231edo: Difference between revisions
Cleanup, +links, +category, and misc |
+more temperaments |
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| Line 49: | Line 49: | ||
! Associated <br> ratio | ! Associated <br> ratio | ||
! Temperaments | ! Temperaments | ||
|- | |||
| 1 | |||
| 27\231 | |||
| 140.26 | |||
| 243/224 | |||
| [[Septichrome]] | |||
|- | |||
| 1 | |||
| 45\231 | |||
| 233.77 | |||
| 8/7 | |||
| [[Slendric]] | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 54: | Line 66: | ||
| 322.08 | | 322.08 | ||
| 135/112 | | 135/112 | ||
| Dee leap week | | [[Dee leap week]] | ||
|- | |||
| 1 | |||
| 73\231 | |||
| 379.22 | |||
| 56/45 | |||
| [[Marthirds]] | |||
|- | |- | ||
| 3 | | 3 | ||
| 61\231<br>(16\231) | | 61\231<br>(16\231) | ||
| 316.88<br>(83.12) | | 316.88<br>(83.12) | ||
| 6/5 | | 6/5<br>(21/20) | ||
| | | [[Tritikleismic]] | ||
|} | |} | ||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:Tritikleismic]] | [[Category:Tritikleismic]] | ||
Revision as of 09:44, 17 March 2022
The 231 equal temperament divides the octave into 231 equal parts of 5.195 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.
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 | 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 |