wk, 
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On the Varicty of Voices. 67 
ihe least term is smallest. Whence it follows, that 
the smoothness of a consonance arises from the sim- 
plicity of its cycle; because the terms of the ratio 
of the single vibrations of the terminating sounds 
are proportionate to the numbers of their times of 
vibrating in the course of one cycle. This is the 
reason why the fifth major makes the best concord 
in the diatonic scale below the octave. But the fifth 
in the tempered scale being diminished by a quarter 
of a comma, the terms of the preceding ratio are 
one and the fourth root of five; consequently it has 
no cycle, as the ratio of incommensurable magni- 
tudes is not that of number to number. Neverthe- 
less this concord is used with advantage, though it 
would be inadmissible in music, on the supposition 
of the ear discriminating with mathematical preci- 
sion. The truth is, an interval so tempered, con- 
sists of an infinite succession of periods, so like the 
simple cycles of the diatonic fifth as to supply its 
place, without doing much violence to the nicest 
sense. The various concords and discords may, 
for the same reason, be produced on a musical ring 
of bells, which can never be made perfect in the 
strict sense of the word; for if any one bell, in the 
best set, be struck separately, its note will be heard 
to undulate or tremble, being manifestly disturbed 
by cycles, which result from a slight inequality in. 
the times of vibrations in different sections of the 
vessel parallel to its mouth. 
