THE MODERN PIANO-FORTE. 



701 



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 diatonie 

 scales, and their respective proportions : 



G. A. 



F. 



510 



G. 



480 



C. D. E. F. 



360 320 288 ii70 



A. B-flat. C. D. E. F. 



432 405 360 324 2S8 2*70 



240 



G. 



240 



G. 



240 



B. 



213^ 192 

 A. B. 



216 192 

 A. B-flat. 



216 202|- 



C. 



180 



C. 

 180 



C. 



180 



D. 

 160 

 D. 

 160 

 D. 

 162 



E. F-sharp. G. 

 144 128 120 



F. 

 135 



F. 



135 



E. 

 144 

 E. 

 144 



G. 



120 

 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.' It is evident that any 



* 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. 

 S ich as 



But the logarithms of the ratios of the intervals are most generally used. The logarith- 

 mic 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 continu- 

 ously-rising tone. 



