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.