That the student of scientific scale building may understand fully the
reasons why the tempered scale is at constant variance with exact
mathematical ratios, we continue this discussion through two more
propositions, No. II, following, demonstrating the result of dividing
the octave into four minor thirds, and Proposition III, demonstrating
the result of twelve perfect fifths. The matter in Lesson XII, if
properly mastered, has given a thorough insight into the principal
features of the subject in question; so the following demonstration
will be made as brief as possible, consistent with clearness.

Let us figure the result of dividing an octave into four minor thirds.
The ratio of the length of string sounding a fundamental, to the
length necessary to sound its minor third, is that of 6 to 5. In other
words, 5/6 of any string sounds a tone which is an exact minor third
above that of the whole string.

Now, suppose we select, as before, a string sounding middle C, as the
fundamental tone. We now ascend by minor thirds until we reach the C,
octave above middle C, which we call 3C, as follows:

Middle C-E♭; E♭-F♯; F♯-A; A-3C.

Demonstrate by figures as follows:--Let the whole length of string
sounding middle C be represented by unity or 1.

E♭ will be sounded by 5/6 of the string 5/6
F♯, by 5/6 of the E♭ segment; that is, by 5/6 of
5/6 of the entire string, which equals 25/36
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