164edo
| ← 163edo | 164edo | 165edo → |
164 equal divisions of the octave (abbreviated 164edo or 164ed2), also called 164-tone equal temperament (164tet) or 164 equal temperament (164et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 164 equal parts of about 7.32 ¢ each. Each step represents a frequency ratio of 21/164, or the 164th root of 2.
Theory
164 = 4 × 41, and 164edo shares its fifth with 41edo. In the 5-limit, 164et tempers out the würschmidt comma, 393216/390625, and the vulture comma, [24 -21 4⟩. It supplies the optimal patent val for the würschmidt temperament.
In the patent val ⟨164 260 381 460 567 607], it tempers out 196/195, 352/351, 385/384, 441/440, 676/675, and supplies the optimal patent val for the 7-limit, 1/41 octave period 41&123 temperament, and the 13-limit momentous temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440.
In the alternative val 164de ⟨164 260 381 461 568 607], it tempers out 243/242, 351/350, 364/363, 640/637, 676/675, 729/728, and 1575/1573. The 164dg val is a good tuning for 7- to 19-limit buzzard temperament, although if harmonic 11 is desired it is only easily accessible through the patent mapping.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.48 | +1.49 | -2.97 | -2.54 | +0.94 | -2.52 | +2.49 | +0.99 | +2.13 | -3.57 |
| Relative (%) | +0.0 | +6.6 | +20.4 | -40.6 | -34.7 | +12.8 | -34.4 | +34.0 | +13.6 | +29.1 | -48.8 | |
| Steps (reduced) |
164 (0) |
260 (96) |
381 (53) |
460 (132) |
567 (75) |
607 (115) |
670 (14) |
697 (41) |
742 (86) |
797 (141) |
812 (156) | |
Subsets and supersets
Since 164 = 22 × 41, 164edo has subset edos 2, 4, 41, 82. 328edo, which doubles it, provides good correction for the approximation to harmonics 7 and 11, and is consistent in the 13-odd-limit.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 393216/390625, [24 -21 4⟩ | [⟨164 260 381]] | −0.316 | 0.262 | 3.58 |
| 2.3.5.13 | 676/675, 256000/255879, 393216/390625 | [⟨164 260 381 607]] | −0.300 | 0.229 | 3.13 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 47\164 | 343.90 | 8000/6561 | Geb |
| 1 | 49\164 | 358.54 | 16/13 | Restles (164) |
| 1 | 53\164 | 387.80 | 5/4 | Würschmidt |
| 1 | 53\164 | 475.61 | 320/243 | Vulture |
| 1 | 69\164 | 504.88 | 104976/78125 | Countermeantone |
| 2 | 17\164 | 124.39 | 275/256 | Semivulture (164) |
| 2 | 25\164 | 182.93 | 10/9 | Unidecmic |
| 4 | 68\164 (14\164) |
497.56 (102.44) |
4/3 (35/33) |
Undim (164deff) / unlit (164f) |
| 41 | 53\164 (1\164) |
387.80 (7.32) |
5/4 (32805/32768) |
Countercomp |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct