hence, it is called the "Philosophical Standard." Below are the actual
vibration numbers of the two pitches in vogue; so you can see that
neither of these pitches would be so favorable to deal with
mathematically.

International--3C--517.3. Concert--3C--540.

(Let us state here that the difference in these pitches is less than a
half-step, but is so near that it is generally spoken of as being just
a half-step.)

Temperament denotes the arrangement of a system of musical sounds in
which _each one_ will form a serviceable interval with _any one_ of
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♭ "
" " 3, " " A♭ " A "
" " 4, " " G " G "
" " 5, " " F " F "
" " 6, " " E♭ " E "
" " 7, " " D♭ " 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.