We believe the foregoing propositions will demonstrate the facts
stated therein, to the student's satisfaction, and that he should now
have a pretty thorough knowledge of the mathematics of the
temperament. That the equal temperament is the only practical
temperament, is confidently affirmed by Mr. W.S.B. Woolhouse, an
eminent authority on musical mathematics, who says:--

"It is very misleading to suppose that the necessity of temperament
applies only to instruments which have fixed tones. Singers and
performers on perfect instruments must all temper their intervals, or
they could not keep in tune with each other, or even with themselves;
and on arriving at the same notes by different routes, would be
continually finding a want of agreement. The scale of equal
temperament obviates all such inconveniences, and continues to be
universally accepted with unqualified satisfaction by the most eminent
vocalists; and equally so by the most renowned and accomplished
performers on stringed instruments, although these instruments are
capable of an indefinite variety of intonation. The high development
of modern instrumental music would not have been possible, and could
not have been acquired, without the manifold advantages of the
tempered intonation by equal semitones, and it has, in consequence,
long become the established basis of tuning."

NUMERICAL COMPARISON OF THE DIATONIC SCALE WITH THE TEMPERED SCALE.

The following table, comparing vibration numbers of the diatonic scale
with those of the tempered, shows the difference in the two scales,
existing between the thirds, fifths and other intervals.

Notice that the difference is but slight in the lowest octave used
which is shown on the left; but taking the scale four octaves higher,
shown on the right, the difference becomes more striking.

|DIATONIC.|TEMPERED.| |DIATONIC.|TEMPERED.|
C|32. |32. |C|512. |512. |
D|36. |35.92 |D|576. |574.70 |
E|40. |40.32 |E|640. |645.08 |
F|42.66 |42.71 |F|682.66 |683.44 |
G|48. |47.95 |G|768. |767.13 |
A|53.33 |53.82 |A|853.33 |861.08 |
B|60. |60.41 |B|960. |966.53 |
C|64. |64. |C|1024. |1024. |

Following this paragraph we give a reference table in which the
numbers are given for four consecutive octaves, calculated for the
system of equal temperament. Each column represents an octave