Page:Popular Science Monthly Volume 11.djvu/721

Rh That a general compromise, or sacrifice of truth to convenience, must be made in instruments having twelve fixed tones to the octave, will be seen by a comparison of three most closely-related diatonic scales, and their respective proportions:

It is clear to the meanest comprehension that the sound "D," the second note of the scale of "C," differs from "D," the sixth note of the scale of "F;" and also that the sound "A," the sixth note of the scale of "C," differs from "A," the second sound of the scale of "G;" and similarly, in the ratio of 80 to 81. The relative speeds of the vibrations of each note of the diatonic scale are here given for the convenience of persons accustomed to calculate by their aid.

The true diatonic scale may be represented in various ways, which may occasionally prove useful in measuring intervals, although the divisions are not exactly correct. Such as—

But the logarithms of the ratios of the intervals are most generally used. The logarithmic or equiangular spiral best illustrates to the eye the return of the octave, the curve



being so drawn that a complete revolution halves the distance from the pole. It is also valuable for other properties besides this geometric periodicity, representing a continuously-rising tone. It is evident that any