But in this theory it was overlooked that Pythagoras himself, who first experimented on the musical intervals, knew nothing about rates of vibration. It was forgotten that the vast majority of those who take delight in music, and who have the sharpest ears for the detection of a dissonance, are in the condition of Pythagoras, knowing nothing whatever about rates or ratios. And it may also be added that the scientific man, who is fully informed upon these points, has his pleasure in no way enhanced by his knowledge. Euler’s explanation, therefore, does not satisfy the mind, and it was reserved for an eminent German investigator of our own day, after a profound analysis of the entire question, to assign the physical cause of consonance and dissonance—a cause which, when once clearly stated, is so simple and satisfactory as to excite surprise that it remained so long without a discoverer.
Various expressions employed in our previous lectures have already, in part, forestalled Helmholtz’s explanation of consonance and dissonance. Let me here repeat an experiment which will, almost of itself, force this explanation upon your attention. Before you are two jets of burning gas, which can be converted into singing-flames by inclosing them within two tubes (represented in [Fig. 118]). The tubes are of the same length, and the flames are now singing in unison. By means of a telescopic slider I lengthen slightly one of the tubes; you hear deliberate beats, which succeed each other so slowly that they can readily be counted. I augment still further the length of the tube. The beats are now more rapid than before: they can barely be counted. It is perfectly manifest that the shocks of which you are now sensible differ only in point of rapidity from the slow beats which you heard a moment ago. There is no breach of continuity here. We begin slowly, we gradually increase the rapidity, until finally the succession of the beats is so rapid as to produce that particular grating effect which every musician that hears it would call dissonance. Let us now reverse the process, and pass from these quick beats to slow ones. The same continuity of the phenomenon is noticed. By degrees the beats separate from each other more and more, until finally they are slow enough to be counted. Thus these singing-flames enable us to follow the beats with certainty, until they cease to be beats and are converted into dissonance.
This experiment proves conclusively that dissonance may be produced by a rapid succession of beats; and I imagine this cause of dissonance would have been pointed out earlier, had not men’s minds been thrown off the proper track by the theory of “resultant tones” enunciated by Thomas Young. Young imagined that, when they were quick enough, the beats ran together to form a resultant tone. He imagined the linking together of the beats to be precisely analogous to the linking together of simple musical impulses; and he was strengthened in this notion by the fact already adverted to, that the first difference-tone, that is to say, the loudest resultant tone, corresponded, as the beats do, to a rate of vibration equal to the difference of the rates of the two primaries. The fact, however, is that the effect of beats upon the ear is altogether different from that of the successive impulses of an ordinary musical tone.
§ 3. Sympathetic Vibrations
But to grasp, in all its fulness, the new theory of musical consonance some preliminary studies will be necessary. And here I would ask you to call to mind the experiments (in Chapter III.) by which the division of a string into its harmonic segments was illustrated. This was done by means of little paper riders, which were unhorsed, or not, according as they occupied a ventral segment or a node upon the string. Before you at present is the sonometer, employed in the experiments just referred to. Along it, instead of one, are stretched two strings, about three inches asunder. By means of a key these strings are brought into unison. And now I place a little paper rider upon the middle of one of them, and agitate the other. What occurs? The vibrations of the sounding string are communicated to the bridges on which it rests, and through the bridges to the other string. The individual impulses are very feeble, but, because the two strings are in unison, the impulses can so accumulate as finally to toss the rider off the untouched string.
Every experiment executed with the riders and a single string may be repeated with these two unisonant strings. Damping, for instance, one of the strings, at a point one-fourth of its length from one of its ends, and placing the red and blue riders formerly employed, not on the nodes and ventral segments of the damped string, but at points upon the second string exactly opposite to those nodes and segments, when the bow is passed across the shorter segment of the damped string, the five red riders on the adjacent string are unhorsed, while the four blue ones remain tranquilly in their places. By relaxing one of the strings, it is thrown out of unison with the other, and then all efforts to unhorse the riders are unavailing. That accumulation of impulses, which unison alone renders possible, cannot here take place, and the consequence is that, however great the agitation of the one string may be, it fails to produce any sensible effect upon the other.
The influence of synchronism may be illustrated in a still more striking manner, by means of two tuning-forks which sound the same note. Two such forks mounted on their resonant supports are placed upon the table. I draw the bow vigorously across one of them, permitting the other fork to remain untouched. On stopping the agitated fork, the sound is enfeebled, but by no means quenched. Through the air and through the wood the vibrations have been conveyed from fork to fork, and the untouched fork is the one you now hear. When, by means of a morsel of wax, a small coin is attached to one of the forks, its power of influencing the other ceases; the change in the rate of vibration, if not very small, so destroys the sympathy between the two forks as to render a response impossible. On removing the coin the untouched fork responds as before.
This communication of vibrations through wood and air may be obtained when the forks, mounted on their cases, stand several feet apart. But the vibrations may also be communicated through the air alone. Holding the resonant case of a vigorously vibrating fork in my hand, I bring one of its prongs near an unvibrating one, placing the prongs back to back, but allowing a space of air to exist between them. Light as is the vehicle, the accumulation of impulses, secured by the perfect unison of the two forks, enables the one to set the other in vibration. Extinguishing the sound of the agitated fork, that which a moment ago was silent continues sounding, having taken up the vibrations of its neighbor. Removing one of the forks from its resonant case, and striking it against a pad, it is thrown into strong vibration. Held free in the air, its sound is audible. But, on bringing it close to the silent mounted fork, out of the silence rises a full mellow sound, which is due, not to the fork originally agitated, but to its sympathetic neighbor.
Various other examples of the influence of synchronism, already brought forward, will occur to you here; and cases of the kind might be indefinitely multiplied. If two clocks, for example, with pendulums of the same period of vibration, be placed against the same wall, and if one of the clocks is set going and the other not, the ticks of the moving clock, transmitted through the wall, will act upon its neighbor. The quiescent pendulum, moved by a single tick, swings through an extremely minute arc; but it returns to the limit of its swing just in time to receive another impulse. By the continuance of this process, the impulses so add themselves together as finally to set the clock a-going. It is by this timing of impulses that a properly-pitched voice can cause a glass to ring, and that the sound of an organ can break a particular window-pane.