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DEFINITION OF THE CONSONANT INTERVALS. 175
If the ratio is not exactly that of 3 : 2, beats are generated. For instance, let the vibration numbers of the two tones be 300 and 201 respectively, then
300  201 = 99 .. Differential of 1st order. 201 99 = 102 ..                          2nd
102 99 = 3 beats per second being produced. The more the tones are out of tune, the greater the rapidity of the beats; so that to tune the interval, one tone must be sharpened or flattened gradually, as the rapidity of the beats decreases, until they vanish altogether.
Fourth. Let 4» and 3n be the vibration numbers of two Simple Tones at this interval. Then
A Fourth between Simple Tones, therefore, is only defined by the coincidence of Differentials of the 1st and 3rd, and of the 2nd and 3rd order. Inasmuch, however, as a 3rd Differential can only be heard under extremely favourable circumstances, this interval can scarcely be said to be defined at all. This is still more the case with the Thirds, the definition of which, in the case of Simple Tones, depends upon the existence of Differentials of the 4th order. Accordingly it is found, as stated before, that in the case of Simple Tones, intervals of any magnitude intermediate between a Minor Third and a Fourth, are usually of equal smoothness. For the same reason, it is impossible without extraneous aid to tune two Simple Tones to the exact interval of a Third, either Major or Minor ; there is no check: they have no definition.
If, however, more than two Simple tones be employed, it becomes easy to tune these intervals. Indeed, it is better to tune the Fifth also by the aid of a third Tone; for, as we have seen, the interval of the Fifth alone, is only guarded by a Differential of the 2nd order; while if the Octave of the lower tone be present, a Differential of the 1 st order becomes available. Suppose for example the vibration numbers of three Simple Tones be 200, 301, and 400 respectively, the 5th, 301, being mistuned, then
301  200 = 101 .. Differential of 1st order, 400 301 = 99 ..                      1st ,,
101 99 = 2 beats per second being thus produced. Thus by