E
, for example, does not occur at all till we come to the key of two sharps, and even then only in occasional modulations, corresponding to the IIId on B in the natural key, whose multiplier is 10. In the key of 3 sharps it becomes another accidental chord, answering to the IIId on E in the key of C, and consequently has 40 for its multiplier. It is only in the key of 6 sharps, that it becomes a constituent chord of the key; when if that key were ever used, it would correspond to the IIId GB on the dominant of the natural key.
After all the products have been taken and reduced to their proper places, in the manner exemplified above, a similar operation must be repeated with the numbers in the second column of Table III. and those in the second columns in the three first divisions of Table II.
The necessity of keeping the major, and its relative minor key, distinct, will be evident, when we consider that the several keys in the minor mode do not follow the same law of frequency as in the major; as is manifest from the observations in Schol. Prop. III. and as clearly appears from an inspection of Table III.
But in order to discover the relative frequency of the different chords on every account, the results of the two foregoing operations must be united. Now, as the numbers in the two columns of Table II. at a medium, are as 3 : 1, and those in Table III. are in the same ratio, although the factors are to each other in only the simple ratio of the relative frequency of the two modes, yet their products will, at a medium, be in the duplicate ratio of that frequency. Hence, to render the two sets of results homologous, so that those which correspond to the same interval may be properly added, to express the general chance of occurrence for that interval in all the major and minor keys in which it is found, this duplicate ratio must be reduced to a simple one, either by dividing the first, or by multiplying the last series of results, by 3. We will do the latter, as it will give the ratios in the largest, and, of course, the most accurate terms. Then adding those results in each which belong to the same interval, and cutting off the three right hand figures, (expressing in the nearest small fractions those results which are under 1000) which will leave a set of ratios abundantly accurate for every purpose; the numbers constituting the final solution of the problem will stand as follows:
TABLE IV.
| Bases. | Vths and 4ths. | IIIds and 6ths. | 3ds and VIths. | Bases. | Vths and 4ths. | IIIds and 6ths. | 3ds and VIths. |
F | 67 | 29 | 1072 | B | —— | —— | 4 |
| F | 639 | 924 | 66 | B | 221 | 135 | 1161 |
| E | —— | —— | 12 | B | 418 | 654 | 5 |
| E | 548 | 323 | 1151 | A | —— | —— | 29 |
| E | 265 | 363 | ½ | A | 870 | 568 | 1085 |
| D | ⅓ | ½ | 144 | A | 52 | 78 | ⅕ |
| D | 1166 | 943 | 569 | G | 5 | 4 | 365 |
| D | 1 | 6 | —— | G | 1207 | 1197 | 567 |
| C | 25 | 12 | 581 | F | —— | —— | ¼ |
| C | 816 | 1131 | 180 | G | —— | ½ | —— |
Note. In this table, as well as the last, the Vths, IIIds, and 3ds are to be taken above, and the 4ths, 6ths, and VIths, their complements to the octave, below the corresponding degrees in the first column. And, in general, whenever the Vths, IIIds, and 3ds are hereafter treated as different classes of concords, each will be understood to include its complement to the octave and its compounds with octaves.