On Musical Temperament. i9 
sonance may be tempered so much as to become better by 
having its temperament increased, in consequence of its ap- 
proaching as near to some other perfect ratio, the terms of 
which are equally small ; or perhaps much nearer some per- 
fect ratio whose terms are not proportionally larger. For ex- 
ample, after we have sharpened the 5th more than $ commas, 
it becomes more harmonious, as approaching much nearer to 
the perfect ratio §. In this, however, and the other concords, 
the value of the nearest perfect ratios in small numbers, varies 
so much from the ratios of these concords, and the consequent 
limits within which the last part of Prop. I. holds true, are so 
wide that there is no hazard in making it a basis of calculation. 
And if there be a few exceptions to this, in some systems, in 
which the temperaments of a few of the concords become so 
large as to approach nearer to some other perfect ratio, whose 
terms are nearly as small as those of the perfect concord, 
although they might become more harmonious, by having their 
temperament increased, yet their effect in melody would be 
still more impaired ; so that the concords may all be considered 
as subjected to the same rule of calculation. 
But the limits within which the second part of Prop. 1. 
holds true, with regard to the more complex consonances, are 
much more limited. We cannot, for instance, sharpen the 
7th, whose ratio is 9; 16 more than 3 a comma, without ren- 
dering it more harmonious, as Approaching nearer another per- 
fect ratio which is simpler; thatof 5:9. Yet the difference 
between these two 7ths is so trifling that they have never re- 
ceived distinct names; and, indeed, their effect on the ear in 
melody would not be sensibly different. 
Again, the 5th, whose perfect ratio has been generally laid 
down as 45: 64, but which is in reality 25: 36,* cannot be 
sharpened more than ; of a comma, before it becomes more. 
* The propriety of making 26: 36, the true ratio of the Sth will be mani- 
fest, when it is considered that this is the value of that interval, as sounded 
by voices and perfect instruments ; when the 3ds which compose it are 
made perfect. This interval, as found in the scale which has the fewest 
tempered concords possible referred to at the beginning of this essay, ought 
to be regarded as the true Sth, flattened by a comma, in the same manner 
as one of its component 3ds, will be allowed by all to be flattened. 
