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′ .
Three successive applications of this theorem to each degree in the scale, in the manner described Prop. VI., will bring them very near to the required position, as appears by the smallness of the corrections in the 3d column below, where the results of the several operations are exhibited at one view.
TABLE X.
| Bases. | First Operation. | Second Operation. | Third Operation. |
| B | -140 | -35 | -2 |
| B | +308 | +33 | -1 |
| A | -8 | -23 | +2 |
| G | -257 | -22 | -2 |
| G | +107 | +24 | -8 |
| F | -264 | -7 | 0 |
| F | +238 | +40 | +6 |
| E | -80 | -34 | -4 |
| E | +157 | +2 | -4 |
| D | +58 | + 8 | 0 |
| C | -352 | -29 | -1 |
| C | +176 | +29 | +4 |
Cor. Hence we may deduce, in the same manner as in Prop. VII., the diatonic and chromatic intervals, the lengths of a string and their vibrations in a second, and the temperaments and beats of all the concords for the scale which results from the foregoing computations. They may be seen in the two following tables:
TABLE XI.
DIATONIC AND CHROMATIC INTERVALS.
| C | ——— | ——— | C |
| 2895 | 2895 | ||
| B | ——— | ——— | B |
| 1991 | |||
| 4869 | ——— | B | |
| 2878 | |||
| A | ——— | ——— | A |
| 2761 | |||
| 4865 | ——— | G | |
| 2104 | |||
| G | ——— | ——— | G |
| 2903 | |||
| 4856 | ——— | F | |
| 1953 | |||
| F | ——— | ——— | F |
| 2911 | 2911 | ||
| E | ——— | ——— | E |
| 2235 | |||
| 4833 | ——— | E | |
| 2598 | |||
| D | ——— | ——— | D |
| 2957 | |||
| 4874 | ——— | C | |
| 1917 | |||
| C | ——— | ——— | C |
TABLE XII.
| Bases. | Temperaments of the | Lengths of Strings. | Vibrations per Second. | Beats in 10 Seconds of the | ||||
| Vths | IIIds | 3ds | Vths | IIIds | 3ds | |||
| B | 143 | 675 | 149 | 53446 | 449,04 | 44,0 | 352,8 | 92,4 |
| B | 105 | 69 | 1114 | 55954 | 428,92 | 30,8 | 34,0 | 155,2 |
| A | 138 | 10. | 154 | 59787 | 401,42 | 38,6 | 4,6 | 85,2 |
| G | 387 | 833 | 288 | 63712 | 376,79 | 98,7 | 360,5 | 155,4 |
| G | 106 | 43 | 175 | 66874 | 358,88 | 26,4 | 17,6 | 86,8 |
| F | 160 | 954 | 150 | 71496 | 335,68 | 37,2 | 372,8 | 69,8 |
| F | 124 | 30 | 957 | 74786 | 320,92 | 27,6 | 10,8 | 143,0 |
| E | 108 | 180 | 151 | 79970 | 300,10 | 22,2 | 66,6 | 62,0 |
| E | 136 | 311 | 818 | 84194 | 285,06 | 26,6 | 102,2 | 186,6 |
| D | 144 | 6 | 174 | 89384 | 268,50 | 26,6 | 2,2 | 64,0 |
| C | 52 | 1009 | 128 | 95682 | 250,83 | 10,9 | 295,3 | 44,8 |
| C | 135 | 16 | 446 | 100000 | 240,00 | 22,4 | 4,0 | 147,0 |
Nothing in the above tables will need explanation, except the anomalous sharp beats of the 3ds, in the last column. These are derived from the perfect ratio 6 : 7, because these 3ds are, in reality, much nearer to the ratio of 6 : 7 than to that of 5 : 6; and hence could their beats be counted, they would be those of the table, and not those which would be derived from considering these 3ds as having flat temperaments of the ratio 5 : 6. But although the beats are slower, the nearer they approach the ratio 6 : 7, this ought not to be regarded as any sufficient reason for admitting so large temperaments into the scale, were it not absolutely necessary, in order to accommodate those 3ds which are of far more frequent occurrence. Although the beats of these 3ds grow slower as their temperaments are increased, yet they are losing their character in melody; and become, in this respect, more and more offensive, the more they are tempered. Hence the harmony and melody of the several intervals, jointly considered, are to be judged of rather from their temperaments, in the three first columns, than from their beats, in the three last.
Scholium 1.
It will be perceived, from a comparison of the temperaments in Table XII. with the corresponding numbers in Table IX., that the harshness of the several concords, especially of the IIIds and 3ds, is, in general, nearly in the inverse ratio of their frequency. The contending claims of the different concords render it impossible that this ratio should hold exactly. Including the Vths, the harmony of the concords is much more nearly equal, than the principle of rendering the temperament of each inversely as its frequency, could it be carried into complete effect, would require.
Scholium 2.
The foregoing system may be put in practice, on the organ, by making the Vths beat flat, with the exception of those on C
, E
, and G
, which must beat sharp, at the rate required in the table; proving the correctness of the temperaments of the Vths, by comparing the beats of the IIIds, as they rise, with those required by column two. Should less accuracy be required, the IIIds on C, D, and A, might be made perfect, without producing any essential change in the system. This would reduce the labour of counting the beats to eight degrees only.
Scholium 3.
To show that the computations of the different frequency of occurrence of the different concords, on which this system of temperament is founded, may be relied on as practically correct, for music in general, it may be proper to state, that a similar series of calculations had been before made, from an enumeration of the concords in fifty scores of music entirely different from that made use of in Prop. IV. They were not, indeed, made with the same accuracy, for the music of which the chords were counted, was too generally of the simpler kind, and the numbers corresponding to those in the two columns under each concord in Table II., and those belonging to the major and to the minor signatures, corresponding to the numbers in Table III., were added, before the products were taken, instead of keeping the modes distinct, which is necessary to perfect accuracy. Yet the resulting scheme of temperament was essentially the same throughout, with the one which has been just described. It had the same anomalous temperaments, viz. the Vths on C
, E
, and G
; and the IIId on A; and these anomalies were similar in degree. The greatest difference between any two corresponding temperaments, was between those of the 3d on E
; the first computation making it only 702, while the last has it 818.