Pe ee ee a ee eee 
On Musical Temperament. : 13 
successive times of vibration of the lower; and a, b, &c. those 
of the higher of the tempered unisons. Let the arch AGN.. 
VA be a part of a circle, representing one period of their 
pulses, and let the points A, a, be the middle points of the 
times of those vibrations which approach the nearest to a co- 
incidence. It is obvious that the distecations 6B, cC, &c. of 
the successive pulses, increase in a ratio which is very nearly 
that of their distances from A, or a. Now if the pulses exactly 
coincided, the unisons would be perfect; and the same would 
be equally true, if the pulses of the one bisected, or divided 
in any other constant ratio, those of the other; as clearly ap- 
pears from observation. It is, therefore, not the absolute 
magnitude, as asserted by Dr. Smith, but the variableness of the 
successive dislocations, Bb, Cc, &c. which renders the imper- 
fect unisons discordant; and the magnitude of the successive 
increments of these dislocations is the measure of the degree 
of discordance heard in the unisons. 
If now the time of vibration in each is doubled, AC ae, &c. 
‘will represent the times of vibration of imperfect unisons an 
octave below, and the successive dislocations will be Ce, Ee, 
&c. only half as frequent as before. But the unisons AK, ae, 
will be equally harmonious with AB ab; because, although 
the successive dislocations are less frequent than before, vet 
the coincidences C’c’ E’e! of the corresponding perfect unisons 
are less frequent in the same ratio. 
Suppose, in the second place, that the time of vibration is 
doubled, in only one of the unisons, a ; and that the times be- 
come AB and ac, or those of imperfect octaves. These will 
also be equally harmonious in their kind with the unisons AB, 
ab. For, although the dislocations Cc, Ee, &c. are but half 
as humerous as before, the coincidences of the corresponding 
perfect octaves will be but half as numerous. The disloca- 
tions which remain are the same as those of the imperfect uni- 
sons; and if some of the dislocations are struck out, and the 
increments of successive ones thus increased, no greater change 
is made in the nature of the imperfect than of the perfect con- 
sonance, * 
