Oil Musical Temperameni, i8i) 



pej'fect, all the Vths on the same letter, in whatever octave 

 they are situated, must have the same temperament. 



The reasoning is precisely the same for the Illds and 3ds, 

 considering the former as forming 4 distinct series of an oc- 

 tave each, beginning with C, C*, D and Eb ; and the latter 

 as forming 3 distinct series of an octave each, beginning with 

 C, C* and D. If the former be made all equal, each will be 

 sharpened 343 ; if the latter be made equal, each will be 

 flattened 392. In every system which renders none of the 

 former flat, and none of the latter sharp, the sum of their 

 temperaments will be 12X343, and 12X392, respectively. 



Cor. The demonstration holds equally true, whatever be 

 the magnitude of a, b, c, Sec. : only if they be such that the 

 difference —a-\-b, —b-{-c, &c. of any two successive ones be 

 greater than the temperament of the corresponding concord in 

 the system of equal semitones, the temperament of that chord 

 must be reckoned negative, and the sum, in the enunciation 

 of the proposition, must be considered as the excess of those 

 temperaments which have the same sign with those of the same 

 concords in the system of equal semitones, above those which 

 have the contrary sign. Hence it is universally true that the 

 excess of the flat above the sharp temperaments of the Vths 

 is equal to 12X49 ; that the excess of the sharp above the flat 

 temperaments of the Illds is equal to 12x343 ; and that the 

 excess of the flat above the sharp temperaments of the 3ds is 

 12X392. Hence likewise we have a very easy method of 

 proving whether the temperaments of any given system have 

 been correctly calculated. It is only to add those which have 

 the same sign ; and if the differences of the sums be equal to 

 the products just stated, the work is right. 



Proposition IX. 



If all the concords of the same name, in a scale of twelve 

 intervals to the octave, were of equally frequent occur- 

 rence, the best system of temperament would be that of 

 equal semitones. 



It is evidently best, so far as the concords of the same name 

 are concerned, that if of equal frequency, they should be 



