166edo
| ← 165edo | 166edo | 167edo → |
166 equal divisions of the octave (abbreviated 166edo or 166ed2), also called 166-tone equal temperament (166tet) or 166 equal temperament (166et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 166 equal parts of about 7.23 ¢ each. Each step represents a frequency ratio of 21/166, or the 166th root of 2.
Theory
166edo is consistent through the 13-odd-limit, yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit amity comma, it tempers out 225/224, 325/324, 385/384, 540/539, and 729/728, hence being an excellent tuning for the rank-3 temperament marvel, in both the 11-limit and in the 13-limit extension hecate, the rank-2 temperament wizard, which also tempers out 4000/3993, and houborizic, which also tempers out 2200/2197, giving the optimal patent val for all of these. In the 13-limit it tempers out 325/324, leading to hecate, and 1573/1568, leading to marvell, and tempering out both gives gizzard, the 72 & 94 temperament, for which 166 is an excellent tuning through the 19-limit.
166edo (as 83edo) contains a very good approximation of the harmonic 7th, of which it is only flat by 0.15121 cent.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.75 | -3.18 | -0.15 | -1.92 | -1.97 | +3.48 | -1.13 | +0.64 | -3.07 | -2.87 |
| Relative (%) | +0.0 | -10.4 | -44.0 | -2.1 | -26.6 | -27.3 | +48.1 | -15.6 | +8.9 | -42.5 | -39.7 | |
| Steps (reduced) |
166 (0) |
263 (97) |
385 (53) |
466 (134) |
574 (76) |
614 (116) |
679 (15) |
705 (41) |
751 (87) |
806 (142) |
822 (158) | |
Subsets and supersets
Since 166 factors into 2 × 83, 166edo contains 2edo and 83edo as subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-263 166⟩ | [⟨166 263]] | +0.237 | 0.237 | 3.27 |
| 2.3.5 | 1600000/1594323, [-31 2 12⟩ | [⟨166 263 385]] | +0.615 | 0.568 | 7.86 |
| 2.3.5.7 | 225/224, 118098/117649, 1250000/1240029 | [⟨166 263 385 466]] | +0.474 | 0.549 | 7.59 |
| 2.3.5.7.11 | 225/224, 385/384, 4000/3993, 322102/321489 | [⟨166 263 385 466 574]] | +0.490 | 0.492 | 6.80 |
| 2.3.5.7.11.13 | 225/224, 325/324, 385/384, 1573/1568, 2200/2197 | [⟨166 263 385 466 574 614]] | +0.498 | 0.449 | 6.21 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 33\166 | 238.55 | 147/128 | Tokko |
| 1 | 47\166 | 339.76 | 243/200 | Houborizic |
| 1 | 81\166 | 585.54 | 7/5 | Merman (7-limit) |
| 2 | 30\166 | 216.87 | 17/15 | Wizard / gizzard |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct