But there are six concords to be accommodated, instead of two; and it is evident that all the pairs cannot have their temperament inversely as their frequency, since the numbers _a_, _b_, &c. and _m_, _n_, &c. have no constant ratio to each other. This, however, will be the case, at a medium, if _x_ be made such, that the _sum_ of the products of the numbers expressing the frequency of those chords whose temperaments are increased by _x_, into their respective temperaments, shall be equal to the sum of the corresponding products belonging to those chords whose temperaments are diminished by _x_. Applying this principle to the system of temperament in Prop. III, which flattens all the concords, it is plain that raising any given degree by _x_ will increase the temperaments of the concords above that degree, and diminish those of the concords below it. Hence it ought to be raised till (m - x) a + (n - x)b + (p - x)c = (m′ + x)a + (n′ + x)b′ + (p + x)c′; from which _x_ is found = (am - a′m′ + bn - b′n′ + cp - c′p′) / (a + a′ + b + b′ + c + c′) Should either of the temperaments be sharp, the sign of that term of the numerator, in which it occurs, must be changed; and should the total value of the expression be negative, _x_ 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 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