fifths perfect (in which case our fourths would also be perfect), our
thirds are so sharp that the ear will not tolerate them; and, if we
tune our thirds low enough to banish all beats, our fifths are
intolerably flat.

The experiment above shows us beautifully the prominent inconsistency
of our scale. We have demonstrated, that if we tune the members of the
chord of C so as to get absolutely pure harmony, we could not use the
chord of A on account of the flat fifth E, which did duty so perfectly
as third in the chord of C.

There is but one solution to this problem: Since we cannot tune either
the fifth or the third perfect, we must compromise, we must strike the
happy medium. So we will proceed by a method that will leave our
fifths flatter than perfect, but not so much as to make them at all
displeasing, and that will leave our thirds sharper than perfect, but
not intolerably so.

We have, thus far, spoken only of the octave, fifth and third. The
inquisitive student may, at this juncture, want to know something
about the various other intervals, such as the minor third, the major
and minor sixth, the diminished seventh, etc. But please bear in mind
that there are many peculiarities in the tempered scale, and we are
going to have you fully and explicitly informed on every point, if you
will be content to absorb as little at a time as you are prepared to
receive. While it may seem to us that the tempered scale is a very
complex institution when viewed as a specific arrangement of tones
from which we are to derive all the various kinds of harmony, yet,
when we consider that the chromatic scale is simply a series of twelve
half-steps--twelve perfectly similar intervals--it seems very simple.