a ee ee Ee Se re CE ee Te a ee TSS ee Ee 
i lil La es 
On Musical Temperament. 15 
quate cause. We may add, that the rapidity of the beats, in 
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 ra- 
tio 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 oc- 
tave, 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 onl y 
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 beara 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 
* See Kollmann’s Harmony, p. 13, &e. 
