On Musical Temperament. 21 



ous in their kinds with these concords respectively ; accord- 

 ing to the corollary of Prop I. 



Hence we have only to find those temperaments of the Vths, 

 Illds, and 3ds, in the compass of one octave, which will ren- 

 der them all, as nearly as possible, equally harmonious. The 

 temperaments of the different concords of the same name 

 ought evidently to be rendered equal ; since, otherwise, their 

 harmony cannot be equal. This can be effected only by ren- 

 dering the major and minor tones equal, and preserving the 

 equality of the two semitones. If this is done, the tempera- 

 ment of all the Illds will be equal, since they will each be the 

 sum of two equal tones. For a similar reason the 3ds, and 

 consequently the Vths, formed by the addition of Illds, and 

 3ds, will be equally tempered. 



In order to reduce the octave to five equal and variable tones, 

 and two equal and variable semitones, we will suppose the 

 intervals of the untempered octave to be represented by 



\ ^"^^ \ ^ — I ^ ! ^ ^ I ^ — 1^ ^ t g i the 

 C DEF GA Be 



parts CD, DE, &c. of the line Cc. Denoting the comma by 

 c, we will suppose the tone DE, which is naturally minor, to 

 be increased by any variable quantity, x ; then, by the fore- 

 going observations, the other minor tone, GA, must be increa- 

 sed by the same quantity. As the major tones must be render- 

 ed equal to the minor, their increment will be x — c. As the 

 octave is to be perfect, the variation of the two semitones must 

 be the same with that of the five tones, with the contrary sign ; 

 and as they are to be equally varied, the decrement of each 



will be "^ ; or what amounts to the same thing, the 



2 



3c — bx 



increment of each will be 



2 



The several concords of the same name in this octave are 

 now affected with equal and variable temperaments. The 

 common increment of the Illds will be 2a;— c ; that of the 3ds 

 ' . c — 3x ; and consequently that of the Vths i. x — c. 



