Now bear in mind, this last fraction, 625/1296, represents the segment of the entire string which should sound the tone 3C, an exact octave above middle C. Remember, our law demands an exact half of a string by which to sound its octave.  How much does it vary?  Divide the denominator (1296) by 2 and place the result over it for a numerator, and this gives 648/1296, which is an exact half.  Notice the comparison.

3C obtained from a succession of exact minor thirds,      625/1296
3C obtained from an exact half of the string              648/1296

Now, the former fraction is smaller than the latter; hence, the segment of string which it represents will be shorter than the exact half, and will consequently yield a sharper tone.  The denominators being the same, we have only to find the difference between the numerators to tell how much too short the former segment is.  This proves the C obtained by the succession of minor thirds to be too short by 23/1296 of the length of the whole string.

If, therefore, all octaves are to remain perfect, it is evident that all minor thirds must be tuned flatter than perfect in the system of equal temperament.

The ratio, then, of 648 to 625 expresses the excess by which the true octave exceeds four exact minor thirds; consequently, each minor third must be flatter than perfect by one-fourth part of the difference between these fractions.  By this means the dissonance is evenly distributed so that it is not noticeable in the various chords, in the major and minor keys, where this interval is almost invariably present. (We find no record of writers on the mathematics of sound giving a name to the above ratio expressing variance, as they have to others.)

PROPOSITION III.

Proposition III deals with the perfect fifth, showing the result from a series of twelve perfect fifths employed within the space of an octave.

METHOD.—­Taking 1C as the fundamental, representing it by unity or 1, the G, fifth above, is sounded by a 2/3 segment of the string sounding C. The next fifth, G-D, takes us beyond the octave, and we find that the D will be sounded by 4/9 (2/3 of 2/3 equals 4/9) of the entire string, which fraction is less than half; so to keep within the bounds of the octave, we must double this segment and make it sound the tone D an octave lower, thus:  4/9 times 2 equals 8/9, the segment sounding the D within the octave.

We may shorten the operation as follows:  Instead of multiplying 2/3 by 2/3, giving us 4/9, and then multiplying this answer by 2, let us double the fraction, 2/3, which equals 4/3, and use it as a multiplier when it becomes necessary to double the segment to keep within the octave.

We may proceed now with the twelve steps as follows:—­

Steps—­