The American journal of science, & c . different consonances, when perfect, are not pleasing to the ear in an equal degree, some approaching nearer to the 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, whenperfect, are as any unequal numbers, a, b, and c ; the best ar-rangement of a tempered scale, other things being equal, wouldbe, not that in which the agreeable
The American journal of science, & c . different consonances, when perfect, are not pleasing to the ear in an equal degree, some approaching nearer to the 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, whenperfect, are as any unequal numbers, a, b, and c ; the best ar-rangement of a tempered scale, other things being equal, wouldbe, not that in which the agreeable effect of the Vth was re-duced to an absolute level with that of the Hid, or 3d, butwhen 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 illustratedin the following manner :. Let the lines AB, ab, represent the times of vibration oftwo tempered unisons. Whatever be the ratio of AB to ab,whether rational or irrational, it is obvious that the successivevibrations will alternately recede from and approach eachother, till they very nearly coincide ; and, that during one ofthese periods, the longer vibration, AB, has gained one of theshorter. Let the points, A, B, &c. represent the middle of the On Musical Temperament. 13 successive times of vibration of the lower ; and a, b, kc. thoseof the higher of the tempered unisons. Let the arch AGN ..VA be a part of a circle, representing one period of theirpulses, and let the points A, a, be the middle points of thetimes of those vibrations which approach the nearest to acoincidence. It is obvious that the dislocations 6B, cC, &c. ofthe successive pulses, increase in a ratio which is very nearlythat of their distances from A, or a. Now if the pulses exactlycoincided, the unisons w
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