stem.

3. That each consonant interval, according to its degree of
consonance, shall lose as little of its original purity as
possible; so that the ear may still acknowledge it as a perfect or
imperfect consonance.

Several ways of adjusting such a system of temperament have been
proposed, all of which may be classed under either the head of equal
or of unequal temperament.

The principles set forth in the following propositions clearly
demonstrate the reasons for tempering, and the whole rationale of the
system of equal temperament, which is that in general use, and which
is invariably sought and practiced by tuners of the present.

PROPOSITION I.

If we divide an octave, as from middle C to 3C, into three major
thirds, each in the perfect ratio of 5 to 4, as C-E, E-G[#] (A[b]),
A[b]-C, then the C obtained from the last third, A[b]-C, will be too
flat to form a perfect octave by a small quantity, called in the
theory of harmonics a _diesis_, which is expressed by the ratio 128 to
125.

EXPLANATION.--The length of the string sounding the tone C is
represented by unity or 1. Now, as we have shown, the major third to
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[b]) to C, which completes the
interval of the octave, middle C to 3C. This last C, being the major
third from the A[b], will be produced as before, by 4/5 of that
segment of the string which sounds A[b]; 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 prod