251edo
| ← 250edo | 251edo | 252edo → |
251 equal divisions of the octave (abbreviated 251edo or 251ed2), also called 251-tone equal temperament (251tet) or 251 equal temperament (251et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 251 equal parts of about 4.78 ¢ each. Each step represents a frequency ratio of 21/251, or the 251st root of 2.
Theory
251et tempers out 1600000/1594323 (amity comma) and [49 -6 -17⟩ (maquila comma) in the 5-limit; 4375/4374, 5120/5103, and 40500000/40353607 in the 7-limit, supporting amity, supermajor, and acrokleismic.
Using the patent val ⟨251 398 583 705 868], it tempers out 1331/1323, 1375/1372, 16896/16807, and 24057/24010 in the 11-limit; 352/351, 676/675, 847/845, and 1573/1568 in the 13-limit.
Using the 251e val ⟨251 398 583 705 869], it tempers out 540/539, 5632/5625, 6250/6237, and 12005/11979 in the 11-limit; 364/363, 676/675, 1716/1715, and 3584/3575 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.83 | +0.94 | +1.69 | +1.67 | -1.52 | +0.91 | +1.77 | +0.22 | -1.10 | -2.26 | -1.98 |
| Relative (%) | +17.4 | +19.6 | +35.4 | +34.9 | -31.7 | +19.0 | +37.0 | +4.7 | -23.0 | -47.2 | -41.4 | |
| Steps (reduced) |
398 (147) |
583 (81) |
705 (203) |
796 (43) |
868 (115) |
929 (176) |
981 (228) |
1026 (22) |
1066 (62) |
1102 (98) |
1135 (131) | |
Subsets and supersets
251edo is the 54th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [398 -251⟩ | [⟨251 398]] | −0.2630 | 0.2630 | 5.50 |
| 2.3.5 | [9 -13 5⟩, [49 -6 -17⟩ | [⟨251 398 583]] | −0.3099 | 0.2247 | 4.70 |
| 2.3.5.7 | 4375/4374, 5120/5103, 40500000/40353607 | [⟨251 398 583 705]] | −0.3830 | 0.2322 | 4.86 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 66\251 | 315.54 | 6/5 | Acrokleismic |
| 1 | 71\251 | 339.44 | 243/200 | Amity |
| 1 | 91\251 | 435.06 | 9/7 | Supermajor |
| 1 | 96\251 | 458.96 | 125/96 | Majvam |
| 1 | 112\251 | 535.46 | 512/375 | Maquila |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct