Since, then, the proposition is true in reference to the Vths, IIIds, and 3ds, when separately considered, it will be equally true when they are considered jointly, that is, as formed into harmonic triads, unless, by rendering the concords of the same name unequal in their temperament, the mean temperament of the Vths could be increased, and that of the IIIds and 3ds proportionally diminished. Could this be done, it might be a question whether the more equal distribution of the temperament among the concords of different names, might not justify the introduction of some inequality among those of the same name. But it is demonstrated in the Lemma, that the sum of the temperaments of each parcel of concords, in the system of equal semitones, is the least possible. Hence no changes in the Vths can diminish the average temperaments of the IIIds and 3ds.
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.
| Bases. | Vths, 4ths, and Octaves. | IIIds, 6ths, and Octaves. | 3ds, VIths, and 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 |
The general theorem of Prop. V. is equally applicable to the determination of the approximate place for any degree in this scale, considering the numbers in the above table as those to be substituted for a, a′, b, &c.; and m, n, and p, in the first instance, as 49, –343 and 392, the uniform temperaments of the Vths, IIIds, and 3ds, in the scale of equal semitones. Since, however, the temperaments of the IIIds in this scale are sharp, which would require the signs of the 3d and 4th terms in the numerator of the general formula to be continually changed, it will be rendered more convenient for practice, if they are changed at first, so that it will stand thus:
x = am – a′m′ – bn + b′n′ + cp – c′p′ a + a′ + b + b′ + c + c′ .