On Musical Temperament, 15 



quate cause. We may add, that the rapidity of the beats, ia 

 a given consonance, increases very nearly in the ratio of the 

 temperament; and universal experience shows, that increas- 

 ing the rapidity of the beats of the same consonance, increases 

 its harshness. This is on the supposition that the consonance 

 is not varied so much as to interfere with any other whose 

 ratio is equally simple. 



Cor. We may hence infer, that in every system of tempera- 

 ment which preserves the octaves perfect, each consonance is 

 equally harmonious, in its kind, with its complement to the 

 octave, and its compounds with octaves. For the tempering 

 ratio of the complement of any concord to the octave, is the 

 same with that of the concord itself, differing only in its sign^ 

 which does not sensibly affect the harmony or the rate of 

 beating ; while the tempering ratio of the compounds with 

 octaves is not only the same, but with the same sign. 



Scholium 1. 



There is no point in harmonics, concerning which theorists 

 have been more divided in opinion than in regard to the true 

 measure of equal harmony, in consonances of different kinds. 

 Euler maintains, that the more simple a consonance is, the 

 less temperament it will bear ; and this seems to have ever 

 been the general opinion of practical musicians.* Dr. Smith, 

 on the contrary, asserts, and has attempted to demonstrate, 

 that the simpler will bear a much greater temperament than the 

 more complex consonances. The foregoing proposition has, 

 at least, the merit of taking the middle ground between these 

 discordant opinions. If admitted, it will greatly simplify the 

 whole subject, and will reduce the labour of rendering all the 

 concords in three octaves as equally harmonious as possible, 

 which occupies so large a portion of Dr. Smith's volume, to a 

 single short proposition. Dr. Smith's measure of equal har- 

 mony, viz. equal numbers of short cycles in the intervals be- 

 tween the successive beats, seems designed, not to render the 

 different consonances proportionally harmonious, but to reduce 



* Sec. Kollmann's Harmony, p, 13, itc. 



