| Bases. | 1st Oper- ation. | 2d. | 3d. | Bases. | 1st Oper- ation. | 2d. | 3d. |
| F | +18 | +5 | +1 | B | +18 | +5 | 0 |
| F | -20 | -6 | -1 | B | +19 | +5 | 0 |
| E | +18 | +5 | 0 | B | -23 | -10 | -1 |
| E | +14 | +5 | 0 | A | +18 | +7 | 0 |
| E | -69 | -8 | -1 | A | +13 | +4 | +1 |
| D | +19 | +5 | +1 | A | -71 | -7 | -2 |
| D | +5 | +2 | +1 | G | +17 | +5 | 0 |
| D | -45 | -7 | -2 | G | -14 | 0 | 0 |
| C | +18 | +6 | 0 | F | +44 | +5 | 0 |
| C | -5 | -5 | -2 | G | -46 | -5 | 0 |
The sign plus denotes that the degree to which it belongs is to be raised, and minus, that it is to be depressed. The corrections in each succeeding operation are to be added to those in the preceding. The errors, in the 3d approximation, are so trifling, that a 4th would be wholly useless.
Note. The foregoing calculations will be rendered much more expeditious and sure, by reducing the theorem, in some sense, to a diagram, as in the first of the following figures; and by applying the successive corrections to the circumference of a circle divided into parts proportioned to the intervals of the enharmonic scale, as in the second.
Proposition VII.
To determine the temperaments and beats of all the concords, together with the values of the diatonic and chromatic intervals, and the lengths and vibrations per second of a string producing all the sounds, of the system resulting from the last proposition.
The temperaments of all the concords are easily deduced from Table V. The Vth CG, for example, has its lower extremity lowered 12, and its upper extremity 14. Hence it is flatter by 2 than at first, and consequently its temperament=156. The temperaments of all the concords, thus calculated, will be found in the 2d, 3d, and 4th columns of Table VII.
Having ascertained the temperaments, the value of the diatonic and chromatic intervals may be found. The Vth CG being flattened 156, and the Vth FC 139, the major tone FG must be diminished 156 + 139, or be = 4820. By thus fixing the extent of one interval after another, from the temperaments of either of the different kinds of concords, as is most convenient, the intervals in question will be found to have the values exhibited in Table VI.
Let the numbers in this table be added successively, beginning at the bottom, to the log. of 240, the number of vibrations per second of the tenor C, (see Rees's Cyc. Art. Concert Pitch,) and the numbers corresponding to these logarithms will be the vibrations in a second, of a string sounding the several degrees of the scale. They are shown in col. 6, Table VII.