331edo
| ← 330edo | 331edo | 332edo → |
331 equal divisions of the octave (abbreviated 331edo or 331ed2), also called 331-tone equal temperament (331tet) or 331 equal temperament (331et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 331 equal parts of about 3.63 ¢ each. Each step represents a frequency ratio of 21/331, or the 331st root of 2.
Theory
331edo is only consistent to the 5-odd-limit and the errors of both harmonics 3 and 5 are quite large, commending itself as a temperament of the 2.9.15.7.11.13.17.19 subgroup.
Using the patent val nonetheless, the equal temperament tempers out 5120/5103, 1959552/1953125 and 78125000/78121827 in the 7-limit; 3025/3024, 12005/11979, 16384/16335, 42875/42768, 43923/43750, 78408/78125, and 180224/180075 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +1.37 | +1.60 | -0.85 | -0.89 | -0.26 | +0.56 | -0.66 | +0.18 | -0.23 | +0.52 | -1.08 |
| relative (%) | +38 | +44 | -23 | -25 | -7 | +15 | -18 | +5 | -6 | +14 | -30 | |
| Steps (reduced) |
525 (194) |
769 (107) |
929 (267) |
1049 (56) |
1145 (152) |
1225 (232) |
1293 (300) |
1353 (29) |
1406 (82) |
1454 (130) |
1497 (173) | |
Subsets and supersets
331edo is the 67th prime edo. 662edo, which doubles it, gives a good correction to the harmonics 3 and 5.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-1049 331⟩ | [⟨331 1049]] | 0.1402 | 0.1402 | 3.87 |
| 2.9.15 | [-7 17 -12⟩, [-81 12 11⟩ | [⟨331 1049 1293]] | +0.1238 | 0.1168 | 3.22 |
| 2.9.15.7 | 65625/65536, 420175/419904, 80387359983/80000000000 | [⟨331 1049 1293 929]] | +0.1685 | 0.1275 | 3.52 |
| 2.9.15.7.11 | 9801/9800, 41503/41472, 137781/137500, 759375/758912 | [⟨331 1049 1293 929 1145]] | +0.1499 | 0.1200 | 3.31 |
| 2.9.15.7.11.13 | 729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875 | [⟨331 1049 1293 929 1145 1225]] | +0.0997 | 0.1568 | 4.33 |
| 2.9.15.7.11.13.17 | 729/728, 833/832, 1089/1088, 2025/2023, 10648/10647, 18816/18785 | [⟨331 1049 1293 929 1145 1225 1353]] | +0.0791 | 0.1537 | 4.24 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 107\331 | 387.92 | 5/4 | Würschmidt (331, 5-limit) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct