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] "     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 if followed progressively.  First, we will consider tone relationship in connection with relative string length.  Students who have small stringed instruments, guitar, violin, or mandolin, may find pleasure in demonstrating some of the following facts thereupon.

One-half of any string will produce a tone exactly an octave above that yielded by its entire length.  Harmonic tones on the violin are made by touching the string lightly with the finger at such points as will cause the string to vibrate in segments; thus if touched exactly in the middle it will produce a harmonic tone an octave above that of the whole string.