In order to reduce the octave to five equal and variable tones, and two equal and variable semitones, we will suppose the intervals of the untempered octave to be represented by the parts CD, DE, &c. of the line C_c_. Denoting the comma by _c_, we will suppose the tone DE, which is naturally minor, to be increased by any variable quantity, _x_; then, by the foregoing observations, the other minor tone, GA, must be increased by the same quantity. As the major tones must be rendered equal to the minor, their increment will be _x_ - _c_. As the octave is to be perfect, the variation of the two semitones must be the same with that of the five tones, with the contrary sign; and as they are to be equally varied, the decrement of each will be (5_x_ - 3_c_)/2; or what amounts to the same thing, the increment of each will be (3_c_ - 5_x_)/2. The several concords of the same name in this octave are now affected with equal and variable temperaments. The common increment of the IIIds will be 2_x_ - _c_; that of the 3ds ½ · (_c_ - 3_x_); and consequently that of the Vths ½ · (_x_ - _c_). In adjusting these variable temperaments, so as to render the harmony of the concords of _different_ kinds, as nearly equal as possible, we immediately discover that, as the Vth is composed of the IIId and 3d, the temperaments of the three cannot all be equal. When the temperaments of the IIId and 3d have the same sign, that of the Vths must be equal to their sum; and, when they have contrary signs, to their difference. Hence the temperament of one of these three concords is necessarily equal to the sum of that of the other two. This being fixed, the temperaments, and consequently, (by Prop. I.) the discordance of the different consonances is the most equably divided possible, when the two smaller temperaments, whose sum is equal to the greater, are made equal to each other. The problem contains three cases. 1. When the temperaments of the IIId and 3d have the same sign, they ought to be equal to each other. Making 2x - c = ½ · (c - 3x), we obtain x = 3/7 c, which, substituted in the general expressions for the temperaments of the Vth, IIId, and 3d, makes their increments equal to -2/7 _c_, -1/7 _c_, -1/7 _c_, respectively. 2. Let the temperaments of the IIId and 3d have contrary signs: and first, let that of the IIIds be the greater. Then the former ought to be double of the latter, in order that the temperament of the Vths and and 3ds may be equal. Hence we have 2x - c = - 2 · ½ · (c - 3x); whence x is found = 0; and by substitution as before, the required temperament of the IIId = - _c_; of the Vth - ½_c_, and of the 3d ½_c_. 3. Let the temperaments of the IIId and 3d have contrary signs, as before; and let that of the 3d be the greater. Making ½ · (c - 3x) = -2 · (2x - c), we obtain x = 3/5 c; which gives, by substitution, the temperaments of the 3d, Vth, and IIId - 2/5 _c_, - 1/5 _c_, and 1/5 _c_, respectively.