179edo
| ← 178edo | 179edo | 180edo → |
179 equal divisions of the octave (abbreviated 179edo or 179ed2), also called 179-tone equal temperament (179tet) or 179 equal temperament (179et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 179 equal parts of about 6.7 ¢ each. Each step represents a frequency ratio of 21/179, or the 179th root of 2.
Theory
179edo does not approximate well any odd harmonic up to 23, best being 21/16 with 22% error. Nonetheless, it is consistent in the 7-odd-limit and there are a number of temperaments to be considered.
The equal temperament tempers out the parakleisma, [8 14 -13⟩ in the 5-limit, and supports parakleismic and its extensions, providing the optimal patent val for 11- and 13-limit parkleismic temperament. In the 7-limit it tempers out 3136/3125, 4375/4374 and 10976/10935, in the 11-limit 176/175 and 1375/1372 and in the 13-limit 169/168, 325/324, 351/350 and 352/351, providing the optimal patent val for 11- and 13-limit ulmo temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.96 | +2.51 | +3.24 | -2.79 | -1.60 | -2.54 | -2.24 | +2.31 | -2.54 | -1.51 | +1.89 |
| Relative (%) | +29.2 | +37.5 | +48.3 | -41.7 | -23.8 | -37.9 | -33.3 | +34.4 | -37.9 | -22.5 | +28.2 | |
| Steps (reduced) |
284 (105) |
416 (58) |
503 (145) |
567 (30) |
619 (82) |
662 (125) |
699 (162) |
732 (16) |
760 (44) |
786 (70) |
810 (94) | |
Subsets and supersets
179edo is the 41st prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [284 -179⟩ | [⟨179 284]] | -0.6169 | 0.6166 | 9.20 |
| 2.3.5 | [20 -17 3⟩, [28 -3 -10⟩ | [⟨179 284 416]] | -0.7718 | 0.5490 | 8.19 |
| 2.3.5.7 | 3136/3125, 4375/4374, 65536/64827 | [⟨179 284 416 503]] | -0.8673 | 0.5034 | 7.51 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 35\179 | 234.64 | 8/7 | Rodan (179d) |
| 1 | 47\179 | 315.08 | 6/5 | Parakleismic |
| 1 | 79\179 | 529.61 | 512/375 | Mabila (5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct