661edo
| ← 660edo | 661edo | 662edo → |
661 equal divisions of the octave (abbreviated 661edo or 661ed2), also called 661-tone equal temperament (661tet) or 661 equal temperament (661et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 661 equal parts of about 1.82 ¢ each. Each step represents a frequency ratio of 21/661, or the 661st root of 2.
Theory
661edo is consistent to the 7-odd-limit and the error of its harmonic 3 is very high. The first prime harmonics have a very sharp tendency, but note the accuracy of its harmonic 13 with an relative error of only 0.9 percent. It can be used in the 2.9.15.21.13.19.23.29 subgroup, tempering out 4375/4374, 875/874, 1863/1862, 1625/1624, 3381/3380, 102600/102557 and 401679/401408. It supports hitch.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.617 | +0.373 | +0.614 | -0.582 | +0.573 | +0.017 | -0.825 | +0.340 | +0.218 | -0.584 | -0.135 |
| Relative (%) | +34.0 | +20.6 | +33.8 | -32.0 | +31.6 | +0.9 | -45.5 | +18.7 | +12.0 | -32.2 | -7.4 | |
| Steps (reduced) |
1048 (387) |
1535 (213) |
1856 (534) |
2095 (112) |
2287 (304) |
2446 (463) |
2582 (599) |
2702 (58) |
2808 (164) |
2903 (259) |
2990 (346) | |
Subsets and supersets
661edo is the 121st prime EDO. 1983edo, which triples it, gives a good correction to the harmonics 3, 7 and 11.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-2095 661⟩ | [⟨661 2095]] | 0.0918 | 0.0918 | 5.06 |