189edo
| ← 188edo | 189edo | 190edo → |
189 equal divisions of the octave (abbreviated 189edo or 189ed2), also called 189-tone equal temperament (189tet) or 189 equal temperament (189et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 189 equal parts of about 6.35 ¢ each. Each step represents a frequency ratio of 21/189, or the 189th root of 2.
189edo is consistent to the 7-odd-limit, but harmonics 3 and 7 are about halfway between its steps. It has good approximations to 5, 9, 11, 19, and 21, making it suitable for a 2.9.5.21.11.19 subgroup interpretation.
Using the full 13-limit patent val nonetheless, the equal temperament tempers out 15625/15552 (kleisma) and [53 -29 -3⟩ in the 5-limit; 4000/3969, 6144/6125, and 537824/531441 in the 7-limit, supporting the hemikleismic temperament. It tempers out 896/891, 1331/1323, 1375/1372, and 16896/16807 in the 11-limit; 169/168, 352/351, 364/363, and 1001/1000 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.81 | +0.99 | +2.60 | -0.74 | +1.06 | -2.43 | -2.55 | +2.98 | +0.90 | -0.94 | +0.30 |
| Relative (%) | +44.2 | +15.6 | +41.0 | -11.6 | +16.7 | -38.3 | -40.2 | +47.0 | +14.2 | -14.8 | +4.7 | |
| Steps (reduced) |
300 (111) |
439 (61) |
531 (153) |
599 (32) |
654 (87) |
699 (132) |
738 (171) |
773 (17) |
803 (47) |
830 (74) |
855 (99) | |
Subsets and supersets
Since 189 factors into 33 × 7, 189edo contains 3, 7, 9, 21, 27, and 63 as its subsets. 378edo, which doubles it, provides a good correction for the approximation of 3 and 7.