330edo
| ← 329edo | 330edo | 331edo → |
330 equal divisions of the octave (abbreviated 330edo or 330ed2), also called 330-tone equal temperament (330tet) or 330 equal temperament (330et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 330 equal parts of about 3.64 ¢ each. Each step represents a frequency ratio of 21/330, or the 330th root of 2.
Theory
330edo has a flat tendency, with its harmonics 3, 5, and 7 tuned progressively flatter. In the 11-limit, the 330e val ⟨330 523 766 926 1141] scores significantly better in TE error than its patent val ⟨330 523 766 926 1142] and allows an extension to the 13-limit.
The equal temperament tempers out 32805/32768 (schisma) in the 5-limit; 250047/250000 (landscape comma), 703125/702464 (meter) and 4802000/4782969 (canousma) in the 7-limit. Using the 330e val, it tempers out 385/384 (keenanisma), 9801/9800 (kalisma), and 14641/14580 (semicanousma) in the 11-limit; 847/845 (cuthbert) and 1001/1000 (sinbadma) in the 13-limit.
It provides a nice tuning for keenanismic, the rank-4 temperament that tempers out 385/384 (even better than its optimal patent val 284edo), and actually a close-to-optimal tuning for 11-limit semicanou, the rank-3 temperament that tempers out 9801/9800 and 14641/14580.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | -0.14 | -0.86 | -1.55 | +1.41 | -0.53 | +0.50 | +0.67 | +0.82 | -0.49 | +0.42 |
| relative (%) | +0 | -4 | -24 | -43 | +39 | -15 | +14 | +18 | +22 | -13 | +12 | |
| Steps (reduced) |
330 (0) |
523 (193) |
766 (106) |
926 (266) |
1142 (152) |
1221 (231) |
1349 (29) |
1402 (82) |
1493 (173) |
1603 (283) |
1635 (315) | |
Subsets and supersets
Since 330 factors into 2 × 3 × 5 × 11, 330edo has subset edos 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-523 330⟩ | [⟨330 523]] | +0.0432 | 0.0432 | 1.19 |
| 2.3.5 | 32805/32768, [-2 -50 35⟩ | [⟨330 523 766]] | +0.1521 | 0.1581 | 4.35 |
| 2.3.5.7 | 32805/32768, 250047/250000, 823543/819200 | [⟨330 523 766 926]] | +0.2524 | 0.2212 | 6.08 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 137\330 | 498.182 | 4/3 | Helmholtz |
| 3 | 137\330 (27\330) |
498.182 (98.182) |
4/3 (200/189) |
Term |
| 6 | 137\330 (27\330) |
498.182 (98.182) |
4/3 (200/189) |
Semiterm (330eff) |
| 22 | 137\330 (2\330) |
498.182 (7.273) |
4/3 ([16 -13 2⟩) |
Major arcana (5-limit) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct