339edo
| ← 338edo | 339edo | 340edo → |
339 equal divisions of the octave (abbreviated 339edo or 339ed2), also called 339-tone equal temperament (339tet) or 339 equal temperament (339et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 339 equal parts of about 3.54 ¢ each. Each step represents a frequency ratio of 21/339, or the 339th root of 2.
Theory
339edo is enfactored in the 3-limit with the same tuning as 113edo, and is only consistent to the 5-odd-limit. Using the 339d val (⟨339 537 787 951]) in the 7-limit and the 339de val (⟨339 537 787 951 1172]) in the 11-limit, it supports tritriple.
It can be used for the 2.3.5.13.19.23.31.41 subgroup, where it tempers out 621/620, 780/779, 1026/1025, 1426/1425, 2945/2944, 14391/14375 and 73853/73800.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -1.07 | -0.47 | +1.09 | +1.40 | +0.89 | -1.59 | -1.54 | +1.24 | -0.17 | +0.02 | -1.73 |
| relative (%) | -30 | -13 | +31 | +40 | +25 | -45 | -44 | +35 | -5 | +0 | -49 | |
| Steps (reduced) |
537 (198) |
787 (109) |
952 (274) |
1075 (58) |
1173 (156) |
1254 (237) |
1324 (307) |
1386 (30) |
1440 (84) |
1489 (133) |
1533 (177) | |
Subsets and supersets
Since 339 factors into 3 × 113, 339edo has 3edo and 113edo as its subsets. 1017edo, which triples it, gives a good correction to the harmonics 7 and 11.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [-13 17 -6⟩, [-44 -3 21⟩ | [⟨339 537 787]] | 0.2930 | 0.2828 | 7.99 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 32\339 | 113.27 | 16/15 | Misneb |
| 1 | 146\339 | 516.81 | 27/20 | Gravity |
| 1 | 158\339 | 559.29 | 864/625 | Tritriple (339d) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct