On Musical Temperament. 177 



Let, for example, a be greater than a, and let x be any 

 Tariable distance to which C is moved, so as tM, diminish the 

 temperament m, of the chord whose frequency is expressed 

 by a. Then the temperament of a will become =myrx, and 

 that of a'=m'-f-x. Hence, as the dissonance head in each, in 

 a given time, is in the compound ratio of its frequency of oc- 

 eurrence and its temperament, their aggregate dissonance will 

 be as a.m>rx-\-d .ni -\-x ; a quantity which, as a is supposed 

 greater than a , evidently becomes a minimum when x=^m, 

 or the chord, whose frequency is a, is made perfect. But 

 in this way we render the harmony of the chords very un- 

 equal, which is, caeteris paribus, a disadvantage. As these 

 considerations are heterogeneous, it must be a matter of 

 judgment, rather than of mathematical certainty, what pre- 

 cise weight is to be given to each. We will give so much 

 weight to the latter consideration, as to make the temperament 

 of each concord inversely as its frequency. We have then 



1 1 , . , . om — a'tn' 



a : a : : : -7-; — ; which gives x= ; — ~ . 



m~x m-\-x^ ° a-j-a 



But there are six concords to be accommodated, instead of 

 two ; and it is evident that all the pairs cannot have their 

 temperament inversely as their frequency, since the num- 

 bers a, h, &.C. and m, n, &c. have no constant ratio to each 

 other. This, however, will be the case, at a medium, if 

 X be made such, that the sum of the products of the numbers 

 expressing the frequency of those chords whose temperaments 

 are increased by x, into their respective temperaments, shall 

 be equal to the sum of the corresponding products belonging 

 to those chords whose temperaments are diminished by x. 

 Applying this principle to the system of temperament in Prop. 

 Ill, which flattens all the concords, it is plain that raising any 

 given degree by x will increase the temperaments of the 

 concords above that degree, and diminish those of the con- 

 cords below it. Hence it ought to be raised till (m — x) a-f- 

 (n—x) b-\-{p — x) c = (m'-\-x) a-{-(n'-\-x) b'-{-(p-\-x) c : from 



,. , . - , am — a'm'-\-bii — b'n4-cp~c'p' 



which X IS found =: ; — , . , , ,, — — ~- — —. Should 



a-f a'-f-6-j-6 -\-c-{-c 



either of the temperaments be sharp, the sign of that term of 



