i2 _ On Musical Temperament. 
nature of discords than others, so a set of tempered consonani- 
ces, ceteris paribus, will be best constituted when their har- 
moniousness is diminished proportionally. Suppose, for exam- 
ple, that the agreeable effects of the Vth, IJd, or 3d, when 
perfect, are as any unequal numbers, a, 6, 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 IId, or 3d, but 
when they were so tempered that their agreeable effects on 
the ear might be expressed by ™ a,” 6, and”. 
mn on nr 
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 im 
the following manner : 
or & 
Let the lines AB, ab, represent the times of vibration of 
two tempered unisons. Whatever be the ratio of AB to ad, 
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 
