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. +--------+----------+-----------+---------+ | | Vths and | IIIds and | 3ds and | | Bases. | 4ths. | 6ths. | VIths. | +--------+----------+-----------+---------+ |F♯ | 67 | 29 | 1072 | |F | 639 | 924 | 66 | |E♯ | ---- | ---- | 12 | |E | 548 | 323 | 1151 | |E♭ | 265 | 363 | ½ | |D♯ | ⅓ | ½ | 144 | |D | 1166 | 943 | 569 | |D♭ | 1 | 6 | ---- | |C♯ | 25 | 12 | 581 | |C | 816 | 1131 | 180 | |B♯ | ---- | ---- | 4 | |B | 221 | 135 | 1161 | |B♭ | 418 | 654 | 5 | |A♯ | ---- | ---- | 29 | |A | 870 | 568 | 1085 | |A♭ | 52 | 78 | ⅕ | |G♯ | 5 | 4 | 365 | |G | 1207 | 1197 | 567 | |F♯♯ | ---- | ---- | ¼ | |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. _Scholium._