22 - On Musical Temperament. 
In adjusting these variable temperaments, so as to render 
the harmony of the concords of different kinds, as nearly equal 
as possible, we immediately discover that, as the Sth is com- 
posed of the IlId and Sd, the temperaments of the three can- 
not allbe equal. When the temperaments of the Iild 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 (by Prop. I.) 
the discordance of the different consonances is the most equably 
divided possible, when the two smaller temperaments, whose 
suin is equal to the greater, are made equal to each other. The 
problem contains three cases. 
1. When the temperaments of the [Id and $d have the 
same sign, they ought to be equal to each other. Making 
28x —c = }.c—3z2, we obtain r= $c, which, substituted in 
the general expressions for the temperaments of the Vth, Illd 
and 3d, makes their increments equal to — 2. ¢,—4 ¢,—4 ¢, 
respectively. — 4 
2. Let the temperaments of the IIId and Sd have contrary 
signs : and first, let that of the II[ds be the greater. Then . 
the former ought to be double of the latter, in order that the 
temperaments of the Vths and 3ds may be equal. Hence we — 
have 2.2 -- ¢ = -- 2.3.c -— 3a; whence z is is found = 03; 
and by substitution as before, the required temperament of the 
Ilid = --¢; of the Vth —- 4 c,and of the 3d3c. 
3. Let the temperaments of the [ifd and 3d have contrary _ 
signs, as before; and let that of the third be the greater: 4 
Making 3 ¢,-- 327 —- 2.2 x-- ¢, we obtain x = %¢; which : 
gives, by substitution, the temperaments of the 3d, Vth, and — 
Illd-- 2 ¢,-- 1 c,and 1 ¢, respectively. 5 
Each of these results makes the harmony of all the conso- 3 
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 propositien constitute the best — 
