20 On Musical Temperament. 
harmonious by having its temperament increased, as approach- 
ing nearer the simpler ratio 7:10. At the same time, the 
effect of this interval in melody would not be sensibly varied. 
The limits, within which the harmoniousness of the IVth is 
inversely as its temperament, are still narrower. 
Hence it appears that no inference can be drawn from the 
temperaments of such consonances as the 7th, 5th, 1Vth, &c. 
respecting their real harmoniousness. The other perfect ra- 
tios which have nearly the same value with those of these 
chords, and which are in equally simple terms, are so nume- 
rous, that by increasing their temperament they alternately be- 
come more and less harmonious; and in a manner so irregu- 
lar, that to attempt to subject them to calculation, with the 
concords, would be in vain. Even when unaltered, they may 
be considered either as greater temperaments of more simple, 
or less temperaments of more complex ratios. Suppose the 
5th, for example, to be flattened + of a comma: shall it be 
considered as deriving its character from the perfect ratio 
25 : 36, and be regarded as flattened 108? or shall it be re- 
ferred to the perfect ratio 7: 10, and considered as sharpened 
239? No one can tell.—On the whole, it is manifest that no 
consonances more complex than those included in the propo- 
sition, can be regarded in adjusting the temperaments of the 
scale. 
Proposition III. 
The best scale of sounds, which renders the harmony of all 
the concords as nearly equal as possible, is that i in which the 
Vths are flattened 2, and the [IIds and Sds, each } of a 
comma. 
The octave must be kept perfect, for reasons hic have 
satisfied all theoretical and practical harmonists, how widely 
soever their opinions have differed in other respects. Ad- 
mitting equal temperament to be the measure of equal harmo- 
ny, the complements of the Vth, Illd, and 3d, to the octave, 
and their compounds with octaves will be equally harmoni- 
