243edo - 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"

2026-03-13T13:38:09Z

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 13:38, 13 March 2026
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	\| [[Ennealimmal]]		\| [[Ennealimmal]]
	\|}		\|}
	<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

	[[Category:Ennealimmal]]		[[Category:Ennealimmal]]
	[[Category:Jove]]		[[Category:Jove]]

FloraC: /* Theory */ + octave stretch, a brief discussion

2025-08-12T20:01:07Z

Theory: + octave stretch, a brief discussion

← Older revision		Revision as of 20:01, 12 August 2025
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|243}}		{{Harmonics in equal\|243}}

			=== Octave stretch ===
			243edo can benefit from slightly [[stretched and compressed tuning\|stretching the octave]], using tunings such as [[385edt]] or [[628ed6]]. This improves most of the approximated harmonics, including the 11 if we use the 243e val.

	=== Subsets and supersets ===		=== Subsets and supersets ===

FloraC: /* Theory */ expand

2025-08-12T19:55:53Z

Theory: expand

← Older revision		Revision as of 19:55, 12 August 2025
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	== Theory ==		== Theory ==
	~~243et~~ [[~~tempering out~~\|~~tempers out~~]] ~~the~~ [[~~semicomma~~]] ~~(i.e~~. the 5-limit ~~orwell comma) 2109375~~/~~2097152 in the~~ 5~~-limit~~, ~~and~~ [[~~2401~~/~~2400~~]] and [[~~4375~~/~~4374~~]] ~~in the 7-limit~~.		243edo is a strong higher-limit system, especially if we skip [[prime harmonic\|prime]] [[11/1\|11]]. It is [[consistent]] to the no-11 [[29-odd-limit]] tending flat, with the [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], [[13/1\|13]], [[17/1\|17]], [[19/1\|19]], [[23/1\|23]], and [[29/1\|29]] all tuned flat.

	~~Using the~~ [[~~patent val]], it~~ tempers out ~~[[243/242~~]], [[~~441/440~~]], and [[~~540/539~~]] in the 11-limit, and ~~provides the~~ [[~~optimal patent val~~]] ~~for the~~ [[~~Ragismic microtemperaments #Ennealimmal\|ennealimnic~~]] ~~temperament. In~~ the 13-limit ~~it tempers out~~ [[~~364/363~~]], [[~~625/624~~]], [[~~729/728]], and [[2080/2079~~]], and ~~provides the optimal temperament for 13-limit ennealimnic and the rank-3~~ [[~~Breed family #Jovial\|jovial~~]] ~~temperament, and in the 17-limit it tempers out 375/374 and 595/594 and provides the optimal patent val for 17-limit ennealimnic~~.		As an equal temperament, it [[tempering out\|tempers out]] the [[semicomma]] (2109375/2097152, the 5-limit orwell comma) and the [[ennealimma]] in the 5-limit, and [[2401/2400]] and [[4375/4374]] in the 7-limit. It [[support]]s [[ennealimmal]], [[quadrawell]], and [[sabric]].

	Using the alternative val 243e {{val\| 241 385 564 682 '''840''' }}, with an lower error, it tempers out [[385/384]], 1375/1372, [[8019/8000]], and [[14641/14580]], and in the 13-limit, 625/624, 729/728, [[847/845]], [[1001/1000]], and [[1716/1715]]. It provides a good tuning for [[fibo]].		Using the [[patent val]], it tempers out [[243/242]], [[441/440]], and [[540/539]] in the 11-limit, and provides the [[optimal patent val]] for the [[Ragismic microtemperaments #Ennealimmal\|ennealimnic]] temperament. In the 13-limit it tempers out [[364/363]], [[625/624]], [[729/728]], and [[2080/2079]], and provides the optimal temperament for 13-limit ennealimnic and the rank-3 [[Breed family #Jovial\|jovial]] temperament, and in the 17-limit it tempers out [[375/374]] and [[595/594]] and provides the optimal patent val for 17-limit ennealimnic.

			Using the alternative val 243e {{val\| 241 385 564 682 '''840''' }}, with an lower error, it tempers out [[385/384]], [[1375/1372]], [[8019/8000]], and [[14641/14580]], and in the 13-limit, 625/624, 729/728, [[847/845]], [[1001/1000]], and [[1716/1715]]. It provides a good tuning for [[fibo]].

	=== Prime harmonics ===		=== Prime harmonics ===
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 243 factors into ~~{{factorization\|243}}~~, 243edo has subset edos {{EDOs\| 3, 9, 27, and 81 }}.		Since 243 factors into primes as 3<sup>5</sup>, 243edo has subset edos {{EDOs\| 3, 9, 27, and 81 }}.

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

FloraC: /* Regular temperament properties */ + more data

2025-08-12T19:40:15Z

Regular temperament properties: + more data

@@ Line 47: / Line 47: @@
 | 0.192
 | 3.90
 |}
 * 243et (243e val) has lower absolute errors than any previous equal temperaments in the 19-, 23-limit, and somewhat beyond, despite inconsistency in the corresponding odd limits. In both the 19- and 23-limit, it beats [[217edo|217]] and is only bettered by [[270edo|270et]].

FloraC: /* Regular temperament properties */ corrections & additions

2025-08-12T19:05:34Z

Regular temperament properties: corrections & additions

← Older revision		Revision as of 19:05, 12 August 2025
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	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| -385 243 }}		\| {{Monzo\| -385 243 }}
	\| {{~~mapping~~\| 243 385 }}		\| {{Mapping\| 243 385 }}
	\| +0.227		\| +0.227
	\| 0.227		\| 0.227
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	\| 2.3.5		\| 2.3.5
	\| 2109375/2097152, {{monzo\| 1 -27 18 }}		\| 2109375/2097152, {{monzo\| 1 -27 18 }}
	\| {{~~mapping~~\| 243 385 564 }}		\| {{Mapping\| 243 385 564 }}
	\| +0.314		\| +0.314
	\| 0.222		\| 0.222
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	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 4375/4374, 2109375/2097152		\| 2401/2400, 4375/4374, 2109375/2097152
	\| {{~~mapping~~\| ~~241~~ 385 564 682 }}		\| {{Mapping\| 243 385 564 682 }}
	\| +0.318		\| +0.318
	\| 0.192		\| 0.192
	\| 3.90		\| 3.90
	\|}		\|}
	* 243et (243e val) has a lower absolute ~~error~~ than any previous equal temperaments in the 19-limit, despite inconsistency in the corresponding odd limit~~. The same subgroup~~ is only ~~better tuned~~ by [[270edo\|270et]]. It is much stronger in the no-11 19-limit~~, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is~~ [[354edo\|354et]] in terms of absolute error and [[935edo\|935et]] in terms of relative error.		* 243et (243e val) has lower absolute errors than any previous equal temperaments in the 19-, 23-limit, and somewhat beyond, despite inconsistency in the corresponding odd limits. In both the 19- and 23-limit, it beats [[217edo\|217]] and is only bettered by [[270edo\|270et]].
			* It is much stronger in the no-11 subgroups of the limits above, holding the record of lowest relative errors until being bettered in the no-11 19-limit by [[354edo\|354et]] in terms of absolute error and [[935edo\|935et]] in terms of relative error, and in the no-11 23-limit by [[422edo\|422]] in terms of absolute error and [[2460edo\|2460]] in terms of relative error.

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
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	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br />per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br />ratio*		! Associated<br>ratio*
	! Temperaments		! Temperaments
	\|-		\|-
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	\|-		\|-
	\| 9		\| 9
	\| 64\243<br />(10\243)		\| 64\243<br>(10\243)
	\| 316.05<br />(49.38)		\| 316.05<br>(49.38)
	\| 6/5<br />(36/35)		\| 6/5<br>(36/35)
	\| [[Ennealimmal]]		\| [[Ennealimmal]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal~~ lists\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[normal lists\|minimal form]] in parentheses if distinct

	[[Category:Ennealimmal]]		[[Category:Ennealimmal]]
	[[Category:Jove]]		[[Category:Jove]]

ArrowHead294 at 14:23, 20 February 2025

2025-02-20T14:23:29Z

← Older revision		Revision as of 14:23, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 18:42, 15 January 2025

2025-01-15T18:42:39Z

← Older revision		Revision as of 18:42, 15 January 2025
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	\| [[Ennealimmal]]		\| [[Ennealimmal]]
	\|}		\|}
	<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

	[[Category:Ennealimmal]]		[[Category:Ennealimmal]]
	[[Category:Jove]]		[[Category:Jove]]

ArrowHead294: Partial undo

2024-11-16T12:29:49Z

Partial undo

← Older revision		Revision as of 12:29, 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		\| 2.3
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	\| 0.192		\| 0.192
	\| 3.90		\| 3.90
	~~{{comma basis end}~~}		\|}
	* 243et (243e val) has a lower absolute error than any previous equal temperaments in the 19-limit, despite inconsistency in the corresponding odd limit. The same subgroup is only better tuned by [[270edo\|270et]]. It is much stronger in the no-11 19-limit, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is [[354edo\|354et]] in terms of absolute error and [[935edo\|935et]] in terms of relative error.		* 243et (243e val) has a lower absolute error than any previous equal temperaments in the 19-limit, despite inconsistency in the corresponding odd limit. The same subgroup is only better tuned by [[270edo\|270et]]. It is much stronger in the no-11 19-limit, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is [[354edo\|354et]] in terms of absolute error and [[935edo\|935et]] in terms of relative error.

	=== 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 />(36/35)		\| 6/5<br />(36/35)
	\| [[Ennealimmal]]		\| [[Ennealimmal]]
	~~{{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

	[[Category:Ennealimmal]]		[[Category:Ennealimmal]]
	[[Category:Jove]]		[[Category:Jove]]

ArrowHead294 at 04:34, 16 November 2024

2024-11-16T04:34:56Z

← Older revision		Revision as of 04:34, 16 November 2024
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	== 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 8ve <br>Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3		\| 2.3
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	\| 0.192		\| 0.192
	\| 3.90		\| 3.90
	\|}		{{comma basis end}}
	* 243et (243e val) has a lower absolute error than any previous equal temperaments in the 19-limit, despite inconsistency in the corresponding odd limit. The same subgroup is only better tuned by [[270edo\|270et]]. It is much stronger in the no-11 19-limit, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is [[354edo\|354et]] in terms of absolute error and [[935edo\|935et]] in terms of relative error.		* 243et (243e val) has a lower absolute error than any previous equal temperaments in the 19-limit, despite inconsistency in the corresponding odd limit. The same subgroup is only better tuned by [[270edo\|270et]]. It is much stronger in the no-11 19-limit, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is [[354edo\|354et]] in terms of absolute error and [[935edo\|935et]] in terms of relative error.

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
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	\|-		\|-
	\| 9		\| 9
	\| 64\243<br>(10\243)		\| 64\243<br />(10\243)
	\| 316.05<br>(49.38)		\| 316.05<br />(49.38)
	\| 6/5<br>(36/35)		\| 6/5<br />(36/35)
	\| [[Ennealimmal]]		\| [[Ennealimmal]]
	\|}		{{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}}

	[[Category:Ennealimmal]]		[[Category:Ennealimmal]]
	[[Category:Jove]]		[[Category:Jove]]

FloraC: /* Theory */ +subsets and supersets

2024-03-24T12:50:49Z

Theory: +subsets and supersets

← Older revision		Revision as of 12:50, 24 March 2024
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|243}}		{{Harmonics in equal\|243}}

			=== Subsets and supersets ===
			Since 243 factors into {{factorization\|243}}, 243edo has subset edos {{EDOs\| 3, 9, 27, and 81 }}.

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