We can determine, with ease, the actual number of vibrations corresponding to any one of those resultant tones. The sound of the flame is that of the open tube which surrounds it, and we have already learned (Chapter III.) that the length of such a tube is half that of the sonorous wave it produces. The wave-length, therefore, corresponding to our 10-3/8-inch tube is 20-3/4 inches. The velocity of sound in air of the present temperature is 1,120 feet a second. Bringing these feet to inches, and dividing by 20-3/4, we find the number of vibrations corresponding to a length of 10-3/8 inches to be 648 per second.
But it must not be forgotten here that the air in which the vibrations are actually executed is much more elastic than the surrounding air. The flame heats the air of the tube, and the vibrations must, therefore, be executed more rapidly than they would be in an ordinary organ-pipe of the same length. To determine the actual number of vibrations, we must fall back upon our siren; and with this instrument it is found that the air within the 10-3/8 inch tube executes 717 vibrations in a second. The difference of 69 vibrations a second is due to the heating of the aërial column. Carbonic acid and aqueous vapor are, moreover, the product of the flame’s combustion, and their presence must also affect the rapidity of the vibration.
Determining in the same way the rate of vibration of the 11·4-inch tube, we find it to be 667 per second; the difference between this number and 717 is 50, which expresses the rate of vibration corresponding to the first deep resultant tone.
But this number does not mark the limit of audibility. Permitting the 11·4-inch tube to remain as before, and lengthening its neighbor, the resultant tone sinks near the limit of hearing. When the shorter tube measures 11 inches, the deep sound of the resultant tone is still heard. The number of vibrations per second executed in this 11-inch tube is 700. We have already found the number executed in the 11·4-inch tube to be 667; hence 700-667=33, which is the number of vibrations corresponding to the resultant tone now plainly heard when the attention is converged upon it. We here come very near the limit which Helmholtz has fixed as that of musical audibility. Combining the sound of a tube 17-3/8 inches in length with that of a 10-3/8-inch tube, we obtain a resultant tone of higher pitch than any previously heard. Now the actual number of vibrations executed in the longer tube is 459; and we have already found the vibrations of our 10-3/8-inch tube to be 717; hence 717-459=258, which is the number corresponding to the resultant tone now audible. This note is almost exactly that of one of our series of tuning-forks, which vibrates 256 times in a second.
And now we will avail ourselves of a beautiful check which this result suggests to us. The well-known fork which vibrates at the rate just mentioned is here, mounted on its case, and I touch it with the bow so lightly that the sound alone could hardly be heard; but it instantly coalesces with the resultant tone, and the beats produced by their combination are clearly audible. By loading the fork, and thus altering its pitch, or by drawing up the paper slider, and thus altering the pitch of the flame, the rate of these beats can be altered, exactly as when we compare two primary tones together. By slightly varying the size of the flame, the same effect is produced. We cannot fail to observe how beautifully these results harmonize with each other.
Standing midway between the siren and a shrill singing-flame, and gradually raising the pitch of the siren, the resultant tone soon makes itself heard, sometimes swelling out with extraordinary power. When a pitch-pipe is blown near the flame, the resultant tone is also heard, seeming, in this case, to originate in the ear itself, or rather in the brain. By gradually drawing out the stopper of the pipe, the pitch of the resultant tone is caused to vary in accordance with the law already enunciated.
The resultant tones produced by the combination of the ordinary harmonic intervals[74] are given in the following table:
| Interval | Ratio of vibrations | Difference | The resultant tone is deeper than the lowest primary tone by |
| Octave | 2 : 3 | 1 | an octave |
| Fourth | 3 : 4 | 1 | a twelfth |
| Major third | 4 : 5 | 1 | two octaves |
| Minor third | 5 : 6 | 1 | two octaves and a major third |
| Major sixth | 3 : 5 | 2 | a fifth |
| Minor sixth | 5 : 8 | 3 | major sixth |
The celebrated Thomas Young thought that these resultant tones were due to the coalescence of rapid beats, which linked themselves together like the periodic impulses of an ordinary musical note. This explanation harmonized with the fact that the number of the beats, like that of the vibrations of the resultant tone, is equal to the difference between the two sets of vibrations. This explanation, however, is insufficient. The beats tell more forcibly upon the ear than any continuous sound. They can be plainly heard when each of the two sounds that produce them has ceased to be audible. This depends in part upon the sense of hearing, but it also depends upon the fact that when two notes of the same intensity produce beats, the amplitude of the vibrating air-particles is at times destroyed, and at times doubled. But by doubling the amplitude we quadruple the intensity of the sound. Hence, when two notes of the same intensity produce beats, the sound incessantly varies between silence and a tone of four times the intensity of either of the interfering ones.
If, therefore, the resultant tones were due to the beats of their primaries, they ought to be heard, even when the primaries are feeble. But they are not heard under these circumstances. When several sounds traverse the same air, each particular sound passes through the air as if it alone were present, each particular element of a composite sound asserting its own individuality. Now, this is in strictness true only when the amplitudes of the oscillating particles are infinitely small. Guided by pure reasoning, the mathematician arrives at this result. The law is also practically true when the disturbances are extremely small; but it is not true after they have passed a certain limit. Vibrations which produce a large amount of disturbance give birth to secondary waves, which appeal to the ear as resultant tones. This has been proved by Helmholtz, and, having proved this, he inferred further that there are also resultant tones formed by the sum of the primaries, as well as by their difference. He thus discovered the summation-tones before he had heard them; and bringing his result to the test of experiment, he found that these tones had a real physical existence. They are not at all to be explained by Young’s theory.