12 



On Musical Temperament, 



nature of discords than others, so a set of tempered consonati- 

 ces, coeteris paribus, will be best constituted when their har- 

 moniousness is diminished proportionally. Suppose, for exam- 

 ple, that the agreeable effects of the Vth, Hid, and 3d, when 

 perfect, are as any unequal numbers, a, b, and c ; the best ar- 

 rangement of a tempered scale, other things being equal, would 

 be, not that in which the agreeable effect of the Vth was re- 

 duced to an absolute level with that of the Hid, or 3d, but 

 when they were so tempered that their agreeable effects on 



. , , I, ^ W* T , *>* 



the ear might be expressed by — a, — o, and — c. 



n n 11 



That different consonances, in this sense, are equally har- 

 monious in their kinds, when equally tempered, or, at least, 

 sufficiently so for every practical purpose, may be illustrated 

 in the following manner : 



Let the lines AB, ab, represent the times of vibration of 

 two tempered unisons. Whatever be the ratio of AB to ab, 

 whether rational or irrational, it is obvious that the successive 

 vibrations will alternately recede from and approach each 

 other, till they very nearly coincide ; and, that during one of 

 these periods, the longer vibration, AB, has gained one of the 

 shorter. Let the points, A, B, &c. represent the middle of the 



