Since the length of a string cæteris paribus is inversely as its number of vibrations, the lengths in col. 5 may be deduced from the vibrations in col. 6; or more expeditiously, by subtracting the numerical distances from C of the several degrees in Table VI. from O, and taking the corresponding numbers, from the table of logarithms. These numbers, when used as logarithms, must be brought back to the decimal form, agreeably to Scholium 2. Prop. I.
To find the number of beats made in a second by any concord, it is only necessary to take from col. 5 the numbers belonging to the degrees which terminate that concord, and to multiply them crosswise into the terms of its perfect ratio. The difference of the products will be the number of beats made in a second. The 3 last columns contain the beats made by each of the concords, in 10 seconds.
TABLE VI.
| C | ——— | ——— | ——— | C |
| 2998 | 2998 | ——— | B | |
| 1772 | ||||
| B | ——— | ——— | ——— | B |
| 1831 | 3033 | |||
B | ——— | 4813 | ||
| ——— | A | |||
| 2982 | 1780 | |||
| A | ——— | ——— | ——— | A |
| 1871 | 3030 | |||
| A | ——— | 4839 | ||
| ——— | G | |||
| 2968 | 1809 | |||
| G | ——— | ——— | ——— | G |
| 1814 | ——— | F | ||
| G | ——— | 1798 | ||
| 4820 | ——— | F | ||
| 3006 | 1824 | |||
| F | ——— | ——— | ——— | F |
| ——— | E | |||
| 2988 | 2988 | |||
| 1777 | ||||
| E | ——— | ——— | ——— | E |
| 1870 | 3028 | |||
| E | ——— | |||
| 4818 | ——— | D | ||
| 2948 | 1790 | |||
| D | ——— | ——— | ——— | D |
| 1835 | 3018 | |||
| D | ——— | 4827 | ||
| ——— | C | |||
| 2992 | 1809 | |||
| C | ——— | ——— | ——— | C |
TABLE VII.
| Bases | Temperaments of the | Lengths of String. | Vibrations in a Second. | Beats in 10 S. of the | ||||
| Vths | IIIds | 3ds | Vths. | IIIds. | 3ds. | |||
| B | 77 | 51431 | 466,64 | 43,4 | ||||
| B | 154 | 76 | 93 | 53574 | 447,98 | 47,4 | 39,0 | 57,8 |
| B | 147 | 35 | 97 | 55880 | 429,49 | 43,5 | 17,7 | 57,4 |
| A | 156 | 78 | 57448 | 417,77 | 45,1 | 46,2 | ||
| A | 153 | 71 | 107 | 59852 | 400,99 | 42,5 | 33,5 | 59,4 |
| A | 154 | 9 | 62487 | 384,08 | 40,4 | 4,0 | ||
| G | 151 | 76 | 75 | 64177 | 373,97 | 39,1 | 32,9 | 39,2 |
| G | 132 | 39 | 97 | 66907 | 358,71 | 32,9 | 16,3 | 48,1 |
| F | 101 | 68778 | 348,95 | 48,5 | ||||
| G | 56 | 69760 | 344,03 | 21,9 | ||||
| F | 154 | 76 | 83 | 71685 | 334,80 | 36,0 | 29,2 | 38,5 |
| F | 139 | 32 | 130 | 74760 | 321,03 | 30,9 | 11,9 | 57,8 |
| E | 154 | 78 | 76874 | 312,20 | 33,2 | 33,5 | ||
| E | 149 | 74 | 110 | 80085 | 299,68 | 30,8 | 25,2 | 45,3 |
| E | 110 | 13 | 54 | 83608 | 287,05 | 21,7 | 4,1 | 21,5 |
| D | 154 | 53 | 78 | 85868 | 279,50 | 29,6 | 17,0 | 30,0 |
| D | 144 | 61 | 112 | 89480 | 268,21 | 26,5 | 18,5 | 41,1 |
| D | 180 | 50 | 93342 | 257,12 | 32,0 | 14,8 | ||
| C | 156 | 78 | 82 | 95920 | 250,20 | 26,6 | 22,0 | 28,0 |
| C | 156 | 46 | 143 | 100000 | 240,00 | 25,8 | 12,8 | 47,5 |