231edo
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.
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) | |
Subsets and supersets
231 = 3 × 7 × 11, with subset edos 3, 7, 11, 21, 33, and 77. Since it contains 77edo, it can be used for playing such a tuning of the Carlos Alpha scale.
Regular temperament properties
Template:Comma basis begin |- | 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 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 1
| 26\231
| 135.06
| 27/25
| Superlimmal
|-
| 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
Template:Rank-2 end
Template:Orf
Music
- Sins of Stoicism (Demo Version, March 2022)