178 On Musical Temperameni. 
term of the numerator, in which it occurs, must be changed; 4 
“and should the total value of the expression be negative, « must 
be taken below C. 
Proposition VI. 
To determine that system of temperaments for the concords of 
~ the changeable scale, which’ will render it, including every 
consideration, the most harmonious possible. 
We can scarcely expect to-find any direct analytical pro- 
cess, which will furnish us with a’ solution of this complicated 
problem, at asingle operation, We shall therefore content 
ourselves with a method which gradually approximates towan 
the desired results. The best position of any given degree, 
as C, supposing all the rest fixed, is determined by the last 
proposition. In the same manner it is evident that the con- 
stitution of the whole scale will be the best possible, when 1# 
degree in it can be elevated or depressed, without rendering 
the sums of the products there referred to, unequal. We ci? 
approximate to this state of the scale, by applying the theorem 
in Prop. V. to each of the degrees successively. It is not 
essential in what order the application is made}; but for the 
sake of uniformity, in the successive approximations, we will 
begin with that degree which has the greatest sum ata’) 
&c. belonging to it, and proceed regularly to that in W 
it is least. Making the equal temperament of Prop. IIL, (@ 
which the Vths, Illds, and Sds are flattened, 154, 77 and 1% 
respectively,) the standard from which to commence the altet- 
ations in the scale required by the unequal frequency of dif 
ferent chords, and beginning with D, the theorem gives 7=*"" 
Hence supposing the rest of the degrees in the scale unaltered, 
it will be in the most harmonious state, when D is raised sit 
of acomma. For by the last proposition, ‘the temperament 
the six concords affected by changing the place of D is 
distributed, and that of the other concords is not at all affected: 
We will now proceed to the second degree in the scale, 1 
A; in which the application of the theorem gives rails. Is 
this application, however, as D was before raised 5, ™ 
temperament of the Vth below A, must be taken 154455 
eee 
