178 On Musical Temperament. 



the numerator, in which it occurs, must be changed ; anfl 

 should the total value of the expression be negative, x must 

 be taken belbw C. 



PROPOSITIOIf 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 a single operation. We shall therefore content 

 ourselves with a method which gradually approximates towards 

 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 no 

 degree in it can be elevated or depressed, without rendering 

 the suras of the products there referred to, unequal. We can 

 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 a+a'4-i+ 

 &c. belonging to it, and proceed regularly to that in which 

 it is least. Making the equal temperament of Prop. III., (in 

 which the Vths, Illds, and 3ds are flattened, 154, 77 and 77. 

 respectively.) the standard from which to commence the alter- 

 ations in the scale required by the unequal frequency of dif- 

 ferent chords, and beginning with D, the theorem gives a:=5- 

 Hence supposing the rest of the degrees in the scale unaltered, 

 it will be in the most harmonious state, when D is raised j^-^ 

 of a comma. For by the last proposition, the temperament of 

 the six concords affected by changing the place of D is best 

 distributed, and that of the other concords is not at all affected. 

 We will now proceed to the second degree in the scale, viz. 

 A ; in which the application of the theorem gives a;=13. In 

 this application, however, as D was before raised 6, m, the 

 temperament of the Vth below A. must be taken 154-|-5 r anci 



