1793edo
| ← 1792edo | 1793edo | 1794edo → |
1793 equal divisions of the octave (abbreviated 1793edo or 1793ed2), also called 1793-tone equal temperament (1793tet) or 1793 equal temperament (1793et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1793 equal parts of about 0.669 ¢ each. Each step represents a frequency ratio of 21/1793, or the 1793rd root of 2.
1793edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. Nonetheless, it is a good 2.3.5.11.13.17 subgroup system, in which it has a comma basis 4225/4224, 6656/6655, 42500/42471, 4787200/4782969, 703125/702559. Higher prime harmonics it supports to <25% error are 23, 31, 41, 53, 61.
If a mapping for 7 is added, this gives two interpretations. First is the patent val, which has a comma basis {2080/2079, 3025/3024, 4225/4224, 5832/5831, 14875/14872, 108086/108085}. The 1793d val has a comma basis {1225/1224, 8624/8619, 12376/12375, 14400/14399, 42500/42471, 29755593/29744000}. It provides the optimal patent val for the luminal temperament.
1793edo tempers out the jacobin comma, which is quite thematic given that 1793 is another notable year of the French Revolution, just as 1789 is. The comma basis for the 1789 & 1793 temperament in the 2.5.11.13 subgroup is {6656/6655, [-176 23 -2 35⟩}.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.109 | -0.145 | +0.276 | +0.217 | +0.160 | +0.075 | -0.037 | +0.120 | +0.312 | -0.285 | +0.170 |
| relative (%) | +16 | -22 | +41 | +32 | +24 | +11 | -5 | +18 | +47 | -43 | +25 | |
| Steps (reduced) |
2842 (1049) |
4163 (577) |
5034 (1448) |
5684 (305) |
6203 (824) |
6635 (1256) |
7005 (1626) |
7329 (157) |
7617 (445) |
7875 (703) |
8111 (939) | |
Subsets and supersets
Since 1793 factors into 11 × 163, 1793edo contains 11edo and 163edo as subsets.