects of Material on Sound Waves Both the tones generated by a musical instrument and those reproduced, as weU as those absorbed or damped, de- pend in a considerable degree upon the material of which the various parts of the instrument are constructed. While 179 THE SCIENCE OF MUSICAL SOUNDS this fact is well known and commonly made use of in connec- tion with certain classes of instruments, its truthfulness is often denied by the devotees of other instruments. The question of the influence of the material of which the body tube of a flute is made has not been settled after more than seventy years of widespread discussion. How does the tone from a gold or silver flute differ from that of a wooden flute ? It was this specific ques- tion that suggested the investigations which, having passed much be- yond the original in- quiry, have furnished the material upon which this course of lec- tures is based. The following experi- ments, suggested by those of Schafhautl (Munich, 1879), ^i indi- cate the great changes in the tone of an organ pipe which may be pro- duced by effects pass- ing through the walls. '^ Three organ pipes are provided, as shown in Fig. 133. The first pipe, of the ordinary type used in physical experiments, is made of wood and sounds the tone 0^= 192. Two pipes having exactly the same internal dimensions as the wooden one are made of sheet zinc about 0.5 millimeter thick. One of the zinc 180 Fig. 133. Organ pipes for demonstrating the influence of the walls on the tone. TONE QUALITIES OF MUSICAL INSTRUMENTS pipes has been placed inside a zinc casing to form a double- walled pipe, with spaces two centimeters wide between the walls ; the outer wall is attached to the inner one only at the extreme bottom on three sides, and just above the upper lip-plate on the front side. These two pipes have exactly the same pitch, giving a tone a little flatter than F2, which is more than two musical semitones lower than that of the wooden pipe of the same dimensions. Using the single-walled zinc pipe one can produce the remarkable effect of choking the pipe till it actually squeals. When the pipe is blown ii; the ordinary manner, its sound has the usual tone quahty. If the pipe is firmly grasped in both hands just above the mouth, it speaks a mixture of three clearly distinguished inharmonic partial tones, the ratios of which are approximately 1 : 2.06 : 2.66. The resulting unmusical sound is so unexpected that it is almost startling, the tone quality having changed from that of a flute to that of a tin horn. Experiments with the double-walled pipe are perhaps more convincing. While the pipe is sounding continuously, the space between the walls is slowly filled with water at room temperature. The pipe, with the dimensions of a wooden pipe giving the tone G2, when empty has the pitch r2, and when the walls are filled with water the pitch is E2 ; during the fiUing the pitch varies more than a semitone, first rising then falling. While the space is filling, the tone quality changes conspicuously thirty or forty times. After the demonstration of these effects, one will surely admit that the quality of a wind-instrument may be affected by the material of its body tube to the comparatively small extent claimed by the player. The flute is perhaps espe- 181 THE SCIENCE OF MUSICAL SOUNDS cially susceptible to this influence because its metal tube is usually only 0.3 millimeter thick. It is conceivable that the presence or absence of a ferrule or of a support for a key might cause the appearance or disappearance of a partial tone, or put a harmonic partial slightly out of tune. The traditional influence of different metals on the flute tone are consistent with the experimental results obtained from the organ pipe. Brass and German silver are usually hard, stiff, and thick, and have but little influence upon the air column, and the tone is said to be hard and trumpet- like. Silver is denser and softer, and adds to the mellow- ness of the tone. The much greater softness and density of gold adds still more to the soft massiveness of the walls, giving an effect like the organ pipe surrounded with water. Elaborate analyses of the tones from flutes of wood, glass, ' silver, and gold prove that the tone from the gold flute is mellower and richer, having a longer and louder series of partials, than flutes of other materials. Mere massiveness of the walls does not fulfill the desired condition ; a heavy tube, obtained from thick walls of brass, has such increased rigidity as to produce an undesir- able result ; the walls must be thin, soft, and flexible, and must be made massive by increasing the density of the ma- terial. The gold flute tube and the organ pipe surrounded with water, are, no doubt, similar to the long strings of the pianoforte, which have a rich quality ; these strings are wound or loaded, making them massive, while the flexi- bility or "softness" is unimpaired. The organ pipe partly fiUed with water is Uke a string unequally loaded, its partials are out of tune and produce a grotesque tone. A flute tube having no tone holes or keys is influenced by the manner of holding ; certain overtones are sometimes difficult to 182 TONE QUALITIES OF MUSICAL INSTRUMENTS produce until the points of support of the tube in the hands have been altered. Beat-Tones When two simple tones are sounding simultaneously, in general, beats are produced, equal in number to the differ- ence of the frequencies. When the beats are few per second, the separate pulsations are easily detected. When the beats are many, the ear does not perceive the sep- arate pulses, and instead the sensation is that of a Fig. 134. Photograph of beats produced by two tuning forks, giving the effect of a third tone, called a beat-tone. third tone, which is as distinct and as musical as the two generating tones, and which has a frequency equal to the difference in the frequencies of the two generators ; that is, its frequency is equal to the number of beats if such rapid beats could be heard. This tone is called a heat-tone.''^ Fig. 134 is a photograph of the waves from two tuning forks having frequencies of Ce = 2048 and De = 2304, respectively, which are in the ratio of 8:9. If the two sets of waves are in like phase at a certain point, they combine and produce a curve of large amplitude, as at a, signifying a loud sound. This condition is repeated at regular intervals along the wave train, as at h, which is 183 THE SCIENCE OF MUSICAL SOUNDS exactly 8 waves of one tone and 9 waves of the other from a. A point c, midway between a and b, is 4 waves of one tone and 4^ waves of the other from a. When the motion at this point due to one wave is upward, that due to the other is downward, and the two neutrahze each other, producing the effect shown in the curve and which corresponds to a minimum of sound. This neutra-lizing effect occurs regu- larly between each two reinforcements. The resultant sound of the two forks waxes and wanes as does the out- line of the curve. When the number of fluctuations is less than 16 per second the ear hears the separate pulsations as beats ; when the number of pulsations is large, the effect upon the ear is that of a continuous simple tone of a fre- quency equal to the number of beats per second ; this effect is the beat-tone. There are 256 beats per second in the instance described, and the ear hears not only the two real fork-tones, Ce = 2048 and Ds = 2304, but also a third beat-tone, of the pitch Cs = 256. The latter sounds just as real as the other two tones, but it has no physical exist- ence as a tone ; there is no vibrating component of motion corresponding to the beat-tone, an analysis of the wave form showing only the two components due to the forks. While beat-tones are purely subjective, yet they affect the ear as do real tones. These subjective partials have great influence on the tone quaUty of many instrumental and vocal sounds as perceived by the ear. This influence has never been fully appreciated. Identification of Instrumental Tones There are many who, listening to a full orchestra, are able to distinguish the tones of a single instrument even 184 TONE QUALITIES OF MUSICAL INSTRUMENTS when all the instruments are being played. We usually think this possibility is dependent on the characteristic quality of the instrument, but investigation indicates that tone quality is only one of several perhaps equally important factors of identification. Other aids in the differentiation are the attendant and characteristic noises of the instru- ment, such as the scratching of the bow, the hissing of the breath, and the snapping of the plucked strings. A further very important help to the observer, especially if he is not a trained musician, is the visual observation of the motions of the performer; the synchronism of these movements with the changes of the melody calls attention to the par- ticular instrument. Every one attending a concert desires to see the musicians as well as to hear them; a seat in a concert hall which allows no view of the musicians is con- sidered most undesirable. It is very difficult to keep the many instruments of an orchestra in perfect tune ; indeed it is almost certain that perfect tuning is unattainable. The imperfections of tun- ing prevent the harmonious blending of the sounds of the various instruments, and an individual instrument may be separated from the mass of sound by its particular pitch, which condition will help to differentiate it and assist the hearer in the identification. The hearer may not be con- scious of the lack of tuning, and, indeed, many persons are not over-critical in this respect. Helmholtz says''* that the proper musical qualities of the tone from a fork and of that produced by blowing across the mouth of a bottle, both being simple, are identical. Certain tones of the flute are also simple, and therefore of the same quality as those of the tuning fork. Experi- mental demonstration has proved that when the auditor 185 THE SCIENCE OF MUSICAL SOUNDS is removed from the sounding bodies so that he is unable either to see them or to hear the attendant noises, he can- not tell whether it is the fork, the bottle, or the flute, that produces the tone. It is a fact that certain tones can be produced on the flute, in the lower register, which cannot be distinguished by the trained musician from certain tones of the violin, and Hke similarities are possible with pairs of other instruments. Of course, any instrument can prob- ably produce certain pecuUar tones which are impossible of imitation by any other instrmnent. The musician and the scientist are interested in the dis- tinguishing features of the tone qualities of the various orchestral instruments and of other sources of musical sounds. The true characteristic tone of an instrument is the sustained and continuable soimd produced after the sound has been started and has reached what may be called the steady state ; this steady sound is usually free from the noises of generation. Systematic analyses covering the entire scale in various degrees of loudness have been made for the flute, violin, horn, and voice, and less complete analyses have been made for other instruments; these studies are to be continued until a general survey has been made of the entire tonal facilities of instrumental music. The analyses which have been completed make it possible to describe the distinguishing characteristics of the tones of the several instruments. The Tuning Fork The musical instrument which gives the simplest and purest tone, as mentioned in Lecture II, is a tuning fork in connection with a resonator. A photograph of the tone from a fork sounding middle C = 256 is shown in Fig. 135 ; 186 TONE QUALITIES OF MUSICAL INSTRUMENTS such a tone needs no other description than the statement that it is simple. This wave form will be recognized as that produced by the simplest possible vibratory motion, Fig. 135. Photograph of the simple tone from a tuning fork. simple harmonic motion. For comparison the analysis of such a simple tone is shown on the diagram of various tones in Fig. 130, page 171 ; since the tone has but one com- ponent thelliagram consists of one line only. Fig. 136. Photograph of the clang-tone from a tuning fork. If a fork is struck a sharp blow with a wooden mallet or other hard body, it can be made to give a ringing sound in which the ear easily distinguishes a high-pitched clang-tone 187 THE SCIENCE OF MUSICAL SOUNDS in addition to the fundamental heard when the fork is sounded with a soft hammer ; this clang-tone is the first natural overtone of the fork. Fig. 136 is a photograph of the tone from a fork struck with a wooden mallet, the kinks in the wave form being due to the overtone thus pro- duced. Inspection shows that the relation of the small wave to the large one occurring at the point a does not recur till Fig. 137. Photograph of the tone of a tuning fork having the octave overtone. the fourth succeeding wave, at fe; in the four large waves there are twenty-five kinks due to the small one, that is, the frequency of the overtone is about 6.25 times that of the fundamental. Since there is not an integral number of the smaller waves to one of the larger, the partial is inharmonic or out of tune, and hence the sound is clanging or metallic rather than musical. When a tuning fork mounted on a resonance box is sounded by vigorous bowing, it sometimes produces a strong octave overtone ; such a tone is not natural to either the fork or the box, and is probably due to some peculiar condition of the combination which has not yet been fully explained."' A 188 TONE QUALITIES OF MUSICAL INSTRUMENTS photograph of this unusual tone from a fork is shown in Fig. 137. Tuning forks have been used as musical instruments in connection with keyboards Uke those of the piano or organ. The tones are remarkably sweet and of greater purity than those obtainable from any other instrument ; but the very fact of purity, that is, the absence of higher partial tones, renders the music monotonous and uninter- esting, and such devices have not survived the experimental stages. The Choralcelo, an instrument of recent design,'^ produces a sustained tone, having the same general characteristics as that of the tuning fork. The vibrations are produced by electromagnets^ through which flow interrupted, direct, elec- tric currents, the pulsations of which are of the same periods as those of the bodies to be set in motion. The sources of sound may be piano strings or ribbons of steel drawn over a soundboard, which are set in vibration by the direct action of the magnets. In other instances, bars of wood, alumi- num, or steel are used in connection with resonators, or dia- phragms of special construction are fastened to the ends of resonant tubes ; soft iron armatures are attached to the bars and diaphragms, which are set in vibration by the pulsations of the magnets, and thus the air in the resonators is moved and the tones are produced. The tones so obtained are nearly simple in quality, consisting mainly of a fundamen- tal. The overtones, naturally absent, are provided by sounding corresponding generators in accordance with a scheme of tone combinations which can be carried out con- veniently by means of stops or controllers operating switches in the electrical apparatus. The Choralcelo produces tones which are very clear and vibrant and of great carrying 189 THE SCIENCE OF MUSICAL SOUNDS vs Fig. 138. The flute. power, due, perhaps, to the strong fundamental component. The combinations of such sounds produce unique tonal effects, of remarkable musical quality, and the possibihties of synthetic tone development are great. The Flute The flute in principle is of the utmost simpUcity ; it con- sists of a cylindrical air column a few inches in length, set into longitudinal vibration by blow- ing across a hole near the end of the tube which incloses the air column. The holes in the body of the flute, with the keys and mechanism. Fig. 138, serve only to control the effective length of the vibrating air column. While the flute is simple acoustically, the manipulation of the instrument in accordance with the requirements of music of the present time, requires a key-mechanism of considerable complexity and of the finest workmanship. The flute has been developed to an acous- 190 \ TONE QUALITIES OF MUSICAL INSTRUMENTS tical and mechanical perfection perhaps not attained by any other orchestral instrument. This is largely due to the artistic and scientific studies of the instrument made by Theobald Boehm, of Munich, who devised the modern system of fingering in 1832, and invented the cyhndrical- bore, metal tube, with large covered finger holes, in 1847.''' The flute gives the simplest sound of any orchestral in- strument, and this is especially true when it is played softly. The paucity of overtones causes its sound to blend more readily with that of other instruments or the voice, and prevents the poignant expressiveness of the stringed and reed instruments ; nevertheless, the flute has an expression pecuUar to itself, and an aptitude for rendering certain sentiments not possessed by any other instrument. Berlioz says: "If it were required to give a sad air an accent of desolation and of humility and resignation at the same time, the feeble sounds of the flute's medium register would certainly produce the desired effect." The flute, because of its agility and ability to play detached and extended passages, arpeggios, and iterated notes, as well as because of its light tone quality, is suited to music of the gayest character. The flute tone is often described as sweet and tender; Sidney Lanier, himself an accomplished flutist, describes this tone in "The Symphony" : " "But presently A velvet flute-note fell down pleasantly Upon the bosom of that harmony, And sailed and sailed incessantly, As if a petal from a wild-rose blown Had fluttered down upon that pool of tone And boatwise dropped o' the convex side And floated down the glassy tide And clarified and glorified ^ The solemn spaces where the shadows bide." 191 THE SCIENCE OF MUSICAL SOUNDS Sound waves from the flute are shown in Fig. 139, in which are three curves for the same tone, G3 = 388, played p, to/, and / ; when played softly the tone is nearly simple, an increase in loudness adds first the octave, and then still higher partials. About a thousand photographs of flute tones have been analyzed, including every note in the scale of the instru- ment, each in several degrees of loudness ; flutes made of various materials have been studied, as wood, silver, gold, '^ ,.'*■ .^ ^ " . "* I V-, ^- ^ ^ ^ ^'^ "'\ " ^-./V s .^ .'S i ~ A ^A ■^A \/-, .• Fig. 139, Three photographs of the tone of a fiute, played p, mf, and /. and glass, and the effects of different sized holes have been investigated. Some of the results of the analyses of the tones of the gold flute will be described; flutes of other materials have the same general characteristics, except that the overtones are fewer and weaker. The average composition of all the tones of the low register of the flute, one octave in range, when played pianissimo, is shown by the lower hne of Fig. 140; these tones are nearly simple, containing about 95 per cent of fundamental, with a very weak octave and just a trace 192 TONE QUALITIES OF MUSICAL INSTRUMENTS of some of the higher partials. The pianissimo tones of the middle register, shown on the second hne of the figure, are simple, without overtones. When the lower register is played forte, it is in effect over- blown, and the first overtone becomes the most prominent partial, as shown on the third line ; the fundamental is weak. 1 i .f.. i i i I ■ \ \ — n — T — I I I I I M I M t 2 3 4 5 6 7 8 9 10 15 Fig. 140. Analyses of flute tones. being just loud enough to characterize the pitch. The player is often conscious of the skill required to prevent the total disappearance of the fundamental and the passing of the tone into the octave. The tones of the low register, when played loudly, have as many as six or eight partials, and at times these sounds suggest the string quality of tone. o 193 THE SCIENCE OF MUSICAL SOUNDS The tones of the middle register played /orie consist mainly of fundamental with traces of the second and third partials. In this respect these flute tones are very similar to those of the soprano voice of like pitch, a fact which is made use of in the duet of the Mad Scene in the opera "Lucia di Lammermoor." The average of all of the tones of the lower and middle registers of the flute, shown in the upper Hne of the figure, leads to the conclusion that the tone of the flute is character- ized by few overtones, with the octave partial predom- inating. The tones of the highest register have been analyzed and found to be practically simple tones. This result is to be expected from the conditions of tone production for the higher tones ; the air column is of a diameter relatively large as compared with its length, and it is difficult to pro- duce loud overtones in such an air column. The Violin The strings of a viohn. Fig. 141, are caused to vibrate by the action of the bow, and these vibrations are trans- mitted through the bridge and body of the instrument to the air ; not only does the body affect the air by its surface movements, but the interior space acts as a resonance chamber. Helmholtz has made a study of the vibrating viohn string, and has developed the mathematical equations defining the motion ; ^' Professor H. N. Davis has investigated the lon- gitudinal vibrations of strings in a manner to throw much light upon the subject ; ^^ Professor E. H. Barton and his colleagues have photographed the movements of the string and body of the instrument ; ^^ and P. H. Edwards and 194 TONE QUALITIES OF MUSICAL INSTRUMENTS C. W. Hewlett have studied the tones of vioHns of differing quality. ^^ By means of the vibration microscope Helmholtz observed the vibrations of the string and plotted the form of its move- ments, point by pointy as shown in Fig. 142. A photograph of a sound wave from a violin is given in Fig. 143; the form is, in general, iden- tical with the Helm- holtz diagram; this identity is remark- able when it is re- membered that the photograph is the wave in air, from the body of the in- strument, while the diagram represents the movements of the string. The photograph shows what may be con- sidered the typical form of a vioUn wave, but it is not the common form; . this particular shape depends upon a critical relation be- tween the pressure, grip, and speed of the bow, and upon the place of bowing and the pitch of the tone. The usual varia- tions in bowing disturb the regularity of the vibrations, and produce a continually changing wave form. This is an 195 Fig. 141. The violin. THE SCIENCE OF MUSICAL SOUNDS indication of the fact that a great variety of tone quaUty can be produced by the usual changes in bowing. Berlioz, in describing the orchestral usefulness of the violin, says : "From them is evolved the greatest power of expression, Fig. 142. Helmholtz's diagram of the vibrations of a violin string. and an incontestable variety of qualities of tone. Violins particularly are capable of a host of apparently inconsistent shades of expression. They possess as a whole force, light- ness, grace, accents both gloomy and gay, thought, and Fig. 143. Photograph of the tone of a viohn. passion. The only point is to know how to make them speak." The tone quality, as well as the wave form, remains con- stant so long as the bowing is constant in pressure, speed, and direction. The direction of bowing may be skillfully 196 TONE QUALITIES OF MUSICAL INSTRUMENTS reversed without changing the tone quality. Fig. 144 is a photograph of the wave form when a change of bowing occurs ; the first part of the curve is for an up-bow, while the other part is produced by the down-bow ; the curve is symmetrically turned over with every change in the direc- tion of })owing, while the confusion caused by the change produces a noise which lasts about two hundredths of a second. Analytically, the turning over of the curve means that the phases of all the components are reversed ; the ear does not detect any change in tone quality due to the 'fff/Yrmrvrm- 'vvvvvvvvi Fig. 144. Photograph of the tone of a violin at the time of reversal of the bowing. reversal of phases, and this fact supports the statement that tone quality is independent of phase. Photographs were taken of a series of tones on each string of the violin, of three degrees of loudness. The average results of the analysis of loud tones from the four strings are shown in Fig. 145. For the lower sounds the funda- mental is weak, as indeed it must be, since these tones are lower than the fundamental resonance of the body of the violin ; the tones from the three higher strings have strong fundamentals. The ear perceives a fundamental in the lower tones of the violin, and this must result from a beat- tone produced by adjacent higher partials which are strong. The tones from the three lower strings seem to be characterized by strong partials as high as the fifth, 197 thp: science of musical sounds while the E string gives a strong third. In general the tone of the violin is characterized by the prominence of the third, fourth, and fifth partials ; and while the vioHn generates a larger series of partials than does the flute, yet it is not equal to the brass and reed instruments in this respect. The great advantage of the vioUn over E Sl.in, 1 * itrino _l I 1 ' ' i J X I 1 U I I I I \ 1 I I I I M I I I 1 2 3 4 6 B 7 6 9 10 IS Fig. 145. Analyses of violin tones. all other orchestral instruments in expressiveness is due to the control which the performer has over the tone production. The Clarinet and the Oboe The study of reed instruments has not been completed, but the analyses of many individual tones show interesting characteristics. The clarinet. Fig. 146, generates sound by means of a single reed of bamboo which vibrates against 198 TONE QUALITIES OP MUSICAL INSTRUMENTS the opening in the mouthpiece; these vibrations are con- trolled and imparted to the air by the body tube. The body has a uniform cylindrical bore, at the lower end of Fig. 146. The clarinet. Fig. 147. The oboe. which is a short, bell-shaped enlargement. The keys vary the resonance of the interior column of air and thus control the pitch. The oboe, Fig. 147, has a mouthpiece consisting of two reeds which vibrate against each other. The body with its ]99 THE SCIENCE OF MUSICAL SOUNDS keys forms a resonance chamber of various pitches, but it differs from the clarinet in that the bore is conical throughout. Fig. 148. Phoix)graph of the tone of an oboe. The photograph of the tone from an oboe, Fig. 148, and that from a clarinet. Fig. 149, both show deep kinks in the wave form ; these kinks indicate the presence of relatively Fig. 149. Photograph of the tone of a clarinet. very loud higher partials which, no doubt, produce the reedy tone quality of these instruments. The presence 200 TONE QUALITIES OF MUSICAL INSTRUMENTS of beats, that is, the recurrence in each wave length of por- tions where the kinks are neutralized, shows that there are two adjacent high partials of nearly equal strength, as was explained under Analysis by Inspection in Lecture IV. The average of several analyses, Fig. 150, shows that the oboe tone has twelve or more partials, the fourth and fifth pre- dominating, with 30 and 36 per cent respectively of the total loudness. The clarinet tone may have twenty or more partials ; the average of several analyses shows twelve of importance, with the seventh, eighth, ninth, and tenth predominating; the seventh partial contains 8 per cent of CLARIKET J . I . I . » i I i I t ■ OBOE ^ . i i 1 . ■ ■ t ■ - I \ \ 1 \ 1 I I I I I I I I I I 2 3 4S6789I0 15 Fig. 150. Analyses of the tones of the oboe and the clarinet. the total loudness, while the eighth, ninth, and tenth con- tain 18, 15, and 18 per cent respectively. The statement is often made that the seventh and ninth partials are "inharmonic" and that their presence renders a musical sound disagreeable. The seventh and ninth partials are just as natural as any others; a partial is not inharmonic because it is the seventh or ninth in the series of natural tones ; any partial whose frequency is an exact multiple of that of the fundamental is truly harmonic; a partial is inharmonic when it is not an exact multiple of the fundamental frequency, whether it is the second or ninth, or any other of the natural series. If the wave form of a sound is periodic, its partials must all be harmonic, 201 THE SCIENCE OF MUSICAL SOUNDS and such a sound is musical. The clarinet gives periodic waves, which, as the analysis shows, contain loud seventh and ninth partials; these partials may almost be said to be the characteristic of the tone, but they are in tune, and are harmonic, and the clarinet tone has a very beautiful musical quality. Lanier in "The Symphony" says : '* "The silence breeds A little breeze among the reeds That seems to blow by sea-marsh weeds ; Then from the gentle stir and fret Sings out the melting clarionet." The adjective "melting" seems to the author not merely a poetic term, but a real description of the clarinet as heard in the orchestra.. The Hoen The horn, Fig. 151, is a brass instrument of extreme sim- plicity, consisting of a slender conical tube, sometimes more than eighteen feet long, with a conical cup-shaped mouthpiece, and a large flaring bell. In its typical form there are no apertures in the walls of the tube, and no valves ; but the modern horn usually has valves, as shown in the figure. The tone of the horn is described by Lavignac as "by turns heroic or rustic, savage or exquisitely poetic ; and it is perhaps in the expression of tenderness and emotion that it best develops its mysterious qualities." The scien- tific analysis shows causes for the variety of musical effects, for the horn produces tones of widely differing composition from one as soft and smooth as a delicate flute tone to a "split" tone that is tonally disrupted by strong higher partials. The low sounds of the horn are rich in overtones, containing the largest number of partials yet found in any musical tone. The analysis of the wave for the tone 202 TONE QUALITIES OF MUSICAL INSTRUMENTS D2 = 162, Fig. 152, shows the presence of the entire series of partials up to thirty, with those from the second to the six- FiG. 151. The horn. teenth about equally loud; a diagram of this analysis is given on the lower line of Fig. 153. The results of analyses of various other tones from the horn are also given in Fig. 153. The second hne shows the Fig. 152. Photograph of the tone of a horn. average composition of loud, medium, and soft tones rang- ing over the entire compass, indicating a strong fundamental followed by a complete series of partials, more than twenty 203 THE SCIENCE OF MUSICAL SOUNDS in number, of gradually diminishing intensity. The tone quality thus indicated approaches more nearly to the ideal described on page 211 than does that of any other instru- ment so far investigated. A "chord tone" which seems to be made by humming with the vocal chords while playing, shown in the third Une of the figure, has the fourth, fifth, and sixth partials the most prominent, which give the conmion chord, do, mi, sol. SPLIT —I— 1-11 1 i i AVERAGE LOW 1 L _i 1 I I i I i 1 I 1 1 \ — I — I I I I I I 1 1 1 1 iiiiiiiiiiiiiii I Z 3 4 5 6 7 B 9 10 IS 20 25 30 Fig. 153. Analyses of tones of the horn. A "smooth" tone is produced when the player muflSies the tone more or less by putting his hand in the bell of the horn. The analysis of such a tone, shown in the fourth Hne, indicates that it is nearly simple and is very much Uke the lower tones of the flute when played softly. ' The "rough" tone is played more loudly and without muffling by the hand ; the analysis, line five, shows an oc- tave overtone which is louder than the fundamental, and a weak third partial. 204 TONE QUALITIES OF MUSICAL INSTRUMENTS Another quality of tone, shown on the top Une, is called a "split tone" ; this tone is literally split into many partials and distributed uniformly from the fundamental to the twelfth. The Voice The sounds of the voice originate in the vibrations of the vocal cords in the larynx, the pitch being controlled largely by muscular tension, while the quahty is dependent mostly upon the resonance effects of the vocal cavities. The tones of the singing voice have not been analyzed except in connection with the vowels, the results of which are described at length in Lectures VII and VIII. Fig. 154 shows the curve for a bass voice (E C) intoning the vowel a in father on the note Fi % = 92, ^^^3- The loud partials in this tone are evidently of a high order, since Fig. 154. Photograph of a bass voice. there are many large kinks in one wave length. A diagram of the analysis of the curve is given in the lower hne of Fig. 156 ; it shows that the seventh, eighth, ninth, and eleventh partials are the strongest. The voice E E M intoned the same vowel on the soprano pitch of Bst*, \l^ |> o || ) producing the curve shown in Fig. 155, the analysis of which is given in the upper hne of 205 THE SCIENCE OF MUSICAL SOUNDS Fig. 156. This curve is simple, and is seen at a glance to con- tain but one strong partial, the second overtone or octave. These two voices, and their corresponding curves, are Fig. 155. Photograph of a soprano voice. very unlike, yet the ear recognizes the same vowel from both. The vowel characteristics of the bass voice, repre- sented by the seventh, eighth, ninth, and eleventh partials, are accompanied by six lower partials, the first of which ■^ 1 " Hi C, D EF G A BC3 D EF G A BC3 D E F G A BC4 D EP G A BCg D EF C A B(^ D EF G A BC7 D E 6B H..n»i.T Of .xnND l29cui»oo.«.»....c'Mi2S9 517 lOSS 2068 4I3B FiQ. 156. Analyses of tones of bass and soprano voices. determines the pitch while the others give the bass quality of the individual voice. The vowel characteristic for the other voice is the second partial, the pitch is determined by the first or fundamental, while a series of five or more higher partials produce the individuality of the soprano tone. 206 TONE QUALITIES OF MUSICAL INSTRUMENTS The Piano The vibrations in a piano originate in the strings which are struck with the felt-covered hammers of the key action, while the sound comes mostly from the soundboard. The relations of strings and soundboard have been considered under Resonators and Resonance in the beginning of this lecture. The lower tones of the piano are found to be very weak in fundamental, but to have many overtones, partials as high as the forty-second having been identified. These high partials are loud enough to be heard by the unaided ear after attention has been directed to them. These characteristics are entirely consistent with the nature of the source, which is a slender metal string struck with a hammer. The higher tones of the piano, originating in much shorter strings imder high tension, have few partials, and the loud- est component is often the second partial or octave. The tones from the middle portion of the scale contain ten or more partials of well distributed intensity. The piano is perhaps the most expressive instrument, and therefore the most musical, upon which one person can play, and hence it is rightly the most popular instru- ment. The piano can produce wonderful; varieties of tone color in chords and groups of notes, and its music is full, rich, and varied. The sounds from any one key are also susceptible of much variation through the nature of the stroke on the key. So skillful does the accom- phshed performer become in producing variety of tone quality in piano music, which expresses his musical moods, that it is often said that something of the personahty of the player is transmitted by the "touch" to the tone 207 THE SCIENCE OF MUSICAL SOUNDS / produced, something which is quite independent of the loudness of the tone. It is also claimed that a variety of tone qualities may be obtained from one key, by a vari- ation in the artistic or emotional touch of the finger upon the key, even when the different touches all pro- duce sounds of the same loudness. This opinion is al- most universal among artistic musicians, and doubtless honestly so. These musicians do in truth produce marvel- ous tone qualities under the direction of their artistic emotions, but they are primarily conscious of their personal feelings and efforts, and seldom thoroughly analyze the principles of physics involved in the compUcated mechanical operations of tone production in the piano. Having investi- gated this question with ample facilities, we are compelled by the definite results to say that, if tones of the same loud- ness are produced by striking a single key of a piano with a variety of touches, the tones are always and necessarily of identical quaUty ; or, in other words, a variation of artis- tic touch cannot produce a variation in tone quality from one key, if the resulting tones are all of the same loudness. From this principle it follows that any tone quahty which can be produced by hand playing can be identically repro- duced by machine playing, it being necessary only that the various keys be struck automatically so as to produce the same loudness as was obtained by the hand, and be struck in the same time relation to one another. There are factors involved in the time relations of beginning the several tones of a chord or combination, which are not often taken into account ; a brief notice of the nature of piano tone will enable us to estabUsh this conclusion. Two photographs of piano tones are shown, the first, Fig. 157, being of the note one octave above middle C 208 TONE QUALITIES OF MUSICAL INSTRUMENTS and the other, Fig. 158, of the note one octave below. The first photograph shows ,two important features : the sound rises to its maximum intensity in about three one- Fig. 157. Photograph of the tone of a piano. hundredths of a second, and in one fifth of a second it has fallen to less than a tenth of its greatest loudness ; it then gradually dies out, but with a progressive change in quality. In the beginning the fundamental is the Fig. 158. Photograph of the tone of a piano. loudest component, but after a tenth of a second, the octave is the loudest part. The second photograph is of a tone two octaves lower and is of a much more complex nature. There are more than p 209 THE SCIENCE OF MUSICAL SOUNDS ten partials of appreciable loudness, which are continually changing in relative intensity, due, no doubt, to peculiarities of piano construction which prolong certain partials and absorb others. Whatever complex tone may be generated by the hammer blow, the quality of tone that enters into combination with that from other strings is dependent upon the parts of the tones from the several strings being simultaneously coexistent. The quality of tone obtained from a piano when a melody note is struck is dependent upon the mass of other tones then existing from other keys previously struck and sustained, and it depends upon the length of time each of these tones has been sounding. It is evident that not only does a piano give great variety of tone by various degrees of hammer blow, but there is pos- sible an almost infinite variety of tone quality in combina- tions of notes struck at intervals of a few hundredths of a second. It is believed that the artistic touch consists in slight variations in the time of striking the different keys, as well as in the strength of the blow, and that tone quality is determined by purely physical and mechanical considera- tions. The correctness of this argument is further supported by the mechanical piano players, which attempt to reproduce the characteristics of individual pianists. The more highly developed such instruments become, the more nearly they imitate hand playing in musical effects ; in many instances the imitation is practically perfect, and I beheve that in the near future the automatic piano wiU reproduce aU of the effects of hand playing. This condition will in no way displace the artist, nor wiU it in the least reduce his prestige; on the contrary, it will enhance his standing, and we shall honor him the more for 210 TONE QUALITIES OF MUSICAL INSTRUMENTS his accomplishments. The machine can never create a musical interpretation, the artist must ever do this. Sextette and Orchestra An illustration of very complex tone quality is obtained with the talking machine reproducing the Sextette from " Lucia di Lammermoor," by six famous voices with orches- tral accompaniment ; photographs of small portions of this music are shown in the frontispiece. The dots on the lower edge of the picture are time signals which are ^^ second apart; each line of the picture represents the vibrations due to music of less than one second's duration. On the scale of the original photograph, which is five inches wide, the length of film required to record the entire selection would be 1000 feet. The effects impressed upon the wave by a particular voice or instrument are clearly reproduced; in the middle of the top line, the increase in the ampUtude of the wave is due to the entrance of the tenor voice ; the second line shows the comparatively simple wave of the solo soprano voice singing high Bb, the smoothness of the curve attesting the pure quality of the voice. The Ideal Musical Tone Neither science nor art furnishes criteria which will define the ideal musical tone; a scientific investigation and analysis of the sound from a violin or a piano cannot determine whether it is the ideal. Musical instruments are used for artistic purposes and their selection is ultimately determined by the aesthetic taste of the artist. When an instrument has been artistically approved, the physicist - can describe its tonal characteristics and select other instru- ments possessing the same qualities ; he can detect defi- 211 THE SCIENCE OF MUSICAL SOUNDS ciencies and defects and, perhaps, can suggest remedies. "The chemist can scrape the paint from a canvas and ana- lyze it, but he cannot thereby select a masterpiece." A musical tone of remarkable quality may be produced by a special set of ten tuning forks shown in Fig. 159 ; these forks are accurately tuned to the pitches of a fundamental tone of 128 vibrations per second and its nine harmonic overtones. When the fundamental alone is sounding, a Fig. 159. Set of tuning forks for demonstrating the quality of composite tones. sweet but dull tone is heard. As the successive overtones are added, the tone grows in richness, until the ten forks are sounding, when the effect is that of one splendid musical tone. One is hardly conscious that the sound is from ten separate sources, the components blend so perfectly into one sound. The tone is vigorous and "Uving" and has a fullness and richness rarely heard in musical instruments. Bearing in mind the quahfications just mentioned, one may speak of an ideal musical tone, meaning the most gratifying single tone which can be produced 212 TONE QUALITIES OF IMUSICAL INSTRUTHENTS from one instrument. Following the above experiment the ideal tone may be arbitrarily described as one having a strong fundamental containing perhaps 50 per cent of the total intensity, accompanied by a complete series of twenty or more overtones of successively dimin- ishing intensity. If, while the forks in the above experiment are sounding, they are silenced in succession from the highest downward, the tone becomes less and less rich, until finally the funda- mental alone is heard. This is a simple tone and is of a dull, droning quality; the experiment demonstrates that a pure tone is a poor tone. It is by no means desirable that all musical instruments should have the quality of tone described. The great variety of musical tone coloring obtained by the modern composers requires instruments of the greatest possible divergence in quality ; the contrasts thus available are very effective. Of the instruments of the orchestra, perhaps the horn, in certain of its lower tones, approaches most nearly to the arbitrary ideal. Demonstration In the oral lectures, the characteristics of various instru- ments as described in the preceding pages and as shown by the photographs, were demonstrated by playing the instru- ments themselves before the phonodeik, which projected the sound waves upon the screen as explained in Lecture III. The sounds so demonstrated were the simple and complex tones from tuning forks, the flute tone as it develops from the simple pianissimo quality to the more complex fortissimo by the addition of successive overtones, the full and vibrant cornet tone having many partials, the string 213 THE SCIENCE OF MUSICAL SOUNDS tone of the violin, with the reversal of phases by changing the direction of bowing, the reedy tone of the clarinet, the varying quaUties of vocal tones, the clanging tone from a bell with its interfering inharmonic overtones, and finally the vocal sextette with orchestra, and the concert band, as reproduced by various types of phonographs. As seen upon the screen, the waves of light, which may be ten feet Avide and forty feet long, stretching across the end of the room, are constantly in motion, and pass from one wave form to another, from simple to most complex shapes, with every change in frequency, loudness, or quality of the sound; the wonderfully changing waves flow with perfect smoothness and reproduce visually the harmoni- ously blending movements of the air, which the ear inter- prets as music. This ability to see the effects of quahtative changes as well as to hear them is certainly advantageous in an analytical study of sounds, and possibly it adds to the musical effectiveness ; it is at least a fascinating and instruc- tive demonstration. 214 LECTURE VII PHYSICAL CHARACTERISTICS OF THE VOWELS The Vowels The vowels have been more extensively investigated than any other subject connected with speech ; the philolo- gist, the physiologist, the physicist, and the vocalist, has each attacked the problem of vowel characteristics from his own separate point of view. The methods of the sev- eral classes of investigators, and the expressions of the re- sults, are so unlike and so highly specialized, that one person is seldom able to appreciate them all. The physicist wishes to interpret the vowels as they exist in the sound waves in air, that is, he wishes to know the nature of the musical tone quality which gives individuality to the several vowels. The tone quality of vowels has been more closely studied than that of all other sounds combined, and yet no single opinion of the cause of vowel quality has prevailed. The first attempt at an explanation of vowel quahty was made in 1829 by Willis, who concluded from experiments with reed organ pipes that it depends upon a fixed charac- teristic pitch; this theory was extended by Wheatstone (1837) and by Grassmann (1854). Bonders (1864) dis- covered that the cavity of the mouth is tuned to different pitches for different vowels. Helmholtz (1862-1877) ex- pounded the theory, a development of those given before, 215 THE SCIENCE OF MUSICAL SOUNDS that each vowel is characterized, not by a single fixed pitch, but by a fixed region of resonance, which is independent of the fundamental tone of the vowel ; this is the so-called fixed-pitch theory. In opposition to this theory, many writers on the sub- ject have held that the quality of a vowel, as well as that of a musical instrument, is characterized by a particular series of overtones accompanying a given fimdamental, the pitches of the overtones varying with that of the funda- mental, so that the ratios remain constant; this is the relative-pitch theory. Auerbach in 1876 developed an intermediate theory, concluding that both characteristics are concerned, and that the pitch of the most strongly reinforced partial alone is not sufiicient to determine the vowel. Hermann (1889) has suggested that the vowels might be characterized by partial tones, the pitches of which are within certain hmits, but which are inharmonic, the partials being independent of the fundamental. Lloyd (1890) considers that the identity of a vowel depends not upon the absolute pitch of one or more resonances, but upon the relative pitches of two or more. Several quotations will indicate the uncertainty existing at the present time in regard to the nature of the vowels. Ellis, the translator of Helmholtz, writes (1885) : "The ex- treme divergence of results obtained by investigators shows the inherent difficulties of the determination." Lord Rayleigh (1896) says : "A general comparison of his results with those obtained by other methods has been given by Hermann, from which it will be seen that much remains to be done before the perplexities involving the subject can be removed." Auerbach (1909) discusses the various theories, but without deciding which is correct.*' 210 PHYSICAL CHARACTERISTICS OF THE VOWELS Two recent publications on this subject arrive at opposite conclusions. Professor Bevier of Rutgers College in one of the most complete studies yet made (1900-1905)/* using the phonograph as an instrument of analysis, arrives at conclusions in accord with Helmholtz's fixed-resonance theory and the method of harmonic analysis. Professor Scripture (1906), formerly of Yale University, says: "the overtone theory of the vowels cannot be correct" ; and he gives extended arguments in support of this opinion and opposed to harmonic analysis of vowels.*^ The results of the work here described are in entire agreement with Helm- holtz's theory, and they are, therefore, out of harmony with Scripture's arguments.** Standard Vowel Tones and Woeds Vowels are speech sounds which can be continuously intoned, separated from the combinations and noises by which they are made into words. A dictionary definition of a vowel is: "one of the openest, most resonant, and continuable sounds uttered by the voice in the process of speaking ; a sound in which the element of tone is pre- dominant ; a tone-sound, as distinguished from a fricative (rusthng sound), from a mute (explosive), and so on." Helmholtz specifies seven vowels, the "Century Diction- ary " gives nineteen vowel sounds in its key to pronunciation, while some writers on phonetics tabulate as many as seventy- two vowel sounds. After preliminary study, eight standard vowels contained in the following words were selected for definitive analysis : father, raw, no, gloom, mat, pet, they, and bee. The particular vowels specified are according to the pro- nunciation of the author. It must be remembered that 217 THE SCIENCE OF MUSICAL SOUNDS any change in the pronunciation produces a different vowel, though we may understand the word to be the same, and that the quantitative results would vary for the slightest change in intonation or inflection. Since individual pro- nunciations vary greatly, even within the range of one language, there seems to be no better method of defining a vowel than by specifying several words, in each of which the author gives the vowel the same sound. Others may disagree with some of the pronunciations, but this does not change the fact that these are the sounds studied and defined in the results. A table of such words follows, while a larger fist is given on page 257. father, far, guard raw, fall, haul no, rode, goal gloom, move, growp mat, add, cat pet, feather, bless they, bait, hate bee, pique, machine Some of these sounds are common to all languages ; the equivalent of father is found in German in vater, and in French in pate; the equivalent of no in German is in wohl, and in French in cdte; but there seems to be no equivalent in either German or French for raw or mat. For the sake of simplicity, instead of using single letters in connection with a multiphcity of signs to designate the several vowels, the writer will give the whole word con- taining the vowel, the latter being indicated by italics; in pronouncing the phrase "a record of the vowel f other," one may emphasize and prolong the vowel as "a record of 218 PHYSICAL CHARACTERISTICS OF THE VOWELS the vowel iah . . . ther," or, better, one may pronounce only the vowel part of the last word, as "a record of the vowel . . .ah " Photographing, Analyzing, and Plotting Vowel Curves The general procedure in the investigation of a vowel is as follows : the speaker begins to pronounce the appropriate word and prolongs the vowel in as natural a manner as \rr:,r^ VV Fig. 160. Photograph of the vowel o in father, for analysis. possible ; by means of the phonodeik a photographic record is taken of the central portion of the vowel, while the zero line and time signals are recorded simultaneously with the voice curve. The vowel curve is then analyzed into its harmonic components, corrections are applied, percentage intensities for the several partials are computed, and the results are diagramed, as explained in Lecture V. A photograph of the vowel father intoned by a baritone voice, at the pitch of F2 = 182, is shown in Fig. 160. The analysis of this curve is given in Fig. 129, page 169, while analyses of other photographs of the same vowel are shown in Figs. 161, 162, and 163. 219 THE SCIENCE OF MUSICAL SOUNDS The ordinates on a vowel diagram indicate the distri- bution of the energy of the sound with reference to its own harmonic partials. A single analysis gives little information as to the distribution of energy for sounds of intermediate pitches, since the sound analyzed can have no intensity whatever for pitches other than those of its own partials. If the vowel is intoned by the same person at a different pitch, its partials may lie between those of the r\ \. S B 7 8 -'^ 5ii^>^^. r I I ' n T ITI' I IlllllTlliril' 6 7 8 9 10 15 20 25 30 I M M 10 IS Cj D EF A BCj D EF G A BC, D EF G A BCj D EF C A BC^ D EP G A BC, D K |)|s|i|s|g|iHS|s|g|S|'|s|;|S|5|S|l|8|;|3|t|i|JI|s|;|5|i|5|£^^ 129 2S8 S17 1035 3069 4138 Fig. 161. Loudness of the several components of the vowel fother, intoned at two different pitches. first sound as shown in Fig. 161. By plotting many analyses to one base line, Fig. 162, D, a curve can be drawn which shows the resonance of the vocal cavities for the particular vowel. For purposes of analytical study it is permissible to show the relations of the separate points of a single analysis to the indicated resonance curve as is done in A, B, and C, Fig. 162. The significance of these 220 PHYSICAL CHARACTERISTICS OF THE VOWELS curves is more fully explained in the section on Classifica- tion of the Vowels, on page 228. Vowels of Various Voices and Pitches Each of the efght vowels has been photographed at several pitches as intoned by each of eight voices, giving about a thousand curves, all of which have been analyzed and ;|g|s|;|;|#|8mis|g|i»|8|^|S|i|s||3|^|s|=|^^ 129 259 517 1035 2069 ■ 4133 Fig. 162. Distribution of energy among the several partials of the vowel father, intoned at various pitches. plotted. There were two bass voices, two baritones, one tenor, one contralto, one boy soprano, and one girl soprano ; the normal pitches of these voices ranged from 106 to 281. The vowel father was intoned by the voice D C M at the pitch 1)2^= 155, its energy distribution curve being as shown in the lower part of Fig. 162 ; the next two curves, B and C, show the same vowel by the same voice intoned at pitches of Fat = 182, and A^^ =227. When the vowel 221 THE SCIENCE OF MUSICAL SOUNDS is intoned at the lowest pitch, the sixth partial having a fre- quency of 930 contains 69 per cent of the total energy of the sound ; in the second case the fifth partial of pitch 910 is loudest with 48 per cent of the energy ; while in the third case the fourth partial of pitch 908 contains 65 per cent of the energy. Fig. 163. Distribution of energy among the several partials of the vowel father, as intoned by eight different voices. The same vowel, father, was intoned by the same voice, D C M, approximately upon each semitone of the octave from Cz = 129 to C3 = 259, at twelve different pitches ; the upper part of the figure, D, shows the location of all the component intensities of the twelve analyses; instead of twelve separate curves, one is drawn showing the average energy distribution. The energy curves of the same vowel, father, intoned 222 PHYSICAL CHARACTERISTICS OF THE VOWELS by eight different voices, at pitches ranging from 106 to 522, are given in Fig. 163. The voices are a bass (0 F E), a bass (E C), a baritone (W R W), a baritone (D C M), a tenor (F P W), a contralto (E E M), a boy soprano (K S, 14 years old), and a girl soprano (H F, 10 years old). The energy curves for the vowel bee, intoned by the same eight voices, at pitches ranging from 111 to 400, are shown ■|:.|;|;|;|g|8|iNg|iMB|s|i|a|a|i|8|;|s|i|iMi|s|5|i|i|-|E|8|s|5|.1|8|j|§|j|i|i|j|3|j|j|in^^ 129 259 S17 1033 2069 MSB FiQ. 164. Distribution of energy among the several partials of the vowel bee, as intoned by eight different voices. in Fig. 164. For this vowel there are two regions of reso- iiance, one at a pitch of about 300, and the other at a pitch of about 3000. While the greater part of the energy is in the lower resonance, yet it may be said that the higher resonance is the characteristic one, since its absence con- verts the vowel bee into gloom, as described on page 231. These diagrams indicate that there is not a fixed partial which characterizes the vowel, neither is there a single, 223 THE SCIENCE OF MUSICAL SOUNDS fixed pitch. The greater part of the energy of the voice is in those partials which fall within certain limits, no matter at what pitch the vowel is uttered, nor by what quality of voice; that is, the vowel is characterized by a fixed region, or regions, of resonance or reinforcement. To estab- lish this theory it is necessary to show that all the different vowels have distinctly different characteristic regions of resonance, which remain the same for all voices. An in- vestigation has been made leading to this conclusion, a detailed account of which is to be published elsewhere; the nature of this study is indicated by the following de- scription, which refers to one voice only. Definitive Investigation of one Voice The study of the vowels of different voices and pitches showed that it is practically impossible to obtain the same vowel from the various voices with sufficient certainty to permit of a definitive study, and even extreme variations in pitch for one voice probably alter the accuracy of pronuncia- tion. Therefore it was decided to make a final study of the principal vowel tones of the English language as spoken by one person in order to determine the physical cause of their differences. Each of the eight vowels previously mentioned was intoned by one voice, D C M ; six photographs were made in succession of the vowel spoken in normal pitch and inflec- tion ; then six more photographs for the same vowel were made, but approximately on six equidistant tones covering one octave, beginning a little below normal pitch. Thus twelve curves were obtained for each vowel, and for some a larger number was made. There are 202 photographs in this second series, all being made under exactly the same 224 PHYSICAL CHARACTERISTICS OF THE VOWELS conditions of the speaker's voice and recording apparatus. The curves were analyzed and reduced in one group, a sepa- rate energy distribution curve was drawn for each analysis, and finally a composite or average curve was made for each vowel. Classification of Vowels The eight final composite vowel curves, drawn on separate j)ieces of paper, were arranged upon a table and their pe- cuharities studied. Many schemes of classification were tried, with the final conclusion that all vowels may be divided into two classes, the first having a single, simple charac- teristic region of resonance, while for the second there are two characteristic regions. The vowels of the first class are represented by father, raw, no, and gloom. The investigations indicate that the most natural vowel sound and the most elemental words used in speech are ma and pa, and one of these may be selected as a starting point for a classification. It adds to the effectiveness if the vowels are indicated by simple syllables of the same general form; for the vowels of the first class the words may be ma, maw, mow, and moo, or po, ipaw, poe, and pooh. The vowels of the second class are represented by mat, pet, they, and bee ; and the new syllables selected are mat, met, mate, and meet, or pat, pet, pate, and peat. This series may be presumed to start from the fundamental vowel ma, which for similarity may be expressed by the words mot or pot. The characteristic curves for vowels of the first class are shown in Fig. 165 ; the vowels are ma, m.aw, mow, and moo, having maximum resonances at pitches of 910, 732, 461, and 362, respectively. The resonance regions overlap Q 225 THE SCIENCE OF MUSICAL SOUNDS but little ; the partials lying within a characteristic region of resonance often contain as much as 90 per cent of the total energy of the sound ; there is a conspicuous total absence of higher tones and all the lower tones are weak; the fundamental is of small intensity, containing only about 4 per cent of the energy, unless its pitch lies within a region of characteristic resonance. The vowel ma seems to have MA m MAW b. MOW MOO C, D EF G A BC3 D EF G A BC, D EF (* I II II I I"* I I I I I . I 1' I . I 1 . EF ,1,1, BC. D EF G A ' ^ ,1,1 -I, I, 129 , 258 517 1035 2069 4138 Fig. 165. Characteristic curves for the distribution of the energy in vowels of Class I, having a single region of resonance. considerable range, as its characteristic may vary from 900 to 1100 ; two curves are shown for this vowel, as the high- est ma is perhaps the initial sound from which all vowels are derived. Sometimes this vowel has two resonances close together, as shown in the curve of the second class. The double peak for this vowel is peculiar to certain voices, and probably there is only one resonance, which is separated into two parts by the absence of a particular partial tone 226 PHYSICAL CHARACTERISTICS OF THE VOWELS from the sound of a particular voice; this condition is indicated by the lower curves in Fig. 163. ' The characteristic curves for the vowels of the second class are shown in Fig. 166, each having two characteristic regions of resonance. The curve for mot (ma), having a single resonance at the pitch 1050, is placed at the top, and next is a curve for the same vowel in which there are MA MA MAT MET MATE m m m m 31 IB 3111 MEET C, D EF G A BC, D EF n A BC, D EF G A BC5 i'^ ''I t I 1 I I 1' I ! ' I . I I . I r. 1 ■ I D EF C A BC. D EF G A BC, ^ I ,. 1 ,1.1 ,. ,1 ...1 ,1, •IsljHaNsm i5|»|i|iNg|8|;|s|s|s| |s|^|; lts?|ti?il!l illiaililll 129 258 517 1035 2089 4138 Fig. 166. Characteristic curves for the distribution of the energy in vowels of Class II, having two regions of resonance. two resonances very close together at pitches of 950 and 1240 ; the other vowels with the pitches of their resonance regions are : mat, 800 and 1840 ; met, 691 and 1953 ; mate, 488 and 2461 ; and meet, 308 and 3100. The lower reso- nances are practically the same as for the vowels of the first class, but contain only about 50 per cent of the energy, while about 25 per cent is in the higher region. The lower and intermediate tones are stronger than in the vowels of 227 THE SCIENCE OF MUSICAL SOUNDS the first class, the fundamental often containing 10 per cent of the energy. Although each vowel is characterized by two regions of resoriance, the distinguishing characteristic is the higher resonance. The characteristic curves show the resonating properties of the vocal cavities when set for the production of the speci- fied vowels, and they have true significance throughout their lengths. These curves may be considered curves of probability, or perhaps they may be called curves of possi- bility, of energy emission when a given vowel is intoned. The mouth is capable of selective tone-emission only, that is, the only frequencies of vibration which can be emitted at one time are in the harmonic ratios. If the harmonic scale (see page 169) is placed upon the characteristic curve of a given vowel with its first line at any designated pitch, then the ordinates of the curve at the several harmonic points show the probable intensities of the various partials when the particular vowel is intoned by any voice at the given pitch. These curves show the probable intensity for that part of the energy which is characteristic of the vowel in general, but, since they are averages of many analyses, they do not show the peculiarities of individual voices aside from the vowel characteristic. Since the pitch region of the maximum emission of energy for a certain vowel is fixed and is independent of the pitch of the fundamental, it follows that the different vowels can- not be represented by characteristic wave forms. When the vowel father is intoned upon the fundamental E2t> = 154, the sixth partial, 6 X 154 = 924, is the loudest, and the wave form has six kinks per wave length. Fig. 167, a, is an actual photograph of the vowel father from a baritone voice. When the same vowel is intoned by a soprano voice 228 PHYSICAL CHARACTERISTICS OF THE VOWELS at the pitch Est? = 462, the second partial, 2 X 462 = 924, is the loudest, and the wave shows two kinks per wave length, as in b. These curves are for the same vowel, but are wholly unlike. When the vowel no is intoned by the baritone at the pitch B2b= 231, the characteristic is the Fig. 167. Photographs a and 6, though unlike, are from the same vowel ; b and c are nearly alike but are from different vowels. second partial, 2 X 231 = 462 ; the wave has two kinks and has the appearance shown at c. Wave forms b and c are alike in general appearance, but are for different vowels. The wave form therefore depends upon the pitch of intona- tion as well as upon the vowel, and one cannot in general determine from inspection alone to what vowel a given curve corresponds. Familiarity with the curves from an 229 THE SCIENCE OF MUSICAL SOUNDS individual voice will, however, often enable one to tell what vowel of this voice is represented. Either one of the word pyramids of Fig. 168 forms an outline for the classification of all vowels; starting at the top and descending to the left are the vowels characterized by single resonances of successively lower pitches ; towards the right are those characterized by two resonances, the first of which descends, while the second ascends for the successive vowels. There is a continuous transition from one vowel to the next through the entire range of each class. The number of possible vowels is indefinitely great, ha\'ing shades of tone quaUty which blend one into another. It pa pot ma I not Tpaw pat pet maw mat met poe pate mow mate pooh peat moo meet Fig. 168. Word pyramids for classification of the vowela. is believed that any other vowel from any language after analysis can be placed upon this classification frame as intermediate between some two of those in the p3rramid. It happens that the pronunciations used in this study cor- respond to nearly uniform distribution of resonances, and the vowels are distinct in sound one from another; they form what may therefore be considered a rational selection of standard vowels and give a scientific pronunciation as a basis for word formation and for phonetic spelling and writing. The continuity of vowel tone, as here described, can be easily demonstrated by intoning the vowel at the top of the pyramid, when, without interrupting the tone, by 230 ma. . maw. as, a ... a ... .. . oo) PHYSICAL CHARACTERISTICS OF THE VOWELS gradually closing the lips one may cause the intoned sound to pass through all possible vowels of the first class, mow . . . moo (pronounce only the vowels, And again by starting with the same vowel at the top of the pyramid, keeping the lips in constant position, but changing the position of the tongue, one can continuously intone all the vowels character- ized by two resonances, ma . . . mat . . . met . . . mote . . . meet (a. . .a. . .e. . .a. . .e). Many photographs have been made which confirm the pyramid classification by simple inspection, showing that Fig. 169. Photographs of the vowels moo and meet. the relations are based upon essential features. Compara- tive curves for the vowels moo and meet are given in Fig. 169. The first is for moo,- and is a very simple curve, the vowel characteristic being the single resonance which is an octave higher than the fundamental and is represented by the wavelets a. The vowel meet has two characteristic resonances, the first of which, 6, is practically identical 231 THE SCIENCE OF MUSICAL SOUNDS with that for moo, while the second is of very high pitch and is represented by the small kinks which are present throughout the curve but show most clearly near the points b; the addition of this high pitch to the sound for moo changes it to meet. The relation of the vowels moo and meet is illustrated by a common difficulty in telephone conversation. Tele- phone lines are purposely so constructed as to damp out vibrations of high frequency ; if the vowel meet is spoken into the transmitter, its high-frequency characteristic is not carried over the wire, and the sound being heard with this part eliminated is interpreted as moo; for this reason the word three is often misunderstood as two, to prevent which the r in three is trilled. Translation of Vowels with the Phonograph The phonograph permits a simple verification of the characteristics of certain vowels, since the pitch of the sounds given out by the machine can be varied by changing the speed of the motor which turns the record. When the vowel ma is recorded, the greater part of the energy is emitted in tones having a frequency of about 925 ; if the record is reproduced at the same speed as that at which it was recorded, one hears the vowel ma ; but when the speed of rotation is reduced so that sounds which previously had the pitch 925 now have the pitch 735, the phonograph speaks the vowel maw ; and still further reduction of speed gives the vowels mow and moo. ^ If maw is recorded, then the record can be made to re- produce ma by increased speed of the motor, and the other vowels mow and moo are obtained by a decreased speed as before. 232 PHYSICAL CHARACTERISTICS OF THE VOWELS In early experiments with the phonograph the vowels ma and Toaaw were recorded several times at various speeds of the cylinder, and afterwards it was impossible to identify the records, because each could be made to reproduce both vowels perfectly. The vowel ma was recorded by the voice D C M on the phonograph, and without stopping the cylinder, the phono- graph was made to speak tliis record into the phonodeik; the sound was photographed and the speed of the phono- graph cylinder was determined at the same time with a stop-watch. Analysis of the photograph showed that the fundamental pitch of intonation was 154, while the maxi- mum energy of the sound was in the sixth partial tone, having a frequency of 924 ; the corresponding speed of the cylinder of the phonograph was one turn in 0.276 second. The speed of the cylinder was reduced, till the ear judged that the vowel laaw was being given by the phonograph; the time for one turn of the cylinder was found to be 0.348 second, corresponding to a frequency of 730 for the tone of maximum energy. Similar trials were made for the vowels mow and moo, and the results of all the experiments are shown in tabular form. Vowel ma maw mow 0.572 444 461 moo Speed, sec. per turn Phonograph, n Analysis, a 0.276 924 922 0.348 730 732 0.814 311 326 Frequencies of vowels obtained by translation with the phonograph. The characteristics from the phonograph experiments for ma, maw, m.ow, and moo, are 924 (the original record), 233 THE SCIENCE OF MUSICAL SOUNDS 730, 444, and 311, respectively, while the photographic analyses give 922, 732, 461, and 326. Since in this experiment the pitch of the fundamental is lowered in the same proportion as is that of the character- istic, to the abnormally low pitch of 54 for moo, it is better to record the first tone at a pitch higher than that of normal speech, or to make the record from a contralto or soprano voice. If the vowel ma is recorded at different pitches or by different voices, when the speed is changed so that any one pitch or voice gives mow, aU other pitches and voices give mow at the same time, since the characteristic for each vowel is the same (approximately) for aU voices and pitches. A similar relation exists when records of any one of the vowels ma, maw, mow, and moo are translated by changed speed of reproduction to any other one of these vowels. In order that the loudness of a soimd may remain con- stant when the pitch is lowered, the amplitude should increase, as was explained in Lectures II and V, in the proportion shown in Fig. 113. In phonographic reproduc- tion, the amplitudes of the several component tones remain constant as the frequency is reduced, since the amplitudes are determined by the depth of the cutting in the wax; this causes a diminution in intensity proportional to the square of the speed reduction, and alters the relative loud- ness of the several component tones ; hence a translated vowel often has an unnatural sound, though it retains the vowel characteristic. This difficulty is somewhat over- come by the method of experimentation described below. When the vowel mow; is intoned by a baritone voice at the normal pitch for speech, Ejl? = 154, the characteristic 234 PHYSICAL CHARACTERISTICS OF THE VOWELS is the third partial of pitch 3 X 154 = 462. If ma is re- corded on the phonograph by the same voice at the same pitch, the characteristic is the sixth partial of pitch. 6 X 154 = 924 ; when the phonograph speed is reduced to sound mow; from this record, the fundamental pitch becomes 77, and the characteristic is the sixth partial of this lower ■ pitch, 6X 77 = 462; while this sound is clearly raow, it is not Uke a natural mow of the baritone voice, being pitched on such a sub-bass fundamental. When ma is recorded by a contralto voice on the fundamental Ejb = 308, the characteristic is the third partial of the pitch 3 X 308 = 924 ; if now this record is reproduced at a slower speed to give mow, the fundamental falls to £2!? = 154, and the characteristic is still the third partial of pitch 3 X 154 = 462 ; this translated ma of the contralto voice becomes m.ow of the baritone voice in general quahty, and has a natural sound. Phonographic translation of the vowels of the second class, mat, met, mate, and meet, is not possible, for each has two regions of resonance, as is shown in Fig. 166, the higher in- creasing in frequency when the lower decreases. Whispered Vowels The vowels can be distinctly whispered without the pro- duction of any larynx tone, that is, without fundamental or pitch and without the series of partials which determine the individuahty of the voice, but these whispered sounds must contain at least the essential characteristics of the vowels. Photographs of such whispered vowels are readily obtained and, the time signals being photographed simul- taneously, they give by direct measurement the absolute pitch of the vowel characteristics. 235 THE SCIENCE OF MUSICAL SOUNDS Many photographs have been taken of nine vowels, whis- pered by several voices ; inspection of these at once proves the general correctness of the classification of the vowels already given. Whisper records for the four vowels of the first class ma, maw, mow, and moo, are shown in Fig. 170, ma being at the top. Since the usual voice tones are entirely absent, each curve consists mainly of one frequency, that charac- /v^/v^i'^A/^^^,A/^A/v^/»»/\AA^A/^r— iV^,^A^ Fio. 170. Photographs of whispered vowels of Class I. teristic of the vowel. These curves are arranged in the or- der of the classification and it is evident that the frequency of the principal vibration in each increases from the lower to the upper record. The second class of whispered vowels, mot, met, mate, and meet, is shown in Fig. 171, mat being at the top. Each curve of this group has two distinct frequencies ; the curve for meet has the lowest and the highest, the high frequency being superposed on the lower Uke beads on a string ; the other curves in order show that the lower tone increases 236 PHYSICAL CHARACTERISTICS OF THE VOWELS in frequency, while the higher one decreases, the two being quite entangled in the upper curve for mat. The variation in the wave form, which is not periodic and therefore is not caused by beats, is perhaps due to the fact that, as the vocal cords are not in motion, the vibra- tion of the air in the mouth cavity is uncontrolled and fluctu- ates in both intensity and pitch within the characteristic limits. Fig. 171. Photographs of whispered vowels of Class II. Comparisons of forty-five curves of whispers show fre- quencies for the characteristics as given in the table ; the Vowel ma maw JXXOW moo mat met mate meet Whisper, n 1019 781 515 383 857 1890 678 1942 488 2385 391 2915 Analysis, n 922 732 461 326 800 1843 691 1953 488 2461 308 3100 Frequencies of the characteristics of whispered vowels. 23 7 THE SCIENCE OF MUSICAL SOUNDS ^ Pi o. d a. 2 o J3 238 PHYSIC AT, CHARACTERISTICS OF THE VOWELS frequencies determined by analysis of the spoken vowels are also given for reference. It appears that the resonance frequency of the mouth in whispering is somewhat higher than when speaking, though the whisper characteristics' are well within the limits of those of speech. Theory op Vowel Quality The analytical studies which have been described lead to the conclusion that intoned vowels are strictly periodic or musical sounds. It is unusual, however, to prolong a vocal sound without variation; in song a sustained tone is usually^ given some emotional expression, and in spoken words the vowels change continuously and are blended with the consonants. The photograph of the soprano voice shown in the frontispiece represents a sustained musical tone which, though simple in quality, is continually chang- ing in intensity. The flowing of speech tones from one quality to another is illustrated by the photographs, Fig. 172, ot the spoken words "Lord Rayleigh," and Fig. 184, of the words " Lowell Institute." The shghtest change in the sound causes a change in the wave form. It requires some practice, but it is not impossible to maintain a pure vowel tone unchanged for several seconds, in which time there may be hundreds of waves which are truly periodic ; Fig. 173 shows the periodicity of the vowel sound mate. Such a photograph is a complete justification of the application of harmonic analysis to the study of vowel curves. The mouth with its adjacent vocal cavities is an adjust- able resonator ; by varying the positions of the jaws, cheeks, tongue, lips, and other parts, this cavity can be tuned to a large range of pitches. When the mouth is wide open and the tongue is low, the cavity responds to a single pitch of 239 THE SCIENCE OP MUSICAL SOUNDS high frequency, and is set for the vowel father, A, Fig. 174 ; when the opening between the lips is small, oo, the pitch is lowered, as for gloom. The mouth cavities may -be ad- justed to reinforce two different pitches at one time, as has been explained by Helmholtz ; when set for the vowel meet. ;!\,':^f;.ft.*;!\> v>^y\^«r|!| I 129 2^9 517 1035 2069 41S8 Fig. 185. Analyses of voice and flute tones. It may be asked what is the effect of raising the pitch of the flute tone a little higher than shown, till its maximum loudness agrees with that of the vowel father. When the flute sounds Asj} = 461, since the second partial is the loudest, the maximum energy is at A4 :(f = 922, and the tone actually has a resemblance to the .vowel. By stopping the breath, somewhat as is done in speaking the word pa-pa, the flute imitates the vowel. 258 WORDS AND MUSIC "Opera in English" The characteristics of the several vowels, which were described in the previous lecture, are shown, superposed, in Fig. 186. The essential part of each vowel is a component tone or tones, the pitch of which is within certain limits in- dicated by the corresponding curve. A characteristic, the i Mi MAW MOW MOO MA MAT MET MATE MEET NASI 3;:6 4111 SIPUIII m m Bill 111 9S( ^i (% D EF a A BC3 D EF G A BC, D EF G A BCs D EF G A BC^ D EF G A BC, |JsN.|=|;|i|i|a|s|gNjs|8|i|s|s|g|sNN|j|s|j3|^|s|s^^ 129 258 sir 1035 2089 4188 Fig. 186. Superposed curves of vowel characteristics showing relations to voice ranges. pitch of which comes in the region where the curves for two vowels overlap, would produce an intermediate vowel which might serve for either of those specified, and, although the pronunciation of such a vowel might be different from that used in defining the characteristics, it would be readily interpreted when used in a word. This shaded pronuncia- tion of a vowel would be accepted in singing, where perfect enunciation is not expected. Many singing tones have 259 THE SCIENCE OF MUSICAL SOUNDS pitches above the characteristic ranges of certain vowels, such as gloom and meet, and these vowels cannot be sung properly at the higher pitches. The vowel father has a characteristic higher than any singing tone and can be sung under any circumstances of voice or pitch. Thus there is a scientific reason for the free use of such syllables as tra-la-la in vocal exercises. Yodeling is probably the easy flowing of varsdng vowel tones to fit the melody. The characteristics of the several vowels are given in musi- cal notation in Fig. 187 ; the notes correspond to the pitches ife *^ J^ 4=^ 1^ 77700 mow maw ma mot mat met mate meet pooh foe -paw' pa pot -pat pet pah peat Fio. 187. Characteristics of the vowels in musical notation. of maximum resonance, as shown by the curves of the pre- vious figure, and each vowel can be most clearly intoned upon the corresponding note. A vowel can also be freely intoned upon any lower note of which the characteristic note is a harmonic, such as notes an octave, a twelfth, or a fifteenth lower (in musical intervals) than the characteristic. A baritone voice can easily intone the vowel gloom, in falsetto, upon the characteristic pitch of E3, shown in the figure. The characteristics of all the vowels can be verified by the test of free enunciation. A consideration of the characteristics of the vowels leads to certain definite conclusions regarding the question, so 260 WORDS AND MUSIC widely discussed, whether grand opera originally written in a foreign language should be sung in English. No doubt every composer sets words to music with some regard for effective rendition, in doing which he conforms, perhaps unconsciously, to the natural requirements. Suppose that in the original the composer set the vowel raw, having a characteristic pitch of about 732, to the melody note F4 1, of the same pitch, the vowel can then be sung and enunciated with ease. If, in the translation, some other vowel, as no, the characteristic pitch of which is 461, falls upon this note its proper enunciation will be difficult, or impossible, since it must be sung at the pitch 732. The vocalist in attempting to sing the vowel will find the result vocally deficient and the effort perhaps physically painful, and will be emphatically of the opinion that translated opera is im- practicable. Furthermore, the auditors will hardly under- stand the English words with the forced and imperfect vowels any better than they understood the foreign language. If the translator arranges the vowels upon the same notes as were used in the original, or upon others equally suitable, the translated opera, so far as this element goes, will be just as satisfactory to both the vocalist and the auditor as was the original. The effectiveness of vocal music is not dependent upon the nationality of its words, but upon the suitability of melody to vowels, a condition which the composer fulfills through his artistic instinct. The translator of an opera must secure this adaptation by his skill ; he needs to be not only a hnguist and a poet, but also a musician and even somewhat of a physicist, since he must constantly be guided by the facts represented in the curves of vowel characteristics ; such a combination of artist and scientist is very rare. 261 THE SCIENCE OF MUSICAL SOUNDS It has been suggested that certain songs and choruses which are especially effective owe this quality to the proper relation of vowel sounds to melody notes, and the Hallelu- jah Chorus from the "Messiah" has been cited as an instance. When one considers the authors of the arguments which have been pubhshed concerning translated opera, it will be found that some soprano singers are opposed to the transla- tion, while many of those who favor it are baritones. The lines at the top of the diagram. Fig. 186, indicate the ranges in pitch of soprano and bass voices. The pitch of the so- prano voice in singing often rises above that of the charac- teristics of certain vowels, which then become diflficult ; the bass voice when highest is still below the lowest vbwel characteristic, and it is thus able to intone any vowel on any note which it can sing ; to one, translated opera seems ineffective, while to the other it causes no difficulty. Relations of the Art and Science of Music In the lectures now brought to a close we have very briefly explained the Science of Musical Sounds and have incom- pletely .described some of the methods and results of Sound Analysis. The science of sound is related to at least three phases of human endeavor, the intellectual, the utilitarian, and the aesthetic. In conclusion we will refer to some of these relations without extended comment since an adequate discussion' would require several lectures. The appreciation of knowledge for its own sake is general ; and what knowledge should be more valued than that con- cerning sound, related as it is to many of the necessities as well as to the luxuries of existence? It is true of the science of sound, as well as of all others, that 262 THE SCIENCE AND ART OF MUSIC "The larger grows the sphere of knowledge The greater becomes its contact with the unknown." Helmholtz, Koenig, and Rayleigh, by observation, experi- ment, and theory, have developed this science to magnifi- cent proportions, yet the realm of nature is so vast and varied that some other indefatigable discoverer may be able to push forward into unknown regions, and chmbing the height of some discovery, see the inspiring prospect of new, though far distant, truths, and be thrilled with the desire for their possession. The challenge of the unknown and the joy of discovery inspire him to devote himself to further exploration. While formerly the regard for science was largely confined to the academic world, within the last few years there has arisen a remarkable and widespread appreciation of scien- tific methods, and now industrial and commercial enterprises are appealing to the scientist for assistance. No sooner is a new scientific fact or process announced than there is an inquiry as to its usefulness. The science of sound will be found ready to satisfy the utilitarian demands which will be made upon it. The science of sound should be of inestimable benefit in the design and construction of musical instruments, and yet with the exception of the important but small work of Boehm in connection with the flute, science has not been extensively employed in the design of any instrument. This can hardly be due to the impossibihty of such applica- tion, but rather to the fact that musical instruments have been mechanically developed from the vague ideas of the artist as to the conditions to be fulfilled. When the artist, the artisan, and the scientist shall all work together in unity of purposes and resources, then unsuspected develop- 263 THE SCIENCE OF MUSICAL SOUNDS ments and perfections will be realized. These possibilities are becoming manifest in relation to the piano, the organ, and some other instruments. The artistic and aesthetic musician has been wont to dis- parage, if not ridicule, the development of mechanical musical instruments, such as the player-piano, yet I believe that, since the estabUshment of the equally tempered musical scale by Bach, nothing else will have contributed so much to the aesthetic development of musical art. The inventor of the Pianola little dreamed that the mechanical operation of the piano would lead to a thorough scientific study of music and musical instruments, but such has been the result. The simple mechanical player has been de- veloped into elaborate devices for the complete reproduction of artistic performances. The success of these synthetic musical instruments depends upon an analytical knowledge of all the factors of sound and music ; that is, the pure science of these subjects must be brought to bear upon the practical problems involved. The mechanical precision of such instruments reacts critically upon the artist-per- former and the composer, resulting in greater artistic per- fection ; the unlimited technical possibiUties of the machine is an incentive to the composer to write music with greater freedom. The marvelous inventions of the telephone and the talking machine could never have been developed without the aid of pure science ; a knowledge of the science of electricity and magnetism and of mechanism is not sufficient for their perfection ; an increased knowledge of the science of sound is also required. The utilitarian application of the science of sound is nowhere better illustrated than in the design of auditoriums ; 264 THE SCIENCE AND ART OF MUSIC for of what avail is a perfected musical instrument con- trolled by a master, or of what effect is an oration pronoxmced with faultless elocution, if the auditors are placed in sur- roundings which distort and confuse the sound waves so that intelUgent perception is impossible ? The artistic world has rather disdainfully held aloof from systematic knowledge and quantitative and formu- lated information ; this is true even of musicians whose art is largely intellectual in its appeal. The student of music is rarely given instruction in those scientific principles of music which are estabUshed. Years are spent in slavish practice in the effort to imitate a teacher, and the mental faculties are driven to exhaustion in learning dogmatic rules and facts. Bach said "music is the greatest of all sciences"; while this comparison may not be true at the present time, yet the construction of the equally tempered scale is clearly scientific, and it is no doubt true that the relations of the major and the minor scales, and the nature of chords and their various forms and progressions, as well as many other fundamental principles, can be explained better by science than by precept. Experience indicates that a month devoted to a study of the science of scales and chords and of melody and harmony, will advance the pupil more than a year spent in the study of harmony as ordinarily presented. Regarding the art of music Helmholtz says : "Music was forced first to select artistically, then to shape for itself, the material on which it works. Painting and sculpture find the fundamental character of their materials, form and color, in nature itself, which they strive to imitate. Poetry finds its niaterial ready formed in the words of language. Archi- tecture has, indeed, also to create its own forms ; but they 265 THE SCIENCE OF MUSICAL SOUNDS are partly forced upon it by technical and not by purely artis- tic considerations. Music alone finds an infinitely rich but totally shapeless plastic material in the tones of the human voice and artificial musical instruments, which must be shaped on purely artistic principles, unfettered by any reference to utility as in architecture, or to the imitation of nature as in painting, or to the existing symbolical meaning of sounds as in poetry. There is a greater and more absolute freedom in the use of the material for music than for any other of the arts. But certainly it is more difficult to make a proper use of absolute freedom, than to advance where external irremovable landmarks limit the width of the path which the artist has to traverse. Hence also the culti- vation of the tonal material of music has, as we have seen, proceeded much more slowly than the development of the other arts." In "Music and the Higher Education," Professor Dickin- son says : "Strange as it may seem that notes 'jangled, out of tune and harsh,' should give pleasure to any one of average intelligence, yet the abundance of evidence that they do so indicates that the training of the youthful ear to discrimina- tion between the pure and the impure is not to be neglected. . . . The guide to musical appreciation need not deem his effort wasted when he preaches upon the need of preparing the auditory sense to catch the finer shades of tone values. . . . Let the music lover not be content with imperfect in- tonation, let him learn to detect all the shades of timbre which instruments and voices afford, let him train himself to per- ceive the multitudinous varieties and contrasts which are due to the relative prominence of overtones . . . and while his ear is invaded by the surge and thunder of the full or- chestra, let him try to analyze the thick and luscious current 266 THE SCIENCE AND ART OF MUSIC into its elements, . . . turning the dense mass of tone color into a huge spectrum of scintillating hues!" For the fullest accomplishment of these ends the musician may well appeal to science. Such mathematical and physi- cal studies as we have described prepare the "infinitely rich, plastic material" out of which music is made, and they provide the methods and instruments for its analytical investigation. Nevertheless, that which converts sound into a grand symphony and exalts it above the experiments of the laboratory is something free and unconstrained and which therefore cannot be expressed by a formula. The creations of fancy and musical inspiration cannot be made according to rule, nor can they be made upon command. Wagner, one of the most inspired musicians that the world has ever known, was offered a great sum of money by the Centennial Commission to compose a Grand March for the opening exercises of the Philadelphia Exposition of 1876. Under the base influence of mere gold which he needed to pay debts, he wrote a very ordinary piece of music, quite unworthy of himself, and of which he was ashamed. In contrast to this, while under the inspiration of the death of a mythical hero in one of his great music dramas, Wagner wrote the Siegfried Death Music, sometimes called the Funeral March, which as performed at Bayreuth is perhaps the greatest and most sublime piece of instrumental music ever heard by man; it is a most profound expression of abstract grief. In "The Mysticism of Music " the late R. Heber Newton says : "Our modern world is not more distinctively the age of science than it is the age of music ; music is the art of the age of knowledge. Music is an emotional symbolism, sug- gesting that which, as feeling, lies beyond all words and 267 THE SCIENCE OF MUSICAL SOUNDS thoughts. 'Where words end, there music begins.' Music can never cease to be emotional, because thought, in pro- portion as it is deep and earnest, always trembles into feelings. The most holy place in the universe is the soul of man. All sciences lead us up to the threshold of this inner creation, this unseen universe, throw the door ajar and point us within. All arts pass through the open door into the vestibule of this inner temple. Music takes us by the hand, boldly leads us within, and closes the door behind us." (These sentences have been selected from the first twenty- nine pages.) Music is indeed a mysterious phenomenon ; tones, noises, rhythms, time values, mathematical ratios, and even silences, all conveyed to the ear by mere variations in air pressure, are its only means of action, yet with these it awakens the deepest emotions. Du Maurier, in "Peter Ibbetsen," describes the impres- siveness of musical sounds as follows : " The hardened soul melts at the tones of the singer, at the unspeakable pathos of the sounds that cannot lie; ... one whose heart, so hopelessly impervious to the written word, so helplesslj- callous to the spoken message, can be reached only by the organized vibrations of a trained larynx, a metal pipe, a reed, a fiddle string — by invisible, impalpable, incompre- hensible little air-waves in mathematical combinations, that beat against a tiny drum at the back of one's ear. And these mathematical combinations and the laws that govern them have existed forever, before Moses, before Pan, long before either a larynx or a tympanum had been evolved. They are absolute! " I would like to quote again from the "Letters of Sidney Lanier," the poet-musician. " 'Twas opening night of 268 THE SCIENCE AND ART OF MUSIC Thomas' orchestra, and I could not resist the temptation to go and bathe in the sweet amber seas of this fine music, and so I went, and tugged me through a vast crowd, and after standing some while, found a seat, and the baton tapped and waved, and I plunged into the sea, and lay and floated. Ah ! the dear flutes and oboes and horns drifted me hither and thither, and the great violins and small violins swayed me upon waves, and overflowed me with strong lavations, and sprinkled glistening foam in [my face, and in among the clarinetti, as among waving water-lilies with flexible stems, I pushed my easy way, and so, even lying in the music-waters, I floated and flowed, nay soul utterly bent and prostrate." And again Lanier writes of an orchestral performance, in which he himself was the principal flutist : "Then came our piece de resistance, the ' Dream of Christmas ' overture, by Ferdinand Hiller. Sweet Heaven — how shall I tell the gentle melodies, the gracious surprises, the frosty glitter of star-light, and flashing of icy spicules and of frozen surfaces, the hearty chanting of peace and good-will to men, the thrilhng pathos of virginal thoughts and trembling anticipa- tions and lofty prophecies, the solemn and tender breathings- about of the coming reign of forgiveness and of love, and the final confusion of innumerable angels flying through the heavens and jubilantly choiring together." A musician must be skilled in the technic of music, he must be trained in musical lore, and above all else he must be an artist ; when to these qualifications is added inspira- tion, the conditions are provided which have given to the world its greatest musicians. But will not the creative mu- sician be a more powerful master if he is also informed in regard to the pure science of the methods and materials of his art? WiU he not be able to mix tone colors with 269 THE SCIENCE OP MUSICAL SOUNDS greater skill if he understands the nature of the ingredients and the effects which they produce? Does not the inter- pretative musician need a knowledge of the capacity, possi- bilities, and limitations of the tonal facilities at his command, as well as a knowledge of the construction of the written music, in order that he may render a composition with consummate effect? While the science of music, because of its incomplete development, has never exerted its full influence on the art, yet it should be appreciated and more generously cultivated for the great assistance which it can be made to yield. Not only will the creative and interpretative artist be the better able to control the purely mechanical means of operation because of complete knowledge, but the receptive musician will derive greater pleasure from this physical phenomenon if he is also cognizant of its marvelous but systematic complexity. The musically uncultivated and scientifically untrained listener may greatly enjoy music, but this enjoyment is a gratification of the senses. If to this pleasure of sensation is added the intellectual satisfac- tion of an understanding of the purposes of the composer, the facilities at the command of the interpreter, and the physical effects received by the hearer, then music truly becomes a source of exquisite deUght which so pervades and thrills one's being that he is carried away "on the golden tides of music's sea." No other art than music, through any sense, can so transport one's whole consciousness with such exalted and noble emotions. 270 APPENDIX REFERENCES General References : H. von Helmholtz, Sensations of Tone, translated by A. J. EUis, 2 English ed., London (1885), 576 pages. The most complete account of the phenomena of sound as related to sensation and to music. H. von Helmholtz, Vorlesungen uber die mathematischen Principien der Ahustik. Liepzig (1898), 256 pages. A concise mathematical treat- ment of certain acoustic phenomena, as treated by Helmholtz in his university lectures. Lord Rayleigh, Theory of Sound, 2 vols., 2 ed., London (1894), 480 + 504 pages. The most comprehensive treatise on the theory of sound, largely mathematical in treatment. E. H. Barton, Text-BooK of Sound, London (1908), 687 pages. An experimental and theoretical treatise on sound in general. R. Koenig, Quelgues Expiriences d' Acoustique, Paris (1882), 248 pages. An account of Koenig's own experimental researches, consisting of a collection of papers published in various scientific journals together with others not published elsewhere. A. Winkelmann, Handbuch der Physik, 2 aufl., Liepzig (1909), Bd. II, Akustik, F. Auerbach, 714 pages. An encyclopedic treatment of the whole field of acoustics, experimental and theoretical ; contains 'thousands of references arranged according to subjects. J. A. Zahm, Sound and Music, Chicago (1892), 452 pages. A popular, yet scientific, account of sound in general, and in particular with reference to music. 271 THE SCIENCE OF MUSICAL SOUNDS Special Rbfbbences : — The number in parenthesis, following a sub- ject, refers to the page of this book where the subject is treated. 1. Velocity of sound (6). Barton, TexUBook of Sound, pp. 513-553. Winkehnann, Akustik, S. 494^588, a fuU discussion of velocity, with more than 200 references. J. VioUe, Congr6s International de physique, I, Paris (1900), pp. 228-250. 2. Simple harmonic motion (7). J. D. Everett, Vibratory Motion and Sound, London (1882), pp. 73-83. 3. New device for producing simple harmonic motion (11). Mr. J. C. Smedley, of Cleveland, in 1912, devised the simple harmonic move- ment shown, as a result of his interest in the harmonic synthesizer de- scribed in Lecture IV. 4. Si(lney Lanier (24). Letters of Sidney Lanier, New York (1899), p. 68. 5. The siren (28). Hehnholtz, Sensations of Tone, p. 161. 6. Determination of pitch (28). Winkehnann, Akustik, S. 178-227, with many references. Barton, Text-Book of Sound, pp. 560-580. 7. Tuning forks (29). Winkelmann, Akustik, S. 345-367, an extended account, with more than 100 references. E. A. Kielhauser, Die. Stimm- gabel, Leipzig, (1907), 188 pages. A. J. EUis, Appendix to HelmkoUz's Sensation of Tone, pp. 443-446. R. Hartmann-Kempf, Elektro-Akus- tische Untersuchungen, Frankfort (1903), 255 pages, with many plates. 8. Tuning-fork resonator (31). R. Koenig, Qudques Experiences, p. 180. 9. Temperature coefficient of tuning fork (31). R. Koenig, Annalen der Physik, 9, 408 (1880) ; Quelques Experiences, p. 182. 10. Relation of amplitude and pitch of tuning fork (32). R. Hart- mann-Kempf, Electro-Akustische Untersuchungen, Frankfort (1903), S. 28-64. 11. Electrically driven tuning fork (33). Rayleigh, Theory of Sound, I, pp. 65-69. Barton, Text-Book of Sound, p. 361. 272 APPENDIX 12. Scheibler's tonometer (37). Helmholtz, Sensations of Tone, pp. 199, 443. 13. Koenig's tonometer (37). Zahm, Sound and Music, p. 74. 14. Lissajous's figures (20, 37). J. Lissajous, Comptes Rendus, Acad. Sci. Paris (1855). Winkelmann, Akustik, S. 42-59. These figures were described by Lissajous in 1855, but they had been previously described by Nathaniel Bowditch, of Salem, in 1815; Mem. American Academy of Arts and Sciences, 3, 413 (1815). J. Lovering, Proc. American Academy of Arts and Sciences, N. S. 8, pp. 292-298. 15. French pitch (38). R. Koenig, Quells Experiences, p. 190; Annalen der Physik, 9, 394r-417 (1880). 16. The clock-fork (38). Niaudet, Comptes Rendus, Acad. Sci. Paris, Dec. 10 (1866). Koenig, Quelques Experiences, p. 173. 17. Flicker (44). S. H. and P. H. Gage, Optic Projection, Ithaca (1914), pp. 423-427. 18. Highest audible sound (45). E. Koenig, Annalen der Physik, 69, 626-660, 721-738 (1899). 19. History of pitch (49). A. J. EUis, Appendix Helmholtz' s Sensa- tions of Tone, pp. 493-513. 20. Musical pitch in America (49). Charles R. Cross, Proc. American Academy of Arts and Sciences, 35, 453-467 (1900). 21. Loudness of sound (53). Winkelmann, Atesfifc, S. 228-254. A. G. Webster, Physical Review, 16, 248 (1903). 22. Acoustic properties of auditoriums (57). W. C. Sabine, American Architect, 68 (a series of papers) (1900) ; Proc. American Academy of Arts and Sciences, 42, 51-84 (1906) ; American Architect, 104, 252-279 (1913). 23. Acoustic properties of auditoriums (58). F. R. Watson, Bulletin No. 73, Engineering Experiment Station, University of Illinois, 32 pages (1914) ; gives references to forty-one papers on architectural acoustics ; Physical Review, 6, 56 (1915). 24. Acoustic properties of auditoriums (58). F. P. Whitman, Science, 38, 707 (1913); 42, 191-193 (1915). T 273 THE SCIENCE OF MUSICAL SOUNDS 25. Model for combinations of waves (59). E. Grimsehl, Zeitschrift f. d. physikalischen u. chemischen Unterricht, 17, 34 (1904) ; Physikalische Zeitschrift (1904) ; Frick's Physikalische Technik, Leipzig (1905), I, S. 1346. 26. Tone quality (62). Helmholtz, Sensations of Tone, p. 33. 27. Phase and tone quality (62). Helmholtz, Sensations of Tone, p. 126. 28. Phase and tone quality (63). R. Koenig, Annalen der Physik, 57, 555-566 (1896). 29. Phase and tone quality (63). F. Lindig, Annalen der Physik, 10, 242-269 (1903). 30. Phase and tone quality (63). M. G. Lloyd and P. G. Agnew, Bulletin of the Bureau of Standards, 6, 255-263 (1909). 31. Phase and tone quaUty (63). Winkehnann, Akustik, S. 268-278. Barton, Text-Book of Sound, pp. 605-607. 32. Resonators (68). Helmholtz, Sensations of Tone, pp. 36-49, 372; Vorlesungen, S. 246. Rayleigh, Theory of Sound, II, pp. 170-235. 33. The phonoautograph (71). Leon Scott, Cosmos, 14, 314 (1859). 34. Manometric flames (73). R. Koenig, Annalen der Physik, 146, 161 (1872) ; Quelques Expiriences, pp. 47-70. 35. Photographing manometric flames (74). Nichols and Merritt, Physical, Review, 7, 93-101 (1908). 36. Vibrating flames (75). J. G. Brown, Physical Review, 33, 442-446 (1911). 37. The oscillograph (75). A. Blondel, Comptes Rendus, Acad. Sci. Paris, 116, 502, 748 (1893). W. Duddell, Proc. British Association for the Advancement of Science, Toronto (1897), p. 575. 38. Oscillograph records (76). D. A. Ramsey, The Electrician, Sept. 21 (1906). 39. The phonograph for acoustical research (77). L. Hermann, Pfluger's Archiv, 45, 282 (1889) ; 47, 42, 44, 347 (1890) ; and others. 274 APPENDIX 40. The phonograph for acoustical research (77). L. Bevier, Physical Review, 10, 193 (1900). 41. Enlarging phonograph records (77). E. W. Scripture, Experi- mental Phonetics, Washington (1906), 204 pages. 42. The phonodeik (79). D. C. Miller, Physical Review, 28, 151 (1909) ; Science, 29, 471 (1909) ; Proc. British Association for the Advance- ment of Science, Winnipeg (1909), p. 414; Proc. British Association for the Advancement of Science, Dundee (1912), p. 419; Engineering, London, 94, 550 (1912). 43. The demonstration phonodeik (85). D. C. Miller, before the American Physical Society and the American Association for the Advance- ment of Science, Boston meeting, Dec. 1909. 44. Photographing waves of compression (88). A. Toepler, Annalen der Physik, 127, 556 (1866); 131, 33, 180 (1867). E. Mach, Wiener Berichte, 77, 78, 92. R. W. Wood, Philosophical Magazine, 48, 218 (1899) ; Physical Optics, 2 ed., New York (1911), page 94. Foley and Souder, Physical Renew, 35, 373-386 (1912). W. C. Sabine, American Architect, 104, 257-279 (1913). 45. Fourier's Series (92, 134). J. B. J. Fourier, La Theorie Analytique de la Chaleur, Paris (1822) ; The Analytical Theory of Heat, English translation by Alexander Freeman, Cambridge (1878). 466 pages. 46. Fourier's Series (93). W. E. Byerly, Fourier's Series and Spheri- cal Harmonics, Boston (1893). H. S. Carslaw, Fourier's Series and Integrals, London (1906). C. P. Steinmetz, Engineering Mathematics, 2 ed.. New York (1915), pp. 94-146. Franklin, McNutt, and Charles, Calculus, South Bethlehem (1913), pp. 199-209. Carse and Shearer, Fourier's Analysis and Periodogram Analysis, London (1915). 47. Fourier's Series (97). CP-Steionietz, Engineering Mathematics, 2 ed., New York (1915), p. 112. 48. Henrici's harmonic analyzer (98). 0. Henrici, Philosophical Magazine, 38, 110 (1894). H. de Morin, Les Appariels d' Integration, Paris (1913), pp. 162, U71. E. M. Horsburgh, Modem Instruments of Calculation, London (1914), p. 223. 275 THE SCIENCE OF MUSICAL SOUNDS 49. Instruments for harmonic analysis and synthesis (100, 104, 114). D. C. MiUer, Journal of the Franklin Institute (1916). 50. Harmonic analyzer and synthesizer (129). Lord Kelvin, Proc. Royal Society, 27, 371 (1878) ; Kelvin and Tait, Natural Philosophy, Part I, Appendix B', Vll, Cambridge, (1896) ; Kelvin, Popular Lectures, Vol. Ill, p. 184. 51. Tide-predicting machine (131). E. G. Fisher, Engineering News, 66, 69-73 (1911). Special Publication No. 32, United States Coast and Geodetic Survey, Washington (1915). 52. Harmonic analyzer and synthesizer (131). Michelson and Strat- ton, American Journal of Science, 5, 1-13 (1898) ; Philosophical Maga- zine, 45, 85 (1898) ; Michelson, Light Waves and Their Uses, Chicago (1903), p. 68. 53. Harmonic analyzer (132). G. R. Rowe, Electrical World, March 25 (1905). O. Mader, Elektrochemische Zeitschrift, Nr. 36 (1909); theory given by A. Schreiber, Physikalische Zeitschrift, 11, 354 (1910). L. W. Chubb, The Electric Journal (Pittsburgh), Feb. 1914, May, 1914. 54. Various harmonic analyzers (132, 134). Carse and Urquhart, Hors- burgh's Modern Instruments of Calculation, London (1914), pp. 220- 253, 337. H. de Morin, Les Appariels d' Integration, Paris (1913), pp. 147-188. W. Dyck, Catalogue (Munich Mathematical Exposition), Munich (1892). E. Orlich, Aufnahme und Analyse von Wechselstrom- kurven, Braunschweig (1906). G. U. Yule, Philosophical Magazine, 39, 367-374 (1895). J. N. LeConte, Physical Review, 7, 27-34 (1898). T. Terada, Zeitschrift fiir Instrumentenkunde, 25, 285-289 (1905) . 55. Harmonic analysis in meteorology (133). Strachey, Proc. Royal Society, 42, 61-79 (1887). Steinmetz, En-gineering Maihemalics, New York (1915), p. 125. 56. Harmonic analysis in astronomy (133). A. Schuster, Terrestrial Magnetism, 3, 13 (1898). H. H. Turner, Tables for Facilitating Harmonic Analysis, Oxford (1913). A. A. Michelson, Astrophysical Journal, 38, 268-274 (1913). A. E. Douglass, Astrophysical Journal, 40, 326-331 (1914). Carse and Shearer, Fourier's Analysis and Periodogram Analysis, London (1915), p. 34. 276 APPENDIX 57. Harmonic analysis in mechanical engineering (133). S.P.Thomp- son, Proc. Physical Society, London, 33, 334r-343 (1911). W. E. Dalby, Valves and Valve-Gear Mechanism, London (1906), pp. 328-353. 58. Periodogram analysis for non-periodic curves (133, 141). Carse and Shearer, Fourier's Analysis and Periodogram Analysis, London (1915), 66 pages. See also references No. 56. 59. Harmonic analysis (134). C. P. Steinmetz, Engineering Mathe- m,aties, 2 ed., New-York (1915), pp. 114r-134. 60. Harmonic analysis (134). C. Runge, Zeitschrift fiXr Mathematik und Physik, 48, 443-456 (1903), 62, 117-123" (1905) ; Erlauterung des Rechnungsformulars, Braunschweig (1913). 61. Harmonic analysis (134). Bedell and Pierce, Direct and Alter- nating Current Manual, 2 ed.. New York (1911), pp. 331-344. 62. Harmonic analysis (135). F. W. Grover, Bulletin of the Bureau of Standards, 9, 567-646 (1913). H. 0. Taylor, Physical Review, 6, 303- 311 (1915). 63. Harmonic analysis (135). S. P. Thompson, Proc. Physical Society, London, 19, 443-450 (1905), 33, 334-343 (1911). 64. Graphical method for harmonic analysis (135). J. Perry, The Electrician, 36, 285 (1895). A. S. Lungsdoii, Physical Review, 12, 184- 190 (1901). W. R. Kelsey, Physical Determinations, London (1907), pp. 86-93. 65. Graphical methods for harmonic analysis (135). Carse and Urquhart, Horsburgh's Modern Instruments of Calculation, London (1914), pp. 247, 248; various articles in The Electrician (1895), (1905), (1911). 66.< Resonance effects in records of sounds (143). D. C. Miller, Proc. Fifth International Congress of Mathematicians, Cambridge (1912), II, pp. 245-249. 67. Vibrating diaphragms and sand figures (151). E. F. F. Chladni, Theorie des Klanges, Leipzig (1787).- Winkehnaim, Akustik, S. 368-401. ^ 68. Resonance (177). E. H. Barton, Text-Book of Sound, pp. 146-148. Helmholtz, Sensations of Tone, pp. 143, 405. 277 THE SCIENCE OF MUSICAL SOUNDS 69. Effect of resonator on tuning fork (178). R. Koenig, Quelques Experiences, p. 180. 70. Damped vibration (179). E. H. Barton, Text-Book of Sound, pp. 91-96. 71. Material and tone quality (180). C. von Schafhautl, Allgemeine Musikalische Zeitung, Leipzig (1879), S. 593-599, 609-632. 72. Material and tone quality (180). D. C. Miller, Science, 29, 161- 171 (1909). 73. Beat-tones (183). Zahm, Sound and Music, pp. 322-340. 74. Simple tones (185). Helmholtz, SensaMons of Tone, p. 70. 75. Octave overtones in tuning forks (189). E. H. Barton, Text-Book of Sound, pp. 395-399. 76. The Choralcelo (189). The Music Trade Review, April 29, 1911. 77. The flute (191). Theobald Boehm, The Flute and Flute-Playing in Acoustical, Technical, and Artistic Aspects, English translation by D. C. Miller, Cleveland (1908), 100 pages. 78. Sidney Lanier (191, 202). Poems of Sidney Lanier, New York, (1903), page 62. 79. Vibration of violin strings (194). Helmholtz, Sensations of Tone, pp. 74-88, 38^387 ; Vorlesungen, S. 121-139. 80. Vibration of violin strings (194). H. N. Davis, Proc. American Academy of Arts and Sciences, 41, 639-727 (1906) ; Physical Review, 22, 121 (1906), 24, 242 (1907). 81. Vibration of violin strings (194). E. H. Barton, Text-Book of Sound, pp. 416-436 ; and numerous papers in the Philosophical Magazine for 1906, 1907, 1910, and 1912. 82. Violin tone quality (195). P. H. Edwards, Physical Review, 32, 23-37 (1911). C. W. Hewlett, Physical Review, 36, 359-372 (1912). 83. Theory of vowels (216). A general discussion of vowel theories, together with references to the published works of a very long list of investigators, will be found in the following books : Helmholtz, Sensations 278 APPENDIX of Tone, pp. 103-126. Lord Rayleigh, Theory of Sound, vol. II, pp. 469- 477. Winkelmann, Akustik, S. 681-705, contains about two hundred references to papers by more than a hundred authors. 84. Analysis of vowels (217). L. Bevier, Physical Review, 10, 193, (1900) ; 14, 171, 214 (1902) ; 15, 44, 271 (1902) ; 21, 80 (1905). 85. Theory of vowel quality (217). E. W. Scripture, Researches in Experimental Phonetics, Washington (1906), pp. 7, 109. 86. Physical characteristics of the vowels (217). D. C. Miller, before the American Physical Society and the American Association for the Advancement of Science, Atlanta meeting (1913-1914). 87. Vowel characteristics (240). R. Koenig, Comptes Rendus, Acad. Sci. Paris, 70, 931 (1870, Quelques ExpSriences, p. 42). 88. Artificial vowels (244). Marage, Physiologie de la Voix, Paris, (1911), p. 92. 89. The wave-siren (245). R. Koenig, Annalen der Physik, 57, 339- 388 (1896). 90. Tuning-fork synthesis of tones (245). Hehnholtz, Sensations of Tones, pp. 123-128, 398-400. 279 INDEX Acoustics of auditoriums, 56, 89, 264. Agnew and Lloyd, phase and tone quality, 63. Alternating current, for phase effect, 63 ; for tuning fork, 32 ; waves, 137. Amplitude, 7, S, 53 ; effect of, on pitch of tuning fork, 32 ; related to loudness, 53, 144. Amplitude and phase calculator, 103, 123. Analysis, arithmetical and graphical, 133 ; harmonic, 92 ; by inspection, 136 ; of organ-pipe curve, 125 ; of violin curve, 101, 103 ; wave method of, 92. Analyzer, harmonic, 97 ; Henrici, 98, 108; various, 128. Arithmetical harmonic analysis, 133. Art and science of music, 1, 262. Art of piano playing, 208, 264. Artificial vowels, 244. Atmospheric vibration, 2, 17, 20. Audibility, limits of, 42. Auditoriums, acoustics of, 56, 89, 264. Auerbach, theory of vowels, 216. Automatic piano, 208, 264. Axis of curve determined, 107, 124 ; photographed, 82. B Bach, equally tempered scale, 64, 264, 265. Barton, violin string, 194. Bass voice, 205 ; vowels, 229. Beats, 33, 87, 138, 183. Beat-tone, 62, 63, 183 ; of violin, 198. Bedell and Pierce, harmonic analysis, 134. Bell, 62 ; glass, 4 ; photograph of sound - of, 141. Bell, telephone, 75. Bell and Tainter, graphophone, 77. Berliner, gramophone, 77. Berlioz, the violin, 196. Beuier, analysis of phonograph records, 77 : theory of vowels, 217. Bibliography, Appendix, 271. Blondell, oscillograph, 75. Boehm, inventor of flute, 191, 263. Boston Symphony Orchestra, adopts inter- national pitch, 50. Bottles, tuned, 22. Bowing, violin, reversal of, 197. Brown, manometric flame, 75. Bugle, 67. Camera for photographing sound waves, 82. Capsule, manometric, 73. Cards for records of harmonic analysis, 122, 123, 165, 167. Carse and Urquhart, harmonic analysis, 134. Centennial Exposition, Philadelphia, Koenig's exhibit, 46 ; Grand March for, 267. Century Dictionary, list of vowels, 217. Characteristic noises, 24, 185. Characteristics of vowels, 215, 225, 259 ; see vowels. Chart for sound analysis, 168. Chladni, sand figures, 151, 153. Choralcelo, 189. Chromatic scale, 35, 48. Clarinet, 176 ; curve, analyzed by in- spection, 138 ; tone quality of, 199, 251. Classification of vowels, 225 ; see vowels. Clifford, harmonic analysis, 134. Clock-fork, 28, 38, SO. Cobb, L. N., phonodeik, 82. Coefficient, in Fourier equation, 94, 97, 123 ; temperature, of fork, 31. Collision balls, 18. Color, tone, 25, 58 ; see tone quality. Compound pendulum, 18. Compression wave, 17 ; photographs of, 88. Convergence of series, 115. Correction of analyses, 162 ; curve, 163 ; of sound waves, 172. Cosine curve, 11. 281 INDEX Costa, Sir Michael, philharmonic pitch, 49. Cross, history of pitch, 50. Curve, axis of determined, 107, 124 ; correction, 163 ; cosine, 11 ; energy, 170 ; enlarging, 108 ; graphical study of, 115; simple harmonic, 11; sine, 11 ; synthesis of harmonic, 110. D Damped vibration, 179. Davis, violin string, 194. Demonstration phonodeik, 85, 214. Diagram of sound analysis, 169. Diapason Normal, 50. Diapason organ pipes, 145. Diaphragm, 70 ; effect of on sound rec- ords, 142 ; response of, 148 ; influence of diameter, 149 ; effect of clamping, , 150 ; modes of vibration, 151 ; free periods of, 153 ; influence of mount- ing, 155. Dickinson, "Music and Higher Educa- tion," 266. Displacement, longitudinal and trans- verse, 16. Donders, theory of vowels, 215. Drum, 71. Duddell, oscillograph, 75. E Ear, 20, 22, 53, 62, 68, 70; pitch de- termined by, 33. Edison, phonograph, 76. Edwards and Hewlett, violin string, 194. Elasticity, 6, 14. Electro-magnetic operation of tuning fork, 32. Ellis, tonometer, 37; history of pitch, 49 ; theory of vowels, 216. Energy of sound, 53, 100, 167, 179 ; dis- tribution of, 170, 220 ; in piano tone, 179 ; of vowels, 226, 227. Engineering, harmonic analysis in, 133 135. Enlarging curves, 108. Epoch of component of curve, 126 ; see pAose. Equally tempered scale, table of fre- quencies, 48; compared with har- monics, 64; invention of, 264, 265; tuning forks for, 34. Equation, Fourier's, 93, 140. Errors in sound records, 142 ; correctine 162. 282 Everett, device for simple harmonic motion, 10. Explosion of skyrocket, photographed, 139. Explosive sounds, 6, 139, 217. Figures, Lissajous's, 20, 28, 37, 38, 41, 249. Fine arts, 1, 265, 268. Fireworks, photographed, 139. Fixed pitch theory, of vowels, 216, 257 ; of instruments, 257. Flame, manometric, 73. Flicker, in moving pictures, 43. Flute, 2, 23, 68, 176 ; analysis of tone, 171 ; effect of material of, 180, 192 ; of gold, 192 ; simple tone of, 185 ; tone compared to tuning fork, 185, to voice, 258; tone quality of, 190, 251. Foley and Souder, compression waves, 88. Forced vibration, 177. Fork, tuning, see tuning fork. Fourier, theorem, 92, 115, 122, 140. Franklin Institute, Journal of, 114. Free period, 71, 143, 153, 158, 177. French horn, tone quality of, 202, 213. French vowels, 218. Frequency, 7, 25, 26 ; and loudness, 53, 144 ; see pitch. Fuller, Gen. Levi K., musical pitch, 50. Fundamental tones, 62. Funeral March, Siegfried, 267. Gallon, whistle, 44. Generator, sound, 175. Geophysics, harmonic analysis in, 133. German vowels, 218. Glass bell, 4. Gramophone, 77, see talking machine. Graphical harmonic analysis, 133, 135. Graphical presentation of analyses, 166, 219. Graphophone, 77, see talking machine. Grassmann, theory of vowels, 215. Grover, harmonic analysis, 134. H Hallelujah Chorus, Messiah, 262. Handel, pitch, 49 ; trumpeter, 29. Harmonic analysis, 92 ; aritlmietical and graphical, 133 ; complete process, 120 ; INDEX example of, 122 ; by inspection, 136 ; limitations, 140 ; by machine, 97 ; verified by synthesis, 128. Harmonic analyzer, Kelvin's, 128 ; Michelson's, 131 ; Rowe's, 132 ; Mader's, 132; Chubb's, 132; Hen- rici's, 98 ; extended to 30 components, 100. Harmonic curves, 11, 92. Harmonic plotting scale, 165, 169, 228. Harmonics, 62 ; tune in, 68. Harmonic synthesizers, 110; Michel- son's, 131. Harmony, study of, 265. Hauptmann, musical critic, 23. Helmholtz, art of music, 265 ; law of tone quality, 62, 63 ; limits of audibility, 42, 43 ; resonance of mouth, 240 ; resonators, 68 ; simple tones, 185 ; theory of vowels, 215 ; violin string, 194 ; vowel apparatus, 245, 263. Henrici, harmonic analyzer, 98, 120. Hermann, analysis of phonograph records, 77 ; theory of vowels, 216. Hewlett and Edwards, violin string, 194. Hitler, Ferdinand, composer, 269. Horn, French, tone quality of, 202, 213. Horn, resonating, effect of, on sound records, 142, 156 ; flare of, 160 ; length of, 159 ; of various materials, 157 ; resonance of, 161. Horsburgh, "Instruments of Calcula- tion," 132, 135. Hiighes, microphone, 75. Ideal musical tone, 204, 212. Inharmonic components, 113, 141. Inharmonic partials, 62, 141, 188, 201. Inspection, harmonic analysis by, 136. Integrator, 97, 98, 108. Intensity of sound, 25, 53 ; of simple sound, 144 ; see energy. International pitch, 49, 50. Interrupter fork, 32. K Kelvin, Lord, 20 ; harmonic analyzer, 128 ; harmonic synthesizer, 110 ; tide predictor, 129. Kintner, harmonic analysis, 134. Koenig, clock-fork, 38, 50 ; limits_ of audibility, 45, 46 ; manometric flame, I 73 ; phase and tone quality, 62 ; phonautograph, 71 ; resonance box for fork, 177 ; resonance of mouth, 240 ; scientific pitch, 51 ; tonometer, 37; tuning forks, 29, 31, 50; vowel characteristics, 240 ; wave siren, 244. Lanier, quotations from writings, 24, 191, 202, 268, 269. Lavignac, the horn, 202. Leyden jar, photographing sound from, 88. Limits of pitch, 42. Lindig, phase and tone quality, 63. Lissajous, figures, 20, 28, 37, 38, 41; method of tuning, 249. Lloyd and Agnew, phase and tone quality, 63. Lloyd, theory of vowels, 216. Logarithmic scale, 145, 147, 168. Longitudinal displacement, 16, 20 ; vi- bration, 3 ; wave, 15. Loudness, 25, 53 ; of simple sound, 144 ; see energy. Lowell Institute, photograph of the words, 239, 255. Lucia di Lammermoor, Mad Scene, 194; Sextette, frontispiece, 211, 239. M Mach, compression waves, 88. Mad Scene from Lucia, 194. Manometric capsule, 73. Manometric flame, 73. Marage, imitates vowels, 244. Marloye, inventor of resonance box, 31. Material affecting sound waves, 179. Maurier, du, "Peter Ibbetsen," 268. Mechanical, harmonic analysis, 97 ; syn- thesis, 110 ; calculation, 103, 168. Merritt and Nichols, manometric flames, 74. Messiah, "Hallelujah Chorus," 262. Meteorology, harmonic analysis in, 133. Michelson, harmonic analyzer and syn- thesizer, 131. Microscope, vibration, 38, 41, 195. Middle C, 49. Molecular vibration, 3. Morin, "Les Appariels d'lnt^gration," 132. ' Motion, of one dimension, 20, 85 ; pen- dular, 6 ; simple harmonic, 6 ; vibra- tory, 6 ; wave, 13. 283 INDEX Mouth, resonance of, 228, 239. Moving picture apparatus, flicker, 43. Mozart, pitch in time of, 49. Musical scale, 47. Music, art of, 1 ; science of, 1 ; science and art of, 262, 269 ; photograph of Sextette, frontispiece. N Naval architecture, harmonic analysis in, 133. Newton, "Mysticism of Music," 267. Niaudel, clock-fork, 38. Nichols and Merritt, manometric flames, 74. Nodes, 4 ; in a string, 67 ; shown by sand figures, 151. Noise, 21, 24 ; characteristic, 24, 185. Non-periodic and periodic curves, 140 ; vibrations, 22. O Oboe, tone quality of, 199, 251. Ohm, law of acoustics, 62, 140. Opera in English, 259. Optical method, Lissajous's, 20, 28, 37, 41, 249. Order, of partials, 63 ; of components, 97, 137. Organ, 33, 42. Organ pipe, analysis and synthesis of sound wave, 122, 127 ; largest, 42 ; open diapason and stopped diapason, 145; smallest, 44; tibia, 247; for sound synthesis, 246 ; of uniform loud- ness, 146 ; of various materials, 180. Oscillograph, 75. Overtones, 62, 64 ; combination of, 212. Pantograph, 10. Paris, Conservatory of Music and Grand Opera, 38. Partials, inharmonic, 62, 141, 201. Partial tones, 62, 140, 168, 175. Pendular motion, 6, 12. Pendulum, simple, 12 ; compound, 19. Period, 7, 8 ; see free period, frequency, pitch. Periodic and nonperiodic curves, 140 ; vibrations, 22, Perry, harmonic analysis, 134. Persistence of vision, 43. Phase, defined, 7, 8 ; does not affect tone quality, 166, 197 ; effect on tone, 62 ; explained, 61, 126 ; relative, deter- mined with synthesizer, 114. Phase and amplitude calculator, 103. Phonautograph, 71. Phonodeik, 28 ; described, 78 ; charac- teristics of, 143 ; for demonstration, 85 ; for determining pitch, 87 ; for tone synthesis, 249 ; for tuning, 249. Phonograph, 76 ; translation of vowels, 232 ; see talking machine. Piano, 33. 42, 70, 176, 178 ; automatic, 208, 264 ; duration of sound, 179 ; tone quality of, 207 ; touch, 208. Pianola, 208, 264. Pierce and Bedell, harmonic analysis, 134. Pin-and-slot device, 7, 110. Pitch, 25, 26 ; American, 49 ; concert, 49 ; determined by beats, 33 ; deter- mined with the phonodeik, 87 ; diapa- son normal, 50 ; French, 50 ; high, 49 ; international, 49 ; Koenig's, 51 ; limits of, 42 ; low, 50 ; philharmonic, 49 ; philosophical, 51 ; scientific, 51 ; Stuttgart, 50 ; see frequency. Planimeter, 97, 107, 124. Player piano, 208, 264. Plotting sound analyses, 166, 219. Portrait, harmonic analysis and syn- thesis of, 118, 141 ; wave form, 120. Prediction of tides, 129. Pressure wave, 17, 70, 88. Profile, harmonic analysis of, 119, 141. Pyramid, classification of vowels, 230. Q Quality of sound, 25, 58 ; see tone quality. R Rayleigh, Lord, photograph of the words, 239, 263 ; theory of vowels, 216. Reed instruments, 199. Reference books, list of. Appendix, 271. Relative pitch theory, of vowels, 216, 257 ; of instruments, 257. Resonance, 176 ; box for tuning forks, 31, 177; curves, 148, 150; of horn, 158; sharpness of, 177 ; of violin body, 197 ; of vocal cavities, 22S, 239. Resonators, 69, 175, ITS. Response to sound, ideal, 144 ; actual, •145. Reverberation, 57, 5S, 91. 284 INDEX Reversal of ^^olin bow, 197. Rosa, harmonic analysis, 134. Runge, harmonic analysis, 134. Rust, effect of, on tuning fork, 33. Sabine, acoustics of auditoriums, 57, 88. Sand figures, Chladni, 151. Scale, chromatic, 35 ; equally tempered, frequencies, 48 ; musical, 47. Scale, harmonic graduated, 165, 169, 228 ; logarithmic, 145, 147, 168. Schafh&utl, influence of material on tone, 180. Scheibler, tonometer, 37, 50. Science and art of music, 1, 262, 269. Scott, phonautograph, 71. Scripture, analysis of phonograph record, 77 ; theory of vowels, 217. Sensation, 1, 2, 43, 53, 270. Senses, 1. Series, Fourier, 93 ; studied with syn- thesizer, 115. Sextettehom Lucia, frontispiece, 211, 239. Shore, inventor of tuning fork, 29. Siegfried, Death Music, 267. Silences, of musical value, 268. Simple harmonic curve, 11. Simple harmonic motion, defined, 6 ; by mechanical movement, 7, 110. Simplified spelling, 256. Sine curves, 11, 137 ; and cosine curves, compounded, 94, 102, 105 ; from fork, 187. Singing tone, 179, 242, 259. Siren, 27, 28 ; wave, 62, 244. Skyrocket, photographed, 139. Smart, Sir George, philharmonic pitch, 49. Smedley, device for simple harmonic motion, 11. Soprano voice, frontispiece, 206, 211, 239; vowels, 228. Souder and Foley, compression waves, 88. Sound analyses, diagram of, 169 ; graph- ical presentation, 166. Soundboard of piano, 70, 176, 178, 207. Sound, defined, 2 ; explosive, 6 ; records, errors in, 142 ; velocity of, 5 ; waves, 5 ; waves made visible, 85, 88, 214. Sounding body, 2. Spectrum, compared to analysis of sound, 172. Staff, 47. Standard pitches, 49. Statistics, studied by harmonic analysis, 133. Steinmetz, harmonic analysis, 134. Sticks of wood, tuned, 22. Strings, vibrating, 64. Stuttgart, standard pitch, 50. Sympathetic vibration, 178. Synthetic vowels, 244, 250. Synthesis, of harmonic analysis, 1 2S ; of harmonic curves, 110; of tones, 244, 251. Tainter and Bell, graphophone, 77. Talking machine, 20, 70, 264 ; records analyzed, 77 ; see phonograph. Tannhduser Overture, 23, 24. Taylor, harmonic analysis, 135. Telephone, 20, 70, 75, 264; for phase experiments, 63 ; transmitting vowels, 232. Telephone siren, 63. Temperature, effect on tuning fork, 31 ; and velocity, 5. Tempered scale, table of frequencies, 48 ; harmonics compared with, 64 ; invention of, 264 ; tuning forks for, 34. Theorem,- Fourier's, 92. Theory of vowels, 215, 239. Thomas, Orchestra, 24. Thompson, S. P., harmonic analysis, 134. Tibia organ pipes, 247. Tidal analysis, 129, 135. Tide predictor, Kelvin's, 129 ; of United States Coast and Geodetic Survey, 129. Time signals on record of sound waves, 82, 87, 139. Time required for harmonic analysis, 120, 136. Toepler, compression waves, 88. Tone, 25, 26 ; and noise, 21 ; ideal, 204, 212 ; pure tone is poor, 213. Tone color, 25, 58, 267, 269; see tone quality. Tone quality, 25, 58, 70, 174, 176 ; law of, 62 ; independent of phase, 166, 197 ; synthetic, 251 ; of vowels, 59, 215, 239. Tonometer, Koenig's and Scheibler's, 37. Torsional wave, 20. Touch, piano, 208. Translation, of Grand Opera, 261 ; of vowels with phonograph, 232. 285 INDEX Transverse, displacement, 16, 20 ; vibra- tion, 4 ; wave motion, 14. Tuning fork, adjusting, 30, 34 ,■ analysis of sound of, 171 ; clock-fork, 28, 38 ; exciting, 32 ; for higher limit of audi- bility, 46 ; invention of, 29 ; for lower limit of audibility, 43 ; as musical in- strument, 189; overtones of, 138, 187, 188 ; pitch affected by amplitude, 32 ; quality of tone, 137, 185, 186 ; resonance box, 31, 177; rust and wear, 33 ; shapes of, 30 ; synthesizer tor vowels, 245 ; effect of temperature on, 31 ; for tuning chromatic scale, 35 ; vibrations of, 3. Tuning, instruments, 34. 51, HI, 177, 185. U Urquhart and Carse, harmonic analysis, 134. Velocity of sound, 5. Vibrating strings, 64. Vibration, atmospheric, 2 forced, 177 ; free, 177 3 ; microscope, 38, 41 thetic, 178 ; transverse, 4, Violin, 33, 58, 68 ; analysis of tone, 100, 171 : resonance of body, 197 ; reversal of bow, 197 ; tone quality of, 194, 251. Vision, persistence of, 43. Voice, analysis of, 171 ; comparison of bass and soprano, 205 ; compared with instruments, 257 ; qualit.v, 58 ; re- lated to vowels, 228, 259. damped, 179 ; longitudinal, 195 ; sympa- Vowel curve, analyzed by inspection, 137 ; periodic curve, 140. Vowels, analyses of, 221 ; artificial, 244 ; characteristics of, 215, 225, 230 ; clas- sification of, 225, 230 : continuity of, 230; defined, 217; diagram of analy- sis, 219; list of,_218, 257; musical quality of, 59 ; ob, ee, 231 ; photo- graphing, 219 ; relation to pitch, 221, 258, 260 ; singing voice related to, 259 ; synthetic, 244; theory of, 215, 239; translation with phonograph, 232 ; from different voices, 224 ; whispered, 235 ; words formed from, 24, 251, 257. Wagner, composer, 23, 24, 267. Watson, acoustics of auditoriums, 58. Waves, of compression, 17 ; longitudinal, 15 ; photographs of compression, 88 ; in solids, liquids, and gases, 17 ; of sound, 5 : torsional, 20 ; transverse, 14 ; portrait, 120. Wave models, 14, 15, 18, 19, 59, 60, 61. Wave motion, 13. Wave siren, 62, 244. Wedmore, harmonic analysis, 134. Wheatstone, theory of vowels, 215. Whispered vowels, 235. Whitman, acoustics of auditoriums, 58. Willis, theory of vowels. 215. Wires and cords in an auditorium, 5S. Wood, compression waves, SS. Words, formation of, 24, 251, 257 ; syn- thetic, 253, 255 ; photographs of, 238, 254. Y Yodeling, nature of, 260. Printed In the United States of America. 286 T HE following pages contain advertisements of Macmillan books on kindred subjects. NEW WORKS ON NATIVE MUSIC AND SINGING The History of American Music By LOUIS C. ELSON New edition, illustrated, cloth, 8vo, $6.00 This has been the standard work on American musical history ever since its first issue in 1904. In the present new edition it is brought completely up to date. All the important and interesting occur- rences of the past ten years are adequately treated, and the scope of the work is expanded to embrace every musical activity of the American people. How to Sing By LILLI LEHMANN Translated from the German by Richard Aldrich. Revised and enlarged edition, with many illustrations. 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