2684edo - 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"

2026-03-13T13:31:56Z

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 13:31, 13 March 2026
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	\| [[Promethium]]		\| [[Promethium]]
	\|}		\|}
	<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

ArrowHead294 at 14:26, 20 February 2025

2025-02-20T14:26:46Z

← Older revision		Revision as of 14:26, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|2684~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 19:25, 15 January 2025

2025-01-15T19:25:54Z

← Older revision		Revision as of 19:25, 15 January 2025
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	\| 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608		\| 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608
	\| {{mapping\| 2684 4254 6232 7535 9825 9932 10971 }}		\| {{mapping\| 2684 4254 6232 7535 9825 9932 10971 }}
	\| -0.0004		\| −0.0004
	\| 0.0136		\| 0.0136
	\| 3.04		\| 3.04
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	\| [[Promethium]]		\| [[Promethium]]
	\|}		\|}
	<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

2024-11-16T12:20:37Z

Partial undo

← Older revision		Revision as of 12:20, 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]] (%)
	\|-		\|-
	\| 2.3.5.7		\| 2.3.5.7
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	\| 0.0150		\| 0.0150
	\| 3.36		\| 3.36
	~~{{comma basis end}~~}		\|}
	* 2684et holds a record for the lowest relative error in the 13-limit, past [[2190edo\|2190]] and is only bettered by [[5585edo\|5585]], which is more than twice its size. In terms of absolute error, it is narrowly beaten by [[3395edo\|3395]].		* 2684et holds a record for the lowest relative error in the 13-limit, past [[2190edo\|2190]] and is only bettered by [[5585edo\|5585]], which is more than twice its size. In terms of absolute error, it is narrowly beaten by [[3395edo\|3395]].
	* 2684et is also notable in the 11-limit, where it has the lowest absolute error, past [[1848edo\|1848]] and before 3395.		* 2684et is also notable in the 11-limit, where it has the lowest absolute error, past [[1848edo\|1848]] and before 3395.
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	Note: 5-limit temperaments supported by [[1342edo]] are not included.		Note: 5-limit temperaments supported by [[1342edo]] are not included.

	{{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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	\| 11907/6875<br />(?)		\| 11907/6875<br />(?)
	\| [[Promethium]]		\| [[Promethium]]
	~~{{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 04:59, 16 November 2024

2024-11-16T04:59:21Z

← Older revision		Revision as of 04:59, 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<br>8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3.5.7		\| 2.3.5.7
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	\| 0.0150		\| 0.0150
	\| 3.36		\| 3.36
	\|}		{{comma basis end}}
	* 2684et holds a record for the lowest relative error in the 13-limit, past [[2190edo\|2190]] and is only bettered by [[5585edo\|5585]], which is more than twice its size. In terms of absolute error, it is narrowly beaten by [[3395edo\|3395]].		* 2684et holds a record for the lowest relative error in the 13-limit, past [[2190edo\|2190]] and is only bettered by [[5585edo\|5585]], which is more than twice its size. In terms of absolute error, it is narrowly beaten by [[3395edo\|3395]].
	* 2684et is also notable in the 11-limit, where it has the lowest absolute error, past [[1848edo\|1848]] and before 3395.		* 2684et is also notable in the 11-limit, where it has the lowest absolute error, past [[1848edo\|1848]] and before 3395.
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	Note: 5-limit temperaments supported by [[1342edo]] are not included.		Note: 5-limit temperaments supported by [[1342edo]] are not included.

	{~~\| 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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	\|-		\|-
	\| 44		\| 44
	\| 1114\2684<br>(16\2684)		\| 1114\2684<br />(16\2684)
	\| 498.063<br>(7.154)		\| 498.063<br />(7.154)
	\| 4/3<br>(18375/18304)		\| 4/3<br />(18375/18304)
	\| [[Ruthenium]]		\| [[Ruthenium]]
	\|-		\|-
	\| 61		\| 61
	\| 557\2684<br>(29\2684)		\| 557\2684<br />(29\2684)
	\| 249.031<br>(12.965)		\| 249.031<br />(12.965)
	\| 11907/6875<br>(?)		\| 11907/6875<br />(?)
	\| [[Promethium]]		\| [[Promethium]]
	\|}		{{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: Adopt template: Factorization

2023-10-30T10:24:03Z

Adopt template: Factorization

← Older revision		Revision as of 10:24, 30 October 2023
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	== Theory ==		== Theory ==
	2684edo is an extremely strong 13-limit system, with a lower 13-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any division until we reach [[5585edo]]. It is ~~distinctly~~ [[consistent]] through the [[17-odd-limit]], and is both a [[~~The Riemann~~ zeta ~~function and tuning #Zeta EDO lists~~\|zeta peak and zeta integral edo]]. It is [[enfactoring\|enfactored]] in the 2.3.5.13 subgroup, with the same tuning as [[1342edo]], tempering out kwazy, {{monzo\| -53 10 16 }}, senior, {{monzo\| -17 62 -35 }} and egads, {{monzo\| -36 52 51 }}. A 13-limit [[comma basis]] is {[[9801/9800]], [[10648/10647]], 140625/140608, 196625/196608, 823680/823543}; it also tempers out [[123201/123200]]. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {4914/4913, [[5832/5831]], 9801/9800, 10648/10647, [[28561/28560]], 140625/140608}.		2684edo is an extremely strong 13-limit system, with a lower 13-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any division until we reach [[5585edo]]. It is [[consistency\|distinctly consistent]] through the [[17-odd-limit]], and is both a [[zeta edo\|zeta peak and zeta integral edo]]. It is [[enfactoring\|enfactored]] in the 2.3.5.13 [[subgroup]], with the same tuning as [[1342edo]], [[tempering out]] kwazy, {{monzo\| -53 10 16 }}, senior, {{monzo\| -17 62 -35 }} and egads, {{monzo\| -36 52 51 }}. A 13-limit [[comma basis]] is {[[9801/9800]], [[10648/10647]], 140625/140608, 196625/196608, 823680/823543}; it also tempers out [[123201/123200]]. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {[[4914/4913]], [[5832/5831]], 9801/9800, 10648/10647, [[28561/28560]], 140625/140608}.

	In higher limits, 2684edo is a good no-19s 31-limit tuning, with errors of 25% or less on all harmonics (except 19).		In higher limits, 2684edo is a good no-19s 31-limit tuning, with errors of 25% or less on all harmonics (except 19).
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 2684 factors ~~as 2<sup>2</sup> × 11 × 61~~, 2684edo has subset edos {{EDOs\| 2, 4, 11, 22, 44, 61, 122, 244, 671, and 1342 }}.		Since 2684 factors into {{factorization\|2684}}, 2684edo has subset edos {{EDOs\| 2, 4, 11, 22, 44, 61, 122, 244, 671, and 1342 }}.

	2684edo tunes the septimal comma, 64/63, to an exact 1/44th of the octave (61 steps). As a corollary, it supports the period-44 [[ruthenium]] temperament.		2684edo tunes the septimal comma, 64/63, to an exact 1/44th of the octave (61 steps). As a corollary, it supports the period-44 [[ruthenium]] temperament.
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	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
			\|+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
	\|-		\|-

FloraC: /* Regular temperament properties */ 11-limit notability

2023-10-15T12:21:01Z

Regular temperament properties: 11-limit notability

← Older revision		Revision as of 12:21, 15 October 2023
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	\| 2.3.5.7		\| 2.3.5.7
	\| 78125000/78121827, {{monzo\| -5 10 5 -8 }}, {{monzo\| -48 0 11 8 }}		\| 78125000/78121827, {{monzo\| -5 10 5 -8 }}, {{monzo\| -48 0 11 8 }}
	\| [{{~~val~~\| 2684 4254 6232 7535 }}]		\| {{mapping\| 2684 4254 6232 7535 }}
	\| +0.0030		\| +0.0030
	\| 0.0085		\| 0.0085
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 9801/9800, 1771561/1771470, 35156250/35153041, 67110351/67108864		\| 9801/9800, 1771561/1771470, 35156250/35153041, 67110351/67108864
	\| [{{~~val~~\| 2684 4254 6232 7535 9825 }}]		\| {{mapping\| 2684 4254 6232 7535 9825 }}
	\| +0.0089		\| +0.0089
	\| 0.0089		\| 0.0089
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543		\| 9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543
	\| [{{~~val~~\| 2684 4254 6232 7535 9825 9932 }}]		\| {{mapping\| 2684 4254 6232 7535 9825 9932 }}
	\| +0.0041		\| +0.0041
	\| 0.0086		\| 0.0086
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608		\| 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608
	\| [{{~~val~~\| 2684 4254 6232 7535 9825 9932 10971 }}]		\| {{mapping\| 2684 4254 6232 7535 9825 9932 10971 }}
	\| -0.0004		\| -0.0004
	\| 0.0136		\| 0.0136
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	\| 2.3.5.7.11.13.17.23		\| 2.3.5.7.11.13.17.23
	\| 4761/4760, 4914/4913, 5832/5831, 8625/8624, 9801/9800, 10648/10647, 28561/28560		\| 4761/4760, 4914/4913, 5832/5831, 8625/8624, 9801/9800, 10648/10647, 28561/28560
	\| [{{~~val~~\| 2684 4254 6232 7535 9825 9932 10971 12141 }}]		\| {{mapping\| 2684 4254 6232 7535 9825 9932 10971 12141 }}
	\| +0.0026		\| +0.0026
	\| 0.0150		\| 0.0150
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	\|}		\|}
	* 2684et holds a record for the lowest relative error in the 13-limit, past [[2190edo\|2190]] and is only bettered by [[5585edo\|5585]], which is more than twice its size. In terms of absolute error, it is narrowly beaten by [[3395edo\|3395]].		* 2684et holds a record for the lowest relative error in the 13-limit, past [[2190edo\|2190]] and is only bettered by [[5585edo\|5585]], which is more than twice its size. In terms of absolute error, it is narrowly beaten by [[3395edo\|3395]].
			* 2684et is also notable in the 11-limit, where it has the lowest absolute error, past [[1848edo\|1848]] and before 3395.

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===

FloraC: Clarify the title row of the rank-2 temp table

2023-10-15T11:58:29Z

Clarify the title row of the rank-2 temp table

← Older revision		Revision as of 11:58, 15 October 2023
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	== Theory ==		== Theory ==
	2684edo is an extremely strong 13-limit system, with a lower 13-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any division until we reach [[5585edo]]. It is distinctly [[consistent]] through the [[17-odd-limit]], and is both a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak and zeta integral edo]]. It is [[enfactoring\|enfactored]] in the 2.3.5.13 subgroup, with the same tuning as [[1342edo]], tempering out kwazy, {{monzo\| -53 10 16 }}, senior, {{monzo\| -17 62 -35 }} and egads, {{monzo\| -36 52 51 }}. A 13-limit [[comma basis]] is {[[9801/9800]], [[10648/10647]], 140625/140608, 196625/196608, 823680/823543}; it also tempers out 123201/123200. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {4914/4913, [[5832/5831]], 9801/9800, 10648/10647, 28561/28560, 140625/140608}.		2684edo is an extremely strong 13-limit system, with a lower 13-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any division until we reach [[5585edo]]. It is distinctly [[consistent]] through the [[17-odd-limit]], and is both a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak and zeta integral edo]]. It is [[enfactoring\|enfactored]] in the 2.3.5.13 subgroup, with the same tuning as [[1342edo]], tempering out kwazy, {{monzo\| -53 10 16 }}, senior, {{monzo\| -17 62 -35 }} and egads, {{monzo\| -36 52 51 }}. A 13-limit [[comma basis]] is {[[9801/9800]], [[10648/10647]], 140625/140608, 196625/196608, 823680/823543}; it also tempers out [[123201/123200]]. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {4914/4913, [[5832/5831]], 9801/9800, 10648/10647, [[28561/28560]], 140625/140608}.

	In higher limits, 2684edo is a good no-19s 31-limit tuning, with errors of 25% or less on all harmonics (except 19).		In higher limits, 2684edo is a good no-19s 31-limit tuning, with errors of 25% or less on all harmonics (except 19).
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	{\| 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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	\| [[Promethium]]		\| [[Promethium]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

BudjarnLambeth at 07:02, 9 July 2023

2023-07-09T07:02:38Z

← Older revision		Revision as of 07:02, 9 July 2023
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	~~{{novelty}}~~{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|2684}}		{{EDO intro\|2684}}

BudjarnLambeth at 04:35, 9 July 2023

2023-07-09T04:35:52Z

← Older revision		Revision as of 04:35, 9 July 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{novelty}}{{Infobox ET}}
	{{EDO intro\|2684}}		{{EDO intro\|2684}}