277edo
| ← 276edo | 277edo | 278edo → |
277 equal divisions of the octave (abbreviated 277edo or 277ed2), also called 277-tone equal temperament (277tet) or 277 equal temperament (277et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 277 equal parts of about 4.33 ¢ each. Each step represents a frequency ratio of 21/277, or the 277th root of 2.
Theory
277edo is a good 5-limit tuning; however, it is inconsistent in the 7-odd-limit. As an equal temperament, it tempers out 32805/32768 (schisma) and [-11 -37 30⟩ in the 5-limit.
The patent val ⟨277 439 643 778] tempers out 4375/4374, 65625/65536, and 829440/823543 in the 7-limit; 540/539, 6250/6237, 15488/15435, and 35937/35840 in the 11-limit; 625/624, 729/728, 1573/1568, 2080/2079, and 2200/2197 in the 13-limit. It supports pontiac.
The 277d val ⟨277 439 643 777] tempers out 1029/1024, 10976/10935, and 48828125/48771072 in the 7-limit; 385/384, 441/440, 19712/19683, and 234375/234256 in the 11-limit; 625/624, 847/845, 1001/1000, and 1575/1573 in the 13-limit. It supports guiron and widefourth.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.15 | -0.75 | +1.57 | -1.14 | -0.09 | -0.98 | +1.40 | -0.12 | +1.47 | -1.35 |
| Relative (%) | +0.0 | -3.5 | -17.4 | +36.3 | -26.3 | -2.2 | -22.7 | +32.4 | -2.7 | +33.9 | -31.2 | |
| Steps (reduced) |
277 (0) |
439 (162) |
643 (89) |
778 (224) |
958 (127) |
1025 (194) |
1132 (24) |
1177 (69) |
1253 (145) |
1346 (238) |
1372 (264) | |
Subsets and supersets
277edo is the 59th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-439 277⟩ | [⟨277 439]] | +0.0473 | 0.0473 | 1.09 |
| 2.3.5 | 32805/32768, [-11 -37 30⟩ | [⟨277 439 643]] | +0.1398 | 0.1364 | 3.15 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 115\277 | 498.19 | 4/3 | Helmholtz |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct