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The Definition of the Consonant Intervals. We have seen in Chap. V, how, by means of the Double Syren, it may be proved, that, for two sounds to be at the exact interval given in the first column below, their vibration numbers must be in the exact ratio of the numbers given in the second column. Interval.                              Eatio.
Octave .. .. .. 2:1
&c.                                      &c.
If the vibration numbers are not in the exact ratio given above, the interval will be perceptibly out of tune. This fact had been ascertained long before the instrument just referred to was invented, by the actual counting of the vibration numbers. Ingenious, but unsatisfactory theories, of a more or less meta­physical nature (among which, that of Euler held sway for many years), were devised to account for this remarkable fact. Its true explanation, as given below, is due to Helmholtz.
We commence as usual with Simple Tones, and first with the Octave. Let two Simple Tones be sounded together, the vibration numbers of which are in the ratio of 2 : 1, say 200 and 100 respectively. They will generate a Differential Tone, the vibration number of which will be 200 — 100 = 100, which Differential Tone will therefore coalesce and be indistinguishable from the lower of the two Simple Tones. This identity in pitch, of the Differential, and the lower of the Simple Tones will always occur, provided the ratio of the two tones is as 2 : 1; for
let 2n be the vibration number of the upper tone, then n will be ,,           ,,           ,, lower ,,
consequently 2n n = n ,,           ,,           ,, Differential ,,