691edo
| ← 690edo | 691edo | 692edo → |
691 equal divisions of the octave (abbreviated 691edo or 691ed2), also called 691-tone equal temperament (691tet) or 691 equal temperament (691et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 691 equal parts of about 1.74 ¢ each. Each step represents a frequency ratio of 21/691, or the 691st root of 2.
Theory
691edo is consistent to the 5-odd-limit, having a flat tendency in the first few harmonics except the harmonic 7. Its harmonic 13 is highly accurate with a relative error of only 0.4 percent. It can be used in the 2.3.5.11.17.19 subgroup, tempering out 1089/1088, 455625/454784, 334611/334400, 374000/373977 and 45125/45056.
Using the 691d val (⟨691 1095 1604 1939 2390]) can also be considered. It tempers out 3025/3024, 42875/42768, 759375/758912 and 6417873/6400000 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.363 | -0.785 | +0.204 | -0.726 | -0.811 | -0.007 | +0.588 | -0.759 | -0.552 | -0.159 | +0.380 |
| Relative (%) | -20.9 | -45.2 | +11.8 | -41.8 | -46.7 | -0.4 | +33.9 | -43.7 | -31.8 | -9.1 | +21.9 | |
| Steps (reduced) |
1095 (404) |
1604 (222) |
1940 (558) |
2190 (117) |
2390 (317) |
2557 (484) |
2700 (627) |
2824 (60) |
2935 (171) |
3035 (271) |
3126 (362) | |
Subsets and supersets
691edo is the 125th prime edo. 2764edo, which quadruples it, gives a good correction to a number of lower harmonics.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1095 691⟩ | [⟨691 1095]] | 0.1145 | 0.1146 | 6.60 |
| 2.3.5 | [-51 19 9⟩, [-20 -24 25⟩ | [⟨691 1095 1604]] | 0.1891 | 0.1410 | 8.12 |