6691edo
| ← 6690edo | 6691edo | 6692edo → |
6691 equal divisions of the octave (abbreviated 6691edo or 6691ed2), also called 6691-tone equal temperament (6691tet) or 6691 equal temperament (6691et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 6691 equal parts of about 0.179 ¢ each. Each step represents a frequency ratio of 21/6691, or the 6691st root of 2.
6691edo is a very strong 11-limit system, with a lower 11-limit relative error than any smaller division until 40006. It is also strong in the 7-limit, where only 3125 is both smaller and with a lesser relative error.
We may note it is a euzenius and parimic system. A basis for the 11-limit commas is {1771561/1771470, 3294225/3294172, 67110351/67108864, 78125000/78121827} and for the 7-limit commas, {78125000/78121827, [-48 0 11 8⟩, [4 -28 -8 21⟩}.
The approximation to harmonic 13 is weaker, though it is still consistent to the 15-odd-limit. In fact, it is consistent to the no-13 or no-17 no-23 29-odd-limit. In the 13-limit we may note it tempers out 10648/10647, and is a good tuning for the corresponding rank-5 temperament. It also tempers out 140625/140608. In the 17-limit, 194481/194480; in the 19-limit, 14080/14079, 23409/23408, 43681/43680, 89376/89375, 165376/165375; and in the 23-limit, 21505/21504, 23276/23275, 25921/25920, 52326/52325, 76545/76544 among others.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0029 | -0.0037 | -0.0021 | -0.0102 | +0.0642 | -0.0384 | +0.0210 | -0.0274 | +0.0447 | +0.0847 |
| Relative (%) | +0.0 | +1.6 | -2.1 | -1.2 | -5.7 | +35.8 | -21.4 | +11.7 | -15.3 | +24.9 | +47.2 | |
| Steps (reduced) |
6691 (0) |
10605 (3914) |
15536 (2154) |
18784 (5402) |
23147 (3074) |
24760 (4687) |
27349 (585) |
28423 (1659) |
30267 (3503) |
32505 (5741) |
33149 (6385) | |
Subsets and supersets
6691edo is the 863rd prime edo.