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.
Each of these results makes the harmony of all the consonances as nearly equal as possible; but as the sum of the temperaments in the first case is much the least, it follows that the temperaments stated in the proposition constitute the best scheme of intervals for the natural scale, in which the harmony of all the different consonances is rendered as nearly equal as possible.
Cor. 1. In the same manner it may be shown that these temperaments are the best, among those which approach as nearly as possible to equal harmony, for the artificial scale; provided that it is furnished with distinct sounds for all the sharps and flats in common use. By inserting a sound between F and G, making the interval F
G equal to either of the semitones found above, the intervals, reckoned from G as a key note, will be exactly the same in respect to their temperaments, as the corresponding ones reckoned from C. The same thing holds, whatever be the number of flats and sharps. It is supposed, however, that the flat of a note is never used for the sharp of that next below, or the contrary; and hence this scheme of temperament would only be adapted to an instrument, furnished with all the degrees of the enharmonic scale; or, at least, with as many as are in common use.
Cor. 2. This scale will differ but little in practice from the one deduced, with so much labour, by Dr. Smith, from his criterion of equal harmony; which flattens the Vths 5/18, the IIIds 1/9, and the 3ds 1/6 of a comma. The several differences are only 1/126, 2/63, and 1/42 of a comma. Hence, as his measure of equal harmony differs so widely from that of Proposition I. we may infer that the consideration of equalizing the harmony of the concords of different names can have very little practical influence on the temperaments of the scale. Should it, therefore, be maintained that the criterion laid down in Prop. I. is not mathematically accurate; yet, as it must be allowed, in the most unfavourable view, to correspond far better with the decisions of experience than that of Doctor Smith, the chance is, that, at the lowest estimate, the temperaments deduced from it approach much more nearly to correctness. Hence it is manifest that equal temperament may be made, without any sensible error in practice, the criterion of equal harmony.