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### Chapter III.

#### ANALYSIS OF MUSICAL TONES BY SYMPATHETIC RESONANCE.

We proceed to shew that the simple partial tones contained in a composite mass of musical tones, produce peculiar mechanical effects in nature, altogether independent of the human ear and its sensations, and also altogether independent of merely theoretical considerations. These effects consequently give a peculiar objective significance to this peculiar method of analysing vibrational forms.

Such an effect occurs in the phenomenon of sympathetic resonance. This phenomenon is always found in those bodies which when once set in motion by any impulse, continue to perform a long series of vibrations before they come to rest. When these bodies are struck gently, but periodically, although each blow may be separately quite insufficient to produce a sensible motion in the vibratory body, yet, provided the periodic time of the gentle blows is precisely the same as the periodic time of the body's own vibrations, very large and powerful oscillations may result. But if the periodic time of the regular blows is different from the periodic time of the oscillations, the resulting motion will be weak or quite insensible.

Periodic impulses of this kind generally proceed from another body which is already vibrating regularly, and in this case the swings of the latter in the course of a little time, call into action the swings of the former. Under these circumstances we have the process called sympathetic oscillation or sympathetic resonance. The essence of the mechanical effect is independent of the rate of motion, which may be fast enough to excite the sensation of sound, or slow enough not to produce anything of the kind. Musicians are well acquainted with sympathetic resonance. When, for example, the strings of two violins are in exact unison, and one string is bowed, the other will begin to vibrate. But the nature of the process is best seen in instances where the vibrations are slow enough for the eye to follow the whole of their successive phases.

Thus, for example, it is known that the largest church-bells may be set in motion by a man, or even a boy, who pulls the ropes attached to them at proper and regular intervals, even when their weight of metal is so great that the strongest man could scarcely move them sensibly, if he did not apply his strength in determinate periodical intervals. When such a bell is once set in motion, it continues, like a struck pendulum, to oscillate for some time, until it gradually returns to rest, even if it is left quite by itself, and no force is employed to arrest its motion. The motion diminishes gradually, as we know, because the friction on the axis and the resistance of the air at every swing destroy a portion of the existing moving force.

As the bell swings backwards and forwards, the lever and rope fixed to its axis rise and fall. If when the lever falls a boy clings to the lower end of the bell-rope, his weight will act so as to increase the rapidity of the existing motion. This increase of velocity may be very small, and yet it will produce a corresponding increase in the extent of the bell's swings, which again will continue for a while, until destroyed by the friction and resistance of the air. But if the boy clung to the bell-rope at a wrong time, while it was ascending, for instance, the weight of his body would act in opposition to the motion of the bell, and the extent of swing would decrease. Now, if the boy continued to cling to the rope at each swing so long as it was falling, and then let it ascend freely, at every swing the motion of the bell would be only increased in speed, and its swings would gradually become greater and greater, until by their increase the motion imparted on every oscillation of the bell to the walls of the belfry, and the external air would become so great as exactly to be covered by the power exerted by the boy at each swing.

The success of this process depends, therefore, essentially on the boy's applying his force only at those moments when it will increase the motion of the bell. That is, he must employ his strength periodically, and the periodic time must be equal to that of the bell's swing, or he will not be successful. He would just as easily bring the swinging bell to rest, if he clung to the rope only during its ascent, and thus let his weight be raised by the bell.

A similar experiment which can be tried at any instant is the following. Construct a pendulum by hanging a heavy body (such as a ring) to the lower end of a thread, holding the upper end in the hand. On setting the ring into gentle pendular vibration, it will be found that this motion can be gradually and considerably increased by watching the moment when the pendulum has reached its greatest departure from the vertical, and then giving the hand a very small motion in the opposite direction. Thus, when the pendulum is furthest to the right, move the hand very slightly to the left; and when the pendulum is furthest to the left, move the hand to the right. The pendulum may be also set in motion from a stat of rest by giving the hand similar very slight motions having the same periodic time as the pendulum's own swings. The displacements of the hand may be so small under these circumstances, that they can scarcely be perceived with the closest attention, a circumstance to which is due the superstitious application of this little apparatus as a divining rod. If namely the observer, without thinking of his hand, follows the swings of the pendulum with his eye, the hand readily follows the eye, and involuntarily moves a little backwards or forwards, precisely in the same time as the pendulum, after this has accidentally begun to move. These involuntary motions of the hand are usually overlooked, at least when the observer is not accustomed to exact observations on such unobtrusive influences. By this means any existing vibration of the pendulum is increased and kept up, and any accidental motion of the ring is readily converted into pendular vibrations, which seem to arise spontaneously without any co-operation of the observer, and are hence attributed to the influence of hidden metals, running streams, and so on.

If on the other hand the motion of the hand is intentionally made in the contrary direction, the pendulum soon comes to rest.

The explanation of the process is very simple. When the upper end of the thread is fastened to an immovable support, the pendulum, once struck, continues to swing for a long time, and the extent of its swings diminishes very slowly. We can suppose the extent of the swings to be measured by the angle which the thread makes with the vertical on its greatest deflection from it. If the attached body at the point of greatest deflection lies to the right, and we move the hand to the left, we manifestly increase the angle between the string and the vertical, and consequently also augment the extent of the swing. By moving the upper end of the string in the opposite direction we should decrease the extent of the swing.

In this case there is no necessity for moving the hand in the same periodic time as the pendulum swings. We might move the hand backwards and forwards only at every third or fifth or other swing of the pendulum, and we should still produce large swings. Thus, when the pendulum is to the right, move the hand to the left, and keep it still, till the pendulum has swung to the left, then again to the right, and then once more to the left, and then return the hand to its first position, afterwards wait till the pendulum has swung to the right, then to the left, and again to the right, and then recommence the first motion of the hand. In this way three complete vibrations, or double excursions of the pendulum, will correspond to one left and right motion of the hand. In the same way one left and right motion of the hand may be made to correspond with seven or more swings of the pendulum. The meaning of this process is always that the motion of the hand must in each case be made at such a time and in such a direction as to be opposed to the deflection of the pendulum and consequently to increase it.

By a slight alteration of the process we can easily make two, four, six, &c., swings of the pendulum correspond to one left and right motion of the hand; for a sudden motion of the hand at the instant of the pendulum's passage through the vertical has no influence on the size of the swings. Hence when the pendulum lies to the right move the hand to the left, and so increase its velocity, let it swing to the left, watch for the moment of its passing the vertical line, and at that instant return the hand to its original position, allow it to reach the right, and then again the left and once more the right extremity of its arc, and then recommence the first motion of the hand.

We are able then to communicate violent motion to the pendulum by very small periodical vibrations of the hand, having their periodic time exactly as great, or else two, three, four, &c., times as great as that of the pendular oscillation. We have here considered that the motion of the hand is backwards. This is not necessary. It may take place continuously in any other way we please. When it moves continuously there will be generally portions of time during which it will increase the pendulum's motion, and others perhaps in which it will diminish the same. In order to create strong vibrations in the pendulum, then, it will be necessary that the increments of motion should be permanently predominant, and should not be neutralised by the sum of the decrements.

Now if a determinate periodic motion were assigned to the hand, and we wished to discover whether it would produce considerable vibrations in the pendulum, we could not always predict the result without calculation. Theoretical mechanics would, however, prescribe the following process to be pursued: Analyse the periodic motion of the hand into a sum of simple pendular vibrations of the hand — exactly in the same way as was laid down in the last chapter for the periodic motions of the particles of air, — then, if the periodic time of one of these vibrations is equal to the periodic time of the pendulum's own oscillations, the pendulum will be set into violent motion, but not otherwise. We might compound small pendular motions of the hand out of vibrations of other periodic times, as much as we liked, but we should fail to produce any lasting strong swings of the pendulum. Hence the analysis of the motion of the hand into pendular swings has a real meaning in nature, producing determinate mechanical effects, and for the present purpose no other analysis of the motion of the hand into any other partial motions can be substituted for it.

In the above examples the pendulum could be set into sympathetic vibration, when the hand moved periodically at the same rate as the pendulum; in this case the longest partial vibration of the hand, corresponding to the prime tone of a resonant vibration, was, so to speak, in unison with the pendulum. When three swings of the pendulum went to one backwards and forwards motion of the hand, it was the third partial swing of the hand, answering as it were to the Twelfth of its prime tone, which set the pendulum in motion. And so on.

The same process that we have thus become acquainted with for swings of long periodic time, holds precisely for swings of so short a period as sonorous vibrations. Any elastic body which is so fastened as to admit of continuing its vibrations for some length of time when once set in motion, can also be made to vibrate sympathetically, when it receives periodic agitations of comparatively small amounts, having a periodic time corresponding to that of its own tone.

Gently touch one of the keys of a pianoforte without striking the string, so as to raise the damper only, and then sing a note of the corresponding pitch forcibly directing the voice against the strings of the instrument. On ceasing to sing, the note will be echoed back from the piano. It is easy to discover that this echo is caused by the string which is in unison with the note, for directly the hand is removed from the key, and the damper is allowed to fall, the echo ceases. The sympathetic vibration of the string is still better shewn by putting little paper riders upon it, which are jerked off as soon as the string vibrates. The more exactly the singer hits the pitch of the string, the more strongly it vibrates. A very little deviation from the exact pitch fails in exciting sympathetic vibration.

In this experiment the sounding board of the instrument is first struck by the vibrations of the air excited by the human voice. The sounding board is well known to consist of a broad flexible wooden plate, which, owing to its extensive surface, is better adapted to convey the agitation of the strings to the air, and of the air to the strings, than the small surface over which string and air are themselves directly in contact. The sounding board first communicates the agitations which it receives from the air excited by the singer, to the points where the string is fastened. The magnitude of any single such agitation is of course infinitesimally small. A very large number of such effects must necessarily be aggregated, before any sensible motion of the string can be caused. And such a continuous addition of effects really takes place, if, as in the preceding experiments with the bell and the pendulum, the periodic time of the small agitations which are communicated to the extremities of the string by the air, through the intervention of the sounding board, exactly corresponds to the periodic time of the string's own vibrations. When this is the case, a long series of such vibrations will really set the string into motion which is very violent in comparison with the exciting cause.

In place of the human voice we might of course use any other musical instrument. Provided only that it can produce the tone of the pianoforte string accurately and sustain it powerfully, it will bring the latter into sympathetic vibration. In place of a pianoforte, again, we can employ any other stringed instrument having a sounding board, as a violin, guitar, harp, &c., and also stretched membranes, bells, elastic tongues or plates, &c., provided only that the latter are so fastened as to admit of their giving a tone of sensible duration when once made to sound.

When the pitch of the original sounding body is not exactly that of the sympathising body, or that which is meant to vibrate in sympathy with it, the latter will nevertheless often make sensible sympathetic vibrations, which will diminish in amplitude as the difference of pitch increases. But in this respect different sounding bodies shew great differences, according to the length of time for which they continue to sound after having been set in action before communicating their whole motion to the air.

Bodies of small mass, which readily communicate their motion to the air, and quickly cease to sound, as, for example, stretched membranes, or violin strings, are readily set in sympathetic vibration, because the motion of the air is conversely readily transferred to them, and they are also sensibly moved by sufficiently strong agitations of the air, even when the latter have not precisely the same periodic time as the natural tone of the sympathising bodies. The limits of pitch capable of exciting sympathetic vibration are consequently a little wider in this case. By the comparatively greater influence of the motion of the air upon light elastic bodies of this kind which offer but little resistance, their natural periodic time can be slightly altered, and adapted to that of the exciting tone. Massive elastic bodies, on the other hand, which are not readily movable, and are slow in communicating their sonorous vibrations to the air, such as bells and plates, and continue to sound for a long time, are also more difficult to move by the air. A much longer addition of effects is required for this purpose, and consequently it is also necessary to hit the pitch of their own tone with much greater nicety, in order to make them vibrate sympathetically. Still it is well known that bell-shaped glasses can be put into violent motion by singing their proper tone into them; indeed it is related that singers with very powerful and pure voices, have sometimes been able to crack them by the agitation thus caused. The principal difficulty in this experiment is in hitting the pitch with sufficient precision, and retaining the tone at that exact pitch for a sufficient length of time.

Tuning-forks are the most difficult bodies to set in sympathetic vibration. To effect this they may be fastened on sounding boxes which have been exactly tuned to their tone, as shewn in fig. 13. If we have two such forks of exactly the same pitch, and excite one by a violin bow, the other will begin to vibrate in sympathy, even if placed at the further end of the same room, and it will continue to sound, after the first has been damped. The astonishing nature of such a case of sympathetic vibration will appear, if we merely compare the heavy and powerful mass of steel set in motion, with the light yielding mass of air which produces the effect by such small motive powers that they could not stir the lightest spring which was not in tune with the fork. With such forks the time required to set them in full swing by sympathetic action, is also of sensible duration, and the lightest disagreement in pitch is sufficient to produce a sensible diminution in the sympathetic effect. By sticking a piece of wax to one prong of the second fork, sufficient to make it vibrate once in a second less than the first — a difference of pitch scarcely sensible to the finest ear — the sympathetic vibration will be wholly destroyed.

After having thus described the phenomenon of sympathetic vibration in general, we proceed to investigate the influence exerted in sympathetic resonance by the different forms of wave of a musical tone.

First, it must be observed that most elastic bodies which have been set into sustained vibration by a gentle force acting periodically, are (with a few exceptions to be considered hereafter) always made to swing in pendular vibrations. But they are in general capable of executing several kinds of such vibration with different periodic times and with a different distribution over the various parts of the vibrating body. Hence to the different lengths of the periodic times correspond different simple tones producible on such an elastic body. These are its so-called proper tones. It is, however, only exceptionally, as in strings and the narrower kinds of organ pipes, that these proper tones correspond in pitch with the harmonic upper partial tones of a musical tone already mentioned. They are for the most part inharmonic in relation to the prime tone.

In many cases the vibrations and their mode of distribution over the vibrating bodies can be rendered visible by strewing a little fine sand over the latter. Take, for example, a membrane (as a bladder or piece of thin india-rubber) stretched over a circular ring. In fig. 14 are shewn the various forms which a membrane can assume when it vibrates. The diameters and circles on the surface of the membrane mark those points which remain at rest during the vibration, and are known as nodal lines. By these the surface is divided into a number of compartments which bend alternately up and down, in such a way that while those marked ( + ) rise, those marked ( - ) fall. Over the figures a, b, c, are shewn the forms of a section of the membrane during vibration. Only those forms of motion are drawn which correspond with the deepest and most easily producible tones of the membrane. The number of circles and diameters can be increased at pleasure by taking a sufficiently thin membrane, and stretching it with sufficient regularity, and in this case the tones would continually sharpen in pitch. By strewing sand on the membrane the figures are easily rendered visible, for as soon as it begins to vibrate the particles of sand collect on the nodal lines.

In the same way it is possible to render visible the nodal lines and forms of vibration of oval and square membranes, and of differently-shaped plane elastic plates, bars, and so on. These form a series of very interesting phenomena discovered by Chladni, but to pursue them would lead us too far from our proper subject. It will suffice to give a few details respecting the simplest case, that of a circular membrane.

In the time required by the membrane to execute 100 vibrations of the form a, fig. 14 (p. 40), the number of vibrations executed by the other forms is as follows : —

 Form of Vibration Pitch Number Cents[1] Notes nearly a   without nodal lines 100 0 $$c$$ b   with one circle 229.6 1439 $$d' \space +$$ c   with two circles 259.9 2217 $$b'\flat \space +$$ d   with one diameter 159 805 $$a\flat$$ e   with one diameter and one circle 292 1858 $$g'-$$ f    with two diameters 214 1317 $$c'\sharp \space +$$

The prime tone has been here arbitrarily assumed as $$c$$, in order to note the intervals of the higher tones. Those simple tones produced by the membrane which are slightly higher than those of the note written, are marked ( + ); those lower, by ( - ). In this case there is no commensurable ratio between the prime tone and the other tones, that is, none expressible in whole numbers.

Strew a very thin membrane of this kind with sand, and sound its prime tone strongly in its neighbourhood; the sand will be driven by the vibrations towards the edge, where it collects. On producing another of the tones of the membrane, the sand collects in the corresponding nodal lines, and we are thus easily able to determine to which of its tones the membrane has responded. A singer who knows how to hit the tones of the membrane correctly, can thus easily make the sand arrange itself at pleasure in one order or the other, by singing the corresponding tones powerfully at a distance. But in general the simpler figures of the deeper tones are more easily generated than the complicated figures of the upper tones. It is easiest of all to set the membrane in general motion by sounding its prime tone, and hence such membranes have been much used in acoustics to prove the existence of some determinate tone in some determinate spot of the surrounding air. It is most suitable for this purpose to connect the membrane with an inclosed mass of air. A, fig. 15, is a glass bottle, having a open mouth a, and in place of its bottom b, a stretched membrane, consisting of wet pig's bladder, allowed to dry after it has been stretched and fastened. At c is attached a single fibre of a silk cocoon, bearing a drop of sealing-wax, and hanging down like a pendulum against the membrane. As soon as the membrane vibrates, the little pendulum is violently agitated. Such a pendulum is very convenient as long as we have no reason to apprehend any confusion of the prime tone of the membrane with any other of its proper tones. There is no scattering of sand, and the apparatus is therefore always in order. But to decide with certainty what tones are really agitating the membrane, we must after all place the bottle with its mouth downwards and strew sand on the membrane. However, when the bottle is of the right size, and the membrane uniformly stretched and fastened, it is only the prime tone of the membrane (slightly altered by that of the sympathetically vibrating mass of air in the bottle) which is easily excited. This prime tone can be made deeper by increasing the size of the membrane, or the volume of the bottle, or by diminishing the tension of the membrane or size of the orifice of the bottle.

A stretched membrane of this kind, whether it is or is not attached to the bottom of a bottle, will not only be set in vibration by musical tones of the same pitch as its own proper tone, but also by such musical tones as contain the proper tone of the membrane among its upper partial tones. Generally, given a number of interlacing waves, to discover whether the membrane will vibrate sympathetically, we must suppose the motion of the air at the given place to be mathematically analysed into a sum of pendular vibrations. If there is one such vibration among them, of which the periodic time is the same as that of any one of the proper tones of the membrane, the corresponding vibrational form of the membrane will be superinduced. But if there are none such, or none sufficiently powerful, the membrane will remain at rest.

In this case, then, we also find that the analysis of the motion of the air into pendular vibrations, and the existence of certain vibrations of this kind, are decisive for the sympathetic vibration of the membrane, and for this purpose no other similar analysis of the motion of the air can be substituted for its analysis into pendular vibrations. The pendular vibrations into which the composite motion of the air can be analysed, here shew themselves capable of producing mechanical effects in external nature, independently of the ear, and independently of mathematical theory. Hence the statement is confirmed, that the theoretical view which first led mathematicians to this method of analysing compound vibrations, is founded in the nature of the thing itself.

As an example take the following description of a single experiment : —

A bottle of the shape shewn in fig. 15 above was covered with a thin vulcanised india-rubber membrane, of which the vibrating surface was 49 millimetres (1.93 inches)[2] in diameter, the bottle being 140 millimetres (5.51 inches) high, and having an opening at the brass mouth of 13 millimetres (.51 inches) in diameter. When blown it gave $$f'\sharp$$, and the sand heaped itself in a circle near the edge of the membrane. The same circle resulted from my giving the same tone $$f'\sharp$$ on an harmonium, or its deeper Octave $$f\sharp$$, or the deeper Twelfth $$B$$. Both $$F\sharp$$ and $$D$$ gave the same circle, but more weakly. Now the $$f'\sharp$$ of the membrane is the prime tone of the harmonium tone $$f'\sharp$$, the second partial tone of $$f\sharp$$, the third of $$B$$, the fourth of $$F\sharp$$ and fifth of $$D$$.[3] All these notes on being sounded set the membrane in the motion due to its deepest tone. A second smaller circle, 19 millimetres (.75 inches) in diameter was produced on the membrane by $$b'$$ and the same more faintly by $$b$$, and there was a trace of it for the deeper Twelfth $$e$$, that is, for simple tones of which vibrational numbers were $$\tfrac{1}{2}$$ and $$\tfrac{1}{3}$$ that of $$b'$$.[4]

Stretched membranes of this kind are very convenient for these and similar experiments on the partials of compound tones. They have the great advantage of being independent of the ear, but they are not very sensitive for the fainter simple tones. Their sensitiveness is far inferior to that of the res'onātors which I have introduced. These are hollow spheres of glass or metal, or tubes, with two openings as shewn in figs. 16 a and 16 b. One opening (a) has sharp edges, the other (b) is funnel-shaped, and adapted for insertion into the ear. This smaller end I usually coat with melted sealing wax, and when the wax has cooled down enough not to hurt the finger on being touched, but is still soft, I press the opening into the entrance of my ear. The sealing wax thus moulds itself to the shape of the inner surface of this opening, and when I subsequently use the resonator, it fits easily and is air-tight. Such an instrument is very like the resonance bottle already described, fig. 15 (p. 42), for which the observer's own tympanic membrane has been made to replace the former artificial membrane.

The mass of air in a resonator, together with that in the aural passage, and with the tympanic membrane or drumskin itself, forms an elastic system which is capable of vibrating in a peculiar manner, and, in especial, the prime tone of the sphere, which is much deeper than any other of its proper tones, can be set into very powerful sympathetic vibration, and then the ear, which is in immediate connection with the air inside the sphere, perceives this augmented tone by direct action. If we stop one ear (which is best done by a plug of sealing wax moulded into the form of the entrance of the ear),[5] and apply a resonator to the other, most of the tones produced in the surrounding air will be considerably damped; but if the proper tone of the resonator is sounded, it brays into the ear most powerfully.

Hence any one, even if he has no ear for music or is quite unpractised in detecting musical sounds, is put in a condition to pick the required simple tone, even if comparatively faint, from out of a great number of others. The proper tone of the resonator may even be sometimes heard cropping up in the whistling of the wind, the rattling of carriage wheels, the splashing of water. For these purposes such resonators are incomparably more sensitive than tuned membranes. When the simple tone to be observed is faint in comparison with those which accompany it, it is of advantage to alternately apply and withdraw the resonator. We thus easily feel whether the proper tone of the resonator begins to sound when the instrument is applied, whereas a uniform continuous tone is not so readily perceived.

A properly tuned series of such resonators is therefore an important instrument for experiments in which individual faint tones have to be distinctly heard, although accompanied by others which are strong, as in observations on the combinational and upper partial tones, and a series of other phenomena to be hereafter described relating to chords. By their means such researches can be carried out even by ears quite untrained in musical observation, whereas it had been previously impossible to conduct them except by trained musical ears, and much strained attention properly assisted. These tones were consequently accessible to the observation of only a very few individuals; and indeed a large number of physicists and even musicians had never succeeded in distinguishing them. And again even the trained ear is now able, with the assistance of resonators, to carry the analysis of a mass of musical tones much further than before. Without their help, indeed, I should scarcely have succeeded in making the observations hereafter described, with so much precision and certainty, as I have been enabled to attain at present.[6]

It must be carefully noted that the ear does not hear the required tone with augmented force, unless that tone attains a considerable intensity within the mass of air enclosed in the resonator. Now the mathematical theory of the motion of the air shews that, so long as the amplitude of the vibrations is sufficiently small, the enclosed air will execute pendular oscillations of the same periodic time as those in the external air, and none other, and that only those pendular oscillations whose periodic time corresponds with that of the proper tone of the resonator, have any considerable strength; the intensity of the rest diminishing as the difference of their pitch from that of the proper tone increases. All this is independent of the connection of the ear and resonator, except in so far as its tympanic membrane forms one of the enclosing walls of the mass of air. Theoretically this apparatus does not differ from the bottle with an elastic membrane, in fig. 15 (p. 42), but its sensitiveness is amazingly increased by using the drumskin of the ear for the closing membrane of the bottle, and thus bringing it in direct connection with the auditory nerves themselves. Hence we cannot obtain a powerful tone in the resonator except when an analysis of the motion of the external air into pendular vibrations, would shew that one of them has the same periodic time as the proper tone of the resonator. Here again no other analysis but that into pendular vibrations would give a correct result.

It is easy for an observer to convince himself of the above-named properties of resonators. Apply one to the ear, and let a piece of harmonised music, in which the proper tone of the resonator frequently occurs, be executed by any instruments. As often as this tone is struck, the ear to which the instrument is held, will hear it violently contrast with all the other tones of the chord.

This proper tone will also often be heard, but more weakly, when deeper musical tones occur, and on investigation we find that in such cases tones have been struck which include the proper tone of the resonator among their upper partial tones. Such deeper musical tones are called the harmonic under tones of the resonator. They are musical tones whose periodic time is exactly 2, 3, 4, 5, and so on, times as great as that of the resonator. Thus if the proper tone of the resonator is $$c''$$, it will be heard when a musical instrument sounds $$c'$$, $$f$$, $$a$$, $$A\flat$$, $$F$$, $$D$$, $$C$$, and so on.[7] In this case the resonator is made to sound in sympathy with one of the harmonic upper partial tones of the compound musical tone which is vibrating in the external air. It must, however, be noted that by no means all the harmonic upper partial tones occur in the compound tones of every instrument, and that they have very different degrees of intensity in different instruments. In the musical tones of violins, pianofortes, and harmoniums, the first five or six are generally very distinctly present. A more detailed account of the upper partial tones of strings will be given in the next chapter. On the harmonium the unevenly numbered partial tones (1, 3, 5, &c.) are generally stronger than the evenly numbered ones (2, 4, 6, &c.). In the same way, the upper partial tones are clearly heard by means of the resonators in the singing tones of the human voice, but differ in strength for the different vowels, as will be shewn hereafter.

Among the bodies capable of strong sympathetic vibration must be reckoned stretched strings which are connected with a sounding board, as on the pianoforte.

The principal mark of distinction between strings and the other bodies which vibrate sympathetically, is that different vibrating forms of strings give simple tones corresponding to the harmonic upper partial tones of the prime tone, whereas the secondary simple tones of membranes, bells, rods, &c., are inharmonic with the prime tone, and the masses of air in resonators have generally only very high upper partial tones, also chiefly inharmonic with the prime tone, and not capable of being much reinforced by the resonator.

The vibrations of strings may be studied either on elastic chords loosely stretched, and not sonorous, but swinging so slowly that their motion may be followed with the hand and eye, or else on sonorous strings, as those of the pianoforte, guitar, monochord, or violin. Strings of the first kind are best made of thin spirals or brass wire, six to ten feet in length. They should be gently stretched, and both ends should be fastened. A string of this construction is capable of making very large excursions with great regularity, which are easily seen by a large audience. The swings are excited by moving the string regularly backwards and forwards by the finger near to one of its extremities.

A string may be first made to vibrate as in fig. 17, a (p. 46), so that its appearance when displaced from its position of rest is always that of a simple half wave. The string in this case gives a single simple tone, the deepest it can produce, and no other harmonic secondary tones are audible.

But the string may also during its motion assume the forms fig. 17, b, c, d. In this case the form of the string is that of two, three, or four half waves of a simple wave-curve. In the vibrational form b the string produces only the upper Octave of its prime tone, in the form c the Twelfth, and in the form d the second Octave. The dotted lines shew the position of the string at the end of half its periodic time. In b the point β remains at rest, in c two points γ1 and γ2 remain at rest, in d three points δ1, δ2, δ3. These points are called nodes. In a swinging spiral wire the nodes are readily seen, and for a resonant string they are shewn by little paper riders, which are jerked off from the vibrating parts and remain sitting on the nodes. When, then, the string is divided by a node into two swinging sections, it produces a simple tone having a pitch number double that of the prime tone. For three sections the pitch number is tripled, for four sections quadrupled, and so on.

To bring a spiral wire into these different forms of vibration, we move it periodically with the finger near one extremity, adopting the period of its slowest swings for a, twice that rate for b, three times for c, and four times for d. Or else we just gently touch one of the nodes nearest the extremity with the finger, and pluck the string half-way between this node and the nearest end. Hence when γ1 in c, or δ1 in d, is kept at rest by the finger, we pluck the string at ε. The other nodes then appear when the vibration commences.

For a sonorous string the vibrational forms of fig. 17 above are most purely produced by applying to its sounding board the handle of a tuning-fork which has been struck and gives the simple tone corresponding to the form required. If only a determinate number of nodes are desired, and it is indifferent whether the individual points of the string do or do not execute simple vibrations, it is sufficient to touch the string very gently at one of the nodes and either pluck the string or rub it with a violin bow. By touching the string with the finger all those simple vibrations are damped which have no node at that point, and only those remain which allow the string to be at rest in that place.

The number of nodes in long thin strings may be considerable. They cease to be formed when the sections which lie between the nodes are too short and stiff to be capable of sonorous vibration. Very fine strings consequently give a greater number of higher tones than thicker ones. On the violin and the lower pianoforte strings it is not very difficult to produce tones with 10 sections; but with extremely fine wires tones with 16 or 20 sections can be made to sound. [Also compare p. 78.]

The forms of vibration here spoken of are those in which each point of the string performs pendular oscillations. Hence these motions excite in the ear the sensation of only a single simple tone. In all other vibrational forms of the strings, the oscillations are not simply pendular, but take place according to a different and more complicated law. This is always the case when the string is plucked in the usual way with the finger (as for guitar, harp, zither) or is struck with a hammer (as on the pianoforte), or is rubbed with a violin bow. The resulting motions may then be regarded as compounded of many simple vibrations, which, when taken separately, correspond to those in fig. 17. The multiplicity of such composite forms of motion is infinitely great, the string may indeed be considered as capable of assuming any given form (provided we confine ourselves in all cases to very small deviations from the position of rest), because, according to what was said in Chapter II., any given form of wave can be compounded out of a number of simple waves such as those indicated in fig. 17, a, b, c, d. A plucked, struck, or bowed string therefore allows a great number of harmonic upper partial tones to be heard at the same time as the prime tone, and generally the number increases with the thinness of the string. The peculiar tinkling sound of very fine metallic strings is clearly due to these very high secondary tones. It is easy to distinguish the upper simple tones up to the sixteenth by means of resonators. Beyond the sixteenth they are too close to each other to be distinctly separable by this means.

Hence when a string is sympathetically excited by a musical tone in its neighbourhood, answering to the pitch of the prime tone of the string, a whole series of different simple vibrational forms will generally be at the same time generated in the string. For when the prime of the musical tone corresponds to the prime of the string all the harmonic upper partials of the first correspond to those of the second, and are hence capable of exciting the corresponding vibrational forms in the string. Generally the string will be brought into as many forms of sympathetic vibration by the motion of the air, as the analysis of that motion shews that it possesses simple vibrational forms, having a periodic time equal to that of some vibrational form, that the string is capable of assuming. But as a general rule when there is one such simple vibrational form in the air, there are several such, and it will often be difficult to determine by which one, out of the many possible simple tones which would produce the effect, the string has been excited. Consequently the usual unweighted strings are not so convenient for the determination of the pitch of any simple tones which exist in a composite mass of air, as the membranes or the inclosed air of resonators.

To make experiments with the pianoforte on the sympathetic vibrations of strings, select a flat instrument, raise its lid so as to expose the strings, then press down the key of the string (for $$c'$$ suppose) which you wish to put into sympathetic vibration, but so slowly that the hammer does not strike, and place a little chip of wood across this $$c'$$ string. You will find the chip put in motion, or even thrown off, when certain other strings are struck. The motion of the chip is greatest when one of the under tones of $$c'$$ (p. 44) is struck, as $$c$$, $$F$$, $$C$$, $$A_{\prime}\flat$$, $$F_{\prime}$$, $$D_{\prime}$$ or $$C_{\prime}$$. Some, but much less, motion also occurs when one of the upper partial tones of $$c'$$ is struck, as $$c''$$, $$g''$$, or $$c'''$$, but in this last case the chip will not move if it has been placed over one of the corresponding nodes of the string. Thus if it is laid across the middle of the string it will be still for $$c''$$ and $$c'''$$, but will move for$$g''$$. Placed at one third the length of the string from its extremity, it will not stir for $$g''$$, but will move for $$c''$$ or $$c'''$$. Finally the string $$c'$$ will also be put in motion when an under tone of one of its upper partial tones is struck; for example, the note $$f$$, of which the third partial tone $$c''$$ is identical with the second partial tone of $$c'$$. In this case also the chip remains at rest when put on to the middle of the string $$c'$$, which is its node for $$c''$$. In the same way the string $$c'$$ will move, with the formation of two nodes, for $$g'$$, $$g$$, or $$e\flat$$, all which notes have $$g''$$ as an upper partial tone, which is also the third partial of $$c'$$.[8]

Observe that on the pianoforte, when one end of the strings is commonly concealed, the position of the nodes is easily found by pressing the string gently on both sides and striking the key. If the finger is at a node the corresponding upper partial tone will be heard purely and distinctly, otherwise the tone of the string is dull and bad.

As long as only one upper partial tone of the string $$c'$$ is excited, the corresponding nodes can be discovered, and hence the particular form of its vibration determined. But this is no longer possible by the above mechanical method when two upper partial tones are excited, such as $$c''$$ and $$g''$$ , as would be the case if both these notes were struck at once on the pianoforte, because the whole string of $$c'$$ would then be in motion.

Although the relations for strings appear more complicated to the eye, their sympathetic vibration is subject to the same law as that which holds for resonators, membranes, and other elastic bodies. The sympathetic vibration is always determined by the analysis of whatever sonorous motions exist, into simple pendular vibrations. If the periodic time of one of these simple vibrations corresponds to the periodic time of one of the proper tones of the elastic body, that body, whether it be a string, a membrane, or a mass of air, will be put into strong sympathetic vibration.

These facts give a real objective value to the analysis of sonorous motion into simple pendular vibration, and no such value would attach to any other analysis. Every individual single system of waves formed by pendular vibrations exists as an independent mechanical unit, expands, and sets in motion other elastic bodies having the corresponding proper tone, perfectly undisturbed by any other simple tones of other pitches which may be expanding at the same time, and which may proceed either from the same or any other source of sound. Each single simple tone, then, can, as we have seen, be separated from the composite mass of tones, by mechanical means, namely by bodies which will vibrate sympathetically with it. Hence every individual partial tone exists in the compound musical tone produced by a single musical instrument, just as truly, and in the same sense, as the different colours of the rainbow exist in the white light proceeding from the sun or any other luminous body. Light is also only a vibrational motion of a peculiar elastic medium, the luminous ether, just as sound is a vibrational motion of the air. In a beam of white light there is a species of motion which may be represented as the sum of many oscillatory motions of various periodic times, each of which corresponds to one particular colour of the solar spectrum. But of course each particle of ether at any particular moment has only one determinate velocity, and only one determinate departure from its mean position, just like each particle of air in a space traversed by many systems of sonorous waves. The really existing motion of any particle of ether is of course only one and individual; and our theoretical treatment of it as compound, is in a certain sense arbitrary. But the undulatory motion of light can also be analysed into the waves corresponding to the separate colours, by external mechanical means, such as by refraction in a prism, or by transmission through fine gratings, and each individual simple wave of light corresponding to a simple colour, exists mechanically by itself, independently of any other colour.

We must therefore not hold it to be an illusion of the ear, or to be mere imagination, when in the musical tone of a single note emanating from a musical instrument, we distinguish many partial tones, as I have found musicians inclined to think, even when they have heard those partial tones quite distinctly with their own ears. If we admitted this, we should have also to look upon the colours of the spectrum which are separated from white light, as a mere illusion of the eye. The real outward existence of partial tones in nature can be established at any moment by a sympathetically vibrating membrane which casts up the sand strewn upon it.

Finally I would observe that, as respects the conditions of sympathetic vibration, I have been obliged to refer frequently to the mechanical theory of the motion of air. Since in the theory of sound we have to deal with well-known mechanical forces, as the pressure of the air, and with motions of material particles, and not with any hypothetical explanation, theoretical mechanics have an unassailable authority in this department of science. Of course those readers who are unacquainted with mathematics, must accept the results on faith. An experimental way of examining the problems in question will be described in the next chapter, in which the laws of the analysis of musical tones by the ear have to be established. The experimental proof there given for the ear, can also be carried out in precisely the same way for membranes and masses of air which vibrate sympathetically, and the identity of the laws in both cases will result from those investigations.[9]

[1][Cents are hundredths of an equal Semitone, and are exceedingly valuable as measures of any, especially unusual, musical intervals. They are fully explained, and the method of calculating them from the Interval Ratios is given in App. XX. sect. C. Here it need only be said that the number of hundreds of cents is the number of equal, that is, pianoforte Semitones in the interval, and these may be counted on the keys of any piano, while the units and tens shew the number of hundredths of a Semitone in excess. Wherever cents are spoken of in the text (as in this table), they must be considered as additions by the translator. In the present case, they give the intervals exactly, and not roughly as in the column of notes. Thus, 1439 cents is sharper than 14 Semitones above $$c$$, that is, sharper than $$d'$$ by 39 hundredths of a Semitone, or about $$\tfrac{1}{3}$$ of a Semitone, and 1858 is flatter than 19 Semitones above $$c$$, that is flatter than $$g'$$ by 42 hundredths of a Semitone, or nearly $$\tfrac{1}{2}$$ a Semitone. — Translator.]
[2][As 10 inches are exactly 254 millimetres and 100 metres, that is, 100,000 millimetres are 3937 inches, it is easy to form little tables for the calculation of one set of measures from the other. Roughly we may assume 25 mm. to be 1 inch. But whenever dimensions are given in the text in mm. (that is, millimetres), they will be reduced to inches and decimals of an inch. — Translator.]
[3][As the instrument was tempered, we should have, approximately, for $$f\sharp$$ the partials $$f\sharp$$, $$f'\sharp$$, \$c.; for $$B$$ the partials $$B$$, $$b$$, $$f'\sharp$$, &c.; for $$F\sharp$$ the partials $$F\sharp$$, $$f\sharp$$, $$c\sharp$$, $$f'\sharp$$, &c.; and for $$D$$ the partials $$D$$, $$d$$, $$a$$, $$d'$$, $$f'\sharp$$, &c. To prevent confusion I have reduced the upper partials of the text to ordinary partials, as suggested in p. 23, note. — Translator.]
[4][Here the partials of $$b$$ are $$b$$, $$b'$$, &c., and of $$e$$ are $$e$$, $$e'$$, $$b'$$, &c., so that both $$b$$ and $$e$$ contain $$b'$$. — Translator.]
[5][For ordinary purposes this is quite enough, indeed it is generally unnecessary to stop the other ear at all. But for such experiments as Mr. Bosanquet had to make on beats (see App. XX. section L. art. 4, b) he was obliged to use a jar as the resonator, conduct the sound from it through first a glass and then an elastic tube to a semicircular metal tube which reached from ear to ear, to each end of which a tube coated with india-rubber, could be screwed into the ear. By this means, when proper care was taken, all sound but that coming from the resonance jar was perfectly excluded. — Translator.]
[6]See Appendix II. for the measures and different forms of these Resonators.
[7] [The $$c''$$ occurs as the 2nd, 3rd, 4th, 5th, 6th, 7th, 8th partials of these notes, the 7th being rather flat. The partials are in fact: —
 $$c$$ $$c''$$ $$f$$ $$f'$$ $$c''$$ $$c$$ $$c'$$ $$f'$$ $$c''$$ $$A\flat$$ $$a\flat$$ $$e\flat$$ $$a'\flat$$ $$c''$$ $$F$$ $$f$$ $$c'$$ $$f'$$ $$a'$$ $$c''$$ $$D$$ $$d$$ $$a$$ $$d'$$ $$f'\sharp$$ $$a$$ $$c''$$ $$C$$ $$c$$ $$f$$ $$c'$$ $$e'$$ $$f'$$ $$b\flat$$ $$c''$$ —Translator.]
[8][These experiments can of course not be conducted on the usual upright cottage piano. But the experimenter can at least hear the tone of $$c'$$, if $$c$$, $$F$$, $$C$$, &c., are struck and immediately damped, or if $$c''$$, $$g''$$, $$c'''$$ are struck and damped. And this sounding of $$c'$$, although unstruck, is itself a very interesting phenomenon. But of course, as it depends on the ear, it does not establish the results of the text. — Translator.]
[9]Optical means for rendering visible weak sympathetic motions of sonorous masses of air, are described in App. II. These means are valuable for demonstrating the facts to hearers unaccustomed to the observing and distinguishing musical tones.
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