Page:Popular Science Monthly Volume 11.djvu/721

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701
THE MODERN PIANO-FORTE.

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:

G.
240
A.
213⅓
B.
192
C.
180
D.
160
E.
144
F-sharp.
128
G.
120
C.
360
D.
320
E.
288
F.
270
G.
240
A.
216
B.
192
C.
180
D.
160
E.
144
F.
135
G.
120
F.
540
G.
480
A.
432
B-flat.
405
C.
360
D.
324
E.
288
F.
270
G.
240
A.
216
B-flat.
202½
C.
180
D.
162
E.
144
F.
135
G.
120

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.[1] It is evident that any

  1. 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.
    264 297 830 352 396 440 495 528

    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—


    ASCENDING SCALE.
     
    1. The Periphery
    of a Circle.
    2. 53 Degrees. 3. 301 Degrees. 4. 730 Degrees. 5. In Mean
    Semitones.
    C. 33° 31" 11’ 10 56 136 1.1173
    B. 611022 27 153 372 2.0391
    A. 544316 16 92 222 1.8240
    G. 611011 27 153 372 2.0391
    F. 333111 10 56 136 1.1173
    E. 544316 16 92 222 1.8240
    D. 611022 27 153 372 2.0391
    C. . . . . . . . . . . . . . . . . . . . . . . . . . .

    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

    PSM V11 D721 Logarithmic spiral of musical tone.jpg

    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.