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174                   HAND-BOOK OF ACOUSTICS.
If, however, the exact ratio be not preserved, the lower tone and the Differential will not coincide, and beats will be heard between them. For example, let the vibration numbers of the two tones be 200 and 99 respectively; then
and therefore 101 99 = 2 beats per second will be heard.
We might therefore define an Octave between two Simple Tones, as that Interval at which the Differential generated by them coincides in pitch with the lower of the two tones; and we see that this perfect coincidence can only occur, when the ratio between the vibration numbers of the two tones is exactly 2:1.
In the example given above, if we had taken 200 and 98 as the respective vibration numbers, that of the Differential would have been 200 98 = 102, which would have given 4 beats per second with the lower tone; from which it is evident, that the more the interval is out of tune, the greater is the number of beats produced. Thus in order to tune two Simple Tones to an exact Octave, after tuning them approximately, one of them must be sharpened or flattened more and more, till the beats becoming less and less, finally vanish. This is an entirely mechanical operation and does not even need a musical ear. For suppose two forks give a false octave, producing beats, and it is required to tune the upper one to a true octave with the lower. Sharpen the former slightly and sound them again; if the beats are more rapid than before, then the higher fork was already too sharp and must be flattened gradually till the beats disappear; if on the other hand they are slower, the fork is too flat, and must be sharpened in a similar manner.
Fifth. Let 3n and 2n be the vibration numbers of two Simple Tones at this interval. Then
Sn In = n.... vib. no. of Differential of 1st order, and 2?i n = n.... ,,            ,,             ,, 2nd ,,
Thus a Fifth between Simple Tones is defined by the coincidence of Differentials of the 1st and 2nd order; and this coincidence can evidently only occur, when the ratio of the vibration numbers of the Simple Tones is as 3 : 2. Differentials of the 2nd order are, however, generally weak, so that this interval between Simple Tones is by no means well defined.