numbers by 4, gives the ratio of 80 to 81). Again, the fifth, A-F,
presents itself in the ratio of 320 to 216, or (dividing each term by
4) 80 to 54; instead of 3 to 2 (=81 to 54--multiplying each term by
27), which is the ratio of the true fifth. Continuing the scale an
octave higher, it will be found that the sixth, F-D, and the fourth,
A-D, will labor under the same imperfections.

The comparison, then, of these ratios of the minor third, D-F, and the
fifth, D-A, with the perfect ratios of these intervals, shows that
each is too small by the ratio expressed by the figures 80 to 81. This
is called, by mathematicians, the _syntonic comma_.

As experience teaches us that the ear cannot endure such deviation as
a whole comma in any fifth, it is easy to see that some tempering must
take place even in such a simple and limited number of sounds as the
above series of eight tones.

The necessity of temperament becomes still more apparent when it is
proposed to combine every sound used in music into a connected system,
such that each individual sound shall not only form practical
intervals with all the other sounds, but also that each sound may be
employed as the root of its own major or minor key; and that all the
tones necessary to form its scale shall stand in such relation to each
other as to satisfy the ear.

The chief requisites of any system of musical temperament adapted to
the purposes of modern music are:--

1. That all octaves must remain perfect, each being divided into
twelve semitones.