NOISE AND MUSICAL TONE. 11

CHAPTEE I.

ON THE SENSATION OF SOUND IN GENERAL.

SENSATIONS result from the action of an external stimulus on thesensitive apparatus of our nerves. Kinds of sensation differ, partlywith the organ of sense excited, and partly with the kind of stim-ulus employed. Each organ of sense produces peculiar sensations,which cannot be excited in any other ; the eye gives sensations oflight, the ear sensations of sound, the skin sensations of touch.Even when the same sunbeams which excite in the eye sensationsof light, impinge on the skin and excite its nerves, they are feltonly as heat, not as light. In the same way the vibration ofelastic bodies heard by the ear, can also be felt by the skin, but inthat case produce only a whirring fluttering sensation, not sound.The sensation of sound is therefore a species of reaction againstexternal stimulus, peculiar to the ear, and excitable in no otherorgan of the body, and is completely distinct from the sensation ofany other sense.

As our problem is to study the laws of the sensation of hearing,our first business will be to examine how many kinds of sensationthe ear can generate, and what differences in the external meansof excitement or sound, correspond to these differences of sensa-tion.

The first and principal difference between various sounds ex-perienced by our ear, is that between noises and musical tones.The soughing, howling, and whistling of the wind, the splashing ofwater, the rolling and rumbling of carriages, are examples of thefirst kind, and the tones of all musical instruments of the second.Noises and musical tones may certainly intermingle in very variousdegrees, and pass insensibly into one another, but their extremesare widely separated.

The nature of the difference between musical tones and noises,can generally be determined by attentive aural observation without

1 2 NOISE AND MUSICAL TONE. PART

artificial assistance. We perceive that generally, a noise is ac-companied by a rapid alternation of different kinds of sensationsof sound. Think, for example, of the rattling of a carriage overgranite paving stones, the splashing or seething of a waterfall orof the waves of the sea, the rustling of leaves in a wood. In allthese cases we have rapid, irregular, but distinctly perceptiblealternations of various kinds of sounds, which crop up fitfully.When the wind howls the alternation is slow, the sound slowlyand gradually rises and then falls again. It is more or less possi-ble to separate restlessly alternating sounds for the greater numberof other noises. We shall hereafter become acquainted with aninstrument, called a resonator, which will materially assist the earin making this separation. On the other hand, a musical tonestrikes the ear as a perfectly undisturbed, uniform sound whichremains unaltered as long as it exists, and it presents no alterna-tion of various kinds of constituents. To this then corresponds asimple, regular kind of sensation, whereas in a noise many varioussensations of musical tone are irregularly mixed up and as it weretumbled about in confusion. We can easily compound noises outof musical tones, as, for example, by striking all the keys con-tained in one or two octaves of a pianoforte at once. Thisshews us that musical tones are the simpler and more regularelements of the sensations of hearing, and that we have conse-quently first to study the laws and peculiarities of this class ofsensations.

Then comes the question : On what difference in the externalmeans of excitement does the difference between noise and musicaltone depend ? The normal and usual means of excitement for thehuman ear is atmospheric vibration. The irregularly alternatingsensation of the ear in the case of noises leads us to conclude thatfor these the vibration of the air must also change irregularly.For musical tones on the other hand we anticipate a regular motionof the air, continuing uniformly, and in its turn excited by anequally regular motion of the sonorous body, whose impulses wereconducted to the ear by the air.

Those regular motions which produce musical tones have beenexactly investigated by physicists. They are oscillations, vibra-tions, or swings, that is, up and down, or to and fro motions ofsonorous bodies, and it is necessary that these oscillations shouldbe regularly periodic. By a periodic inotion we mean one which

CHAP. I. PROPAGATION OF SOUND. 13

constantly returns to the same condition after exactly equal inter-vals of time. The length of the equal intervals of time betweenone state of the motion and its next exact repetition, we call thelength of the oscillation vibration or swing, or the 'period of themotion. The kind of motion of the moving body during oneperiod, is perfectly indifferent. As illustrations of periodicalmotion, take the motion of a clock pendulum, of a stone attachedto a string and whirled round in a circle with uniform velocity, ofa hammer made to rise and fall uniformly by its connection witha water wheel. All these motions, however different be their form,are periodic in the sense here explained. The length of theirperiods, which in the cases adduced is generally from one toseveral seconds, is relatively long in comparison with the muchshorter periods of the vibrations producing musical tones, thelowest or deepest of which makes at least 30 in a second, whilein other cases their number may increase to several thousand in asecond.

Our definition of periodic motion then enables us to answer thequestion proposed as follows :—The sensation of a musical tone isdue to a rapid periodic motion of the sonorous body ; the sensa-tion of a noise to non-periodic motions.

The musical vibrations of solid bodies are often visible. Al-though they may be too rapid for the eye to follow them singly,we easily recognise that a sounding string, or tuning fork, or thetongue of a reed-pipe, is rapidly vibrating between two fixedlimits, and the regular, apparently immovable image that we see,notwithstanding the real motion of the body, leads us to concludethat the backward and forward motions are quite regular. Inother cases we can feel the swinging motions of sonorous solids.Thus, the player feels the trembling of the reed in the mouthpieceof a clarinet, oboe, or bassoon, or of his own lips in the mouth-pieces of trumpets and trombones.

The motions proceeding from the sounding bodies are usuallyconducted to our ear by means of the atmosphere. The particlesof air must also execute periodically recurrent vibrations, in orderto excite the sensation of a musical tone in our ear. This isactually the case, although in daily experience sound at first seemsto be some agent, which is constantly advancing through the air,and propagating itself further and further. We must, however,here distinguish between the motion of the individual particles of

14 PROPAGATION OF SOUND. PAET I.

air—which takes place periodically backwards and forwards withinvery narrow limits—and the propagation of the sonorous tremor.The latter is constantly advancing by the constant attraction offresh particles into its sphere of tremor.

This is a peculiarity of all so-called undulatory motions.Suppose a stone to be thrown into a piece of calm water.Eound the spot struck there forms a little ring of wave, which,advancing equally in all directions, expands to a constantly in-creasing circle. Corresponding to this ring of wave, sound alsoproceeds in the air from the excited point and advances in alldirections as far as the limits of the mass of air extend. The pro-cess in the air is essentially identical with that on the surface ofthe water. The principal difference consists in the spherical pro-pagation of sound in all directions through the atmosphere whichfills all surrounding space, whereas the waves of the water can onlyadvance in rings or circles on its surface. The crests of the wavesof water correspond in the waves of sound to spherical shells wherethe air is condensed, and the troughs to shells of rarefaction. Onthe free surface of the water, the mass on compression can slipupwards and so form ridges, but in the interior of the sea of air,the mass must be condensed, as there is no unoccupied spot for itsescape.

The waves of water, therefore, continually advance withoutreturning. But we must not suppose that the particles of waterof which the waves are composed advance in a similar manner tothe waves themselves. The motion of the particles of water on thesurface can easily be rendered visible by floating a chip of woodupon it. This will perfectly share the motion of the adjacent par-ticles. Now, such a chip is not carried on by the rings of wave.It only bobs up and down and finally rests on its original spot.The adjacent particles of water move in the same manner. Whenthe ring of wave reaches them they are set bobbing; when it haspassed over them they are still in their old place, and remain thereat rest, while the ring of wave continues to advance towards freshspots on the surface of the water, and sets new particles of waterin motion. Hence the waves which pass over the surface of thewater, are constantly built up of fresh particles of water. Whatreally advances as a wave is only the tremor, the altered form ofthe surface, while the individual particles of water themselvesmerely move up and down transiently, and never depart far fromtheir original position.

CHA.P. I. PROPAGATION OF SOUND. 15

The same relation is seen still more clearly in the wavesof a rope or chain. Take a flexible string of several feet inlength, or a thin metal chain, hold it at one end and let the otherhang down, stretched by its own weight alone. Now, move thehand by which you hold it quickly to one side and back again.The excursion which we have caused in the upper end of the stringby moving the hand, will run down it as a kind of wave, so thatconstantly lower parts of the string will make a sidewards excursionwhile the upper return again into the straight position of rest.But it is evident that while the wave runs down, each individualparticle of the string can have moved only horizontal backwardsand forwards, and can have taken no share at all in the advance ofthe wave.

The experiment succeeds still better with a long elastic line,such as a thick piece of india-rubber, or a brass-wire spiral spring,from eight to twelve feet in length, fastened at one end, and slightlystretched by being held with the hand at the other. The hand isthen easily able to excite waves which will run very regularly tothe other end of the line, be there reflected and return. In thiscase it is also evident that it can be no part of the line itself whichruns backwards and forwards, but that the advancing wave is com-posed of continually fresh particles of the line. By these examplesthe reader will be able to form a mental image of the kind ofmotion to which sound belongs, where the material particles ofthe body merely make periodical oscillations, while the tremor it-self is constantly propagated forwards.

Now let us return to the surface of the water. We have sup-posed that one of its points has been struck by a stone and set inmotion. This motion has spread out in the form of a ring of waveover the surface of the water, and having reached the chip of woodhas set it bobbing up and down. Hence by means of the wave, themotion which the stone first excited in one point of the surface ofthe water has been communicated to the chip which was at anotherpoint of the same surface. The process which goes on in the at-mospheric ocean about us, is of a precisely similar nature. Forthe stone substitute a sounding body, which shakes the air; forthe chip of wood substitute the human ear, on which impinge thewaves of air excited by the shock, setting its movable parts invibration. The waves of air proceeding from a sounding body,transport the tremor to the human ear exactly in the same way as

16 DIFFERENCES IN MUSICAL TONES. PART I.

the water transports the tremor produced by the stone to the float-ing chip.

In this way also it is easy to see how a body which itselfmakes periodical oscillations, will necessarily set the particles ofair in periodical motion. A falling stone gives the surface of thewater a single shock. Now replace the stone by a regular seriesof drops falling from a vessel with a small orifice. Every sepa-rate drop will excite a ring of wave, each ring of wave will advanceover the surface of the water precisely like its predecessor, andwill be in the same way followed by its successors. In this mannera regular series of concentric rings will be formed and propagatedover the surface of the water. The number of drops which fallinto the water in a second will be the number of waves whichreach our floating chip in a second, and the number of times thatthis chip will therefore bob up and down in a second, thus execut-ing a periodical motion, the period of which is equal to the in-terval of time between the falling of consecutive drops. Similarlyin the atmosphere, a periodically oscillating sonorous body pro-duces a similar periodical motion, first in the mass of air, andthen in the drum of our ear, and the period of these vibrationsmust be the same as that of the vibration in the sonorous body.

Having thus spoken of the principal division of sound intoNoise and Musical Tones, and then described the general motionof the air for these tones, we pass on to the peculiarities whichdistinguish such tones one from the other. We are acquaintedwith three points of difference in musical tones, confining our-selves in the first place to such tones as are isolatedly producedby our usual musical instruments, and excluding the simultaneoussounding of the tones of different instruments. Musical tones aredistinguished:—

1. By their force or loudness.2. By their pitch, or relative height.3. By their quality.It is unnecessary to explain what we mean by the force

and pitch of a tone. By the quality of a tone we mean thatpeculiarity which distinguishes the musical tone of a violin fromthat of a flute or that of a clarinet, or that of the human voice,when all these instruments produce the same note at the samepitch.

We have now to explain what peculiarities of the motion of

OKA?. I. PITCH AND VlBEATIONAL NUMBER. 17

sound correspond to these three principal differences betweenmusical tones.

First, We easily recognise that the force or loudness of amusical tone increases and diminishes with the extent or so-calledamplitude of the oscillations of the particles of the soundingbody. When we strike a string, its vibrations are at first suffi-ciently large for us to see them, and its corresponding tone isloudest. The visible vibrations become smaller and smaller, andat the same time the loudness diminishes. The same observationcan be made on strings excited by a violin bow, and on the reedsof reed-pipes, and on many other sonorous bodies. The same con-clusion results from the diminution of the loudness of a tone whenwe increase our distance from the sounding body in the open air,although the pitch and quality remain unaltered; for it is onlythe amplitude of the oscillations of the particles of air whichdiminishes as their distance from the sounding body increases.Hence loudness must depend on this amplitude, and none other ofthe properties of sound do so.1

The second essential difference between different musicaltones consists in their pitch or relative height. Daily ex-perience shews us that musical tones of the same pitch can beproduced upon most diverse instruments by means of mostdiverse mechanical contrivances, and with most diverse degreesof loudness. All the motions of the air thus excited mustbe periodic, because they would not otherwise excite in us thesensation of a musical tone. But the motion within each singleperiod may be of any kind whatever, and yet if the lengthof the periodic time of two musical tones is the same, they havethe same pitch. Hence: Pitch depends solely on the length oftime in which each single vibration is executed, or, which comesto the same thing, on the vibrational number of the tone. Weare accustomed to take a second of time as the unit, and conse-quently mean by vibrational number the number of vibrationswhich the particles of a sounding body perform in one second oftime. I t is self-evident that we find the periodic time or vibra-

1 Mechanically the force of the oscillations for tones of different pitch is measuredby their vis viva, that is, by the square of the greatest velocity attained by the oscil-lating particles. But the ear has different degrees of sensibility for tones of differentpitch, so that no measure can be found for the intensity of the sensation of sound,that is, for the loudness of sound at all pitches.

C

18 PITCH AND VIBRATIONAL DUMBER. PART I.

tional period, that is length of time which is occupied in per-forming a single vibration backwards and forwards, by dividingone second of time by the vibrational number.

Musical tones are higher, the greater their vibrational number,that is, the shorter is their vibrational period.

The exact determination of the vibrational number for suchelastic bodies as produce audible tones, presents considerable diffi-culty, and physicists had to contrive many comparatively compli-cated processes in order to solve this problem for each particularcase. Mathematical theory and numerous experiments had torender mutual asistance. I t is consequently very convenient forthe demonstration of the fundamental facts in this department ofknowledge, to be able to apply a peculiar instrument for produc-ing musical tones—the so-called siren—which is constructed insuch a manner as to determine the vibrational number of the toneproduced, by a direct observation. The principal parts of thesimplest form of the siren are shewn in fig. 1, after Seebeck.

A is a thin disc of cardboard or tinplate, which can be set inrapid rotation about its axle b by means of a string f f, whichpasses over a larger wheel. On the margin of the disc there ispunched a set of holes at equal intervals: of these there are twelve

in the figure ; one or more similar series of holes at equal dis-tances are introduced on concentric circles, (there is one such ofeight holes in the figure), c is a pipe which is directed over oneof the holes. Now, on setting the disc in rotation and blowingthrough the pipe c, the air will pass freely whenever one of theholes passes the end of the pipe, but will be checked whenever anunpierced portion of the disc comes before it. Each hole of thedisc, then, that passes the end of the pipe lets a single puff of airescape. Supposing the disc to make a single revolution and the

CHAP. I. PITCH AND VIBKATIONAL NUMBER. 19

pipe to be directed to the outer circle of holes, we have twelvepuffs corresponding to the twelve holes ; but if the pipe is directedto the inner circle we have only eight puffs. If the disc is madeto revolve ten times in one second, the outer circle would produce120 puffs in one second, which would give rise to a weak and deepmusical tone, and the inner circle eighty puffs. Generally, if weknow the number of revolutions which the disc makes in a second,and the number of holes in the series to which the tube isdirected, the product of these two numbers evidently gives thenumber of puffs in a second. This number is consequently far

FIG. 2. FIG. 3.

easier to determine exactly than in any other musical instrument,and sirens are accordingly extremely well adapted for studying allchanges in musical tones resulting from the alterations and ratiosof the vibrational numbers.

c 2

2 0 PITCH AND VIBRATIONAL NUMBEE. PAST 1.

The form of siren here described gives only a weak tone. Ihave placed it first because its action can be most readily under-stood, and, by changing the disc, it can be easily applied to expe-riments of very different descriptions. A stronger tone is producedin the siren of Cagniard de la Tour, shewn in figures 2, 3, and 4,p. 19. Here s s is the rotating disc, of which the upper surface isshewn in fig. 3, and the side is seen in figs. 2 and 4. It is placedover a windchest A A, which is connected with a bellows by thepipe BB. The cover of the windchest A A, which lies immediatelyunder the rotating disc s s, is pierced with precisely the samenumber of holes as the disc, and the direction of the holes piercedin the cover of the chest is oblique to that of the holes in the disc,as shewn in fig. 4, which is a vertical section of the instrumentthrough the line nn in fig. 3. This position of the holes enablesthe wind escaping from A A to set the disc ss in rotation, and byincreasing the pressure on the bellows, as much as 50 or 60 rota-tions in a second can be produced. Since all the holes of onecircle are blown through at the same time in this siren, a muchmore powerful tone is produced than in Seebeck's, fig. 1. Torecord the revolutions, a counter zz is introduced, connectedwith a toothed wheel which works in the screw t, and advancesone tooth for each revolution of the disc s s. By the handle hthis counter may be moved slightly to one side, so that the wheel-work and screw may be connected or disconnected at pleasure.If they are connected at the beginning of one second, and discon-nected at the beginning of another, the hand of the counter shewshow many revolutions of the disc have been made in the corre-sponding number of seconds.1

Dove2 introduced into this siren several rows of holes intowhich the wind might be directed, or from which it might becut off, at pleasure. A polyphonic siren of this description withother peculiar arrangements will be figured and described inChapter VIII.

It is clear that when the pierced disc of one of these sirens ismade to revolve with a uniform velocity, and the air escapesthrough the holes in puffs, the motion of the air thus producedmust be periodic in the sense already explained. The holes standat equal intervals of space, and hence on rotation follow each

1 See Appendix I. 2 [Pronounce Bo-ve, in two syllables.—Translator.]

CHAP. I. PITCH AND VIBRATIONAL NUMBER. 2 1

other at equal intervals of time. Through every hole there ispoured, as it were, a drop of air into the external atmosphericocean, exciting waves in it, which succeed each other at uniformintervals of time, just as was the case when regularly falling dropsimpinged upon a surface of water (p. 16). Within each separateperiod, each individual puff will have considerable variations ofform in sirens of different construction, depending on the differentdiameters of the holes, their distance from each other, and theshape of the extremity of the pipe which conveys the air ; but inevery case, as long as the velocity of rotation and the position ofthe pipe remain unaltered, a regularly periodic motion of theair must result, and consequently the sensation of a musical tonemust be excited in the ear, and this is actually the case.

It results immediately from experiments with the siren thattwo series of the same number of holes revolving with the samevelocity, give musical tones of the same pitch, quite independentlyof the size and form of the holes, or of the pipe. We even obtaina musical tone of the same pitch if we allow a metal point to strikein the holes as they revolve instead of blowing. Hence it followsfirstly that the musical height of a tone depends only on thenumber of puffs or swings, and not on their form, force, or methodof production. Further it is very easily seen with this instrumentthat on increasing the velocity of rotation, and consequently thenumber of puffs produced in a second, the pitch becomes sharperor higher. The same result ensues if, maintaining a uniformvelocity of rotation, we first blow into a series with a smaller andthen into a series with a greater number of holes. The lattergives the sharper or higher pitch.

With the same instrument we also very easily find theremarkable relation which the vibrational numbers of twomusical tones mu3t possess in order to form a consonant interval.Make a series of 8 and another of 16 holes on a disc, and blowinto both sets while the disc is kept at uniform velocity of rotation.Two tones will be heard which stand to one another in the exactratio of an Octave. Increase the velocity of rotation; both toneswill become sharper, but both will continue at the new pitch toform the interval of an Octave. Hence we conclude that a musicaltone which is an Octave higher than another, makes exactly twiceas many vibrations in a given time as the latter.

The disc shewn in fig. 1, p. 18, has two circles of 8 and 12 holes

22 PITCH AND VIBKATIONAL NUMBER. PAET I.

respectively. Each, blown successively, gives two tones which formwith each other a perfect Fifth, independently of the velocity ofrotation of the disc. Hence, two musical tones stand in the rela-tion of a so-called Fifth when the higher tone makes three vibra-tions in the same time as the lower makes two.

If we obtain a musical tone by blowing into a circle of 8 holes,we require a circle of 16 holes for its Octave, and 12 for its Fifth.Hence the ratio of the vibrational numbers of the Fifth and theOctave is 12 T 16 or 3 I 4. But the interval between the Fifthand the Octave is the Fourth, so that we see that when twomusical tones form a Fourth, the higher makes four vibrationswhile the lower makes three.

The polyphonic siren of Dove has usually four circles of 8, 10,12 and 16 holes respectively. The series of 16 holes gives theOctave of the series of 8 holes, and the Fourth of the series of 12holes. The series of 12 holes gives the Fifth of the series of 8holes, and the minor Third of the series of 10 holes. While theseries of 10 holes gives the major Third of the series of 8 holes.The four series consequently give the constituent musical tones ofa major chord.

By these and similar experiments we find the following rela-tions of the vibrational numbers:—

123 :45 .

: 2

456

OctaveFifthFourthmajor Thirdminor Third

When the fundamental tone of a given interval is takenan Octave higher, the interval is said to be inverted. Thus aFourth is an inverted Fifth, a minor Sixth an inverted majorThird, and a major Sixth an inverted minor Third. The corre-sponding ratios of the vibrational numbers are consequently ob-tained by doubling the smaller number in the original interval.

From 2 : 3 the Fifth, we thus have 3 : 4 the Fourth„ 4 : 5 the major Third . . . . 5 : 8 the minor Sixth„ 5:6 the minor Third, 6 : 10 = 3 : 5 the major Sixth.

These are all the consonant intervals which lie within thecompass of an Octave. With the exception of the minor Sixth,

CHAP. I. PITCH AND VIBKATIONAL NUMBER. 23

which is really the most imperfect of the above consonances, theratios of their vibrational numbers are all expressed by means ofthe whole numbers, 1, 2, 3, 4, 5, 6.

Comparatively simple and easy experiments with the siren,therefore, corroborate that remarkable law mentioned in theintroduction (p. 2), according to which the vibrational numbers ofconsonant musical tones bear to each other ratios expressible bysmall whole numbers. In the course of our investigation weshall employ the same instrument to verify more completely thestrictness and exactness of this law.

Long before anything was known of vibrational numbers, orthe means of counting them, Pythagoras had discovered that if astring be divided into two parts by a bridge, in such a way as to givetwo consonant musical tones when struck, the lengths of these partsmust be in the ratio of these whole numbers. If the bridge is soplaced that § of the string lie to the right, and ^ on the left, sothat the two lengths are in the ratio of 2 *. 1, they produce theinterval of an Octave, the greater length giving the deeper tone.Placing the bridge so that J- of the string lie on the right and fon the left, the ratio of the two lengths is 3 : 2, and the intervalis a Fifth.

These measurements had been executed with great precisionby the Greek musicians, and had given rise to a system of tones,contrived with considerable art. For these measurements theyused a peculiar instrument, the monochord, consisting of asounding board and box on which a single string was stretchedwith a scale below, so as to set the bridge correctly.

It was not till much later that, through the investigations ofGralileo (1638), Newton, Euler (1729). and Daniel Bernouilli(1771), the law governing the motions of strings becameknown, and it was thus found that the simple ratios of thelengths of the strings existed also for the vibrational numbersof the tones they produced, and that they consequently belongedto the musical intervals of the tones of all instruments, and werenot confined to the strings through which the law had been firstdiscovered.

This relation of whole numbers to musical consonances wasfrom all time looked upon as a wonderful mystery of deep signifi-cance. The Pythagoreans themselves made use of it in theirspeculations OD the harmony of the spheres. From that time it

24 PITCH AND VIBRATIONAL NUMBEE. PAHT I.

remained partly the goal and partly the starting point of the strangestand most venturesome, fantastic or philosophic combinations, tillin modern times the majority of investigators adopted the notionaccepted by Euler himself, that the human mind had a peculiarpleasure in simple ratios, because it could better understand themand comprehend their bearings. But it remained uninvestigatedhow the mind of a listener not versed in physics, who perhaps wasnot even aware that musical tones depended on periodical vibra-tions, contrived to recognise and compare these ratios of thevibrational numbers. To shew what processes take place in theear to render sensible the difference between consonance anddissonance, will be one of the principal problems in the secondpart of this work.

Calculation of the Vibrational Numbers for all the Tones of theMusical Scale.

By means of the ratios of the vibrational numbers alreadyassigned for the consonant intervals, it is easy, by pursuing theseintervals throughout, to calculate the ratios for the whole extentof the musical scale.

The major triad or chord of three tones, consists of a majorThird and a Fifth. Hence its ratios are :

Cl1

or 4

E

If we associate with this triad that of its dominant 0 '. B '. D,and that of its sub-dominant F '. A '. C, each of which has onetone in common with the triad of the tonic G '. E '. 0, we obtainthe complete series of tones for the major scale of C, with thefollowing ratios of the vibrational numbers :

C1

[or 24

DSL8

27

Ei30

F : GA.3

40

BV45

c2.48]32 : 36

_ In order to extend the calculation to other octaves, we shalladopt the following notation of musical tones, marking the higheroctaves by accents as is usual in Germany,1 as follows :

1 [English works use strokes above and below the letters, which is typographicallyinconvenient, and would interfere with a notation introduced in Chap. XIV. Hencethe German notation, which is typographically more convenient, is retained. Onnotation of pitch generally see Appendix XIX.—Translator.]

CHAP. I. PITCH AND VIBRATIONAL NUMBER. 25

1. The Unaccented or Small Octave (the 4-foot Octave on theOrgan):—

. *=> a.>-, —g?-

c rf e / g a b

2. The Once-accented Octave (2-foot) :—

3. T&<? Twice-accented Octave (1-foot):—

And so on for higher Octaves. Below the small Octave lies thegreat Octave, written with unaccented capital letters ; its C re-quires an organ pipe of eight feet in length, and hence it is calledthe 8-foot Octave.

4. Great or 8-foot Octave.

-p ^ -

T G 5 J5

Below this follows the 16-foot or Contra-Octave; the loweston the pianoforte and most organs, the tones of which may berepresented by C, D / JE/ F, Qt At 1?,, with an inverted accent.On great organs there is a still deeper, 32-foot Octave, the tonesof which may be written G/t D/t E,, Flt Gy/ Ay/ Bn, with twoinverted accents, but they scarcely retain the character of musicaltones. (See Chap. IX.)

Since the vibrational numbers of any Octave are always twiceas great as those for the next deeper, we find the vibrational num-bers of the higher tones by multiplying those of the small or un-accented Octave as many times by 2 as its symbol has upper accents.And on the contrary the vibrational numbers for the deeper

26 PITCH AND VIBKATIONAL NUMBEE. PART I.

Octaves are found by dividing those of the great Octave, as oftenas its symbol has lower accents.

Thus c" = 2 x 2 x c = 2 x 2 x 2 C

For the pitch of the musical scale German physicists havegenerally adopted that proposed by Scheibler, and adopted subse-quently by the German Association of Natural Philosophers (diedeutsche Naturforscherversammlung) in 1834. This makes theonce accented a' perform 440 vibrations in a second.1 Henceresults the following table for the scale of C major, which willserve to determine the pitch of all tones that are defined by theirvibrational numbers in the following work.

Notes

CDEPGAB

Contra OctaveC, to B,

3337-12541-254449-55561-875

Great OctaveCto.0

6674-2582-58899

110123-75

TXnaceentedOctave

t W) C

132148-5165176198220247-5

OnceaccentedOctaved\nV

264297330352396440495

TwiceaccentedOe< avec" to b"

528594660704792880990

TliriceaccentedOctavec"' to V"

1056118813201408158417601980

Four timesaccentedOctave

c"" to b""

2112237626402816316835203960

1 The Paris Academy has lately fixed the vibrational number of the same note at435. This is called 870 by the Academy, because Prench physicists have adopted theinconvenient habit of counting the forward motion of a swinging body as one vibration,and the backward as another, so that the whole vibration is counted as two. [ThisFrench method of counting is used everywhere for the seconds pendulum. In thiswork and in England generally, the backwards and forwards motion are countedmusically as one vibration, hence sometimes called ' a double vibration.' In Englandthe theoretical pitch gives 512 as the vibrational number of c", and most tuning forks,such as those issued by Mr. Hullah, and the Tonic Sol-faists, profess to give thatpitch. The Society of Arts adopted 528 for c", as in this table. Various otherpitches are also in use. It should be observed that when ' equal temperament' isemployed, and a' is taken as 440, c" is only 523-25, and if c" is taken as 528, a' in• equal temperament' is 44399 or nearly 444, that is, considerably sharper (nearly |of a comma) than Scheibler's pitch. In the table in the text 'just intonation' isadopted. The difference between the ' just ' and the 'equally tempered' scale, thelatter of which is now almost universally employed for pianofortes and organs, will beexplained in Chap. XVI. As many calculations are founded on the numbers given inthis table, it was thought best not to reduce th^m to the English theoretical pitch inthis translation. All the acoustical apparatus of Koenig (Appendix II.) is, however,now constructed fore" = 512, unless otherwise ordered; see the note prefixed to hiscatalogue of apparatus published in 1865. See also Appendix XIX.—Translator.]

CHAP. I. QUALITY OF TONE AND FORM OF VIBEATION. 27

The lowest tone on orchestral instruments is the Et of thedouble bass, making 41 \ vibrations in a second. Modern piano-fortes and organs usually go down to O/ with 33 vibrations, andthe latest grand pianos even down to A/y with 27^ vibrations.On larger organs as already mentioned, there is also a deeperOctave reaching to On with 16-| vibrations. But the musicalcharacter of all these tones below Et is imperfect, because we arehere near to the limit of the power of the ear to combine vibra-tions into musical tones. These lower tones cannot therefore beused musically except in connection with their higher Octaves towhich they impart a character of greater depth without renderingthe conception of the pitch indeterminate.

Upwards, pianofortes generally reach a"" with 3520, or evencv with 4224 vibrations. The highest tone in the orchestra isprobably the five times accented ds/ of the piccolo flute with 4752vibrations. Despretz asserts that by means of small tuning-forksexcited by a violin bow he has even reached the eight timesaccented ds"" with 38016 vibrations in a second. These hightones were very painfully unpleasant, and the pitch of thosewhich exceed the boundaries of the musical scale was very imper-fectly discriminated. More on these limits in Chap. IX.

The musical tones which can be used with advantage, andhave clearly distinguishable pitch, have therefore between 40 and4000 vibrations in a second, extending over 7 Octaves. Thosewhich are audible at all have from 20 to 38000 vibrations,extending over 11 Octaves. This shews what a great variety ofdifferent vibrational numbers can be perceived and distinguishedby the ear. In this respect the ear is far superior to the eye,which likewise distinguishes light of different periods of vibrationby the sensation of different colours, for the compass of the vibra-tions of light distinguishable by the eye little exceeds an Octave.

Loudness and 'pitch were the two first differences which wefound between musical tones ; the third was quality of tone, whichwe now have to investigate. If the same note is sounded succes-sively on a pianoforte, a violin, clarinet, oboe, or trumpet, or bythe human voice, notwithstanding its having the same force andthe same pitch, the musical tone of each of these instruments isdifferent, and by means of this tone we recognise with the greatestease which of these instruments was used. Varieties of quality oftone appear to be infinitely numerous. Not only do we know a

28 QUALITY OF TONE AND FORM OF VIBRATION. PAST I.

long series of musical instruments which could each produce thesame note; not only do different individual instruments of thesame form, and the voices of different individual singers shewcertain more delicate shades of quality of tone, which our ear isable to distinguish ; but the same note can sometimes be soundedon the same instrument with several qualitative varieties. In thisrespect the " bowed" instruments (i.e. those of the violin kind)are distinguished above all other. But the human voice is stillricher, and human speech employs these very qualitative varietiesof tone, in order to distinguish different letters. The differentvowels, namely, belong to the class of sustained tones which canbe used in music, while the character of consonants mainlydepends upon brief and transient noises.

On inquiring to what external physical difference in the wavesof sound the different qualities of tone correspond, we must re-member that the amplitude of the vibration determines the forceor loudness, and the period of vibration the pitch. Quality oftone can therefore depend upon neither of these. The only pos-sible hypothesis, therefore, is that the quality of tone shoulddepend upon the manner in which the motion is performed withinthe period of each single vibration. For the generation of amusical tone we have only required that the motion should beperiodic, that is, that in any one single period of vibration exactlythe same state should occur, in the same order of occurrence as itpresents itself in any other single period. As to the kind of motionthat should take place within any single period, no hypothesis wasmade. In this respect then an endless variety of motions mightbe possible for the production of sound.

Observe instances, taking first such periodic motions as areperformed so slowly that we can follow them with the eye. Takea pendulum, which we can at any time construct by attaching aweight to a thread and setting it in motion. The pendulumswings from right to left with a uniform motion, uninterrupted byjerks. Near to either end of its path it moves slowly, and in themiddle fast. Among sonorous bodies, which move in the sameway, only very much faster, we may mention tuning forks. Whena tuning fork is struck or is excited by a violin bow, and itsmotion is allowed to die away slowly, its two prongs oscillate^backwards and forwards in the same way and after the same law

CHAP. I. QUALITY OF TONE AND FORM OF VIBRATION. 29

as a pendulum, only they make many hundred swings for eachsingle swing of the pendulum.

As another example of a periodic motion, take a hammermoved by a water wheel. I t is slowly raised by the millwork,then released, and falls down suddenly, is then again slowly raised,and so on. Here again we have a periodical backwards and for-wards motion; but it is manifest that this kind of motion istotally different from that of the pendulum. Among motionswhich produce musical sounds, that of a violin string, excited bya bow, would most nearly correspond with this, as will be seenfrom the detailed description in Chap. V. The string clings fora time to the bow, and is carried along by it, then suddenlyreleases itself, like the hammer in the mill, and, like the latter,retreats somewhat with much greater velocity than it advanced,and is again caught by the bow and carried forward.

Again, imagine a ball thrown up vertically, and caught on itsdescent with a blow which sends it up again to the same height,and suppose this operation to be performed at equal intervals oftime. Such a ball would occupy the same time in rising as infalling, but at the lowest point its motion would be suddenly in-terrupted, whereas above it would pass through gradually dimin-ishing speed of ascent into a gradually increasing speed of descent.This then would be a third kind of alternating periodic motion,and would take place in a manner essentially different from theother two.

To render the law of such motions more comprehensible tothe eye than is possible by lengthy verbal descriptions, mathe-maticians and physicists are in the habit of applying a graphicalmethod, which must be frequently employed in this work, andshould therefore be well understood.

To render this method intelligible suppose a drawing point b,fig. 5, p. 30, to be fastened to the prong A of a tuning fork in sucha manner as to mark a surface of paper B B. Let the tuningfork be moved with a uniform velocity in ihe direction of theupper arrow, or else the paper be drawn under it in the oppositedirection, as shewn by the lower arrow. When the fork is notsounding, the point will describe the dotted straight line d c. Butif the prongs have been first set in vibration, the point willdescribe the undulating line d c, for as the prong vibrates, theattached point b will constantly move backwards and forwards,

30 QUALITY OF TONE AND FOEM OF VIBRATION. PART I.

and hence be sometimes on the right and sometimes on the leftof the dotted straight line d c, as is shewn by the wavy line inthe figure. This wavy line once drawn, remains as a permanentimage of the kind of motion performed by the end of the forkduring its musical vibrations. As the point b is moved in thedirection of the straight line d c with a constant velocity, equal

FIG. 5.

sections of the straight line d c will correspond to equal sectionsof the time during which the motion lasts, and the distance of thewavy line on either side of the straight line will shew how far thepoint b has moved from its mean position to one side or the otherduring those sections of time.

In actually performing such an experiment as this, it is bestto wrap the paper over a cylinder which is made to rotate uni-formly by clockwork. The paper is wetted, and then passed overa turpentine flame which coats it with lampblack, on which afine and somewhat smooth steel point will easily trace delicatelines. Fig. 6 is the copy of a drawing actually made in this way

FIG. 6.

on the rotating cylinder of Messrs. Scott and Koenig's Phonauto-grajph.

Fig. 7 shews a portion of this curve on a larger scale. I t iseasy to see the meaning of such a curve. The drawing point haspassed with a uniform velocity in the direction c h. Suppose thatit has described the section e g in -^ of a second. Divide e g

CHIP. I. QUALITY OF TONE AND FORM OF VIBRATION. 31

into 12 eqnal parts, as in the figure, then the point has beenof a second in describing the length of any such section horizon-tally, and the curve shews us on what side and at what distancefrom the position of rest the vibrating point will be at the endof T|~^, J^-Q, and so on, of a second, or, generally, at any givenshort interval of time since it left the point e. We see, in thefigure, that after -^~ of a second it had reached the height 1, andthat it rose gradually till the end of j^-g- of a second ; then, how-ever, it began to descend gradually till, at the end of TI-O^TO

second, it had reached its mean position f, and then it continueddescending on the opposite side till the end of jf^- of a secondand so on. We can also easily determine where the vibratingpoint was to be found at the end of any fraction of this hundred-and-twentieth of a second. A drawing of this kind consequently

FIG. 7.

I/"io

shews immediately at what point of its path a vibrating particleis to be found at any given instant, and hence gives a completeimage of its motion. If the reader wishes to reproduce the motionof the vibrating point, he has only to cut a narrow vertical slit ina piece of paper, and place it over fig. 6 or fig. 7, so as to shew avery small portion of the curve through the vertical slit, and drawthe book slowly but uniformly under the slit, from right to left;the white or black point in the slit will then appear to movebackwards and forwards in precisely the same manner as the ori-ginal drawing point attached to the fork, only of course muchmore slowly.

We are not yet able to make all vibrating bodies describe theirvibrations directly on paper, although much progress has recentlybeen made in the methods required for this purpose. But we areable ourselves to draw such curves for all sounding bodies, whenthe law of their motion is known, that is, when we know how farthe vibrating point will be from its mean position at any given

32 QUALITY OF TONE AND FORM OF VIBEATION. PABT I.

moment of time. We then set off on a horizontal line, such ase f, fig. 7, lengths corresponding to the interval of time, and letfall perpendiculars to it on either side, making their lengthsequal or proportional to the distance of the vibrating point fromits mean position, and then by joining the extremities of theseperpendiculars we obtain a curve such as the vibrating body wouldhave drawn if it had been possible to make it do so.

Thus fig. 8 represents the motion of the hammer raised by awater wheel, or of a point in a string excited by a violin bow.For the first 9 intervals it rises slowly and uniformly, and duringthe 10th it falls suddenly down.

FIG. 8. FIG. 9.

10

Fig. 9 represents the motion of the ball which is struck upagain as soon as it comes down. Ascent and descent are per-formed with equal rapidity, whereas in fig. 8 the ascent takesmuch longer time. But at the lowest point the blow suddenlychanges the kind of motion.

Physicists, then, having in their mind such curvilinear forms,representing the law of the motion of sounding bodies, speakbriefly of the form of vibration of a sounding body, and assertthat the quality of tone depends on the form of vibration. Thisassertion, which physicists hitherto based simply on the fact oftheir knowing that the quality of the tone could not possiblydepend on the periodic time of a vibration, or on its amplitude,will be strictly examined hereafter. I t will be shewn to be so farcorrect that every different quality of tone requires a differentform of vibration, but on the other hand it will also appear thatdifferent forms of vibration may correspond to the same quality oftone.

On exactly and carefully examining the effect produced on theear by different forms of vibration, as for example that in fig. 8,corresponding nearly to a violin string, we meet with a strangeand unexpected phenomenon, long known indeed to individualmusicians and physicists, but commonly regarded as a mere

CHAP. I. HARMONIC UPPER PARTIAL TONES. 3 3

curiosity, its generality and its great significance for all mattersrelating to musical tones not having been recognised. Theear when its attention has been properly directed to the effectof the vibrations which strike it, does not hear merely that onemusical tone whose pitch is determined by the period of thevibrations in the manner already explained, but in addition tothis it becomes aware of a whole series of higher musical tones,which we will call the harmonic upper partial tones, and some-times simply the upper partials of that musical tone, in contra-distinction to that first tone, the fundamental or prime partialtone or simply the prime, which is the lowest and generally theloudest of all, and by whose pitch we judge of the pitch of thewhole compound musical tone, or simply the compound. Theseries of these upper partial tones is precisely the same for allcompound musical tones which correspond to a uniformly periodicalmotion of the air. It is as follows:—

The first upper partial tone is the upper Octave of the primetone, and makes double the number of vibrations in the sametime. If we call the prime c, this upper octave will be c'.

The second upper partial tone is the Fifth of this Octave, or(f, making three times as many vibrations in the same time asthe prime.

The third upper partial tone is the second higher Octave orc"', making four times as many vibrations as the prime in thesame time.

The fourth upper partial tone is the major Third of this secondhigher Octave, or e", with five times as many vibrations as theprime in the same time.

The fifth upper partial tone is the Fifth of the second higherOctave, or g", making six times as many vibrations as the primein the same time.

And thus they go on, becoming continually fainter, to tonesmaking 7, 8, 9, &c, times as many vibrations in the same time,as the prime tone. Or in musical notation

c"4

e"5

I)

9"6

¥'\>7

c'"8

d"

91 2 3 4 5 6 7 8 9 10

34 HARMONIC UPPER PARTIAL TONES. PART I.

where the figures beneath shew how many times the correspondingvibrational number is greater than that of the prime tone.

The whole sensation excited in the ear by a periodic vibrationof the air we have called a musical to tie. We now find that thisis compound, containing a series of different tones, which wedistinguish as the constituents or partial tones of the compound.The first of these constituents is the prime partial tone of thecompound, and the rest its harmonic upper partial tones. Thenumber which shews the order of any partial tone in the seriesshews how many times its vibrational number exceeds that of theprime tone.1 Thus, the second partial tone makes twice as many,the third three times as many vibrations in the same time asthe prime tone, and so on.

Gr. S. Ohm was the first to declare that there is only one formof vibration which will give rise to no harmonic upper partial tones,and which will therefore consist solely of the prime tone. This isthe form of vibration which we have described above as peculiar tothe pendulum and tuning-forks, and drawn in figs. 6 and 7. Wewill call these pendular vibrations, or, since they cannot be ana-lysed into a compound of different tones, simple vibrations. Inwhat sense not merely other musical tones, but all other formsof vibration, may be considered as compound, will be shewnhereafter (Chap. IV.) The terms simple ox pendular vibration*

1 [The ordinal number of a partial tone generally, must be distinguished from theordinal number of an ujrper partial tone in particular. For the same tone the formernumber is always greater by unity than the latter, because the' partials generallyinclude the prime, which is reckoned as the first, and the vpper partials exclude theprime, which being the lowest partial is of course not an tipper partial at all. Thusthe partials generally numbered 2 3 4 5 6 7 8 9 are the same as theupper partials numbered 1 2 3 4 5 6 7 8. As even the authorhas occasionally failed to note this distinction in the original German text, and otherwriters have constantly neglected it, too much weight cannot be here laid upon it.The presence or absence of the word upper before the word partial must always becarefully observed.—Translator.]

2 The-law of these vibrations may be popularly explained by means of the con-struction in fig. 10. Suppose a point to describe the circle of which c is the centrewith a uniform velocity, and that an observer stands at a considerable distance in theprolongation of the line eh, so that he does not see the surface of the circle but onlyits edge, in which case the point will appear merely to move up and down along itsdiameter a b. This up and down motion would take place exactly according to thelaw of pendular vibration. To represent this motion graphically by means of a curve,divide the length e g, supposed to correspond to the time of a single period, into asmany (here 12) equal parts as the circumference of the circle, and draw the perpen-diculars 1, '2, 3, &c, on the dividing points of the line eg, in order, equal in length

CHAP. I. HARMONIC UPPER PARTIAL TONES. 35

will therefore be used as synonymous. We have hitherto used theexpression tone and musical tone indifferently. It is absolutelynecessary to distinguish in acoustics first, a musical tone, that is,the impression made by any periodical vibration of the air;secondly, a simple tone, that is, the impression produced by asimple or pendular vibration of the air; and thirdly, a compoundtone, that is, the impression produced by the simultaneous actionof several simple tones with certain definite ratios of pitch asalready explained. A musical tone may be either simple or com-pound. For the sake of brevity, tone will be used in the generalsense of a musical tone, leaving the context or a prefixed qualifi-cation to determine whether it is simple or compound. A com-pound tone will often be briefly called a compound, and a simpletone will also be frequently called a partial, when used in connec-tion with a compound ; otherwise, the full expression simple tonewill be employed. A compound has, properly speaking, no singlepitch, as it is made of various partials having each its own pitch.By the pitch of a compound tone we shall therefore mean thepitch of its loivest partial or prime tone. By a chord or combina-tion of tones we mean several musical tones (whether simple orcompound) produced by different instruments so as to be heard atth3 same time. The facts here adduced shew us then that everymusical tone in which harmonic upper partial tones can be dis-tinguished, although produced by a single instrument, may really

to and in the same direction "with, those drawn in the circle from the correspondingpoints 1, 2, 3, &c. In this way we obtain the curve drawn in fig. 10, which agrees inform with that drawn by the tuning fork, fig. 6, p. 30, but is of a larger size.

FIG. 1Q.

Mathematically expressed, the distance of the vibrating point from its mean positionat any time is equal to the sine of an arc proportional to the corresponding time, andhence the form of simple vibration is also called the curve of sines.

D 2

36 HARMONIC UPPER PARTIAL TONES. 1>ABT I.

be considered as in itself a chord or combination of varioussimple tones.1

1 [The above paragraph relating to the English terms used in this translation,necessarily differs in many renpectB from the original, in which a justification is givenof the use made by the author of certain German expressions. It has been my objectto employ terms which should be thoroughly English, and should not in any wayrecall the German words. The word tone in English is extremely ambiguous. Prof.Tyndall {Lectures on Sound, 2nd ed. 1869, p. 117) has ventured to define a tone as asimple tone, in agreement with Prof. Helmholtz, who in the present passage limits theGorman word Ton in the same way. But I felt that an English reader could not besafely trusted to keep this very peculiar and important class of musical tones, whichlie has very rarely or never heard separately, invariably distinct from those musicaltones with which he is familiar, unless the word tone were uniformly qualified by theepithet simple. The only exception I could make was in the case of a partial tone,which is received at once as a new conception. Even Prof. Helmholt2 himself has notsucceeded in using his word Ton consistently for a simple tone only, and this was anadditional warning to me. English musicians have been also in the habit of usingtone to signify a certain musical interval, and semitone for half of that interval, onthe equally tempered scale. In this case I write Tone and Semitone with capitalinitials, a practice which I have found convenient for the names of all intervals, asThirds, Fifths, &c. Prof. Helmholtz uses the word Klang for a musical tone, whichgenerally, but not always, means a compound tone. Prof. Tyndall (ibid. p. 118) pro-poses to use the English word clang in the same sense. But clang has already ameaning in English, thus defined by Webster: ' a sharp shrill sound, made by strikingtogether metallic substances, or sonorous bodies, as the clang of arms, or any likesound, as the clang of trumpets. This word implies a degree of harshness in thesound, or more harshness than clink? Interpreted scientifically, then, clang accordingto this definition, is either noise or one of those musical tunes with inharmonic tipper•partials, which will be subsequently explained. It is therefore totally unadapted torepresent a musical tone in general, for which the simple word tone seems eminentlysuited, being of course originally the tone produced by a stretched string. Of course,if dang could not be used, Prof. Tyndall's suggestion to translate Prof. Helmholtz'sKlangfarbe by clangtint (ibid.) fell to the ground. I can find no valid reason forsupplanting the time-honoured expression quality of tone. Prof. Tyndall (ibid.) quotesPr Young to the effect that ' this quality of sound is sometimes called its register,colour, or timbre." Register has a distinct meaning in vocal music which must not bedisturbed. Timbre, properly a kettle drum, then a helmet, then the coat of armssurmounted with a helmet, then the official stamp bearing that coat of arms (nowused in France for a postage label), and then the mark which declared a thing to bewhat it pretends to be, Burns's ' guinea's stamp,' is a foreign word, often odiouslymispronounced, and not worth preserving. Colour I have never met with, except atmost as a passing metaphorical expression when applied to music. But the differenceof tones in quality is familiar to our language. Then as to the Upper Partial Tones,Prof. Helmholtz uses Thciltoi/c, and Purtialtb'ne, which are aptly Englished by partialsimple wnes. The word simple, however, may be omitted when partial is employed,as partial tones are necessarily simple. The constituent tones of a chord may be eithersimple or compound. The Grundton of a compound tone then becomes its prime tone,or briefly its -prime. The Grundton of a chord will be further explained hereafter.Upper partial (simple) tones, that is, the partials exclusive of the prime, "even whenharmonic, that is, for the most part, belonging to the first six partial tones, must

CHAP. I. HARMONIC UPPEE PARTIAL TONES. 37

'Now, since quality of tone, as we have seen, depends on theform of vibration, which also determines the occurrence of upperpartial tones, we have to inquire how far differences in quality oftone depend on different force or loudness of upper partials. Thisinquiry will be found to give a means of clearing up our concep-tions of what has hitherto been a perfect enigma,— the nature ofquality of tone. And we must then, of course, attempt to explainhow the ear manages to analyse every musical tone into a series ofpartial tones, and what is the meaning of this analysis. These in-vestigations will engage our attention in the following chapters.

be distinguished from the sounds usually called harmonics when produced on a violinor harp for instance, for such harmonics are not necessarily single tones, but are moregenerally compounds of some of the complete series of partial tones belonging to themusical tone of the whole string, produced by damping the remainder. The harmonicsheard in listening to the sound of a pianoforte string, struck and undamped, as thesound dies away,are also compound and not simple partial tones, but as they have thesuccessive partials for their successive primes, they have the pitch of those partials.Both sets of harmonics serve to give representations of upper partial tones, but theyare no more those upper partial tones themselves, than the original compound tone ofthe string is its own prime. Great confusion of thought having, to my own knowledge,arisen from confounding such harmonics with upper partial tones, I have generallyavoided using the ambiguous substantive harmonic. Prof. Helmholtz's term Obcrtonc,is merely a contraction for Oberpartialtone, but has led Prof. Tyndall to the trans-lation overtones, which, on many accounts, I prefer avoiding. The continual recur-rence of such words as clang, clangtint, overtone, would combine to give a strangeun-English appearance to a work like the present, which on the contrary I have en-deavoured to put into as straightforward English as possible. But for those acquaintedwith the original and with Prof. Tyndall's work, this explanation seemed necessary,— Translator.]