1277edo
| ← 1276edo | 1277edo | 1278edo → |
1277 equal divisions of the octave (abbreviated 1277edo or 1277ed2), also called 1277-tone equal temperament (1277tet) or 1277 equal temperament (1277et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1277 equal parts of about 0.94 ¢ each. Each step represents a frequency ratio of 21/1277, or the 1277th root of 2.
Theory
1277edo is consistent to the 11-odd-limit. The equal temperament tempers out 4375/4374, 52734375/52706752, 645700815/645657712 (starscape comma) and [51 -13 -1 -10⟩ (technologisma) in the 7-limit; 151263/151250, 759375/758912, and 2097152/2096325 in the 11-limit. It supports monzismic, supermajor, revopent, as well as the rank-3 temperament bragi.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.003 | -0.096 | +0.007 | +0.287 | -0.434 | +0.291 | +0.373 | +0.387 | +0.337 | +0.462 |
| Relative (%) | +0.0 | +0.3 | -10.2 | +0.8 | +30.6 | -46.2 | +31.0 | +39.7 | +41.1 | +35.8 | +49.1 | |
| Steps (reduced) |
1277 (0) |
2024 (747) |
2965 (411) |
3585 (1031) |
4418 (587) |
4725 (894) |
5220 (112) |
5425 (317) |
5777 (669) |
6204 (1096) |
6327 (1219) | |
Subsets and supersets
1277edo is the 206th prime edo. 2554edo, which divides the edostep in two, is the smallest edo distinctly consistent through the 41-odd-limit, and provides correction for harmonics 11 through 41.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [2024 -1277⟩ | [⟨1277 2024]] | -0.0009 | 0.0009 | 0.10 |
| 2.3.5 | [54 -37 2⟩, [-67 -9 35⟩ | [⟨1277 2024 2965]] | +0.0132 | 0.0199 | 2.12 |
| 2.3.5.7 | 4375/4374, 52734375/52706752, [51 -13 -1 -10⟩ | [⟨1277 2024 2965 3585]] | +0.0093 | 0.0186 | 1.98 |
| 2.3.5.7.11 | 4375/4374, 151263/151250, 759375/758912, 2097152/2096325 | [⟨1277 2024 2965 3585 4418]] | -0.0092 | 0.0405 | 4.31 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 265\1277 | 249.021 | [-27 11 3 1⟩ | Monzismic |
| 1 | 380\1277 | 357.087 | 768/625 | Dodifo |
| 1 | 463\1277 | 435.082 | 9/7 | Supermajor |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct