A, by 5/6 of 25/36 of entire string, which equals 125/216
3C, by 5/6 of 125/216 of entire string, which equals 625/1296

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 l