231edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
In the 5-limit, 231et [[tempering out|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. | In the 5-limit, 231et [[tempering out|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. | 231 years is the number of years in a 41 out of 231 leap week cycle, which corresponds to a {{nowrap|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 === | === Odd harmonics === | ||
| Line 28: | Line 28: | ||
| 15625/15552, {{monzo| -64 36 3 }} | | 15625/15552, {{monzo| -64 36 3 }} | ||
| {{mapping| 231 366 536 }} | | {{mapping| 231 366 536 }} | ||
| 0.410 | | +0.410 | ||
| 0.334 | | 0.334 | ||
| 6.43 | | 6.43 | ||
| Line 35: | Line 35: | ||
| 1029/1024, 15625/15552, 823543/820125 | | 1029/1024, 15625/15552, 823543/820125 | ||
| {{mapping| 231 366 536 648 }} | | {{mapping| 231 366 536 648 }} | ||
| 0.539 | | +0.539 | ||
| 0.365 | | 0.365 | ||
| 7.01 | | 7.01 | ||
| Line 42: | Line 42: | ||
| 385/384, 441/440, 4000/3993, 823543/820125 | | 385/384, 441/440, 4000/3993, 823543/820125 | ||
| {{mapping| 231 366 536 648 799 }} | | {{mapping| 231 366 536 648 799 }} | ||
| 0.469 | | +0.469 | ||
| 0.354 | | 0.354 | ||
| 6.81 | | 6.81 | ||
| Line 99: | Line 99: | ||
| [[Tritikleismic]] | | [[Tritikleismic]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Music == | == Music == | ||