Before explaining the relation that sound has to telephony, it will be necessary to make quite plain what sound is, and how it affects the substance of the body through which it moves. If I strike my pencil upon the table, I hear a snap that appears to the ear to be simultaneous with the stroke: if, however, I see a man upon a somewhat distant hill strike a tree with an axe, the sound does not reach me until some appreciable time has passed; and it is noted, that, the farther away the place where a so-called sound originates, the longer time does it take to reach any listener. Hence sound has in air a certain velocity which has been very accurately measured, and found to be 1,093 feet per second when the temperature of the air is at the freezing point of water. As the temperature increases, the velocity of sound will increase a little more than one foot for every Fahrenheit degree; so that at 60° the velocity is 1,125 feet per second. This is the velocity in air. In water the velocity is about four times greater, in steel sixteen times, in pine-wood about ten times.

CONSTITUTION OF A SINGLE SOUND-WAVE.

If a person stands at the distance of fifteen or twenty rods from a cannon that is fired, he will first see the flash, then the cloud of smoke that rushes from the cannon's mouth, then the ground will be felt to tremble, and lastly the sound will reach his ear at the same time that a strong puff of air will be felt. This puff of air is the sound-wave itself, travelling at the rate of eleven hundred feet or more per second. At the instant of explosion of the gunpowder, the air in front of the cannon is very much compressed; and this compression at once begins to move outwards in every direction, so as to be a kind of a spherical shell of air constantly increasing in diameter; and, whenever it reaches an ear, the sound is perceived. Whenever such a sound-wave strikes upon a solid surface, as upon a cliff or a building, it is turned back, and the reflected wave may be heard; in which case we call it an echo. When a cannon is fired, we generally hear the sound repeated, so that it apparently lasts for a second or more; but when, as in the first case, we hear the sound of a pencil struck upon the table, but a single short report is noticed, and this, as may be supposed, consists of a single wave of condensed air.

Imagine a tuning-fork that is made to vibrate. Each of the prongs beats the air in opposite directions at the same time. Look at the physical condition of the air in front of one of these prongs. As the latter strikes outwards, the air in front of it will be driven outwards, condensed; and, on account of the elasticity of the air, the condensation will at once start to travel outwards in every direction,—a wave of denser air; but directly the prong recedes, beating the air back in the contrary direction, which will obviously rarefy the air on the first side. But the disturbance we call rarefaction moves in air with the same velocity as a condensation. We must therefore remember, that just behind the wave of condensation is the wave of rarefaction, both travelling with the same velocity, and therefore always maintaining the same relative position to each other. Now, the fork vibrates a great many times in a second, and will consequently generate as many of these waves, all of them constituted alike, and having the same length; by length meaning the sum of the thicknesses of the condensation and the rarefaction. Suppose a fork to make one hundred vibrations per second: at the end of the second, the wave generated by the vibration at the beginning of the second would have travelled, say, eleven hundred feet; and evenly distributed between the fork and the outer limit, would be ranged the intermediate waves occupying the whole distance: that is to say, in eleven hundred feet there would be one hundred sound-waves, each of them evidently being eleven feet long. If the fork made eleven hundred vibrations per second, each of these waves would be one foot long; for sound-waves of all lengths travel in air with the same rapidity. Some late experiments seem to show that the actual amplitude of motion of the air, when moved by such a high sound as that from a small whistle, is less than the millionth of an inch.

PITCH.

The pitch of a sound depends wholly upon the number of vibrations per second that produce it; and if one of two sounds consists of twice as many vibrations per second as the other one, they differ in pitch by the interval called in music an octave, this latter term merely signifying the number of intervals into which the larger interval is divided for the ordinary musical scale. The difference between a high and a low sound is simply in the number of vibrations of the air reaching the ear in a given time. The smaller intervals into which the octave is divided stand in mathematical relations to each other when they are properly produced, and are represented by the following fractions:—

These numbers are to be interpreted thus: Suppose that we have a tuning-fork giving 256 vibrations per second: the sound will be that of the standard or concert pitch for the C on the added line as shown on the staff. Now, D when properly tuned will make 9 vibrations while C makes but 8; but, as C in this case makes 256, D must make 256×⁹/₈=288. In like manner G is produced by 256׳/₂=384, and C above by 256×2=512, and so on for any of the others. If other sounds are used in the octave above or below this one, the number of vibrations of any given note may be found by either doubling or halving the number for the corresponding note in the given octave. Thus G below will consist of ³⁸⁴/₂=192, and G above of 384×2=768.

During the past century there has been a quite steady rise in the standard pitch, and this has been brought about in a very curious and unsuspected way. The tuning-fork has been the instrument to preserve the pitch, as it is the best available instrument for such a purpose, it being convenient to use, and does not vary as most other musical instruments do. But a tuning-fork is brought to its pitch with a file, which warms it somewhat, so that at the moment when it is in tune with the standard that is being duplicated it is above its normal temperature; and when it cools its tone rises. When another is made of like pitch with this one, the same thing is repeated; and so it has continued until the standard pitch has risen nearly a tone higher than it was in HÄndel's time.

The common A and C tuning-forks to be had in music stores, often vary a great deal from the accepted concert pitch. Such as the writer has measured have been generally too high; sometimes being ten or more vibrations per second beyond the proper number. The tuning-forks made by M. KÖenig of Paris are accurate within the tenth of one vibration, the C making 256 vibrations in one second.

LIMITS OF AUDIBILITY.

Numerous experiments have been made to determine the limits of audible sounds; and here it is found that there is a very great difference in individuals in their ability to perceive sounds. Helmholtz states that about 23 vibrations per second is the fewest in number that can be heard as continuous sound; if they are fewer in number than that, the vibrations are heard as separate distinct noises, as when one knocks upon a door four or five times a second. If one could knock evenly 23 times per second, he would be making a continuous musical sound of a very low pitch. But this limit of 23 is not the limit for all: some can hear a continuous sound with as few as 16 or 18 vibrations per second, while others are as far above the medium as this is below it. The limits of sound in musical instruments are about all included in the range of a 7-octave pianoforte from F to F, say from 42 to 5,460 vibrations per second. But this high number is not anywhere near the upper limit of audible sounds for man.

Very many of the familiar sounds of insects, such as crickets and mosquitoes, have a much higher pitch. Helmholtz puts this upper limit at 38,000 vibrations per second, and Despraetz at 36,850. The discrepancy of results is due solely to the marked difference in individuals as to acoustic perception.

For the production of high musical tones, KÖenig of Paris makes a set of steel rods. A steel rod of a certain length, diameter, and temper, will give a musical sound which may be determined. The proper length for other rods for giving higher tones may be determined by the rule that the number of vibrations is inversely proportional to the square of the length of the rod.

The dimensions of these rods when made 2 c. m. in diameter are as follows:—

These rods need to be suspended upon loops of silk, and they are struck with a piece of steel so short as to be wholly beyond the ability of any ear to hear its ring. Nothing but a short thud is to be heard from it when it strikes, while from the others comes a distinct ringing sound. In experimenting with such a set of steel rods I have not found any one yet who could hear as many as 25,000 per second, my own limit being about 21,000. But it has been experimentally found that children and youth have a perceptive power for high sounds considerably above adults. Dr. Clarence Blake of Boston reports a case in his aural practice, of a woman whose hearing had been gradually diminishing for some years until she could not hear at all with one ear, and the ticking of a watch could only be heard with the other when the watch was held against the ear. After treatment it was discovered that the sensibility to high sounds was very great, and that she could hear the steel rod having a tone of 40,000 vibrations.

Last year Mr. F. Galton, F.R.S., exhibited before the Science Conference an instrument in the shape of a very small whistle, which he had devised for producing a very high sound. The whistle had a diameter less than the one twenty-fifth of an inch. The length could be varied by moving a plug at the end of the whistle. It was easy to make a sound upon such an instrument that was altogether out of hearing-range of any person. Mr. Galton tried some very interesting experiments upon animals, by using these whistles. He went through the ZoÖlogical Gardens, and produced such high sounds near the ears of all the animals. Some of them would prick up their ears, showing that they heard the sound; while others apparently could not hear it. He declares that among all the animals the cat was found to hear the sharpest sound. Small dogs can also hear very shrill notes, while larger ones can not. Cattle were found to hear higher sounds than horses. The squeak of bats and of mice cannot be heard by many persons who can hear ordinary sounds as well as any; sharpness of hearing having nothing to do with the limits of hearing.

EFFECTS OF SOUND UPON OTHER BODIES.

If a vibrating tuning-fork be held close to a delicately suspended body, the latter will approach the fork, as if impelled by some attractive force. The experiment can be made by fastening a bit of paper about an inch square to a straw five or six inches long, and then suspending the straw to a thread, so that it is balanced horizontally. Bring the vibrating tuning-fork within a quarter of an inch of the paper. In this case the motion of approach is due to the fact that the pressure of the air is less close to a vibrating body than at a distance from it; there is therefore a slightly greater pressure on the side of the paper away from the fork than on the side next to it.

If a vibrating tuning-fork be held near to the ear, and turned around, there may be found four places in one rotation where the sound will be heard but very faintly, while in every other position it can be heard plainly enough. The extinction of the sound is due to what is called interference. Each of the prongs of the fork is giving out a sound-wave at the same time, but in opposite directions, each wave advancing outwards in every direction. Where the rarefied part of one wave exactly balances the condensed part of the other, there of course the sound will be extinguished; and these lines of interference are found to be hyperbolas, or, if considered with reference to both entire waves, two hyperbolic surfaces.

SYMPATHETIC VIBRATIONS.

When it is once understood that a musical sound is caused by the vibrations more or less frequent which only make the difference we call pitch, it might at once be inferred, that if we have a body that is capable of vibrating say a hundred times a second, and it receives a hundred pulses or pushes a second, it would in this way be made to vibrate. Suppose, then, that we take two tuning-forks, each capable of vibrating 256 times a second: if one be struck while the other is left free, the former one will be giving to the air 256 impulses per second, which will reach the other fork, each pulse tending to move it a little, the cumulative result being to make it move perceptibly, that is, to give out a sound. The principle is just the same as that employed in the common swing. One push makes the swing to move a little, upon its return another is given, in like manner a third, and so on until a person may be swung many feet high. If a glass tumbler be struck, it gives out a musical sound of a certain pitch, which will set a piano-string sounding that is tuned to the same pitch, provided that the damper be raised. It is said that some persons' voices have broken tumblers by singing powerfully near them the same note which the tumblers could give out, the vibrations of the tumblers being so great as to overcome cohesion of the molecules.

There are very many interesting effects due to sympathetic vibrations.

Large trees are sometimes uprooted by wind that comes in gusts timed to the rate of vibration of the tree. When troops of soldiers are to cross a bridge, the music ceases, and the ranks are broken, lest the accumulated strain of timed vibrations should break the structure; indeed, such accidents have several times occurred. There is not so much danger to a bridge when it is heavily loaded with men or with cattle, as when a few men go marching over it. "When the iron bridge at Colebrooke Dale was building, a fiddler came along, and said to the workmen that he could fiddle their bridge down. The builders thought this boast a fiddle-de-dee, and invited the musician to fiddle away to his heart's content. One note after another was struck upon the strings, until one was found with which the bridge was in sympathy. When the bridge began to shake violently, the workmen were alarmed at the unexpected result, and ordered the fiddler to stop."

Some halls and churches are wretchedly adapted to hear either speaking or singing in. If wires be stretched across such halls, between the speaker's stand and the opposite end, they will absorb the passing sound-waves, and will be made to sympathetically vibrate, thus preventing in a good degree the interfering echoes. The wire should be rather fine piano-wire, and it should be stretched so tightly as to give out a low musical sound when plucked with the fingers. In a large hall there should be twenty or more such wires.

RESONANCE.

When a tuning-fork is struck, and held out in the air, the vibrations can be felt for a time by the fingers; but the sound is hardly audible unless the fork be placed close to the ear. Let the stem of the fork rest upon the table, a chair, or any solid body of considerable size, and the sound is so much increased in loudness as to be heard in every part of a large room. The reason appears to be, that in the first case the vibrations are so slight that the air is not much affected. Most of the force of the vibration is absorbed by the hand that holds it; but when the stem rests upon a hard body of considerable extent, the vibrations are given up to it, and every part of its surface is giving off the vibrations to the air. In other words, it is a much larger body that is now vibrating, and consequently the air is receiving the amplified sound-waves.

If the stem of the fork had been made to rest upon a bit of rubber, the sound would not only not have been re-enforced in such a way, but the fork would very soon have been brought to rest; for India rubber absorbs sound vibrations, and converts them into heat vibrations, as is proved by placing such a combination upon the face of a thermo-pile.

If one will but put his hand upon a table or a chair-back in any room where a piano or an organ is being played, or where voices are singing, especially in church, he cannot fail to feel the sound; and if he notices carefully he will perceive that some sounds make such table or seat to shake much more vigorously than others,—a genuine case of sympathetic vibrations.

It is for this reason that special materials and shapes are given to parts of musical instruments, so that they may respond to the various vibrations of the strings or reeds. For instance, the piano has an extensive thin board of spruce underneath all the strings, which is called the sounding-board. This board takes up the vibrations of the strings; but, unlike the rubber, gives them all out to the air, greatly re-enforcing their strength, and changing somewhat their quality. But the air itself may act in like manner. In almost any room or hall not more than fifteen or twenty feet long, a person can find some tone of the voice that will seem to meet some response from the room. Some short tunnels will from certain positions yield very powerful, responsive, resonant tones. There is certainly one such in Central Park, New York. It is forty or fifty feet long. To a person standing in the middle of this, and speaking or making any kind of a noise on a certain pitch, the resonance is almost deafening. It is easy to understand. When a column of air enclosed in a tube is made to vibrate by any sound whose wave-length is twice the length of the tube, we have such column of air now filled with the condensed part of the wave, and now with the rarefied part; and as these motions cannot be conducted laterally, but must move in the direction of the length of the tube, the air has a very great amplitude of motion, and the sound is very loud. If one end of the tube be closed, then the length must be but one-fourth of the wave-length of the sound. Take a tuning-fork of any convenient pitch, say a C of 512 vibrations per second: hold it while vibrating over a vertical test-tube about eight inches long. No response will be heard; but, if a little water be carefully poured into the tube to the depth of about two inches, the tube will respond loudly, so that it might be heard over a large hall. In this case the length of the air-column that was responding, being one-fourth the wave-length, would give twenty-four inches as the wave-length of that fork.

It is easy in this way to measure approximately the number of vibrations made by a fork.

When a vibrating tuning-fork is placed opposite the embouchure of an organ-pipe of the same pitch, the pipe will resound to it, giving quite a volume of sound. In 1872 it occurred to me, that the action of an organ-pipe might be quite like that of a vibrating reed in front of the embouchure. As the air is driven past it from the bellows, the form of the escaping air will evidently be like a thin, elastic strip; and, having considerable velocity, it will carry off by friction a little of the air in the tube: this will of course rarefy the air in the tube somewhat, and a wave of condensation will travel down the tube. At the bottom, being suddenly stopped, its re-action will be partly outwards, and so will drive the strip of air away from the tube. After this will follow, for a like reason, the other phase of the wave, the rarefaction, which will swing the strip of air towards the tube. This theory I verified by filling the bellows with smoke, and watching the motion of the escaping air and smoke with a stroboscope. This view is now advocated by an organ-builder in England, Herman Smith; but whether he discovered it before or after me, I do not know.

When a membrane vibrates, its motion is generally perceptible to the eye; and it may have a very great amplitude of motion, as in the case of the drum; and various instruments have been devised for the study of vibrations, using membranes like rubber, gold-beater's skin, or even tissue paper, to receive the vibrations. One of the musical instruments of a former generation of boys was the comb. A strip of paper was placed in front of it, and placed at the mouth, and sung through, the paper responding to the pitch with a loose nasal sound. KÖenig fixed a membrane across a small capsule, one side of which was connected by a tube to any source of sound, and the other side to a gas-pipe and a small burner. A sound made in the tube would shake the flame, and a mirror moving in front of the flame would show a zigzag outline corresponding to the sound vibrations.

In like manner if a thin rubber be stretched over the end of a tube one or two inches in diameter and four or five inches long, and a bit of looking-glass one-fourth of an inch square be made fast to the middle of the membrane, the motions of the latter can be seen by letting a beam of sunlight fall upon the mirror so as to be reflected upon a white wall or screen a few feet away. (Fig. 8.)

When a sound is made in this tube, the spot of light will at once assume some peculiar form,—either a straight line with some knots of light in it, or some curve simple or compound, and such as are known as Lissajous curves. If, while some of these forms are upon the screen, the instrument be moved sideways, the forms will change to undulating lines with or without loops, varying with the pitch and intensity, but being alike for the same pitch and intensity. (Fig. 9)

This instrument I called the opeidoscope.

The vibration of a membrane and that of a solid differ chiefly in the amplitude of such vibration. The scratch of a pin at one end of a long log can be heard by an ear applied to the other end of the log; but every molecule in the log must move slightly; and there are all degrees of movement between that visible to the eye, which we call mass motion, and that called molecular simply because we cannot measure the amplitude of the motion. We may, then, roughly divide all bodies into two classes, as to their relations to sound,—such as re-enforce it, and such as distribute it: the first depending upon the form of the body, as related to a particular sound; the second independent of form, and responding to all orders of vibrations. Air, wood, and metals belong in this latter class. The common toy-string telegraph, or lovers' telegraph, is an example of this class. Two tin boxes are connected by a string passing through the middle of the bottom of each. When the string is stretched, and a person speaks in one box, what is said can be heard by an ear applied at the other. If the speaking-tubes be made about four inches in diameter, and about four inches deep, they are capable of doing much more service than is generally supposed to be possible. I know of two lines, one of five hundred feet and the other of a thousand feet in length, over which one can talk, and be heard with distinctness. In the line of a thousand feet, the end of the tube is made of sheepskin tightly stretched, and the line is made of No. 8 cotton thread. The greater the tension, the better is the sound transmitted. The thread is supported at intervals by running through a loop on the ends of cords not less than three feet long, attached to supports. The thread pierces the membrane, and is attached to a small button which is in contact with the membrane. Wind and rain affect this line disadvantageously. The other line of five hundred feet, between a passenger and a freight depot, has the tube end covered with stretched calfskin. Instead of thread, a copper-relay wire is employed (any small uninsulated wire will do as well). This permits a good tension, and is unaffected by the weather. One may stand in front of it about three feet, and converse with ease, and in an ordinary tone. The wire is supported in loops of string, as in the other.

Musicians have in all times employed various instruments for the production of musical effects. Whistles made of bone were used by pre-historic men, some of them having finger-holes so that different tones could be produced. A stag-horn that was blown like a flageolet, and having three finger-holes, has also been found; while on the old monuments of Egypt are pictured harps, pipes with seven finger-holes, a kind of flute, drums, tambourines, cymbals, and trumpets. In later times these primeval forms have been modified into the various instruments in use in the modern orchestra. It seems as if no musician had ever been interested in the question as to why one instrument should give out a sound so different from another one, even though it was sounding upon the same pitch. No one can ever mistake the sound of a violin, or a horn, or a piano, for any other instrument; and no two persons have voices alike. This difference in tone, which enables us to identify an instrument by its sound or a friend by his voice, is called quality of tone, or timbre.

About twenty years ago, that great German physicist Helmholtz undertook the investigation of this subject, and succeeded in unravelling the whole mystery of the qualities of sound.

He discovered first, that a musical sound is very rarely a simple tone, but is made up of several tones, sometimes as many as ten or fifteen, having different degrees of intensity and pitch. The lowest sound, which is also the strongest, is called the fundamental; and it is this tone we mean when we speak of the pitch of a sound, as the pitch of middle C upon a piano, or the pitch of the A string on a violin. The higher sounds that accompany the fundamental are called sometimes harmonics, sometimes upper partial tones, but generally overtones. The character or quality of a sound depends altogether upon the number and intensity of these overtones associated with the fundamental. If a sound can be made upon a pipe and a violin, that consists wholly of the fundamental with no overtones, the two instruments sound absolutely alike. It is exceedingly difficult to do this; and such sound when produced is smooth, but without character, and unpleasing.

Second, Helmholtz discovered that the overtones always stand in the simplest mathematical relation to the fundamental tone,—in fact, are simple multiples of that tone, being two, three, four, and so on, times the number of vibrations of it.

This will be readily understood by considering the position of such related sounds when they are written upon the staff.

If we start with C in the bass as indicated in the staff, calling that the fundamental, then the notes that will represent the above ratios are those indicated by smaller notes, which are the overtones up to the ninth. The first overtone, being produced by twice the number of vibrations, must be the octave; the second, the fifth of the second octave; the third will be two octaves from the first, and so on: the number of vibrations of each of these notes being the number of the fundamental multiplied by its order in the series.

Taking C with 128 vibrations, we have for this series:—

This series is continued up to the limits of hearing. Now, it appears that all instruments do not give the complete series: indeed, it is not possible to obtain them all upon some instruments. Each of them, however, when present helps in the general effect which we call quality. Sometimes the overtones are more prominent than the fundamental, as when a piano-wire is struck with a nail. It has always been noticed that it does not give out the sound that is wanted when it is struck in this way. Hence it is the art of an instrument-maker to so construct the instrument as to develop and re-enforce such tones as are pleasing, and to suppress the interfering and disagreeable overtones. Piano-makers learned by trial where was the proper place to strike the stretched wire in order to develop the most musical sound upon it; but no reason could be given until it was observed that striking it at a point about one-seventh or one-ninth its length from either end prevented the development of the objectionable overtones, the seventh and the ninth. Hence they can scarcely be heard in a properly constructed instrument. These overtones are very discordant with the lower sounds.

Organ-pipes have their specific qualities given to them by making them wide-mouthed, narrow-mouthed, conical, and so on; shapes which experience has determined give pleasing sounds with different qualities.

The violin is an instrument that seems to puzzle makers more than almost any other. Some of the old violins made two hundred years ago by the Amati family at Cremona are worth many times their weight in gold. Recent makers have tried in vain to equal them; but, when their ingenuity and skill have failed, they declare that age has much to do with such instruments, that age mellows the sounding quality of the violin. But the Cremona violins were just as extraordinary instruments when they left the hands of the makers as they are now; and the fame of the Amati family as violin-makers was over all Europe while they were living.

A good violin when well played gives an exquisite musical effect, and on account of its range and quality of tones it is the leading orchestral instrument, always pleasing and satisfying; but in unskilled hands even the best Cremona will give forth sounds that make one grieve that it was ever invented. Overtones of all sorts and with all degrees of prominence may be easily developed upon it: therefore the skilful player draws the bow at such a place upon the strings as to develop the overtones he wants, and suppress the ones not wanted. The usual rule is to draw the bow about an inch below the bridge; but the place for the bow depends upon where the fingers are that stop the strings, and also the pressure upon it. It requires an almost incredible amount of practice to be able to play a violin very well.

In the accompanying table will be found the component parts of tones upon a few instruments in common use.

TONE COMPOSITION.

The components of the tones are indicated by lines in the column underneath the figures representing the series. Thus the narrow-stopped organ-pipe gives a sound composed of a fundamental, and overtones three, five, seven, and nine times the number of vibrations of it.

TONE COMPOSITION.

It must not be inferred that all of the overtones are of equal strength: they are very far from that; but these differ in different instruments, and it is this that constitutes the difference between a good instrument and a poor one of the same name.

In a few of the spaces very light lines are made for the purpose of indicating that such overtones are quite weak. For instance: the piano has the sixth, seventh, and eighth thus marked; these tones being suppressed by the mechanism, as described on a former page.

Only a few of the many forms of organ-pipes are given; but these are sufficient to show what a physical difference there is between the musical tones in such pipes.

As for the human voice, it is very rich in overtones; but no two voices are alike, therefore it would be impossible to tabulate the components of it in the manner they are tabulated for musical instruments.

In Helmholtz's experiments in the analysis of sounds, use was made of the principle of resonance of a body of air enclosed in a vessel. In the experiment with the tuning-fork to determine the wave-length, p. 78, it is remarked that no response came until the volume of the air in the tube was reduced to a certain length, which depended upon the vibration number of the fork. If instead of a test-tube a bottle had been taken, the result would have been the same. Every kind of a vessel can respond to some tone of a definite wave-length, and a sphere has been found to give the best results. These are made with a hole on one side for the sound-wave to enter, and a projection on the opposite side, through which a hole about the one-eighth of an inch is made, this to be placed in the ear. Any sound that is made in front of the large orifice will not meet any response, unless it be that particular one which the globe can naturally re-enforce, when it will be plainly heard. Suppose, then, one has a series of twenty or more of these, graduated to the proper size for re-enforcing sounds in the ratio of one, two, three, four, and so on. Take any instrument, say a flute: have one to blow it upon the proper pitch to respond to the largest sphere, then take each of the spheres in their order, applying them to the ear while the flute is being sounded. When the overtones are present they will be heard plainly and distinct from the fundamental sound. In like manner any or all other sounds may be studied.

But Helmholtz did not stop after analyzing sounds of so many kinds: he invented a method of synthesis, by which the sounds of any kind of an instrument could be imitated. A tuning-fork, when made to vibrate by an electric current, gives out a tone without harmonics or overtones. So if a series of forks with vibration periods equal to the numbers of the series of overtones given on p. 86 be so arranged that any of them may be made to vibrate at will, it is evident that the resulting compound tone would be comparable with that from an instrument having such overtones. Thus, if with a tuning-fork giving a fundamental C, other forks giving two, three, and four times the number of the fundamental were associated, each one giving a simple tone, we should have for a resultant the tone of a flute, as shown on p. 91. If one, three, five, seven, and nine, were all sounded, the resulting tone would be that of the clarionet, and so on. This he actually accomplished, and now makers of physical apparatus advertise just such instruments.

Helmholtz also contrived a set of tuning-forks, which, when bowed, will give out the vowel sounds like the voice.

It was remarked upon p. 89 that it has generally been considered that age has a mellowing effect upon the sound of a violin. Once in possession of the facts concerning sound that have been alluded to on the preceding pages, it is easy to see how such an opinion should arise, and also the fallacy of it. It is proved conclusively that the ability to hear high sounds decreases as one grows older. As the violin gives a very great number of overtones, even up to the limits of audibility, it is plain that if such an instrument should not change in its quality of tone in the least degree, yet to a man who played upon it for a number of years it would seem to change by subtracting some of the higher overtones from the sound; that is, it would seem to become mellower. There is no evidence that such a physical change takes place in the instrument. It is not here affirmed that no change does take place. It may be probable; but all the evidence we have is the opinions of individuals whose hearing we know does change; and this change is competent to modify the judgment as to the quality of the sound in the same direction. Before it can be affirmed that such a physical change does take place in the violin as to make a perceptible difference in the quality of its tone, it will be needful to determine accurately the number and intensity of the overtones at intervals during many years, and then to compare them. This has not yet been done.

FORM OF A COMPOUND SOUND-WAVE IN AIR.

Upon p. 63 is given a picture of the form of a simple sound-wave in air, which, as described, consists of two parts, a condensation and a rarefaction. All simple sound-waves have such a form; but when two or more sound-waves that stand in some simple ratio to each other, as do the sounds of musical instruments, are formed in air, the resulting wave is more or less complex in structure; and where there are many components, as there are where a number of different kinds of instruments are all sounding at once, it is well-nigh impossible to figure even approximately the form of such wave-combinations. It is generally given in treatises upon sound with ordinates representing the factors with their relative intensities. When the extremities of the ordinates are connected, there is drawn a curved line with regularly recurring loops. This cannot give a correct idea of the form of the wave, because the motion of a particle of air is not up and down like a floating body upon waving water, but it is forward and back, in the direction of the motion of the wave.

In Fig. 10 three simple sound-waves are thus represented at 1, 2, and 3, these having the wave-length 1, 2, and 3. In 4, the three are combined into one compound wave, and better show the form of a transverse section of such a sound-wave in the air. The organ-pipe called the principal gives out such a compound wave as is seen by referring to the table on p. 91. The second overtone, however, is quite weak in that pipe, which would so modify the form as to lessen somewhat the density at b, and increase it at a.

In like manner the space in the length of the fundamental sound, whatever it may be, is divided up into a number of minor condensations and rarefactions, which may strengthen each other, or so interfere as to change the position of both; as is seen in the figure at b, where the condensation due to wave 2 interferes with the rarefaction of 3.

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