On Musical Temperament. 19 



sonance may be tempered so much cis 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 Vth more than 3 commas, 

 it becomes more harmonious, as approaching much nearer to 

 the perfect ratio f , 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 con- 

 sidered as subjected to the same rule of calculation. 



But the limits within which the second part of Prop. I. 

 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 i a comma, without ren- 

 dering it more harmonious, as approaching nearer another per- 

 fect ratio which is simpler ; that of 5 : 9. Yet the difference 

 between these two 7ths is so trifling that they have never 

 received 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 y of a comma, before it becomes more 



* The propriety of making 25 : 36 the true ratio of the 5th will be manifest, 

 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 5th, 

 flattened by a comma, in the same manner as one of its component 3ds will be 

 allowed by all to be flattened. 



2* 



