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, 6th, IVth, &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 

 6th, for example, to be flattened ^ of a comma : shall it be 

 considered as deriving its character from the perfect ratio 

 26 : 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 in which the 

 Vths are flattened f , and the I lids and 3ds, each i of a 

 comma. 



The octave must be kept perfect, for reasons which have 

 satisfied all theoretical and practical harmonists, how widely 

 soever their ©pinions have differed in other respects. Ad- 

 mitting equal temperament to be the measure of equal harmo- 

 ny, the complements of the Vth, Hid, and 3d, to the octave, 

 and their compounds with octaves will be equally harmoni- 



