§ 7. Beats and Dissonance.—"If two tuningforks" sounded together "are not of the same pitch, but so related that the period of vibration of the one is not an exact multiple of that of the other, the sensation which we experience has certain marked features. We hear a sound which is the effect on our ear of the compound wave formed out of the two waves; but the sound is not uniform in intensity. As we listen the sound is heard now to grow louder and then to grow fainter or even to die away, but soon to revive again, and once more to fall away, thus rising and falling at regular intervals, the rhythmic change being either from sound to actual silence or from a louder sound to a fainter one. Such variations of intensity are due to the fact that, owing to the difference of pitch, the vibratory impulses of the two sounds do not exactly correspond in time. Since the vibration period, the time during which a particle is making an excursion, moving a certain distance in one direction and then returning, is shorter in one sound than in the other, it is obvious that the vibrations belonging to one sound will, so to speak, get ahead of those belonging to the other: hence a time will come when, while the impulse of one sound is tending to drive a particle in one direction, say forwards, the impulse of the other sound is tending to drive the same particle in the other direction, i.e. backwards. The result is that the particle will not move, or will not move so much as if it were subject to one impulse only, still less to both impulses acting in the same direction; the vibrations of the particle will be stopped or lessened, and the sensation of sound to which its vibrations are giving rise will be wanting or diminished: the one sound has more or less completely neutralised or 'interfered' with the other, the crest of the wave of one sound has more or less coincided with the trough of the wave of the other sound. Conversely, at another time, the two impulses will be acting in the same direction on the same particle, the movements of the particle will be intensified, and the sound will be augmented. And the one condition will pass gradually into the other. The repetitions of increased intensity thus brought about are spoken of as beats."* Beats are separately discernible when the difference between the vibration frequency of the concurrent tones is very small. As the difference becomes greater, the beats occur more rapidly, and are not so clearly discernible. They then give rise to a rattling or whirring effect. This ceases somewhere between thirty and sixty beats in the second. But even then the beats still manifest their presence by imparting to the notes which produce them a certain roughness. This experience may persist even when there are hundreds of beats in the second. When the beats occur with sufficient rapidity, the roughness or harshness ceases. Before this point is reached, the notes, because of their harsh effect, are said to be dissonant. The number of beats produced by two notes which approach each other in vibration frequency, is equal to the mathematical difference between the number of vibrations per second of each. "Thus two...tuningforks vibrating respectively at sixtyfour or seventytwo a second, will give eight beats a second,"+ because the shorter wave overtakes the longer eight times, so as to give to the vibrating particles opposite impulses, which neutralise each other.

* Op. cit., pp. 1367, 1368.          +Ibid.

We have seen that as the interval between the combined tones becomes increased, the beats become so rapid that they are no longer appreciable; but they recur again when the interval is sufficiently increased. They recur when the interval is somewhat greater or less than the octave, and again, when it is somewhat greater or less than the twelfth, the double octave, etc. Two tones of 200 and 396 vibrations in a second give four beats; four beats are also produced by tones of 200 and 404 vibrations in a second. The number of beats is equal to the difference between the vibration number of the higher tone and that multiple of the vibration number of the lower tone which comes nearest to the vibration number of the higher tone. Thus if the notes are 200 and 596 the number of beats is 3 x 200 596 = 4. This explains why a small deviation from the octave or other musical interval produces a dissonant effect.