ower, 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--

1. 1C to 1G segment 2/3 for 1G
2. 1G " 1D Multiply 2/3 by 4/3, gives segment 8/9 " 1D
3. 1D " 1A " 8/9 " 2/3 " " 16/27 " 1A
4. 1A " 1E " 16/27 " 4/3 " " 64/81 " 1E
5. 1E " 1B " 64/81 " 2/3 " " 128/243 " 1B
6. 1B " 1F[#] " 128/243 " 4/3 " " 512/729 " 1F[#]
7. 1F[#] " 1C[#] " 512/729 " 4/3 " " 2048/2187 " 1C[#]
8. 1C[#] " 1G[#] " 2048/2187 " 2/3 " " 4096/6561 " 1G[#]
9. 1G[#] " 1D[#] " 4096/6561 " 4/3 " " 16384/19683 " 1D[#]
10. 1D[#] " 1A[#] " 16384/19683 " 2/3 " " 32768/59049 " 1A[#]
11. 1A[#] " 1F " 32768/59049 " 4/3 " " 131072/177147 " 1F
12. 1F " 2C " 131072/177147 " 2/3 " " 262144/531441 " 2C

Now, this last fraction should be equivalent to 1/2, when reduced to
its lowest terms, if it is destined to produce a true octave; but,
using this numerator, 262144, a half would be expressed by
262144/524288, the segment producing the true octave; so the fraction
262144/531441, which represents the segment for 2C, obtained by the
circle of fifths, being evidently less than 1/2, this segment will
yield a tone somewhat sharper than the true octave. The two
denominators are taken in this case to show the ratio of the variance;
so the octave obtained from the circle of fifths is sharper than the
true octave in the ratio expressed by 531441 to 524288, which ratio is
called the _ditonic comma_. This comma is equal to one-fifth of a
half-step.

We are to conclude, then, that if octaves are to remain perfect, and
we desire to establish an equal temperament, the above-named
difference is best disposed of by dividing it into twelve equal parts
and depressing each of the fifths one-twelfth part of the ditonic
comma; thereby dispersing the dissonance so that it will allow perfect
octaves, and yet, but slightly impair the consonance of the fifths.