18 On Musical Temperament. 



multiply each of these values by 100,000, which will give a 

 set of integral values having the same ratio. The octave will 

 now become 30103, the comma 540, &c. ; and, in general, when 

 temperaments are hereafter expressed by numbers, they are 

 to be considered as so many 540ths of a comma. Had more 

 logarithmic places been taken, the intervals would have been 

 expressed with greater accuracy ; but it was supposed that the 

 additional accuracy would not compensate for the increased la- 

 bour of computation which it would occasion. This notation 

 has been adopted by Dr. Robinson, in the article Temperament, 

 (Encyc. Brit. Supplement ;) and for every practical purpose, 

 is as. much superior to that proposed by Mr. Farey, in parts 

 of the Schisma, lesser fraction and minute,* as all decimal 

 measures necessarily are, to those which consist of diflferent 

 denominations. 



Proposition II. 



In adjusting the imperfections of the scale, so as to render all 

 the consonances as equally harmonious as possible, only the 

 simple consonances, such as the Vth, Hid, and 3d, with their 

 complements to and compounds with the octave, can be re- 

 garded. 



It has been generally assigned as the reason for neglecting 

 the consonances, usually termed discords, in ascertaining the 

 best scheme of temperament, that they are of less frequent oc- 

 currence than the concords. This, however, if it were the 

 only reason, would lead us, not to neglect them entirely, but 

 merely to give them a less degree of influence than the con- 

 cords, in proportion as they are less used. 



A consideration which seems not to have been often noticed, 

 renders it impossible to pay them any regard in harmonical 

 computations. All such computations must proceed on the 

 supposition that within the limits to which the temperaments 

 of the different consonances extend, they become harsher as 

 their temperaments are increased. It is evident that any con.- 



* Tilloch's Phil. Mag. Vol. XXVUI. p. UO. 



