22 On Musical Temperammt. 



In adjusting these variable temperaments, so as to render 

 the harmony of the concords oidiJJ'ercnt kinds, as nearly equal 

 as possible, we immediately discover that, as the Vth is com- 

 posed of the Illd and 3d, the temperaments of the three can- 

 not all be equal. When the temperaments of the Hid and 

 3d have the same sign, that of the Vths must be equal to their 

 sum ; and, when they have contrary signs, to their difference. 

 Hence the temperament of one of these three concords is 

 necessarily equal to the sum of that of the other two. This 

 being fixed, the temperaments, and consequently, Cby Prop. I.) 

 the discordance of the different consonances is the most equably 

 divided possible, when the two smaller temperaments, whose 

 sum is equal to the greater, are made equal to each other. 

 The problem contains three cases. 



1. When the temperaments of the Illd and 3d have the 

 same sign, they ought to be equal to each other. Making 

 2a; — c ■=.^. c — Sx, we obtain x = ^ c, which, substituted in 

 the general expressions for the temperaments of the Vth, Illd, 

 and 3d, makes their increments equal to — f c, ~ ^ c, — ^ c, 

 respectively. 



2. Let the temperaments of the Illd and 3d have contrary 

 signs : and first, let that of the Illds be the greater. Then 

 the former ought to be double of the latter, in order that the 

 temperament of the Vths and and 3ds may be equak Hence we 

 have 2a; — c = — 2, i. c — 3a; ; whence a; is found = o ; and 

 by substitution as before, the required temperament of the 

 Hid = - c ; of the Vth ~-\c, and of the 3d i c. 



3. Let the temperaments of the Illd and 3d have contrary 

 signs, as before ; and let that of the ^d be the greater. 

 Making A. c — 3a; = — 2. 2 a; — c, we obtain a; = | c; which 

 gives, by substitution, the temperaments of the 3d, Vth, and 

 Hid — ^c,— J c, and j c, respectively. 



Each of these results makes the harmony of all the conso- 

 nances as nearly equal as possible ; but as the sum of the 

 temperaments in the first case is much the least, it follows that 

 the temperaments stated in the proposition constitute the best 



