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. +------+-----------+-----+-----+ | | 1st | 2d. | 3d. | |Bases.| Operation.| | | +------+-----------+-----+-----+ | F♯ | +18 | +5 | +1 | | F | -20 | -6 | -1 | | E♯ | +18 | +5 | 0 | | E | +14 | +5 | 0 | | E♭ | -69 | -8 | -1 | | D♯ | +19 | +5 | +1 | | D | +5 | +2 | +1 | | D♭ | -45 | -7 | -2 | | C♯ | +18 | +6 | 0 | | C | -5 | -5 | -2 | | B♯ | +18 | +5 | 0 | | B | +19 | +5 | 0 | | B♭ | -23 | -10 | -1 | | A♯ | +18 | +7 | 0 | | A | +13 | +4 | +1 | | A♭ | -71 | -7 | -2 | | G♯ | +17 | +5 | 0 | | G | -14 | 0 | 0 | | F♯♯ | +44 | +5 | 0 | | G♭ | -46 | -5 | 0 | +------+-----------+-----+-----+ 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. [Illustration] PROPOSITION VII.