_Cor._ Hence we derive an important practical conclusion: that whatever irregularities are introduced into the scale, must be such as are demanded by the different frequency of occurrence of the several concords. If we make any alterations in the scale of equal semitones, this must be our sole criterion. A given system of temperament is eligible, in proportion to the accuracy with which it is deduced from the different frequency of the different concords. And those who maintain that the frequency of different intervals does not sensibly vary, or that it is of such a nature as not to be susceptible of calculation, must, to be consistent, adhere to the scale of equal semitones. PROPOSITION X. To determine the best distribution of the temperaments of the concords in the Douzeave Scale. As the scale of equal semitones has been demonstrated to be the best, on supposition that all the concords of the same name occurred equally often, it ought to be made the standard from which all the variations, required by their unequal frequency, are to be reckoned. To find a set of numbers expressing the relative frequency of the several concords in the common scale, we have only to unite the numbers in Table IV. standing against those adjacent degrees which have but one sound in this scale. They will then stand as in the following table: TABLE IX. +------+------------+-------------+------------+ | | Vths, 4ths,| IIIds, 6ths,| 3ds, VIths,| |Bases.| and | and | and | | | Octaves. | Octaves. | Octaves. | +------+------------+-------------+------------+ | B | 221 | 135 | 1161 | +------+------------+-------------+------------+ | B♭ | 418 | 654 | 34 | +------+------------+-------------+------------+ | A | 870 | 568 | 1085 | +------+------------+-------------+------------+ | G♯ | 57 | 82 | 365⅕ | +------+------------+-------------+------------+ | G | 1207 | 1197 | 567¼ | +------+------------+-------------+------------+ | F♯ | 67 | 29½ | 1072 | +------+------------+-------------+------------+ | F | 639 | 924 | 78 | +------+------------+-------------+------------+ | E | 548 | 323 | 1151 | +------+------------+-------------+------------+ | E♭ | 265⅓ | 363½ | 144½ | +------+------------+-------------+------------+ | D | 1166 | 943 | 569 | +------+------------+-------------+------------+ | C♯ | 26 | 18 | 581 | +------+------------+-------------+------------+ | C | 816 | 1131 | 184 | +------+------------+-------------+------------+