2. That each sound of the system may be employed as the root of a
major or minor scale, without increasing the number of sounds in
the system.

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♭), A♭-C,
then the C obtained from the last third, A♭-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