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. _Scholium 3._