that C, which is E, is produced by 4/5 of its length.

In like manner, G♯, the major third to E, will be produced by 4/5 of
that segment of the string which sounds the tone E; that is, G♯ will
be produced by 4/5 of 4/5 (4/5 multiplied by 4/5) which equals 16/25
of the entire length of the string sounding the tone C.

We come, now, to the last third, G♯ (A♭) to C, which completes the
interval of the octave, middle C to 3C. This last C, being the major
third from the A♭, will be produced as before, by 4/5 of that segment
of the string which sounds A♭; that is, by 4/5 of 16/25, which equals
64/125 of the entire length of the string. Keep this last fraction,
64/125, in mind, and remember it as representing the segment of the
entire string, which produces the upper C by the succession of three
perfectly tuned major thirds.

Now, let us refer to the law which says that a perfect octave is
obtained from the exact half of the length of any string. Is 64/125 an
exact half? No; using the same numerator, an exact half would be
64/128.

Hence, it is clear that the octave obtained by the succession of
perfect major thirds will differ from the true octave by the ratio of
128 to 125. The fraction, 64/125, representing a longer segment of the
string than 64/128 (1/2), it would produce a flatter tone than the
exact half.

It is evident, therefore, that _all major thirds must be tuned
somewhat sharper than perfect_ in a system of equal temperament.