176edo
| ← 175edo | 176edo | 177edo → |
176 equal divisions of the octave (abbreviated 176edo or 176ed2), also called 176-tone equal temperament (176tet) or 176 equal temperament (176et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 176 equal parts of about 6.82 ¢ each. Each step represents a frequency ratio of 21/176, or the 176th root of 2.
Theory
176edo is consistent to the 11-odd-limit. The equal temperament tempers out 78732/78125 (sensipent comma) and [41 -20 -4⟩ (undim comma) in the 5-limit; 6144/6125, 10976/10935, and 50421/50000 in the 7-limit; 441/440, 3388/3375, 6912/6875, 8019/8000, 9801/9800, and 16384/16335 in the 11-limit. Using the patent val, 351/350, 364/363, 2080/2079, 2197/2187, and 4096/4095 in the 13-limit.
176edo tempers the Archytas' comma to 1/44th of the octave (4 steps) and as a corollary supports the ruthenium temperament. It supports the bison temperament and the bicommatic temperament, and provides the optimal patent val for countermiracle in the 7- and 11-limit, and countermanna, one of the extensions, in the 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.32 | +2.32 | -0.64 | +0.95 | -1.89 | -2.68 | +2.49 | -1.00 | -0.03 | +0.42 |
| Relative (%) | +0.0 | +4.7 | +34.1 | -9.4 | +14.0 | -27.7 | -39.3 | +36.5 | -14.7 | -0.5 | +6.1 | |
| Steps (reduced) |
176 (0) |
279 (103) |
409 (57) |
494 (142) |
609 (81) |
651 (123) |
719 (15) |
748 (44) |
796 (92) |
855 (151) |
872 (168) | |
Subsets and supersets
Since 176 factors into 24 × 11, 176edo has subset edos 2, 4, 8, 11, 16, 22, 44, and 88.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [279 -176⟩ | [⟨176 279]] | −0.100 | 0.100 | 1.47 |
| 2.3.5 | 78732/78125, [41 -20 -4⟩ | [⟨176 279 409]] | −0.400 | 0.432 | 6.34 |
| 2.3.5.7 | 6144/6125, 10976/10935, 50421/50000 | [⟨176 279 409 494]] | −0.243 | 0.463 | 6.79 |
| 2.3.5.7.11 | 441/440, 3388/3375, 6144/6125, 8019/8000 | [⟨176 279 409 494 609]] | −0.250 | 0.414 | 6.08 |
| 2.3.5.7.11.13 | 351/350, 364/363, 441/440, 2197/2187, 3146/3125 | [⟨176 279 409 494 609 651]] | −0.123 | 0.473 | 6.93 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 17\176 | 115.91 | 77/72 | Mercy / countermiracle / counterbenediction / countermanna |
| 1 | 35\176 | 238.64 | 147/128 | Tokko |
| 1 | 65\176 | 443.18 | 162/125 | Sensipent |
| 1 | 73\176 | 497.73 | 4/3 | Gary / cotoneum |
| 1 | 83\176 | 565.91 | 13/9 | Tricot / trident |
| 2 | 23\176 | 20.45 | 81/80 | Bicommatic |
| 2 | 23\176 | 156.82 | 35/32 | Bison |
| 4 | 73\176 (15\176) |
497.73 (102.27) |
4/3 (35/33) |
Undim |
| 8 | 73\176 (7\176) |
497.73 (47.73) |
4/3 (36/35) |
Twilight |
| 8 | 83\176 (5\176) |
565.91 (34.09) |
168/121 (55/54) |
Octowerck (176f) / octowerckis (176) |
| 11 | 73\176 (7\176) |
497.73 (47.73) |
4/3 (36/35) |
Hendecatonic |
| 22 | 73\176 (1\176) |
497.73 (6.82) |
4/3 (385/384) |
Icosidillic / major arcana |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct