| 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