660edo
| ← 659edo | 660edo | 661edo → |
660 equal divisions of the octave (abbreviated 660edo or 660ed2), also called 660-tone equal temperament (660tet) or 660 equal temperament (660et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 660 equal parts of about 1.82 ¢ each. Each step represents a frequency ratio of 21/660, or the 660th root of 2.
660edo is enfactored in the 5-limit, with the same tuning as 330edo, tempering out the schisma and the 22nd-octave major arcana comma, [-193 154 -22⟩.
However, it is far better viewed as a no-5 system. It does tune well the 2.3.7.11.13 subgroup, with errors less than 50% on all pairs of intervals in that subgroup that also belong to the 15-odd-limit. In addition, in the no-5s 17-odd-limit, it only misses the pair {17/13, 26/17}.
Nonetheless, patent val does have some use. It tunes the undecimal dimcomp temperament and also provides the optimal patent val for the quadrant temperament in the 11-limit as well as the 13-limit. Furthermore, in 2.3.7 it is a septiruthenian system, and the patent val mapping for 5 allows the tuning of ruthenium temperament.
Other mappings can be considered. Taking a different mapping for 5, the 660c val tunes qintosec and atomic. In the 7-limit, it tunes decoid. In addition, one can combine the mappings for five to produce a 2.3.25 subgroup interpretation. There, 660edo has less error than on either vals and it tempers out the kwazy comma, as well as the landscape comma in the 2.3.25.7 subgroup.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -0.137 | -0.859 | +0.265 | -0.274 | -0.409 | -0.528 | +0.822 | +0.499 | +0.669 | +0.128 | +0.817 |
| relative (%) | -8 | -47 | +15 | -15 | -22 | -29 | +45 | +27 | +37 | +7 | +45 | |
| Steps (reduced) |
1046 (386) |
1532 (212) |
1853 (533) |
2092 (112) |
2283 (303) |
2442 (462) |
2579 (599) |
2698 (58) |
2804 (164) |
2899 (259) |
2986 (346) | |
Subsets and supersets
Since 660 factors as 22 × 3 × 5 × 11, it has subset edos 1, 2, 3, 4, 5, 6, 10, 11, 12, 15, 20, 22, 30, 33, 44, 55, 60, 66, 110, 132, 165, 220, 330.