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 process, 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 constitution of the whole scale will be the best possible, when no degree in it can be elevated or depressed, without rendering the sums 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′ + b + &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, IIIds, and 3ds are flattened, 154, 77 and 77, respectively.) the standard from which to commence the alterations in the scale required by the unequal frequency of different chords, and beginning with D, the theorem gives x = 5. Hence supposing the rest of the degrees in the scale unaltered, it will be in the most harmonious state, when D is raised 5/540 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 x = 13. In this application, however, as D was before raised 5, m, the temperament of the Vth below A, must be taken 154 + 5; and in all the succeeding operations, when the exterior termination of any concord has been already altered, we must take its temperament, not what it was at first, but what it has become, by such previous alteration. In this manner, the scale is becoming more harmonious at every step, till we have completed the whole succession of degrees which it contains.

Let us now revert to D, the place where we began. As each of the outer extremities of the chords which are terminated by D has been changed, a new application of the theorem will give a second correction for the place of D; although, as the numbers a, a', b, &c. continue the same, it will be less than before. Continue the process through the whole scale, and a second approximation to the most harmonious state will be obtained. In this manner let the theorem be applied, till the value of x is exhausted, for every degree; and it will then be in the most harmonious state possible. Three operations gave the following results:

TABLE V.

Bases.1st Oper-
ation.		2d.3d.Bases.1st Oper-
ation.		2d.3d.
F+18+5+1B+18+50
F-20-6-1B+19+50
E+18+50B-23-10-1
E+14+50A+18+70
E-69-8-1A+13+4+1
D+19+5+1A-71-7-2
D+5+2+1G+17+50
D-45-7-2G-1400
C+18+60F+44+50
C-5-5-2G-46-50

The sign plus denotes that the degree to which it belongs is to be raised, and minus, that it is to be depressed. The corrections in each succeeding operation are to be added to those in the preceding. The errors, in the 3d approximation, are so trifling, that a 4th would be wholly useless.

Note. The foregoing calculations will be rendered much more expeditious and sure, by reducing the theorem, in some sense, to a diagram, as in the first of the following figures; and by applying the successive corrections to the circumference of a circle divided into parts proportioned to the intervals of the enharmonic scale, as in the second.