436edo - 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"

2026-03-13T13:33:04Z

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 13:33, 13 March 2026
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	\| [[Quadrant]]		\| [[Quadrant]]
	\|}		\|}
	<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

Lériendil: Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"

2025-04-17T02:29:14Z

Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"

← Older revision		Revision as of 02:29, 17 April 2025
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	\| 498.17		\| 498.17
	\| 4/3		\| 4/3
	\| [[Helmholtz temperament\|Helmholtz]]		\| [[Helmholtz (temperament)\|Helmholtz]]
	\|-		\|-
	\| 4		\| 4

Sintel: fix helmholtz link

2025-04-16T15:40:17Z

fix helmholtz link

← Older revision		Revision as of 15:40, 16 April 2025
Line 99:		Line 99:
	\| 498.17		\| 498.17
	\| 4/3		\| 4/3
	\| [[Helmholtz]]		\| [[Helmholtz temperament\|Helmholtz]]
	\|-		\|-
	\| 4		\| 4

ArrowHead294 at 16:36, 20 February 2025

2025-02-20T16:36:03Z

← Older revision		Revision as of 16:36, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|436~~}}		{{ED intro}}

	== Theory ==		== Theory ==
	436edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, [[harmonic]]s from 3 to 37 are all flat.		436edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, [[harmonic]]s from 3 to 37 are all flat.

	~~The equal temperament~~ [[tempering out\|tempers out]] [[32805/32768]] and {{monzo\| 1 -68 46 }} in the 5-limit; [[390625/388962]], 420175/419904, and [[2100875/2097152]] in the 7-limit; 1375/1372, [[6250/6237]], [[41503/41472]], and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], [[1275/1274]], [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit. It [[support]]s and gives a good tuning to [[quadrant]]. It also supports [[tsaharuk]], but [[171edo]] is better suited for that purpose.		It [[tempering out\|tempers out]] [[32805/32768]] and {{monzo\| 1 -68 46 }} in the 5-limit; [[390625/388962]], 420175/419904, and [[2100875/2097152]] in the 7-limit; 1375/1372, [[6250/6237]], [[41503/41472]], and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], [[1275/1274]], [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit. It [[support]]s and gives a good tuning to [[quadrant]]. It also supports [[tsaharuk]], but [[171edo]] is better suited for that purpose.

	436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]].		436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]].

ArrowHead294 at 16:47, 15 January 2025

2025-01-15T16:47:35Z

← Older revision		Revision as of 16:47, 15 January 2025
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	\| [[Quadrant]]		\| [[Quadrant]]
	\|}		\|}
	<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

ArrowHead294: Partial undo

2024-11-16T12:21:47Z

Partial undo

← Older revision		Revision as of 12:21, 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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	\| 0.1443		\| 0.1443
	\| 5.24		\| 5.24
	~~{{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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	\| 4/3<br>(35/33)		\| 4/3<br>(35/33)
	\| [[Quadrant]]		\| [[Quadrant]]
	~~{{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

ArrowHead294 at 04:56, 16 November 2024

2024-11-16T04:56:23Z

← Older revision		Revision as of 04:56, 16 November 2024
Line 18:		Line 18:

	== 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		\| 2.3
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	\| 0.1443		\| 0.1443
	\| 5.24		\| 5.24
	\|}		{{comma basis end}}

	=== 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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	\| 4/3<br>(35/33)		\| 4/3<br>(35/33)
	\| [[Quadrant]]		\| [[Quadrant]]
	\|}		{{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}}

FloraC: Rework; cleanup; clarify the title row of the rank-2 temp table

2023-11-03T08:09:07Z

Rework; cleanup; clarify the title row of the rank-2 temp table

← Older revision		Revision as of 08:09, 3 November 2023
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	== Theory ==		== Theory ==
	436edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, ~~harmonics~~ from 3 to 37 are all flat. It tempers out [[32805/32768]] and {{monzo\| 1 -68 46 }} in the 5-limit; [[390625/388962]], 420175/419904, and [[2100875/2097152]] in the 7-limit; 1375/1372, [[6250/6237]], [[41503/41472]], and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], 1275/1274, [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit.		436edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, [[harmonic]]s from 3 to 37 are all flat.

	436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]].		The equal temperament [[tempering out\|tempers out]] [[32805/32768]] and {{monzo\| 1 -68 46 }} in the 5-limit; [[390625/388962]], 420175/419904, and [[2100875/2097152]] in the 7-limit; 1375/1372, [[6250/6237]], [[41503/41472]], and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], [[1275/1274]], [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit. It [[support]]s and gives a good tuning to [[quadrant]]. It also supports [[tsaharuk]], but [[171edo]] is better suited for that purpose.

			436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]].

	=== Prime harmonics ===		=== Prime harmonics ===
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 436 factors into ~~2<sup>2</sup> × 109~~, 436edo has subset edos {{EDOs\| 2, 4, 109, and 218 }}.		Since 436 factors into {{factorization\|436}}, 436edo has subset edos {{EDOs\| 2, 4, 109, and 218 }}.

	[[1308edo]], which divides ~~the~~ edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit.		[[1308edo]], which divides its edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit.

	== Regular temperament properties ==		== Regular temperament properties ==
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	\| 2.3		\| 2.3
	\| {{monzo\| -691 436 }}		\| {{monzo\| -691 436 }}
	\| [{{~~val~~\| 436 691 }}]		\| {{mapping\| 436 691 }}
	\| +0.0379		\| +0.0379
	\| 0.0379		\| 0.0379
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	\| 2.3.5		\| 2.3.5
	\| 32805/32768, {{monzo\| 1 -68 46 }}		\| 32805/32768, {{monzo\| 1 -68 46 }}
	\| [{{~~val~~\| 436 691 1012 }}]		\| {{mapping\| 436 691 1012 }}
	\| +0.1678		\| +0.1678
	\| 0.1863		\| 0.1863
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	\| 2.3.5.7		\| 2.3.5.7
	\| 32805/32768, 390625/388962, 420175/419904		\| 32805/32768, 390625/388962, 420175/419904
	\| [{{~~val~~\| 436 691 1012 1224 }}]		\| {{mapping\| 436 691 1012 1224 }}
	\| +0.1275		\| +0.1275
	\| 0.1758		\| 0.1758
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 1375/1372, 6250/6237, 32805/32768, 41503/41472		\| 1375/1372, 6250/6237, 32805/32768, 41503/41472
	\| [{{~~val~~\| 436 691 1012 1224 1508 }}]		\| {{mapping\| 436 691 1012 1224 1508 }}
	\| +0.1517		\| +0.1517
	\| 0.1645		\| 0.1645
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 625/624, 1375/1372, 2080/2079, 10648/10647, 15379/15360		\| 625/624, 1375/1372, 2080/2079, 10648/10647, 15379/15360
	\| [{{~~val~~\| 436 691 1012 1224 1508 1613 }}]		\| {{mapping\| 436 691 1012 1224 1508 1613 }}
	\| +0.1749		\| +0.1749
	\| 0.1589		\| 0.1589
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 625/624, 715/714, 1089/1088, 1225/1224, 2431/2430, 10648/10647		\| 625/624, 715/714, 1089/1088, 1225/1224, 2431/2430, 10648/10647
	\| [{{~~val~~\| 436 691 1012 1224 1508 1613 1782 }}]		\| {{mapping\| 436 691 1012 1224 1508 1613 1782 }}
	\| +0.1628		\| +0.1628
	\| 0.1501		\| 0.1501
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	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 625/624, 715/714, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1729/1728		\| 625/624, 715/714, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1729/1728
	\| [{{~~val~~\| 436 691 1012 1224 1508 1613 1782 1852 }}]		\| {{mapping\| 436 691 1012 1224 1508 1613 1782 1852 }}
	\| +0.1503		\| +0.1503
	\| 0.1443		\| 0.1443
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	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! 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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	\| [[Quadrant]]		\| [[Quadrant]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

FloraC: Cleanup

2023-07-31T07:15:30Z

Cleanup

← Older revision		Revision as of 07:15, 31 July 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	~~The '''436 equal divisions of the octave''' ('''436edo'''), or the '''436(-tone) equal temperament''' ('''436tet''', '''436et''') when viewed from a [[regular temperament]] perspective, is the [[~~EDO\|~~equal division of the octave]] into~~ 436 ~~parts of about 2.75 [[cent]]s each.~~		{{EDO intro\|436}}

	== Theory ==		== Theory ==
	The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, harmonics from 3 to 37 are all flat. It ~~is consistent to the~~ [[~~23-odd-limit]], tempering out~~ 32805/32768 and {{monzo\| 1 -68 4 }} in the 5-limit; [[390625/388962]], 420175/419904, and 2100875/2097152 in the 7-limit; 1375/1372, 6250/6237, 41503/41472, and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], 1275/1274, [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit.		436edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, harmonics from 3 to 37 are all flat. It tempers out [[32805/32768]] and {{monzo\| 1 -68 46 }} in the 5-limit; [[390625/388962]], 420175/419904, and [[2100875/2097152]] in the 7-limit; 1375/1372, [[6250/6237]], [[41503/41472]], and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], 1275/1274, [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit.

	436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 subgroup.		436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 11:		Line 11:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	~~406edo~~ has subset edos {{EDOs\|2, 4, 109, 218}}.		Since 436 factors into 2<sup>2</sup> × 109, 436edo has subset edos {{EDOs\| 2, 4, 109, and 218 }}.

	[[1308edo]], which divides edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit.		[[1308edo]], which divides the edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit.

	== 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]] (¢)
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	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! Periods<br>per ~~octave~~		! Periods<br>per 8ve
	! Generator<br>(~~reduced~~)		! Generator<br>(Reduced)
	! Cents<br>(~~reduced~~)		! Cents<br>(Reduced)
	! Associated<br>~~ratio~~		! Associated<br>Ratio
	! Temperaments		! Temperaments
	\|-		\|-
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	\| [[Quadrant]]		\| [[Quadrant]]
	\|}		\|}

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

Eliora: /* Theory */

2023-07-30T14:49:30Z

Theory

← Older revision		Revision as of 14:49, 30 July 2023
Line 9:		Line 9:
	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|436}}		{{Harmonics in equal\|436}}

			=== Subsets and supersets ===
			406edo has subset edos {{EDOs\|2, 4, 109, 218}}.

			[[1308edo]], which divides edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit.

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