308edo
| ← 307edo | 308edo | 309edo → |
308 equal divisions of the octave (abbreviated 308edo or 308ed2), also called 308-tone equal temperament (308tet) or 308 equal temperament (308et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 308 equal parts of about 3.9 ¢ each. Each step represents a frequency ratio of 21/308, or the 308th root of 2.
Theory
308edo only is consistent in the 5-odd-limit. Ignoring the harmonics 7, 11 and 13, it is strong in the 2.3.5.17.19.23.29.31 subgroup.
Using the patent val nonetheless, the equal temperament tempers out 19683/19600, 65625/65536, and 390625/388962 in the 7-limit, and 243/242, 1375/1372, 6250/6237, 9801/9800, and 14700/14641 in the 11-limit.
Using the 308d val, it supports unidec and gammic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.66 | -0.60 | +1.30 | +1.93 | +1.03 | +0.24 | -1.41 | -1.00 | -1.01 | +0.42 |
| Relative (%) | +0.0 | -16.8 | -15.4 | +33.5 | +49.5 | +26.5 | +6.1 | -36.2 | -25.7 | -25.8 | +10.8 | |
| Steps (reduced) |
308 (0) |
488 (180) |
715 (99) |
865 (249) |
1066 (142) |
1140 (216) |
1259 (27) |
1308 (76) |
1393 (161) |
1496 (264) |
1526 (294) | |
Subsets and supersets
Since 308 factors into 22 × 7 × 11, 308edo has subset edos 2, 4, 7, 11, 14, 22, 28, 44, 77, and 154.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-122 77⟩ | [⟨308 488]] | 0.2070 | 0.2071 | 5.32 |
| 2.3.5 | [-36 11 8⟩, [-7 22 -12⟩ | [⟨308 488 715]] | 0.2241 | 0.1708 | 4.38 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 9\308 | 35.06 | 128/125 | Gammic (308d) |
| 28 | 128\308 (4\308) |
498.70 (15.58) |
4/3 (126/125) |
Oquatonic (308) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct