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--