231edo - Revision history

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:32:32Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:32, 13 March 2026
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 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 ==		== Music ==

Overthink: /* Subsets and supersets */ note 1848edo

2025-12-24T01:09:47Z

Subsets and supersets: note 1848edo

← Older revision		Revision as of 01:09, 24 December 2025
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	=== Subsets and supersets ===		=== 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.		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. [[1848edo]], which divides its step into eight, provides a near-just representation of the 11-limit.

	== Regular temperament properties ==		== Regular temperament properties ==

ArrowHead294 at 14:20, 20 February 2025

2025-02-20T14:20:37Z

← Older revision		Revision as of 14:20, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|231~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 18:10, 15 January 2025

2025-01-15T18:10:15Z

← Older revision		Revision as of 18:10, 15 January 2025
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	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 ===
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	\| 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
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	\| [[Tritikleismic]]		\| [[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 lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

ArrowHead294: Partial undo

2024-11-16T12:40:30Z

Partial undo

← Older revision		Revision as of 12:40, 16 November 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[Comma list]]
			! rowspan="2" \| [[Mapping]]
			! rowspan="2" \| Optimal<br />8ve stretch (¢)
			! colspan="2" \| Tuning error
			\|-
			! [[TE error\|Absolute]] (¢)
			! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\| 2.3.5		\| 2.3.5
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	\| 0.354		\| 0.354
	\| 6.81		\| 6.81
	~~{{comma basis end}~~}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{rank-2 ~~begin}}~~		{\| class="wikitable center-all left-5"
			\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
			\|-
			! Periods<br />per 8ve
			! Generator*
			! Cents*
			! Associated<br />ratio*
			! Temperaments
	\|-		\|-
	\| 1		\| 1
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	\| 6/5<br />(21/20)		\| 6/5<br />(21/20)
	\| [[Tritikleismic]]		\| [[Tritikleismic]]
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Music ==		== Music ==

ArrowHead294 at 04:11, 16 November 2024

2024-11-16T04:11:59Z

← Older revision		Revision as of 04:11, 16 November 2024
Line 14:		Line 14:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal <br> 8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3.5		\| 2.3.5
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	\| 0.354		\| 0.354
	\| 6.81		\| 6.81
	\|}		{{comma basis end}}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center~~-~~all left-5"~~		{{rank-2 begin}}
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
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	\|-		\|-
	\| 3		\| 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]]		\| [[Tritikleismic]]
	\|}		{{rank-2 end}}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

	== Music ==		== Music ==
Line 102:		Line 89:
	* [https://www.youtube.com/watch?v=-bgUQ5BYnqM ''Sins of Stoicism''] (Demo Version, March 2022)		* [https://www.youtube.com/watch?v=-bgUQ5BYnqM ''Sins of Stoicism''] (Demo Version, March 2022)

			[[Category:Listen]]
	[[Category:Tritikleismic]]		[[Category:Tritikleismic]]
	~~[[Category:Listen]]~~

FloraC: Cleanup

2024-03-26T10:02:49Z

Cleanup

← Older revision		Revision as of 10:02, 26 March 2024
Line 3:		Line 3:

	== 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.		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 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 = 3 × 7 × 11, with subset 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.~~

	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|231}}		{{Harmonics in equal\|231}}

			=== 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.

	== Regular temperament properties ==		== Regular temperament properties ==
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	\| 2.3.5		\| 2.3.5
	\| 15625/15552, {{monzo\| -64 36 3 }}		\| 15625/15552, {{monzo\| -64 36 3 }}
	\| [{{~~val~~\| 231 366 536 }}]		\| {{mapping\| 231 366 536 }}
	\| 0.410		\| 0.410
	\| 0.334		\| 0.334
Line 32:		Line 33:
	\| 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 }}]		\| {{mapping\| 231 366 536 648 }}
	\| 0.539		\| 0.539
	\| 0.365		\| 0.365
Line 39:		Line 40:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 385/384, 441/440, 4000/3993, 823543/820125		\| 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		\| 0.354
Line 48:		Line 49:
	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	! Periods<br>per 8ve		! Periods<br>per 8ve
	! Generator~~<br>(Reduced)~~		! Generator*
	! Cents~~<br>(Reduced)~~		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
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	\| [[Tritikleismic]]		\| [[Tritikleismic]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

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

	~~[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->~~
	[[Category:Tritikleismic]]		[[Category:Tritikleismic]]
	[[Category:Listen]]		[[Category:Listen]]

FloraC: /* Regular temperament properties */ +superlimmal

2022-11-18T10:40:13Z

Regular temperament properties: +superlimmal

← Older revision		Revision as of 10:40, 18 November 2022
Line 14:		Line 14:
	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	! rowspan="2" \| Subgroup		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list\|Comma List]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \| Optimal <br> 8ve ~~stretch~~ (¢)		! rowspan="2" \| Optimal <br> 8ve Stretch (¢)
	! colspan="2" \| Tuning ~~error~~		! colspan="2" \| Tuning Error
	\|-		\|-
	! [[TE error\|Absolute]] (¢)		! [[TE error\|Absolute]] (¢)
Line 47:		Line 47:
	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	! Periods <br> per ~~octave~~		! Periods<br>per 8ve
	! Generator		! Generator<br>(Reduced)
	! Cents		! Cents<br>(Reduced)
	! Associated <br> ~~ratio~~		! Associated<br>Ratio
	! Temperaments		! Temperaments
			\|-
			\| 1
			\| 26\231
			\| 135.06
			\| 27/25
			\| [[Superlimmal]]
	\|-		\|-
	\| 1		\| 1

FloraC: Fix

2022-11-15T18:43:43Z

Fix

← Older revision		Revision as of 18:43, 15 November 2022
Line 89:		Line 89:
	\| [[Tritikleismic]]		\| [[Tritikleismic]]
	\|}		\|}

	~~[[Category:Equal divisions of the octave\|###]]~~

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

	* [~~https~~:~~//www.youtube.com/watch?v=-bgUQ5BYnqM Sins~~ of ~~Stoicism (Demo Version, March 2022)] by [[Mercury Amalgam~~]]<!-- 3-digit number -->		[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->

	[[Category:Tritikleismic]]		[[Category:Tritikleismic]]
	[[Category:Listen]]		[[Category:Listen]]

Eliora: added music

2022-11-15T16:54:40Z

added music

← Older revision		Revision as of 16:54, 15 November 2022
Line 90:		Line 90:
	\|}		\|}

	[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->		[[Category:Equal divisions of the octave\|###]]

			== Music ==

			* [https://www.youtube.com/watch?v=-bgUQ5BYnqM Sins of Stoicism (Demo Version, March 2022)] by [[Mercury Amalgam]]<!-- 3-digit number -->

	[[Category:Tritikleismic]]		[[Category:Tritikleismic]]
			[[Category:Listen]]