460edo - 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"

2026-03-13T13:32:18Z

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
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	\| [[Soviet ferris wheel]]		\| [[Soviet ferris wheel]]
	\|}		\|}
	<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 14:59, 20 February 2025

2025-02-20T14:59:57Z

← Older revision		Revision as of 14:59, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|460~~}}		{{ED intro}}

	== Theory ==		== Theory ==
	460edo is a very strong 19-limit system and is [[distinctly consistent]] to the [[21-odd-limit]], with [[harmonic]]s of 3 to 19 all tuned flat.		460edo is a very strong 19-limit system and is [[distinctly consistent]] to the [[21-odd-limit]], with [[harmonic]]s of 3 to 19 all tuned flat.

	~~The equal temperament~~ [[tempering out\|tempers out]] the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and [[65536/65625]] in the 7-limit, so that it [[support]]s [[pontiac]], the 171 & 289 temperament. In the 11-limit it tempers of [[3025/3024]] and [[9801/9800]], and 43923/43904; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]], so that it supports [[deca]], the 190 & 270 temperament; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2058/2057]], [[2431/2430]], [[2601/2600]] and [[4914/4913]]; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank-5 temperament tempering out 833/832 and 1001/1000.		It [[tempering out\|tempers out]] the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and [[65536/65625]] in the 7-limit, so that it [[support]]s [[pontiac]], the {{nowrap\|171 & 289}} temperament. In the 11-limit it tempers of [[3025/3024]] and [[9801/9800]], and 43923/43904; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]], so that it supports [[deca]], the {{nowrap\|190 & 270}} temperament; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2058/2057]], [[2431/2430]], [[2601/2600]] and [[4914/4913]]; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank-5 temperament tempering out 833/832 and 1001/1000.

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

ArrowHead294 at 19:52, 15 January 2025

2025-01-15T19:52:49Z

← Older revision		Revision as of 19:52, 15 January 2025
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	== Theory ==		== Theory ==
	460edo is a very strong 19-limit system and is [[~~consistency\|~~distinctly consistent]] to the [[21-odd-limit]], with [[harmonic]]s of 3 to 19 all tuned flat.		460edo is a very strong 19-limit system and is [[distinctly consistent]] to the [[21-odd-limit]], with [[harmonic]]s of 3 to 19 all tuned flat.

	The equal temperament [[tempering out\|tempers out]] the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and [[65536/65625]] in the 7-limit, so that it [[support]]s [[pontiac]], the 171 & 289 temperament. In the 11-limit it tempers of [[3025/3024]] and [[9801/9800]], and 43923/43904; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]], so that it supports [[deca]], the 190 & 270 temperament; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2058/2057]], [[2431/2430]], [[2601/2600]] and [[4914/4913]]; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank-5 temperament tempering out 833/832 and 1001/1000.		The equal temperament [[tempering out\|tempers out]] the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and [[65536/65625]] in the 7-limit, so that it [[support]]s [[pontiac]], the 171 & 289 temperament. In the 11-limit it tempers of [[3025/3024]] and [[9801/9800]], and 43923/43904; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]], so that it supports [[deca]], the 190 & 270 temperament; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2058/2057]], [[2431/2430]], [[2601/2600]] and [[4914/4913]]; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank-5 temperament tempering out 833/832 and 1001/1000.
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	\| [[Soviet ferris wheel]]		\| [[Soviet ferris wheel]]
	\|}		\|}
	<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

2024-11-16T12:18:38Z

Partial undo

← Older revision		Revision as of 12:18, 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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	\| 0.1082		\| 0.1082
	\| 4.15		\| 4.15
	~~{{comma basis end}~~}		\|}
	* 460et has lower absolute errors in the 17- and 19-limit than any previous equal temperaments. It beats [[422edo\|422]] in either subgroup, and is bettered by [[494edo\|494]] in the 17-limit, and [[525edo\|525]] in the 19-limit.		* 460et has lower absolute errors in the 17- and 19-limit than any previous equal temperaments. It beats [[422edo\|422]] in either subgroup, and is bettered by [[494edo\|494]] in the 17-limit, and [[525edo\|525]] in the 19-limit.

	=== 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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	\| 238/165<br />(?)		\| 238/165<br />(?)
	\| [[Soviet ferris wheel]]		\| [[Soviet ferris wheel]]
	~~{{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 05:05, 16 November 2024

2024-11-16T05:05:21Z

← Older revision		Revision as of 05:05, 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		\| 2.3
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	\| 0.1082		\| 0.1082
	\| 4.15		\| 4.15
	\|}		{{comma basis end}}
	* 460et has lower absolute errors in the 17- and 19-limit than any previous equal temperaments. It beats [[422edo\|422]] in either subgroup, and is bettered by [[494edo\|494]] in the 17-limit, and [[525edo\|525]] in the 19-limit.		* 460et has lower absolute errors in the 17- and 19-limit than any previous equal temperaments. It beats [[422edo\|422]] in either subgroup, and is bettered by [[494edo\|494]] in the 17-limit, and [[525edo\|525]] in the 19-limit.

	=== 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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	\|-		\|-
	\| 10		\| 10
	\| 121\460<br>(17\460)		\| 121\460<br />(17\460)
	\| 315.65<br>(44.35)		\| 315.65<br />(44.35)
	\| 6/5<br>(40/39)		\| 6/5<br />(40/39)
	\| [[Deca]]		\| [[Deca]]
	\|-		\|-
	\| 20		\| 20
	\| 66\460<br>(20\460)		\| 66\460<br />(20\460)
	\| 172.173<br>(52.173)		\| 172.173<br />(52.173)
	\| 169/153<br>(?)		\| 169/153<br />(?)
	\| [[Calcium]]		\| [[Calcium]]
	\|-		\|-
	\| 20		\| 20
	\| 217\460<br>(10\460)		\| 217\460<br />(10\460)
	\| 566.086<br>(26.086)		\| 566.086<br />(26.086)
	\| 238/165<br>(?)		\| 238/165<br />(?)
	\| [[Soviet ferris wheel]]		\| [[Soviet ferris wheel]]
	\|}		{{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: 17- and 19-limit notability

2023-11-06T16:23:46Z

17- and 19-limit notability

← Older revision		Revision as of 16:23, 6 November 2023
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	\| 4.15		\| 4.15
	\|}		\|}
			* 460et has lower absolute errors in the 17- and 19-limit than any previous equal temperaments. It beats [[422edo\|422]] in either subgroup, and is bettered by [[494edo\|494]] in the 17-limit, and [[525edo\|525]] in the 19-limit.

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

FloraC: Cleanup; clarify the title row of the rank-2 temp table; -"commatose" (requires 500+ gensteps), replaced with more notable temps

2023-11-03T07:01:49Z

Cleanup; clarify the title row of the rank-2 temp table; -"commatose" (requires 500+ gensteps), replaced with more notable temps

← Older revision		Revision as of 07:01, 3 November 2023
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	{{Infobox ET}}		{{Infobox ET}}
	~~The '''~~460 equal divisions of the octave''' ('''460edo'''), or the '''460(-tone) equal temperament''' ('''460tet''', '''460et''') when viewed from a [[regular temperament]] perspective, divides the octave into 460 equal parts of about 2.61 [[cent]]s each.		{{EDO intro\|460}}

	== Theory ==		== Theory ==
	460edo is a very strong 19-limit system and is ~~uniquely~~ [[consistent]] to the [[21-odd-limit]], with ~~harmonics~~ of 3 to 19 all tuned flat. It tempers out the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and 65536/65625 in the 7-limit, so that it [[support]]s [[pontiac]]. In the 11-limit it tempers of ~~43923/43904,~~ [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]]; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2058/2057]], 2431/2430, [[2601/2600]] and 4914/4913; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank-5 temperament tempering out 833/832 and 1001/1000~~. 460edo supports the 460 & [[1789edo\|1789]] temperament in the 2.9.5.7.11.13 subgroup called [[commatose]]~~.		460edo is a very strong 19-limit system and is [[consistency\|distinctly consistent]] to the [[21-odd-limit]], with [[harmonic]]s of 3 to 19 all tuned flat.

			The equal temperament [[tempering out\|tempers out]] the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and [[65536/65625]] in the 7-limit, so that it [[support]]s [[pontiac]], the 171 & 289 temperament. In the 11-limit it tempers of [[3025/3024]] and [[9801/9800]], and 43923/43904; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]], so that it supports [[deca]], the 190 & 270 temperament; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2058/2057]], [[2431/2430]], [[2601/2600]] and [[4914/4913]]; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank-5 temperament tempering out 833/832 and 1001/1000.

	=== Prime harmonics ===		=== Prime harmonics ===
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 460 factors into ~~2<sup>2</sup> × 5 × 23~~, 460edo has subset edos 2, 4, 5, 10, 20, 23, 46, 92, 115, and 230.		Since 460 factors into {{factorization\|460}}, 460edo has subset edos {{EDOs\| 2, 4, 5, 10, 20, 23, 46, 92, 115, and 230 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
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	\| 2.3		\| 2.3
	\| {{monzo\| -729 460 }}		\| {{monzo\| -729 460 }}
	\| [{{~~val~~\| 460 729 }}]		\| {{mapping\| 460 729 }}
	\| +0.0681		\| +0.0681
	\| 0.0681		\| 0.0681
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	\| 2.3.5		\| 2.3.5
	\| 32805/32768, {{monzo\| 6 68 -49 }}		\| 32805/32768, {{monzo\| 6 68 -49 }}
	\| [{{~~val~~\| 460 729 1068 }}]		\| {{mapping\| 460 729 1068 }}
	\| +0.0780		\| +0.0780
	\| 0.0573		\| 0.0573
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	\| 2.3.5.7		\| 2.3.5.7
	\| 4375/4374, 32805/32768, {{monzo\| -4 -2 -9 10 }}		\| 4375/4374, 32805/32768, {{monzo\| -4 -2 -9 10 }}
	\| [{{~~val~~\| 460 729 1068 1291 }}]		\| {{mapping\| 460 729 1068 1291 }}
	\| +0.1475		\| +0.1475
	\| 0.1303		\| 0.1303
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 3025/3024, 4375/4374, 32805/32768, 184877/184320		\| 3025/3024, 4375/4374, 32805/32768, 184877/184320
	\| [{{~~val~~\| 460 729 1068 1291 1591 }}]		\| {{mapping\| 460 729 1068 1291 1591 }}
	\| +0.1691		\| +0.1691
	\| 0.1243		\| 0.1243
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 1001/1000, 3025/3024, 4225/4224, 4375/4374, 26411/26364		\| 1001/1000, 3025/3024, 4225/4224, 4375/4374, 26411/26364
	\| [{{~~val~~\| 460 729 1068 1291 1591 1702 }}]		\| {{mapping\| 460 729 1068 1291 1591 1702 }}
	\| +0.1647		\| +0.1647
	\| 0.1139		\| 0.1139
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 833/832, 1001/1000, 1089/1088, 1225/1224, 1701/1700, 4225/4224		\| 833/832, 1001/1000, 1089/1088, 1225/1224, 1701/1700, 4225/4224
	\| [{{~~val~~\| 460 729 1068 1291 1591 1702 1880 }}]		\| {{mapping\| 460 729 1068 1291 1591 1702 1880 }}
	\| +0.1624		\| +0.1624
	\| 0.1056		\| 0.1056
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	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 833/832, 1001/1000, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1617/1615		\| 833/832, 1001/1000, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1617/1615
	\| [{{~~val~~\| 460 729 1068 1291 1591 1702 1880 1954 }}]		\| {{mapping\| 460 729 1068 1291 1591 1702 1880 1954 }}
	\| +0.1457		\| +0.1457
	\| 0.1082		\| 0.1082
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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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	\| 566.086<br>(26.086)		\| 566.086<br>(26.086)
	\| 238/165<br>(?)		\| 238/165<br>(?)
	\| [[Soviet ~~Ferris~~ wheel]]		\| [[Soviet ferris wheel]]
	\|}		\|}
			<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 06:35, 9 July 2023

2023-07-09T06:35:44Z

← Older revision		Revision as of 06:35, 9 July 2023
Line 1:		Line 1:
	~~{{novelty}}{{stub}}~~{{Infobox ET}}		{{Infobox ET}}
	The '''460 equal divisions of the octave''' ('''460edo'''), or the '''460(-tone) equal temperament''' ('''460tet''', '''460et''') when viewed from a [[regular temperament]] perspective, divides the octave into 460 equal parts of about 2.61 [[cent]]s each.		The '''460 equal divisions of the octave''' ('''460edo'''), or the '''460(-tone) equal temperament''' ('''460tet''', '''460et''') when viewed from a [[regular temperament]] perspective, divides the octave into 460 equal parts of about 2.61 [[cent]]s each.

BudjarnLambeth at 05:47, 9 July 2023

2023-07-09T05:47:29Z

← Older revision		Revision as of 05:47, 9 July 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{novelty}}{{stub}}{{Infobox ET}}
	The '''460 equal divisions of the octave''' ('''460edo'''), or the '''460(-tone) equal temperament''' ('''460tet''', '''460et''') when viewed from a [[regular temperament]] perspective, divides the octave into 460 equal parts of about 2.61 [[cent]]s each.		The '''460 equal divisions of the octave''' ('''460edo'''), or the '''460(-tone) equal temperament''' ('''460tet''', '''460et''') when viewed from a [[regular temperament]] perspective, divides the octave into 460 equal parts of about 2.61 [[cent]]s each.

FloraC: /* Regular temperament properties */ cleanup

2023-03-08T13:51:38Z

Regular temperament properties: cleanup

← Older revision		Revision as of 13:51, 8 March 2023
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	== 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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	\|+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<br>(Reduced)
	! Cents<br>(~~reduced~~)		! Cents<br>(Reduced)
	! Associated<br>~~ratio~~		! Associated<br>Ratio
	! Temperaments		! Temperaments
	\|-		\|-
	\|1		\| 1
	\|9\460		\| 9\460
	\|23.48		\| 23.48
	\|531441/524288		\| 531441/524288
	\|[[Commatose]]		\| [[Commatose]]
	\|-		\|-
	\| 1		\| 1
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	\| 498.26		\| 498.26
	\| 4/3		\| 4/3
	\| [[~~Helmholtz]] / [[pontiac~~]]		\| [[Pontiac]]
	\|-		\|-
	\| 10		\| 10
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	\| [[Deca]]		\| [[Deca]]
	\|-		\|-
	\|20		\| 20
	\|66\460<br>(20\460)		\| 66\460<br>(20\460)
	\|172.173<br>(52.173)		\| 172.173<br>(52.173)
	\|169/153<br>(?)		\| 169/153<br>(?)
	\|[[Calcium]]		\| [[Calcium]]
	\|-		\|-
	\|20		\| 20
	\|217\460<br>(10\460)		\| 217\460<br>(10\460)
	\|566.086<br>(26.086)		\| 566.086<br>(26.086)
	\|238/165<br>(?)		\| 238/165<br>(?)
	\|[[Soviet Ferris wheel]]		\| [[Soviet Ferris wheel]]
	\|}		\|}

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