On Musical Temperament. 13 



successive times of vibration of the lower ; and a, b, kc. those 

 of the higher of the tempered unisons. Let the arch AGN .. 

 VA be a part of a circle, representing one period of their 

 pulses, and let the points A, a, be the middle points of the 

 times of those vibrations which approach the nearest to a 

 coincidence. It is obvious that the dislocations 6B, cC, &c. of 

 the successive pulses, increase in a ratio which is very nearly 

 that of their distances from A, or a. Now if the pulses exactly 

 coincided, the unisons would be perfect ; and the same would 

 be equally true, if the pulses of the one bisected, or divided 

 in any other constant ratio, those of the other ; as clearly 

 appears from observation. It is, therefore, not the absolute 

 magnitude, as asserted by Dr. Smith, but the variableness of the 

 successive dislocations, Bb, Cc, &c. which renders the imper- 

 fect unisons discordant ; and the magnitude of the successive 

 increments of these dislocations is the measure of the degree 

 of discordance heard in the unisons. 



If now the time of vibration in each is doubled, AC, ac, &c. 

 will represent tlie times of vibration of imperfect unisons an 

 octave below, and the successive dislocations will be Cc, Ee, 

 &.C. only half as frequent as before. But the unisons AE, ac, 

 will be equally harmonious with AB, ab ; because, although 

 the successive dislocations are less frequent than before, yet 

 the coincidences Cc , E'e' of the corresponding perfect unisons 

 are less frequent in the same ratio. 



Suppose, in the second place, that the time of vibration is 

 doubled, in only one of the unisons, ab; and that the times 

 become AB and ac, or those of imperfect octaves. These will 

 also be equally harmonious in their kind with the unisons AB, 

 ah. For, although the dislocations Cc, Ee, &c. are but half 

 as numerous as before, the coincidences of the corresponding 

 perfect octaves will be but half as numerous. The disloca- 

 tions which remain are the same as those of the imperfent uni- 

 sons ; and if some of the dislocations are struck out, and the 

 increments of successive ones thus increased, no greater 

 change is made in the nature of the imperfect than of the 

 perfect consonance. 



