1381edo
| ← 1380edo | 1381edo | 1382edo → |
1381 equal divisions of the octave (abbreviated 1381edo or 1381ed2), also called 1381-tone equal temperament (1381tet) or 1381 equal temperament (1381et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1381 equal parts of about 0.869 ¢ each. Each step represents a frequency ratio of 21/1381, or the 1381st root of 2.
Theory
1381edo is consistent to the 9-odd-limit, tempering out 2401/2400, 29360128/29296875 and [33 -37 5 5⟩ in the 7-limit. It is strong in the 2.3.7.23.29 subgroup, tempering out 60817408/60761421, 5888/5887, 121025149/120932352 and 661153497088/660379746861. Using the 2.3.7.23.37 subgroup, it tempers out 1702/1701. The equal temperament supports quasiorwell.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.145 | +0.363 | +0.037 | +0.290 | -0.413 | -0.267 | -0.361 | +0.186 | -0.337 | +0.182 | -0.034 |
| Relative (%) | +16.7 | +41.7 | +4.3 | +33.4 | -47.5 | -30.7 | -41.6 | +21.4 | -38.8 | +21.0 | -3.9 | |
| Steps (reduced) |
2189 (808) |
3207 (445) |
3877 (1115) |
4378 (235) |
4777 (634) |
5110 (967) |
5395 (1252) |
5645 (121) |
5866 (342) |
6066 (542) |
6247 (723) | |
Subsets and supersets
1381edo is the 221st prime edo. 2762edo, which doubles it, gives a good correction to the harmonic 11.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [2189 -1381⟩ | [⟨1381 2189]] | −0.0457 | 0.0457 | 5.26 |
| 2.3.5 | [-16 35 -17⟩, [93 -3 -38⟩ | [⟨1381 2189 3207]] | −0.0825 | 0.0641 | 7.38 |
| 2.3.5.7 | 2401/2400, 29360128/29296875, [33 -37 5 5⟩ | [⟨1381 2189 3207 3877]] | −0.0652 | 0.0631 | 7.26 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 210\1381 | 182.476 | 10/9 | Minortone |
| 1 | 312\1381 | 271.108 | 1024/875 | Quasiorwell |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct