231edo: Difference between revisions

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== 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.

231et [[tempering out|tempers out]] the kleisma, [[15625/15552]] in the [[5-limit]], and [[1029/1024]] in the [[7-limit]], 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 ===

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=== Subsets and supersets ===

231 = 3 × 7 × 11, ~~with~~ subset edos {{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.

Since 231 factors into primes as {{nowrap| 3 × 7 × 11 }}, 231edo has subset edos {{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. [[1848edo]], which divides its step into eight, provides a near-just representation of the 11-limit.

== Regular temperament properties ==

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| 15625/15552, {{monzo| -64 36 3 }}

| {{mapping| 231 366 536 }}

| 0.410

| +0.410

| 0.334

| 6.43

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| 1029/1024, 15625/15552, 823543/820125

| {{mapping| 231 366 536 648 }}

| 0.539

| +0.539

| 0.365

| 7.01

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| 385/384, 441/440, 4000/3993, 823543/820125

| {{mapping| 231 366 536 648 799 }}

| 0.469

| +0.469

| 0.354

| 6.81

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| [[Tritikleismic]]

|}

<nowiki />* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if ~~it is~~ distinct

<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Music ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|231}}
+{{ED intro}}
 == 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.
+et [[tempering out|tempers out]] the kleisma, [[15625/15552]] in the [[5-limit]], and [[1029/1024]] in the [[7-limit]], 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.
-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.
+years is the number of years in a 41 out of 231 leap week cycle, which corresponds to a {{nowrap|41 &amp; 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 ===
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-= 3 × 7 × 11, with subset edos {{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.
+Since 231 factors into primes as {{nowrap| 3 × 7 × 11 }}, 231edo has subset edos {{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. [[1848edo]], which divides its step into eight, provides a near-just representation of the 11-limit.
 == Regular temperament properties ==
@@ Line 28: / Line 28: @@
 | 15625/15552, {{monzo| -64 36 3 }}
 | {{mapping| 231 366 536 }}
-| 0.410
+| +0.410
 | 0.334
 | 6.43
@@ Line 35: / Line 35: @@
 | 1029/1024, 15625/15552, 823543/820125
 | {{mapping| 231 366 536 648 }}
-| 0.539
+| +0.539
 | 0.365
 | 7.01
@@ Line 42: / Line 42: @@
 | 385/384, 441/440, 4000/3993, 823543/820125
 | {{mapping| 231 366 536 648 799 }}
-| 0.469
+| +0.469
 | 0.354
 | 6.81
@@ Line 99: / Line 99: @@
 | [[Tritikleismic]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Music ==