282edo - 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"

2026-03-13T13:38:12Z

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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	\| [[Sextile]]		\| [[Sextile]]
	\|}		\|}
	<nowiki>*</nowiki> [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[normal ~~lists~~\|minimal form]] in parentheses if distinct		<nowiki>*</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

	== Scales ==		== Scales ==

Contribution at 19:18, 11 November 2025

2025-11-11T19:18:06Z

← Older revision		Revision as of 19:18, 11 November 2025
Line 142:		Line 142:
	\| [[Sextile]]		\| [[Sextile]]
	\|}		\|}
	<nowiki/>* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[normal lists\|minimal form]] in parentheses if distinct		<nowiki>*</nowiki> [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[normal lists\|minimal form]] in parentheses if distinct

			== Scales ==
			[[Overtone scale#Over-3 scales\|Mode 12]]: 33 30 28 26 25 23 22 21 20 19 18 17
	[[Category:Jupiter]]		[[Category:Jupiter]]
	[[Category:Septisuperfourth]]		[[Category:Septisuperfourth]]

FloraC: Cleanup and update

2025-03-16T15:31:22Z

Cleanup and update

← Older revision		Revision as of 15:31, 16 March 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	282edo is the smallest edo [[consistency\|distinctly consistent]] through to the [[23-odd-limit]], and also the smallest consistent to the [[29-odd-limit]]. It shares the same 3rd, 7th, and 13th harmonics with [[94edo]] (282 = 3 × 94), as well as [[11/10]] and [[20/17]] (~~supporting~~ the [[Stearnsmic clan #Garistearn\|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29.		282edo is the smallest edo [[consistency\|distinctly consistent]] through to the [[23-odd-limit]], and also the smallest consistent to the [[29-odd-limit]]. It shares the same 3rd, 7th, and 13th harmonics with [[94edo]] ({{nowrap\| 282 {{=}} 3 × 94 }}), as well as [[11/10]] and [[20/17]] ([[support]]ing the [[Stearnsmic clan #Garistearn\|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29.

	The equal temperament [[tempering out\|tempers out]] [[6144/6125]] (porwell), 118098/117649 (stearnsma), and [[250047/250000]] (landscape comma) in the 7-limit, and [[540/539]] and [[5632/5625]] in the 11-limit, so that it provides the [[optimal patent val]] for the [[jupiter]] temperament; it also tempers out [[4000/3993]] and 234375/234256, providing the optimal patent val for [[septisuperfourth]] temperament. In the 13-limit, it tempers out [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[10648/10647]].		The equal temperament [[tempering out\|tempers out]] [[6144/6125]] (porwell comma), 118098/117649 (stearnsma), and [[250047/250000]] (landscape comma) in the 7-limit, and [[540/539]] and [[5632/5625]] in the 11-limit, so that it provides the [[optimal patent val]] for the [[jupiter]] temperament; it also tempers out [[4000/3993]] and 234375/234256, providing the optimal patent val for [[septisuperfourth]] temperament. In the 13-limit, it tempers out [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[10648/10647]].

	It allows [[essentially tempered chord]]s including [[swetismic chords]], [[squbemic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit.		It allows [[essentially tempered chord]]s including [[swetismic chords]], [[squbemic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit.
Line 13:		Line 13:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 282 factors into {{~~factorization~~\|~~282~~}}, 282edo has subset edos {{EDOs\| 2, 3, 47, 94, and 141 }}.		Since 282 factors into primes as {{nowrap\| 2 × 3 × 47 }}, 282edo has subset edos {{EDOs\| 2, 3, 47, 94, and 141 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 21:		Line 21:
	! 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 28:		Line 28:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| 32 -7 -9 }}, {{monzo\| -7 22 -12 }}		\| {{Monzo\| 32 -7 -9 }}, {{monzo\| -7 22 -12 }}
	\| {{~~mapping~~\| 282 447 655 }}		\| {{Mapping\| 282 447 655 }}
	\| −0.1684		\| −0.1684
	\| 0.1671		\| 0.1671
Line 36:		Line 36:
	\| 2.3.5.7		\| 2.3.5.7
	\| 6144/6125, 118098/117649, 250047/250000		\| 6144/6125, 118098/117649, 250047/250000
	\| {{~~mapping~~\| 282 447 655 792 }}		\| {{Mapping\| 282 447 655 792 }}
	\| −0.2498		\| −0.2498
	\| 0.2020		\| 0.2020
Line 43:		Line 43:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 540/539, 4000/3993, 5632/5625, 137781/137500		\| 540/539, 4000/3993, 5632/5625, 137781/137500
	\| {{~~mapping~~\| 282 447 655 792 976 }}		\| {{Mapping\| 282 447 655 792 976 }}
	\| −0.3081		\| −0.3081
	\| 0.2151		\| 0.2151
Line 50:		Line 50:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575		\| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575
	\| {{~~mapping~~\| 282 447 655 792 976 1044 }}		\| {{Mapping\| 282 447 655 792 976 1044 }}
	\| −0.3480		\| −0.3480
	\| 0.2156		\| 0.2156
Line 57:		Line 57:
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197		\| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197
	\| {{~~mapping~~\| 282 447 655 792 976 1044 1153 }}		\| {{Mapping\| 282 447 655 792 976 1044 1153 }}
	\| −0.3481		\| −0.3481
	\| 0.1996		\| 0.1996
Line 64:		Line 64:
	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573		\| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573
	\| {{~~mapping~~\| 282 447 655 792 976 1044 1153 1198 }}		\| {{Mapping\| 282 447 655 792 976 1044 1153 1198 }}
	\| −0.3152		\| −0.3152
	\| 0.2061		\| 0.2061
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	\| 2.3.5.7.11.13.17.19.23		\| 2.3.5.7.11.13.17.19.23
	\| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287		\| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287
	\| {{~~mapping~~\| 282 447 655 792 976 1044 1153 1198 1276 }}		\| {{Mapping\| 282 447 655 792 976 1044 1153 1198 1276 }}
	\| −0.3173		\| −0.3173
	\| 0.1944		\| 0.1944
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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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	\| 565.96		\| 565.96
	\| 4096/2835		\| 4096/2835
	\| [[~~Trident~~]] (~~282ef~~)		\| [[Alphatrident]] (7-limit)
	\|-		\|-
	\| 2		\| 2
Line 125:		Line 125:
	\|-		\|-
	\| 6		\| 6
	\| 51\282<br />(4\282)		\| 51\282<br>(4\282)
	\| 217.02<br />(17.02)		\| 217.02<br>(17.02)
	\| 17/15<br />(105/104)		\| 17/15<br>(105/104)
	\| [[Stearnscape]]		\| [[Stearnscape]]
	\|-		\|-
	\| 6		\| 6
	\| 80\282<br />(14\282)		\| 80\282<br>(14\282)
	\| 340.43<br />(59.57)		\| 340.43<br>(59.57)
	\| 162/133<br />(88/85)		\| 162/133<br>(88/85)
	\| [[Semiseptichrome]]		\| [[Semiseptichrome]]
	\|-		\|-
	\| 6		\| 6
	\| 117\282<br />(23\282)		\| 117\282<br>(23\282)
	\| 497.87<br />(97.87)		\| 497.87<br>(97.87)
	\| 4/3<br />(128/121)		\| 4/3<br>(128/121)
	\| [[Sextile]]		\| [[Sextile]]
	\|}		\|}
	<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:Jupiter]]		[[Category:Jupiter]]
	[[Category:Septisuperfourth]]		[[Category:Septisuperfourth]]

ArrowHead294 at 14:28, 20 February 2025

2025-02-20T14:28:35Z

← Older revision		Revision as of 14:28, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|282~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 18:16, 15 January 2025

2025-01-15T18:16:48Z

← Older revision		Revision as of 18:16, 15 January 2025
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	\| {{monzo\| 32 -7 -9 }}, {{monzo\| -7 22 -12 }}		\| {{monzo\| 32 -7 -9 }}, {{monzo\| -7 22 -12 }}
	\| {{mapping\| 282 447 655 }}		\| {{mapping\| 282 447 655 }}
	\| ~~−0~~.1684		\| −0.1684
	\| 0.1671		\| 0.1671
	\| 3.93		\| 3.93
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	\| 6144/6125, 118098/117649, 250047/250000		\| 6144/6125, 118098/117649, 250047/250000
	\| {{mapping\| 282 447 655 792 }}		\| {{mapping\| 282 447 655 792 }}
	\| ~~−0~~.2498		\| −0.2498
	\| 0.2020		\| 0.2020
	\| 4.75		\| 4.75
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	\| 540/539, 4000/3993, 5632/5625, 137781/137500		\| 540/539, 4000/3993, 5632/5625, 137781/137500
	\| {{mapping\| 282 447 655 792 976 }}		\| {{mapping\| 282 447 655 792 976 }}
	\| ~~−0~~.3081		\| −0.3081
	\| 0.2151		\| 0.2151
	\| 5.06		\| 5.06
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	\| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575		\| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575
	\| {{mapping\| 282 447 655 792 976 1044 }}		\| {{mapping\| 282 447 655 792 976 1044 }}
	\| ~~−0~~.3480		\| −0.3480
	\| 0.2156		\| 0.2156
	\| 5.07		\| 5.07
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	\| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197		\| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197
	\| {{mapping\| 282 447 655 792 976 1044 1153 }}		\| {{mapping\| 282 447 655 792 976 1044 1153 }}
	\| ~~−0~~.3481		\| −0.3481
	\| 0.1996		\| 0.1996
	\| 4.69		\| 4.69
Line 65:		Line 65:
	\| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573		\| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573
	\| {{mapping\| 282 447 655 792 976 1044 1153 1198 }}		\| {{mapping\| 282 447 655 792 976 1044 1153 1198 }}
	\| ~~−0~~.3152		\| −0.3152
	\| 0.2061		\| 0.2061
	\| 4.84		\| 4.84
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	\| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287		\| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287
	\| {{mapping\| 282 447 655 792 976 1044 1153 1198 1276 }}		\| {{mapping\| 282 447 655 792 976 1044 1153 1198 1276 }}
	\| ~~−0~~.3173		\| −0.3173
	\| 0.1944		\| 0.1944
	\| 4.57		\| 4.57
Line 142:		Line 142:
	\| [[Sextile]]		\| [[Sextile]]
	\|}		\|}
	<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:Jupiter]]		[[Category:Jupiter]]
	[[Category:Septisuperfourth]]		[[Category:Septisuperfourth]]

ArrowHead294 at 12:13, 16 November 2024

2024-11-16T12:13:24Z

← Older revision		Revision as of 12:13, 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		\| 2.3.5
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	\| 0.1944		\| 0.1944
	\| 4.57		\| 4.57
	~~{{comma basis end}~~}		\|}
	* 282et has a lower relative error than any previous equal temperaments in the 23-limit, past [[270edo\|270]] and before [[311edo\|311]].		* 282et has a lower relative error than any previous equal temperaments in the 23-limit, past [[270edo\|270]] and before [[311edo\|311]].

	=== 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 />(128/121)		\| 4/3<br />(128/121)
	\| [[Sextile]]		\| [[Sextile]]
	~~{{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:Jupiter]]		[[Category:Jupiter]]
	[[Category:Septisuperfourth]]		[[Category:Septisuperfourth]]

ArrowHead294 at 05:19, 16 November 2024

2024-11-16T05:19:40Z

← Older revision		Revision as of 05:19, 16 November 2024
Line 16:		Line 16:

	== 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		\| 2.3.5
	\| {{monzo\| 32 -7 -9 }}, {{monzo\| -7 22 -12 }}		\| {{monzo\| 32 -7 -9 }}, {{monzo\| -7 22 -12 }}
	\| {{mapping\| 282 447 655 }}		\| {{mapping\| 282 447 655 }}
	\| -0.1684		\| −0.1684
	\| 0.1671		\| 0.1671
	\| 3.93		\| 3.93
Line 36:		Line 28:
	\| 6144/6125, 118098/117649, 250047/250000		\| 6144/6125, 118098/117649, 250047/250000
	\| {{mapping\| 282 447 655 792 }}		\| {{mapping\| 282 447 655 792 }}
	\| -0.2498		\| −0.2498
	\| 0.2020		\| 0.2020
	\| 4.75		\| 4.75
Line 43:		Line 35:
	\| 540/539, 4000/3993, 5632/5625, 137781/137500		\| 540/539, 4000/3993, 5632/5625, 137781/137500
	\| {{mapping\| 282 447 655 792 976 }}		\| {{mapping\| 282 447 655 792 976 }}
	\| -0.3081		\| −0.3081
	\| 0.2151		\| 0.2151
	\| 5.06		\| 5.06
Line 50:		Line 42:
	\| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575		\| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575
	\| {{mapping\| 282 447 655 792 976 1044 }}		\| {{mapping\| 282 447 655 792 976 1044 }}
	\| -0.3480		\| −0.3480
	\| 0.2156		\| 0.2156
	\| 5.07		\| 5.07
Line 57:		Line 49:
	\| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197		\| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197
	\| {{mapping\| 282 447 655 792 976 1044 1153 }}		\| {{mapping\| 282 447 655 792 976 1044 1153 }}
	\| -0.3481		\| −0.3481
	\| 0.1996		\| 0.1996
	\| 4.69		\| 4.69
Line 64:		Line 56:
	\| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573		\| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573
	\| {{mapping\| 282 447 655 792 976 1044 1153 1198 }}		\| {{mapping\| 282 447 655 792 976 1044 1153 1198 }}
	\| -0.3152		\| −0.3152
	\| 0.2061		\| 0.2061
	\| 4.84		\| 4.84
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	\| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287		\| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287
	\| {{mapping\| 282 447 655 792 976 1044 1153 1198 1276 }}		\| {{mapping\| 282 447 655 792 976 1044 1153 1198 1276 }}
	\| -0.3173		\| −0.3173
	\| 0.1944		\| 0.1944
	\| 4.57		\| 4.57
	\|}		{{comma basis end}}
	* 282et has a lower relative error than any previous equal temperaments in the 23-limit, past [[270edo\|270]] and before [[311edo\|311]].		* 282et has a lower relative error than any previous equal temperaments in the 23-limit, past [[270edo\|270]] and before [[311edo\|311]].

	=== 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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	\|-		\|-
	\| 6		\| 6
	\| 51\282<br>(4\282)		\| 51\282<br />(4\282)
	\| 217.02<br>(17.02)		\| 217.02<br />(17.02)
	\| 17/15<br>(105/104)		\| 17/15<br />(105/104)
	\| [[Stearnscape]]		\| [[Stearnscape]]
	\|-		\|-
	\| 6		\| 6
	\| 80\282<br>(14\282)		\| 80\282<br />(14\282)
	\| 340.43<br>(59.57)		\| 340.43<br />(59.57)
	\| 162/133<br>(88/85)		\| 162/133<br />(88/85)
	\| [[Semiseptichrome]]		\| [[Semiseptichrome]]
	\|-		\|-
	\| 6		\| 6
	\| 117\282<br>(23\282)		\| 117\282<br />(23\282)
	\| 497.87<br>(97.87)		\| 497.87<br />(97.87)
	\| 4/3<br>(128/121)		\| 4/3<br />(128/121)
	\| [[Sextile]]		\| [[Sextile]]
	\|}		{{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:Jupiter]]
	[[Category:Septisuperfourth]]		[[Category:Septisuperfourth]]
	~~[[Category:Jupiter]]~~

FloraC: /* Regular temperament properties */ 23-limit notability

2024-04-19T03:48:25Z

Regular temperament properties: 23-limit notability

← Older revision		Revision as of 03:48, 19 April 2024
Line 75:		Line 75:
	\| 4.57		\| 4.57
	\|}		\|}
			* 282et has a lower relative error than any previous equal temperaments in the 23-limit, past [[270edo\|270]] and before [[311edo\|311]].

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

FloraC: /* Regular temperament properties */ +semiseptichrome

2023-12-05T11:59:05Z

Regular temperament properties: +semiseptichrome

← Older revision		Revision as of 11:59, 5 December 2023
Line 124:		Line 124:
	\| 51\282<br>(4\282)		\| 51\282<br>(4\282)
	\| 217.02<br>(17.02)		\| 217.02<br>(17.02)
	\| 17/15<br>(~~245~~/~~243~~)		\| 17/15<br>(105/104)
	\| [[Stearnscape]]		\| [[Stearnscape]]
			\|-
			\| 6
			\| 80\282<br>(14\282)
			\| 340.43<br>(59.57)
			\| 162/133<br>(88/85)
			\| [[Semiseptichrome]]
	\|-		\|-
	\| 6		\| 6

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

2023-11-10T10:26:15Z

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

← Older revision		Revision as of 10:26, 10 November 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	282edo is the smallest edo ~~distinctly~~ [[consistent]] through to the [[23-odd-limit]], and also the smallest consistent to the [[29-odd-limit]]. It shares the same 3rd, 7th, and 13th harmonics with [[94edo]] (282 = 3 × 94), as well as [[11/10]] and [[20/17]] (supporting the [[Stearnsmic clan #Garistearn\|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29. It tempers out [[6144/6125]] (porwell), 118098/117649 (stearnsma), and [[250047/250000]] (landscape comma) in the 7-limit, and [[540/539]] and [[5632/5625]] in the 11-limit, so that it provides the [[optimal patent val]] for the [[jupiter]] temperament; it also tempers out [[4000/3993]] and 234375/234256, providing the optimal patent val for [[septisuperfourth]] temperament. In the 13-limit, it tempers out [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[10648/10647]]. It allows [[essentially tempered chord]]s including [[swetismic chords]], [[squbemic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit.		282edo is the smallest edo [[consistency\|distinctly consistent]] through to the [[23-odd-limit]], and also the smallest consistent to the [[29-odd-limit]]. It shares the same 3rd, 7th, and 13th harmonics with [[94edo]] (282 = 3 × 94), as well as [[11/10]] and [[20/17]] (supporting the [[Stearnsmic clan #Garistearn\|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29.

			The equal temperament [[tempering out\|tempers out]] [[6144/6125]] (porwell), 118098/117649 (stearnsma), and [[250047/250000]] (landscape comma) in the 7-limit, and [[540/539]] and [[5632/5625]] in the 11-limit, so that it provides the [[optimal patent val]] for the [[jupiter]] temperament; it also tempers out [[4000/3993]] and 234375/234256, providing the optimal patent val for [[septisuperfourth]] temperament. In the 13-limit, it tempers out [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[10648/10647]].

			It allows [[essentially tempered chord]]s including [[swetismic chords]], [[squbemic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 282 factors into ~~2 × 3 × 47~~, it has subset edos {{EDOs\| 2, 3, 47, 94, and 141 }}.		Since 282 factors into {{factorization\|282}}, 282edo has subset edos {{EDOs\| 2, 3, 47, 94, and 141 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
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	\| 2.3.5		\| 2.3.5
	\| {{monzo\| 32 -7 -9 }}, {{monzo\| -7 22 -12 }}		\| {{monzo\| 32 -7 -9 }}, {{monzo\| -7 22 -12 }}
	\| [{{~~val~~\| 282 447 655 }}]		\| {{mapping\| 282 447 655 }}
	\| -0.1684		\| -0.1684
	\| 0.1671		\| 0.1671
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	\| 2.3.5.7		\| 2.3.5.7
	\| 6144/6125, 118098/117649, 250047/250000		\| 6144/6125, 118098/117649, 250047/250000
	\| [{{~~val~~\| 282 447 655 792 }}]		\| {{mapping\| 282 447 655 792 }}
	\| -0.2498		\| -0.2498
	\| 0.2020		\| 0.2020
Line 38:		Line 42:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 540/539, 4000/3993, 5632/5625, 137781/137500		\| 540/539, 4000/3993, 5632/5625, 137781/137500
	\| [{{~~val~~\| 282 447 655 792 976 }}]		\| {{mapping\| 282 447 655 792 976 }}
	\| -0.3081		\| -0.3081
	\| 0.2151		\| 0.2151
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575		\| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575
	\| [{{~~val~~\| 282 447 655 792 976 1044 }}]		\| {{mapping\| 282 447 655 792 976 1044 }}
	\| -0.3480		\| -0.3480
	\| 0.2156		\| 0.2156
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197		\| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197
	\| [{{~~val~~\| 282 447 655 792 976 1044 1153 }}]		\| {{mapping\| 282 447 655 792 976 1044 1153 }}
	\| -0.3481		\| -0.3481
	\| 0.1996		\| 0.1996
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	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573		\| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573
	\| [{{~~val~~\| 282 447 655 792 976 1044 1153 1198 }}]		\| {{mapping\| 282 447 655 792 976 1044 1153 1198 }}
	\| -0.3152		\| -0.3152
	\| 0.2061		\| 0.2061
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	\| 2.3.5.7.11.13.17.19.23		\| 2.3.5.7.11.13.17.19.23
	\| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287		\| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287
	\| [{{~~val~~\| 282 447 655 792 976 1044 1153 1198 1276 }}]		\| {{mapping\| 282 447 655 792 976 1044 1153 1198 1276 }}
	\| -0.3173		\| -0.3173
	\| 0.1944		\| 0.1944
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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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	\| 565.96		\| 565.96
	\| 4096/2835		\| 4096/2835
	\| [[~~Tricot]] / [[trident~~]] (282ef)		\| [[Trident]] (282ef)
	\|-		\|-
	\| 2		\| 2
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	\| 157.45		\| 157.45
	\| 35/32		\| 35/32
	\| [[Nessafof]]		\| [[Nessafof]] (7-limit)
	\|-		\|-
	\| 6		\| 6
	\| 51\282<br>(4\282)		\| 51\282<br>(4\282)
	\| 217.02<br>(17.02)		\| 217.02<br>(17.02)
	\| ~~567~~/~~500~~<br>(245/243)		\| 17/15<br>(245/243)
	\| [[Stearnscape]]		\| [[Stearnscape]]
	\|-		\|-
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	\| [[Sextile]]		\| [[Sextile]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	[[Category:Septisuperfourth]]		[[Category:Septisuperfourth]]
	[[Category:Jupiter]]		[[Category:Jupiter]]