3361edo
| ← 3360edo | 3361edo | 3362edo → |
3361 equal divisions of the octave (abbreviated 3361edo or 3361ed2), also called 3361-tone equal temperament (3361tet) or 3361 equal temperament (3361et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3361 equal parts of about 0.357 ¢ each. Each step represents a frequency ratio of 21/3361, or the 3361st root of 2.
Theory
3361edo is consistent to the 5-odd-limit, with an almost perfect harmonic 5, its relative error is only 0.1%. Its harmonic 7 is about halfway its steps though, hence the consistency of this edo. As a temperament, 3361edo is strong in the 2.3.5.11.13.17.31 subgroup, tempering out 37180/37179, 10881/10880, 196625/196608, 3025269/3025000, 492804/492745 and 131769/131750. Using the 2.3.5.13.17.37.43 subgroup, it tempers out 7956/7955.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.021 | -0.000 | +0.171 | -0.053 | -0.064 | +0.013 | -0.102 | +0.110 | +0.116 | -0.019 |
| Relative (%) | +0.0 | -5.9 | -0.0 | +48.0 | -15.0 | -17.8 | +3.7 | -28.4 | +30.8 | +32.6 | -5.4 | |
| Steps (reduced) |
3361 (0) |
5327 (1966) |
7804 (1082) |
9436 (2714) |
11627 (1544) |
12437 (2354) |
13738 (294) |
14277 (833) |
15204 (1760) |
16328 (2884) |
16651 (3207) | |
Subsets and supersets
3361edo is the 474th prime edo. 6722edo, which doubles it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-5327 3361⟩ | [⟨3361 5327]] | 0.0066 | 0.0066 | 1.85 |
| 2.3.5 | [37 25 -33⟩, [-176 92 13⟩ | [⟨3361 5327 7804]] | 0.0044 | 0.0063 | 1.76 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 1570\3361 | 560.547 | 864/625 | Whoosh |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct