231edo: Difference between revisions
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== Theory == | == Theory == | ||
In the 5-limit, 231et tempers out the [[kleisma]], 15625/15552, and in the 7-limit [[1029/1024]], so that it [[support]]s 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 === | |||
{{Harmonics in equal|231}} | {{Harmonics in equal|231}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 19: | Line 21: | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| 15625/15552, | | 15625/15552, {{monzo| -64 36 3 }} | ||
| [{{val|231 366 536}}] | | [{{val| 231 366 536 }}] | ||
| 0.410 | | 0.410 | ||
| 0.334 | | 0.334 | ||
| Line 27: | Line 29: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 1029/1024, 15625/15552, 823543/820125 | | 1029/1024, 15625/15552, 823543/820125 | ||
| [{{val|231 366 536 648}}] | | [{{val| 231 366 536 648 }}] | ||
| 0.539 | | 0.539 | ||
| 0.365 | | 0.365 | ||
| Line 33: | Line 35: | ||
|- | |- | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 385/384, 441/440, | | 385/384, 441/440, 4000/3993, 823543/820125 | ||
| [{{val|231 366 536 648 799}}] | | [{{val| 231 366 536 648 799 }}] | ||
| 0.469 | | 0.469 | ||
| 0.354 | | 0.354 | ||
| Line 40: | Line 42: | ||
|} | |} | ||
== Rank | == Rank-2 temperaments == | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
! Periods <br> per octave | ! Periods <br> per octave | ||
| Line 51: | Line 53: | ||
| 62\231 | | 62\231 | ||
| 322.08 | | 322.08 | ||
| | | 135/112 | ||
| Dee leap week | | Dee leap week | ||
|- | |- | ||
| 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 | ||
| Tritrikleismic | | Tritrikleismic | ||
|} | |} | ||
[[Category: | [[Category:Equal divisions of the octave]] | ||
[[Category:Tritikleismic]] | |||
Revision as of 09:36, 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 | 62\231 | 322.08 | 135/112 | Dee leap week |
| 3 | 61\231 (16\231) |
316.88 (83.12) |
6/5 | Tritrikleismic |