Now the rotation of the handle is so related to the rotation of the wind-chest C′ that when the handle turns through half a right angle the wind-chest turns through one-sixth of a right angle, or through the one-twenty-fourth of its whole circumference. But in the case now before us, where the circle is perforated by 12 orifices, the rotation through one-twenty-fourth of its circumference causes the apertures of the upper wind-chest to be closed at the precise moments when those of the lower one are opened, and vice versa. It is plain, therefore, that the intervals between the puffs of the lower siren, which correspond to the rarefactions of its sonorous waves, are here filled by the puffs, or condensations, of the upper siren. In fact, the condensations of the one coincide with the rarefactions of the other, and the absolute extinction of the sounds of both sirens is the consequence.
I may seem to you to have exceeded the truth here; for when the handle is placed in the position which corresponds to absolute extinction, you still hear a distinct sound. And, when the handle is turned continuously, though alternate swellings and sinkings of the tone occur, the sinkings by no means amount to absolute silence. The reason is this: The sound of the siren is a highly composite one. By the suddenness and violence of its shocks, not only does it produce waves corresponding to the number of its orifices, but the aërial disturbance breaks up into secondary waves, which associate themselves with the primary waves of the instrument, exactly as the harmonics of a string, or of an open organ-pipe, mix with their fundamental tone. When the siren sounds, therefore, it emits, besides the fundamental tone, its octave, its twelfth, its double octave, and so on. That is to say, it breaks the air up into vibrations which have twice, three times, four times, etc., the rapidity of the fundamental one. Now, by turning the upper siren through one-twenty-fourth of its circumference, we extinguish utterly the fundamental tone. But we do not extinguish its octave.[75] Hence, when the handle is in the position which corresponds to the extinction of the fundamental tone, instead of silence we have the full first harmonic of the instrument.
Helmholtz has surrounded both his upper and his lower siren with circular brass boxes, B, B′, each composed of two halves, which can be readily separated (one-half of each box is removed in the figure). These boxes exalt by their resonance the fundamental tone of the instrument, and enable us to follow its variations much more easily than if it were not thus reinforced. It requires a certain rapidity of rotation to reach the maximum resonance of the brass boxes; but when this speed is attained, the fundamental tone swells out with greatly augmented force, and, if the handle be then turned, the beats succeed each other with extraordinary power.
Still, as already stated, the pauses between the beats of the fundamental tone are not intervals of absolute silence, but are filled by the higher octave; and this renders caution necessary when the instrument is employed to determine rates of vibration. It is not without reason that I say so. Wishing to determine the rate of vibration of a small singing-flame, I once placed a siren at some distance from it, sounded the instrument, and after a little time observed the flame dancing in synchronism with audible beats. I took it for granted that unison was nearly attained, and, under this assumption, determined the rate of vibration. The number obtained was surprisingly low—indeed not more than half what it ought to be. What was the reason? Simply this: I was dealing, not with the fundamental tone of the siren, but with its higher octave. This octave and the flame produced beats by their coalescence; and hence the counter of the instrument, which recorded the rate, not of the octave, but of the fundamental, gave a number which was only half the true one. The fundamental tone was afterward raised to unison with the flame. On approaching unison beats were again heard, and the jumping of the flame proceeded with an energy greater than that observed in the case of the octave. The counter of the instrument then recorded the accurate rate of the flame’s vibration.
The tones first heard in the case of the siren are always overtones. These attain sonorous continuity sooner than the fundamental, flowing as smooth musical sounds while the fundamental tone is still in a state of intermittence. The siren is, however, so delicately constructed that a rate of rotation which raises the fundamental tone above its fellows is almost immediately attained. And if we seek, by making the blast feeble, to keep the speed of rotation low, it is at the expense of intensity. Hence the desirability, if we wish to examine the overtones, of devising some means by which a strong blast and slow rotation shall be possible.
Helmholtz caused a spring to press as a light brake against the disk of the siren. Thus raising by slow degrees the speed of rotation, he was able deliberately to notice the predominance of the overtones at the commencement, and the final triumph of the fundamental tone. He did not trust to the direct observation of pitch, but determined the tone by the number of beats corresponding to one revolution of the handle of the upper siren. Supposing 12 orifices to be opened above and 12 below, the motion of the handle through 45° produces interference, and extinguishes the fundamental tone. The coincidences of that tone occur at the end of every rotation of 90°. Hence, for the fundamental tone, there must be four beats for every complete rotation of the handle. Now Helmholtz, when he made the arrangement just described, found that the first beats numbered, not 4, but 12, for every revolution. They were, in fact, the beats, not of the fundamental tone, not even of the first overtone, but of the second overtone, whose rate of vibration is three times that of the fundamental. These beats continued as long as the number of air-shocks did not exceed 30 or 40 per second. When the shocks were between 40 and 80 per second, the beats fell from 12 to 8 for every revolution of the handle. Within this interval the first overtone, or the octave of the fundamental tone, was the most powerful, and made the beats its own. Not until the impulses exceeded 80 per second did the beats sink to 4 per revolution. In other words, not until the speed of rotation had passed this limit was the fundamental tone able to assert its superiority over its companions.
This premised, we will combine the tones in definite order, while the cultivated ears here present shall judge of their musical relationship. The flow of perfect unison when the two series of 12 orifices each are opened has been already heard. I now open a series of 8 holes in the upper and of 16 in the lower siren. The interval you judge at once to be an octave. If a series of 9 holes in the upper and of 18 holes in the lower siren be opened, the interval is still an octave. This proves that the interval is not disturbed by altering the absolute rates of vibration, so long as the ratio of the two rates remains the same. The same truth is more strikingly illustrated by commencing with a low speed of rotation, and urging the siren to its highest pitch; as long as the orifices are in the ratio of 1:2, we retain the constant interval of an octave. Opening a series of 10 holes in the upper and of 15 in the lower siren, the ratio is as 2:3, and every musician present knows that this is the interval of a fifth. Opening 12 holes in the upper and 18 in the lower siren does not change the interval. Opening two series of 9 and 12, or of 12 and 16, we obtain an interval of a fourth; the ratio in both these cases being as 3:4. In like manner two series of 8 and 10, or of 12 and 15, give us the interval of a major third; the ratio in this case being as 4:5. Finally, two series of 10 and 12, or of 15 and 18, yield the interval of a minor third, which corresponds to the ratio 5:6.
These experiments amply illustrate two things: First, that a musical interval is determined, not by the absolute number of vibrations of the two combining notes, but by the ratio of their vibrations. Secondly, and this is of the utmost significance, that the smaller the two numbers which express the ratio of the two rates of vibration, the more perfect is the consonance of the two sounds. The most perfect consonance is the unison 1:1; next comes the octave 1:2; after that the fifth 2:3; then the fourth 3:4; then the major third 4:5; and finally the minor third 5:6. We can also open two series numbering, respectively, 8 and 9 orifices: this interval corresponds to a tone in music. It is a dissonant combination. Two series which number respectively 15 and 16 orifices make the interval of a semi-tone: it is a very sharp and grating dissonance.
§ 2. The Theory of Musical Consonance. Pythagoras and Euler
Whence, then, does this arise? Why should the smaller ratio express the more perfect consonance? The ancients attempted to solve this question. The Pythagoreans found intellectual repose in the answer “All is number and harmony.” The numerical relations of the seven notes of the musical scale were also thought by them to express the distances of the planets from their central fire; hence the choral dance of the worlds, the “music of the spheres,” which, according to his followers, Pythagoras alone was privileged to hear. And might we not in passing contrast this glorious superstition with the grovelling delusion which has taken hold of the fantasy of our day? Were the character which superstition assumes in different ages an indication of man’s advance or retrogression, assuredly the nineteenth century would have no reason to plume itself, in comparison with the sixth B.C. A more earnest attempt to account for the more perfect consonance of the smaller ratios was made by the celebrated mathematician, Euler, and his explanation, if such it could be called, long silenced, if it did not satisfy, inquirers. Euler analyzes the cause of pleasure. We take delight in order; it is pleasant to us to observe means “co-operant to an end.” But then, the effort to discern order must not be so great as to weary us. If the relations to be disentangled are too complicated, though we may see the order, we cannot enjoy it. The simpler the terms in which the order expresses itself, the greater is our delight. Hence the superiority of the simpler ratios in music over the more complex ones. Consonance, then, according to Euler, was the satisfaction derived from the perception of order without weariness of mind.