231edo: Difference between revisions

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{{Infobox ET

~~| Prime factorization = 3 × 7 × 11~~

~~| Step size = 5.19481¢~~

~~| Fifth = 135\231 (701.30¢) (→ [[77edo|45\77]])~~

~~| Semitones = 21:18 (109.09¢ : 93.51¢)~~

~~| Consistency = 11~~

}}

The '''231 equal divisions of the octave''' ('''231edo'''), or the '''231(-tone) equal temperament''' ('''231tet''', '''231et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 231 [[equal]] parts of about 5.19 [[cent]]s each.

== 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.

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.

~~Since the patent val mapping of fifth in 231edo is divisible by 9, it can be used for playing the [[Carlos Alpha]] scale~~.

=== Odd harmonics ===

=== Subsets and supersets ===

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 ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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| 2.3.5

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

| [{{~~val~~| 231 366 536 }}]

| {{mapping| 231 366 536 }}

| 0.410

| +0.410

| 0.334

| 6.43

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| 2.3.5.7

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

| [{{~~val~~| 231 366 536 648 }}]

| {{mapping| 231 366 536 648 }}

| 0.539

| +0.539

| 0.365

| 7.01

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| 2.3.5.7.11

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

| [{{~~val~~| 231 366 536 648 799 }}]

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

| 0.469

| +0.469

| 0.354

| 6.81

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=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

! Periods per ~~octave~~

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Generator

|-

! Cents

! Periods per 8ve

! Associated ratio

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 26\231

| 135.06

| 27/25

| [[Superlimmal]]

|-

| 1

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|-

| 3

| 61\231 (16\231)

| 61\231 (16\231)

| 316.88 (83.12)

| 316.88 (83.12)

| 6/5 (21/20)

| 6/5 (21/20)

| [[Tritikleismic]]

|}

<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 ==

; [[Mercury Amalgam]]

* [https://www.youtube.com/watch?v=-bgUQ5BYnqM ''Sins of Stoicism''] (Demo Version, March 2022)

[[Category:~~Equal divisions of the octave~~]]

[[Category:Listen]]

[[Category:Tritikleismic]]

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 3 × 7 × 11
+{{ED intro}}
-| Step size = 5.19481¢
-| Fifth = 135\231 (701.30¢) (→ [[77edo|45\77]])
-| Semitones = 21:18 (109.09¢ : 93.51¢)
-| Consistency = 11
-}}
-The '''231 equal divisions of the octave''' ('''231edo'''), or the '''231(-tone) equal temperament''' ('''231tet''', '''231et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 231 [[equal]] parts of about 5.19 [[cent]]s each.
 == 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.
+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.
-Since the patent val mapping of fifth in 231edo is divisible by 9, it can be used for playing the [[Carlos Alpha]] scale.
 === Odd harmonics ===
 {{Harmonics in equal|231}}
+=== Subsets and supersets ===
+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 ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal <br> 8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 31: / Line 27: @@
 | 2.3.5
 | 15625/15552, {{monzo| -64 36 3 }}
-| [{{val| 231 366 536 }}]
+| {{mapping| 231 366 536 }}
-| 0.410
+| +0.410
 | 0.334
 | 6.43
@@ Line 38: / Line 34: @@
 | 2.3.5.7
 | 1029/1024, 15625/15552, 823543/820125
-| [{{val| 231 366 536 648 }}]
+| {{mapping| 231 366 536 648 }}
-| 0.539
+| +0.539
 | 0.365
 | 7.01
@@ Line 45: / Line 41: @@
 | 2.3.5.7.11
 | 385/384, 441/440, 4000/3993, 823543/820125
-| [{{val| 231 366 536 648 799 }}]
+| {{mapping| 231 366 536 648 799 }}
-| 0.469
+| +0.469
 | 0.354
 | 6.81
@@ Line 53: / Line 49: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-! Periods <br> per octave
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Generator
+|-
-! Cents
+! Periods<br />per 8ve
-! Associated <br> ratio
+! Generator*
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
+|-
+| 1
+| 26\231
+| 135.06
+| 27/25
+| [[Superlimmal]]
 |-
 | 1
@@ Line 90: / Line 94: @@
 |-
 | 3
-| 61\231<br>(16\231)
+| 61\231<br />(16\231)
-| 316.88<br>(83.12)
+| 316.88<br />(83.12)
-| 6/5<br>(21/20)
+| 6/5<br />(21/20)
 | [[Tritikleismic]]
 |}
+<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 ==
+; [[Mercury Amalgam]]
+* [https://www.youtube.com/watch?v=-bgUQ5BYnqM ''Sins of Stoicism''] (Demo Version, March 2022)
-[[Category:Equal divisions of the octave]]
+[[Category:Listen]]
 [[Category:Tritikleismic]]