HANDBOOK OF ACOUSTICS - online book

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ON DISSONANCE.                                157
very faint tones. For if another tone of about the same intensity, but differing very slightly in pitch, be sounded with it, the intensity of the resulting sound, as we saw in the last chapter, will alternate between silence and four times the intensity of the original sound, and this increase of intensity will combine with the alternation to render it audible.
With regard to the second question, " why should the beats cease to be unpleasant, when they become sufficiently rapid''? we must again have recourse to the analogous phenomenon of light. If a carriage wheel be revolved slowly, we can see each of the spokes separately; on revolving more quickly, they merge together into a shadowy circle. Again the singing flame of fig. 3 is all but extinguished two or three hundred times per second, but to the unaided eye it appears stationary. When the alternations between irritation and rest follow one another too quickly, they cease to be perceived, and the sensation becomes continuous. So in the case of sound, after the exciting cause has ceased to act, a certain minute interval of time is necessary for the excited nerve to lose its excitement; and, when the beats succeed one another so rapidly, that there is not this interval between them, then the cessations and reinforcements, that is, the beats, become imperceptible.
In our first experiment, we began with two C forks in unison, and on gradually increasing the interval between them, we found that the harshest discord was obtained, when they produced about 32 beats per second, and that, when their vibration numbers differed by about 78, the two tones were just beyond beating distance : that is the 78 beats so coalesced as to be imperceptible. Now these numerical results apply only to this region of pitch. If we select another pair of tones in a different part of the musical realm, the general result will be the same, but the numbers will not be those above: that is to say, the discord will become harsher and harsher as the beats increase up to a certain point, but the number of beats per second at this point will not be 32 ; and similarly, the discord will become less and less after this, and finally vanish, but the number of beats per second at the beating distance, will not be 78.
The harshness of a dissonance therefore, does not depend upon the rapidity of beats alone: it depends also upon the position of the beating tones in the musical scale. This will be evident from the following examples—