581edo - Revision history

FloraC: /* Regular temperament properties */ countritonic -> garitritonic

2026-05-20T11:46:34Z

Regular temperament properties: countritonic -> garitritonic

← Older revision		Revision as of 11:46, 20 May 2026
Line 130:		Line 130:
	\| 285\581		\| 285\581
	\| 588.64		\| 588.64
	\| ~~{{Monzo\| -14 15 -4 }}~~		\| 351/250
	\| [[~~Countritonic~~]] ~~(5-limit)~~		\| [[Garitritonic]]
	\|}		\|}
	<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		<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:Newt]]		[[Category:Newt]]

Overthink: /* Theory */ note monotonicity limit

2026-03-29T00:46:56Z

Theory: note monotonicity limit

← Older revision		Revision as of 00:46, 29 March 2026
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	As an equal temperament, it [[tempering out\|tempers out]] [[2401/2400]], [[33554432/33480783]], and [[48828125/48771072]] in the 7-limit; [[3025/3024]], [[19712/19683]], [[234375/234256]] in the 11-limit; [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the {{nowrap\| 270 & 311 }} microtemperament, which features a neutral-third generator.		As an equal temperament, it [[tempering out\|tempers out]] [[2401/2400]], [[33554432/33480783]], and [[48828125/48771072]] in the 7-limit; [[3025/3024]], [[19712/19683]], [[234375/234256]] in the 11-limit; [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the {{nowrap\| 270 & 311 }} microtemperament, which features a neutral-third generator.

			It notably achieves [[diamond monotone]] in the 71-odd-limit with the 581jks [[val]] (s is the [[wart]] for prime 67), which is a large improvement from the previous record of the 59-odd-limit, held by [[571edo]] using its [[patent val]].

	=== Prime harmonics ===		=== Prime harmonics ===

FloraC: Expand

2026-03-28T14:34:22Z

Expand

← Older revision		Revision as of 14:34, 28 March 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	581edo is a very strong 19- and 23-limit system, [[consistency\|distinctly consistent]] to the [[25-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], ~~151263~~/~~151250~~ in the 11-limit~~, and~~ [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the {{nowrap\|270 & 311}} microtemperament, which features a neutral-third generator.		581edo is a very strong [[17-limit\|17-]], [[19-limit\|19-]] and [[23-limit]] system, [[consistency\|distinctly consistent]] to the [[25-odd-limit]], and except for [[27/23]] and its [[octave complement]], it is consistent to the [[27-odd-limit]].

			As an equal temperament, it [[tempering out\|tempers out]] [[2401/2400]], [[33554432/33480783]], and [[48828125/48771072]] in the 7-limit; [[3025/3024]], [[19712/19683]], [[234375/234256]] in the 11-limit; [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the {{nowrap\| 270 & 311 }} microtemperament, which features a neutral-third generator.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|581}}		{{Harmonics in equal\|581\|columns=11}}
			{{Harmonics in equal\|581\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 581edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 581 factors into 7 × 83, 581edo contains [[7edo]] and [[83edo]] as subsets.		Since 581 factors into primes as {{nowrap\| 7 × 83 }}, 581edo contains [[7edo]] and [[83edo]] as subsets.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 17:		Line 20:
	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[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
	\|-		\|-
Line 24:		Line 27:
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| 921 -581 }}		\| {{Monzo\| 921 -581 }}
	\| {{~~mapping~~\| 581 921 }}		\| {{Mapping\| 581 921 }}
	\| −0.0891		\| −0.0891
	\| 0.0891		\| 0.0891
Line 31:		Line 34:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| -29 -11 20 }}, {{monzo\| 33 -34 9 }}		\| {{Monzo\| -29 -11 20 }}, {{monzo\| 33 -34 9 }}
	\| {{~~mapping~~\| 581 921 1349 }}		\| {{Mapping\| 581 921 1349 }}
	\| −0.0475		\| −0.0475
	\| 0.0936		\| 0.0936
Line 39:		Line 42:
	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 33554432/33480783, 48828125/48771072		\| 2401/2400, 33554432/33480783, 48828125/48771072
	\| {{~~mapping~~\| 581 921 1349 1631 }}		\| {{Mapping\| 581 921 1349 1631 }}
	\| −0.0222		\| −0.0222
	\| 0.0922		\| 0.0922
Line 46:		Line 49:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 2401/2400, 3025/3024, 19712/19683, 234375/234256		\| 2401/2400, 3025/3024, 19712/19683, 234375/234256
	\| {{~~mapping~~\| 581 921 1349 1631 2010 }}		\| {{Mapping\| 581 921 1349 1631 2010 }}
	\| −0.0261		\| −0.0261
	\| 0.0828		\| 0.0828
Line 53:		Line 56:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078		\| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078
	\| {{~~mapping~~\| 581 921 1349 1631 2010 2150 }}		\| {{Mapping\| 581 921 1349 1631 2010 2150 }}
	\| −0.0259		\| −0.0259
	\| 0.0756		\| 0.0756
Line 60:		Line 63:
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913		\| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913
	\| {{~~mapping~~\| 581 921 1349 1631 2010 2150 2375 }}		\| {{Mapping\| 581 921 1349 1631 2010 2150 2375 }}
	\| −0.0355		\| −0.0355
	\| 0.0738		\| 0.0738
Line 67:		Line 70:
	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913		\| 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913
	\| {{~~mapping~~\| 581 921 1349 1631 2010 2150 2375 2468 }}		\| {{Mapping\| 581 921 1349 1631 2010 2150 2375 2468 }}
	\| −0.0283		\| −0.0283
	\| 0.0717		\| 0.0717
Line 74:		Line 77:
	\| 2.3.5.7.11.13.17.19.23		\| 2.3.5.7.11.13.17.19.23
	\| 1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079		\| 1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079
	\| {{~~mapping~~\| 581 921 1349 1631 2010 2150 2375 2468 2628 }}		\| {{Mapping\| 581 921 1349 1631 2010 2150 2375 2468 2628 }}
	\| −0.0155		\| −0.0155
	\| 0.0800		\| 0.0800
Line 86:		Line 89:
	\|+ 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
	\|-		\|-
Line 101:		Line 104:
	\| 64\581		\| 64\581
	\| 132.19		\| 132.19
	\| {{~~monzo~~\| -38 5 13 }}		\| {{Monzo\| -38 5 13 }}
	\| [[Astro]]		\| [[Astro]]
	\|-		\|-
Line 125:		Line 128:
	\| 285\581		\| 285\581
	\| 588.64		\| 588.64
	\| {{~~monzo~~\| -14 15 -4 }}		\| {{Monzo\| -14 15 -4 }}
	\| [[Countritonic]] (5-limit)		\| [[Countritonic]] (5-limit)
	\|}		\|}
	<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		<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:Newt]]		[[Category:Newt]]

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"

2026-03-13T13:32:13Z

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 128:		Line 128:
	\| [[Countritonic]] (5-limit)		\| [[Countritonic]] (5-limit)
	\|}		\|}
	<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:Newt]]		[[Category:Newt]]

ArrowHead294 at 15:37, 20 February 2025

2025-02-20T15:37:53Z

← Older revision		Revision as of 15:37, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|581~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T16:27:18Z

Theory: cleanup

← Older revision		Revision as of 16:27, 17 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	581edo is a very strong 19- and 23-limit system, [[consistency\|distinctly consistent]] to the [[25-odd-limit]]. It [[tempers out]] [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the {{nowrap\|270 &~~amp;~~ 311}} microtemperament, which features a neutral-third generator.		581edo is a very strong 19- and 23-limit system, [[consistency\|distinctly consistent]] to the [[25-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the {{nowrap\|270 & 311}} microtemperament, which features a neutral-third generator.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|581~~\|columns=11~~}}		{{Harmonics in equal\|581}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 581 factors into ~~{{factorization\|581}}~~, 581edo contains [[7edo]] and [[83edo]] as subsets.		Since 581 factors into 7 × 83, 581edo contains [[7edo]] and [[83edo]] as subsets.

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

ArrowHead294 at 19:55, 15 January 2025

2025-01-15T19:55:56Z

← Older revision		Revision as of 19:55, 15 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	581edo is a very strong 19- and 23-limit system, [[consistency\|distinctly consistent]] to the [[25-odd-limit]]. ~~The equal temperament~~ [[~~tempering out\|~~tempers out]] [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the 270 & 311 microtemperament, which features a neutral-third generator.		581edo is a very strong 19- and 23-limit system, [[consistency\|distinctly consistent]] to the [[25-odd-limit]]. It [[tempers out]] [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the {{nowrap\|270 & 311}} microtemperament, which features a neutral-third generator.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 26:		Line 26:
	\| {{monzo\| 921 -581 }}		\| {{monzo\| 921 -581 }}
	\| {{mapping\| 581 921 }}		\| {{mapping\| 581 921 }}
	\| ~~−0~~.0891		\| −0.0891
	\| 0.0891		\| 0.0891
	\| 4.32		\| 4.32
Line 33:		Line 33:
	\| {{monzo\| -29 -11 20 }}, {{monzo\| 33 -34 9 }}		\| {{monzo\| -29 -11 20 }}, {{monzo\| 33 -34 9 }}
	\| {{mapping\| 581 921 1349 }}		\| {{mapping\| 581 921 1349 }}
	\| ~~−0~~.0475		\| −0.0475
	\| 0.0936		\| 0.0936
	\| 4.53		\| 4.53
Line 40:		Line 40:
	\| 2401/2400, 33554432/33480783, 48828125/48771072		\| 2401/2400, 33554432/33480783, 48828125/48771072
	\| {{mapping\| 581 921 1349 1631 }}		\| {{mapping\| 581 921 1349 1631 }}
	\| ~~−0~~.0222		\| −0.0222
	\| 0.0922		\| 0.0922
	\| 4.46		\| 4.46
Line 47:		Line 47:
	\| 2401/2400, 3025/3024, 19712/19683, 234375/234256		\| 2401/2400, 3025/3024, 19712/19683, 234375/234256
	\| {{mapping\| 581 921 1349 1631 2010 }}		\| {{mapping\| 581 921 1349 1631 2010 }}
	\| ~~−0~~.0261		\| −0.0261
	\| 0.0828		\| 0.0828
	\| 4.01		\| 4.01
Line 54:		Line 54:
	\| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078		\| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078
	\| {{mapping\| 581 921 1349 1631 2010 2150 }}		\| {{mapping\| 581 921 1349 1631 2010 2150 }}
	\| ~~−0~~.0259		\| −0.0259
	\| 0.0756		\| 0.0756
	\| 3.66		\| 3.66
Line 61:		Line 61:
	\| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913		\| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913
	\| {{mapping\| 581 921 1349 1631 2010 2150 2375 }}		\| {{mapping\| 581 921 1349 1631 2010 2150 2375 }}
	\| ~~−0~~.0355		\| −0.0355
	\| 0.0738		\| 0.0738
	\| 3.58		\| 3.58
Line 68:		Line 68:
	\| 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913		\| 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913
	\| {{mapping\| 581 921 1349 1631 2010 2150 2375 2468 }}		\| {{mapping\| 581 921 1349 1631 2010 2150 2375 2468 }}
	\| ~~−0~~.0283		\| −0.0283
	\| 0.0717		\| 0.0717
	\| 3.47		\| 3.47
Line 75:		Line 75:
	\| 1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079		\| 1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079
	\| {{mapping\| 581 921 1349 1631 2010 2150 2375 2468 2628 }}		\| {{mapping\| 581 921 1349 1631 2010 2150 2375 2468 2628 }}
	\| ~~−0~~.0155		\| −0.0155
	\| 0.0800		\| 0.0800
	\| 3.87		\| 3.87
Line 128:		Line 128:
	\| [[Countritonic]] (5-limit)		\| [[Countritonic]] (5-limit)
	\|}		\|}
	<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:Newt]]		[[Category:Newt]]

ArrowHead294: Partial undo

2024-11-16T12:19:01Z

Partial undo

← Older revision		Revision as of 12:19, 16 November 2024
Line 12:		Line 12:

	== 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
Line 69:		Line 78:
	\| 0.0800		\| 0.0800
	\| 3.87		\| 3.87
	~~{{comma basis end}~~}		\|}
	* 581et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 19- and 23-limit. It is the first after [[270edo\|270]] with a lower 19-limit relative error, and the first after [[311edo\|311]] with a lower 23-limit relative error. It is only bettered by [[742edo\|742]] in terms of either 19-limit absolute error or 19-limit relative error, by [[718edo\|718]] in terms of 23-limit absolute error, and not until [[1578edo\|1578]] do we reach a lower 23-limit relative error.		* 581et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 19- and 23-limit. It is the first after [[270edo\|270]] with a lower 19-limit relative error, and the first after [[311edo\|311]] with a lower 23-limit relative error. It is only bettered by [[742edo\|742]] in terms of either 19-limit absolute error or 19-limit relative error, by [[718edo\|718]] in terms of 23-limit absolute error, and not until [[1578edo\|1578]] do we reach a lower 23-limit relative error.
	* 581et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past [[494edo\|494]] and followed by 742.		* 581et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past [[494edo\|494]] and followed by 742.

	=== 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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	\| {{monzo\| -14 15 -4 }}		\| {{monzo\| -14 15 -4 }}
	\| [[Countritonic]] (5-limit)		\| [[Countritonic]] (5-limit)
	~~{{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:Newt]]		[[Category:Newt]]

ArrowHead294 at 05:03, 16 November 2024

2024-11-16T05:03:52Z

← Older revision		Revision as of 05:03, 16 November 2024
Line 12:		Line 12:

	== 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
	\| {{monzo\| 921 -581 }}		\| {{monzo\| 921 -581 }}
	\| {{mapping\| 581 921 }}		\| {{mapping\| 581 921 }}
	\| -0.0891		\| −0.0891
	\| 0.0891		\| 0.0891
	\| 4.32		\| 4.32
Line 32:		Line 24:
	\| {{monzo\| -29 -11 20 }}, {{monzo\| 33 -34 9 }}		\| {{monzo\| -29 -11 20 }}, {{monzo\| 33 -34 9 }}
	\| {{mapping\| 581 921 1349 }}		\| {{mapping\| 581 921 1349 }}
	\| -0.0475		\| −0.0475
	\| 0.0936		\| 0.0936
	\| 4.53		\| 4.53
Line 39:		Line 31:
	\| 2401/2400, 33554432/33480783, 48828125/48771072		\| 2401/2400, 33554432/33480783, 48828125/48771072
	\| {{mapping\| 581 921 1349 1631 }}		\| {{mapping\| 581 921 1349 1631 }}
	\| -0.0222		\| −0.0222
	\| 0.0922		\| 0.0922
	\| 4.46		\| 4.46
Line 46:		Line 38:
	\| 2401/2400, 3025/3024, 19712/19683, 234375/234256		\| 2401/2400, 3025/3024, 19712/19683, 234375/234256
	\| {{mapping\| 581 921 1349 1631 2010 }}		\| {{mapping\| 581 921 1349 1631 2010 }}
	\| -0.0261		\| −0.0261
	\| 0.0828		\| 0.0828
	\| 4.01		\| 4.01
Line 53:		Line 45:
	\| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078		\| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078
	\| {{mapping\| 581 921 1349 1631 2010 2150 }}		\| {{mapping\| 581 921 1349 1631 2010 2150 }}
	\| -0.0259		\| −0.0259
	\| 0.0756		\| 0.0756
	\| 3.66		\| 3.66
Line 60:		Line 52:
	\| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913		\| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913
	\| {{mapping\| 581 921 1349 1631 2010 2150 2375 }}		\| {{mapping\| 581 921 1349 1631 2010 2150 2375 }}
	\| -0.0355		\| −0.0355
	\| 0.0738		\| 0.0738
	\| 3.58		\| 3.58
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	\| 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913		\| 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913
	\| {{mapping\| 581 921 1349 1631 2010 2150 2375 2468 }}		\| {{mapping\| 581 921 1349 1631 2010 2150 2375 2468 }}
	\| -0.0283		\| −0.0283
	\| 0.0717		\| 0.0717
	\| 3.47		\| 3.47
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	\| 1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079		\| 1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079
	\| {{mapping\| 581 921 1349 1631 2010 2150 2375 2468 2628 }}		\| {{mapping\| 581 921 1349 1631 2010 2150 2375 2468 2628 }}
	\| -0.0155		\| −0.0155
	\| 0.0800		\| 0.0800
	\| 3.87		\| 3.87
	\|}		{{comma basis end}}
	* 581et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 19- and 23-limit. It is the first after [[270edo\|270]] with a lower 19-limit relative error, and the first after [[311edo\|311]] with a lower 23-limit relative error. It is only bettered by [[742edo\|742]] in terms of either 19-limit absolute error or 19-limit relative error, by [[718edo\|718]] in terms of 23-limit absolute error, and not until [[1578edo\|1578]] do we reach a lower 23-limit relative error.		* 581et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 19- and 23-limit. It is the first after [[270edo\|270]] with a lower 19-limit relative error, and the first after [[311edo\|311]] with a lower 23-limit relative error. It is only bettered by [[742edo\|742]] in terms of either 19-limit absolute error or 19-limit relative error, by [[718edo\|718]] in terms of 23-limit absolute error, and not until [[1578edo\|1578]] do we reach a lower 23-limit relative error.
	* 581et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past [[494edo\|494]] and followed by 742.		* 581et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past [[494edo\|494]] and followed by 742.

	=== 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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	\| {{monzo\| -14 15 -4 }}		\| {{monzo\| -14 15 -4 }}
	\| [[Countritonic]] (5-limit)		\| [[Countritonic]] (5-limit)
	\|}		{{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:Newt]]		[[Category:Newt]]

FloraC: +subsets and supersets; misc. cleanup

2023-11-02T13:38:28Z

+subsets and supersets; misc. cleanup

← Older revision		Revision as of 13:38, 2 November 2023
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	== Theory ==		== Theory ==
	581edo is a very strong 19- and 23-limit system, [[consistency\|distinctly consistent]] to the [[25-odd-limit]]. It tempers out [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the 270 & 311 microtemperament, which features a neutral-third generator.		581edo is a very strong 19- and 23-limit system, [[consistency\|distinctly consistent]] to the [[25-odd-limit]]. The equal temperament [[tempering out\|tempers out]] [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the 270 & 311 microtemperament, which features a neutral-third generator.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|581\|columns=11}}		{{Harmonics in equal\|581\|columns=11}}

			=== Subsets and supersets ===
			Since 581 factors into {{factorization\|581}}, 581edo contains [[7edo]] and [[83edo]] as subsets.

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