401edo
| ← 400edo | 401edo | 402edo → |
401 equal divisions of the octave (abbreviated 401edo or 401ed2), also called 401-tone equal temperament (401tet) or 401 equal temperament (401et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 401 equal parts of about 2.99 ¢ each. Each step represents a frequency ratio of 21/401, or the 401st root of 2.
Theory
401edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise, it has a reasonable approximation to harmonics 5, 7, 9, 11, and 13, making it suitable for a 2.9.5.7.11.13 subgroup interpretation.
Using the patent val nonetheless, the equal temperament tempers out 283115520/282475249 and 703125/702464 in the 7-limit; 35156250/35153041, 2097152/2096325, 117649/117612, 226492416/226474325, 9765625/9732096, 42875/42768, 1375/1372, 5632/5625, 15488/15435, 202397184/201768035, 102487/102400 and 805255/802816 in the 11-limit. It provides the optimal patent val for diatessic, the 140 & 261 temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.29 | -0.28 | +0.75 | -0.42 | -0.69 | +0.37 | +1.01 | -0.22 | -1.25 | -0.96 | +0.15 |
| Relative (%) | +43.0 | -9.3 | +25.1 | -14.0 | -23.2 | +12.4 | +33.7 | -7.3 | -41.9 | -31.9 | +5.2 | |
| Steps (reduced) |
636 (235) |
931 (129) |
1126 (324) |
1271 (68) |
1387 (184) |
1484 (281) |
1567 (364) |
1639 (35) |
1703 (99) |
1761 (157) |
1814 (210) | |
Subsets and supersets
401edo is the 79th prime edo
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [636 -401⟩ | [⟨401 636]] | −0.4060 | 0.4058 | 13.56 |
| 2.3.5 | 15625/15552, [107 -66 -1⟩ | [⟨401 636 931]] | −0.2307 | 0.4139 | 13.83 |
| 2.3.5.7 | 10976/10935, 15625/15552, 67108864/66706983 | [⟨401 636 931 1126]] | −0.2400 | 0.3588 | 11.99 |
| 2.3.5.7.11 | 2200/2187, 1375/1372, 5632/5625, 102487/102400 | [⟨401 636 931 1126 1387 ]] | −0.1518 | 0.3661 | 12.23 |
| 2.3.5.7.11.13 | 325/324, 352/351, 625/624, 1375/1372, 3276800/3270267 | [⟨401 636 931 1126 1387 1484]] | −0.1431 | 0.3348 | 11.19 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 169\401 | 505.74 | 75/56 | Diatessic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Scales
Music
- Diana Tessa (2023) – diatessic in 403edo tuning