f
the others. Any given tone must do duty as the initial or key-note of
a major or of a minor scale and also as any other member; thus:

C must serve as 1, in the key of C major or C minor.
" " 2, " " B[b] " B[b] "
" " 3, " " A[b] " A "
" " 4, " " G " G "
" " 5, " " F " F "
" " 6, " " E[b] " E "
" " 7, " " D[b] " C[#] "

Likewise, all other tones of the instrument must be so stationed that
they can serve as _any member_ of _any scale_, major or minor.

This is rendered necessary on account of the various modulations
employed in modern music, in which every possible harmony in every key
is used.

RATIONALE OF THE TEMPERAMENT.

Writers upon the mathematics of sound tell us, experience teaches us,
and in previous lessons we have demonstrated in various ways, that if
we tune all fifths perfect up to the seventh step (see diagram, pages
82, 83) the last E obtained will be too sharp to form a major third to
C. In fact, the third thus obtained is so sharp as to render it
offensive to the ear, and therefore unfit for use in harmony, where
this interval plays so conspicuous a part. To remedy this, it becomes
necessary to tune each of the fifths a very small degree flatter than
perfect. The E thus obtained will not be so sharp as to be offensive
to the ear; yet, if the fifth be properly altered or tempered, the
third will still be sharper than perfect; for if the fifths were
flattened enough to render the thirds perfect, they (the fifths) would
become offensive. Now, it is a fact, that the third will bear greater
deviation from perfect consonance than the fifth; so the compromise
is made somewhat in favor of the fifth. If we should continue the
series of perfect fifths, we will find the same defect in all the
major thirds throughout the scale.

We must, therefore, flatten each fifth of the complete circle,
C-G-D-A-E-B-F[#]-C[#]-G[#] or A[b]-E[b]-B[b]-F-C, successively in a
very small degree; the depression, while it will not materially impair
the consonant quality of the fifths, will produce a series of somewhat
sharp, though still agreeable and harmonious major thirds.

We wish, now, to demonstrate the cause of the foregoing by
mathematical calculation, which, while it is somewhat lengthy and
tedious, is not difficult