231edo
| ← 230edo | 231edo | 232edo → |
The 231 equal divisions of the octave (231edo), or the 231(-tone) equal temperament (231tet, 231et) when viewed from a regular temperament perspective, divides the octave into 231 equal parts of about 5.19 cents each.
Theory
In the 5-limit, 231et tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it supports the tritikleismic temperament, and in fact provides the optimal patent val. In the 11-limit it tempers out 385/384, 441/440 and 4000/3993, leading to 11-limit tritikleismic for which it also gives the optimal patent val.
231 years is the number of years in a 41 out of 231 leap week cycle, which corresponds to a 41 & 149 temperament tempering out 132055/131072, 166375/165888, and 2460375/2458624. This type of solar calendar leap rule scale may actually be of more use to harmony, since a 41 note subset mimics 41edo, a rather useful edo harmonically, and it preserves the simple commas mentioned above.
Since the patent val mapping of fifth in 231edo is divisible by 9, it can be used for playing the Carlos Alpha scale.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.66 | -1.90 | -2.59 | -1.31 | -0.67 | +1.03 | -2.55 | -1.06 | -1.41 | +1.95 | +0.30 |
| Relative (%) | -12.6 | -36.5 | -49.9 | -25.3 | -12.9 | +19.8 | -49.2 | -20.4 | -27.1 | +37.5 | +5.7 | |
| Steps (reduced) |
366 (135) |
536 (74) |
648 (186) |
732 (39) |
799 (106) |
855 (162) |
902 (209) |
944 (20) |
981 (57) |
1015 (91) |
1045 (121) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 15625/15552, [-64 36 3⟩ | [⟨231 366 536]] | 0.410 | 0.334 | 6.43 |
| 2.3.5.7 | 1029/1024, 15625/15552, 823543/820125 | [⟨231 366 536 648]] | 0.539 | 0.365 | 7.01 |
| 2.3.5.7.11 | 385/384, 441/440, 4000/3993, 823543/820125 | [⟨231 366 536 648 799]] | 0.469 | 0.354 | 6.81 |
Rank-2 temperaments
| Periods per octave |
Generator | Cents | Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 27\231 | 140.26 | 243/224 | Septichrome |
| 1 | 45\231 | 233.77 | 8/7 | Slendric |
| 1 | 61\231 | 316.88 | 6/5 | Hanson |
| 1 | 62\231 | 322.08 | 135/112 | Dee leap week |
| 1 | 73\231 | 379.22 | 56/45 | Marthirds |
| 3 | 61\231 (16\231) |
316.88 (83.12) |
6/5 (21/20) |
Tritikleismic |