If, for example, the best temperaments were required for a scale of 15 degrees to the octave, such as is that of some European organs, or in other words, having no Enharmonic intervals except D♯ E♭, and G♯ A♭,--the numbers in Table IV. belonging to C♯ and D♭, E♯ and F, F♯ and G♭, &c. must be united, and their sums substituted when they occur, for _a_, _a′_, _b_, &c. in the theorem; while the temperament, for example, of the IIId on C♯ must not be reckoned 77 as in the complete scale, but 1261 - 77 sharp, since its upper termination has become F, instead of E♯. With these variations let the same theorem be applied as before, till no value of _x_ can be obtained, and the temperaments for that scale will be the best adjusted possible. But as the scale which contains but 13 degrees, or 12 intervals, to the octave, is in much more general use than every other, we shall content ourselves with stating _how_ the problem may be solved for scales containing any intermediate number of degrees, and proceed directly to the consideration of that which is so much the most practically important. LEMMA. No arrangement of the intervals in the common scale of 12 degrees, which renders none of the Vths or 3ds sharp, and none of the IIIds flat, can make any change in the aggregate temperaments of all the concords of the same name. We will conceive the 12 Vths of the Douzeave scale to be arranged in succession, as CG, GD, DA, &c. embracing 7 octaves. Let them at first be all equal: they will each be flattened 49. I say that no change in these Vths which preserves the two extreme octaves perfect, and renders none of them sharp, can alter the sum of their temperaments. Let _a_, _b_, _c_, &c. be any quantities, positive or negative, by which the points C, G, D, &c. may be conceived to be raised above the corresponding points, belonging to the scheme of equal Vths. Then as the mean temperament Vth = V - 49, the first Vth in the supposed arrangement will be V - 49 + _a_. The distance from C to D will be, in like manner, 2 · (V - 49) + _b_; and consequently the Vth GD will be V - 49 + _b_ - _a_. In the same manner the third Vth DE will be V - 49 + _c_ - _b_, &c. Hence the temperament of CG = -49 + _a_, of GD = -49 + _b_ - _a_, of DA = -49 + _c_ - _b_, &c. Adding the 12 temperaments together, we find their sum = -12 × 49 + a + b + &c. - a - b - &c. in which all the terms except the first destroy each other, and leave their sum = -12 × 49 which is the aggregate temperament of the twelve equal Vths in the scheme of equal semitones. The same reasoning holds good if we bring these Vths within the compass of an octave; since, if the octave be kept perfect, all the Vths on the same letter, in whatever octave they are situated, must have the same temperament.