331edo
Jump to navigation
Jump to search
Prime factorization
331 (prime)
Step size
3.62538¢
Fifth
194\331 (703.323¢)
Semitones (A1:m2)
34:23 (123.3¢ : 83.38¢)
Dual sharp fifth
194\331 (703.323¢)
Dual flat fifth
193\331 (699.698¢)
Dual major 2nd
56\331 (203.021¢)
Consistency limit
5
Distinct consistency limit
5
| ← 330edo | 331edo | 332edo → |
331 equal divisions of the octave (331edo), or 331-tone equal temperament (331tet), 331 equal temperament (331et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 331 equal parts of about 3.63 ¢ each.
Theory
331et tempers out 78125000/78121827, 5120/5103 and 1959552/1953125 in the 7-limit; 806736/805255, 1835008/1830125, 1019215872/1019046875, 12005/11979, 16384/16335, 2359296/2358125, 42875/42768, 180224/180075, 1684375/1679616, 968000/964467, 3025/3024, 78408/78125, 1362944/1361367, 4108797/4096000 and 43923/43750 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⟩, [-74 -5 23⟩ | ⟨331 1049 1293] | 0.1494 | 0.1152 | 3.18 |
| 2.9.15.7 | 65625/65536, 420175/419904, 80387359983/80000000000 | ⟨331 1049 1293 929] | 0.1878 | 0.1199 | 3.31 |
| 2.9.15.7.11 | 9801/9800, 41503/41472, 137781/137500, 759375/758912 | ⟨331 1049 1293 929 1145] | 0.1653 | 0.1163 | 3.21 |
| 2.9.15.7.11.13 | 729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875, 53361/53248, 20336647/2028000 | ⟨331 1049 1293 929 1145 1225] | 0.1125 | 0.1587 | 4.38 |
| 2.9.15.7.11.13.17 | 729/728, 833/832, 1089/1088, 2025/2023, 10648/10647, 14161/14157, 14175/14144, 43904/43875, 18816/18785, 92823/95744 | ⟨331 1049 1293 929 1145 1225 1353] | 0.0901 | 0.1568 | 4.33 |
Rank-2 temperaments
| Periods per 8ve |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 89\331 | 322.66 | 6/5 | Magicaltet |
| 1 | 107\331 | 387.92 | 5/4 | Würschmidt |
| 1 | 137\331 | 496.68 | 5457/4096 | Edson |