451edo - 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"

2026-03-13T13:32:37Z

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 74:		Line 74:
	\| [[Tertiaseptal]]		\| [[Tertiaseptal]]
	\|}		\|}
	<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:Quartonic]]		[[Category:Quartonic]]

Francium: changed EDO intro to ED intro

2025-02-21T06:28:41Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:28, 21 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|451~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 18:18, 15 January 2025

2025-01-15T18:18:06Z

← Older revision		Revision as of 18:18, 15 January 2025
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	== Theory ==		== Theory ==
	~~451 = 11 × 41, and~~ 451edo shares its [[3/2\|fifth]] with [[41edo]]. Unlike 41, however, 451 is only [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation up to the [[13-limit]] using the [[patent val]]. The equal temperament [[tempering out\|tempers out]] 390625000/387420489 ([[quartonic comma]]) in the 5-limit; [[2401/2400]], [[65625/65536]], [[703125/702464]], [[2100875/2097152]], in the 7-limit; [[6250/6237]], 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and [[625/624]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]], 4459/4455, and 17303/17280 in the 13-limit. It [[support]]s [[tertiaseptal]], [[tertiseptisix]], and [[hemermacomp]], providing the [[optimal patent val]] for 5-limit [[quartonic]].		451edo shares its [[3/2\|fifth]] with [[41edo]]. Unlike 41, however, 451 is only [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation up to the [[13-limit]] using the [[patent val]]. The equal temperament [[tempering out\|tempers out]] 390625000/387420489 ([[quartonic comma]]) in the 5-limit; [[2401/2400]], [[65625/65536]], [[703125/702464]], [[2100875/2097152]], in the 7-limit; [[6250/6237]], 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and [[625/624]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]], 4459/4455, and 17303/17280 in the 13-limit. It [[support]]s [[tertiaseptal]], [[tertiseptisix]], and [[hemermacomp]], providing the [[optimal patent val]] for 5-limit [[quartonic]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 9:		Line 9:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 451 factors into ~~11 × 41~~, 451edo has [[11edo]] and [[41edo]] as its subsets.		Since 451 factors into {{factorisation\|451}}, 451edo has [[11edo]] and [[41edo]] as its subsets.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 74:		Line 74:
	\| [[Tertiaseptal]]		\| [[Tertiaseptal]]
	\|}		\|}
	<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:Quartonic]]		[[Category:Quartonic]]

ArrowHead294: Partial undo

2024-11-16T12:57:06Z

Partial undo

← Older revision		Revision as of 12:57, 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
Line 41:		Line 50:
	\| 0.1736		\| 0.1736
	\| 6.52		\| 6.52
	~~{{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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	\| 256/245		\| 256/245
	\| [[Tertiaseptal]]		\| [[Tertiaseptal]]
	~~{{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:Quartonic]]		[[Category:Quartonic]]

ArrowHead294 at 03:08, 16 November 2024

2024-11-16T03:08:33Z

← Older revision		Revision as of 03:08, 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]] (%)~~
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| 390625000/387420489, {{monzo\| -59 5 22 }}		\| 390625000/387420489, {{monzo\| -59 5 22 }}
	\| {{mapping\| 451 715 1047 }}		\| {{mapping\| 451 715 1047 }}
	\| -0.0294		\| −0.0294
	\| 0.2144		\| 0.2144
	\| 8.06		\| 8.06
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	\| 0.1736		\| 0.1736
	\| 6.52		\| 6.52
	\|}		{{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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	\| 256/245		\| 256/245
	\| [[Tertiaseptal]]		\| [[Tertiaseptal]]
	\|}		{{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:Quartonic]]		[[Category:Quartonic]]

FloraC: +quartonic

2024-09-18T10:20:31Z

+quartonic

← Older revision		Revision as of 10:20, 18 September 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	451 = 11 × 41, and 451edo shares its [[3/2\|fifth]] with [[41edo]]. Unlike 41, however, 451 is only [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation up to the [[13-limit]] using the [[patent val]]. The equal temperament [[tempering out\|tempers out]] [[2401/2400]], [[65625/65536]], [[703125/702464]], [[2100875/2097152]], ~~and 390625000/387420489~~ in the 7-limit; [[6250/6237]], 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and [[625/624]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]], 4459/4455, and 17303/17280 in the 13-limit. It [[support]]s [[tertiaseptal]], [[tertiseptisix]], and [[hemermacomp]].		451 = 11 × 41, and 451edo shares its [[3/2\|fifth]] with [[41edo]]. Unlike 41, however, 451 is only [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation up to the [[13-limit]] using the [[patent val]]. The equal temperament [[tempering out\|tempers out]] 390625000/387420489 ([[quartonic comma]]) in the 5-limit; [[2401/2400]], [[65625/65536]], [[703125/702464]], [[2100875/2097152]], in the 7-limit; [[6250/6237]], 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and [[625/624]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]], 4459/4455, and 17303/17280 in the 13-limit. It [[support]]s [[tertiaseptal]], [[tertiseptisix]], and [[hemermacomp]], providing the [[optimal patent val]] for 5-limit [[quartonic]].

	=== Prime harmonics ===		=== Prime harmonics ===
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	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| ~~{{monzo\| 3 -18 11 }}~~, {{monzo\| -59 5 22 }}		\| 390625000/387420489, {{monzo\| -59 5 22 }}
	\| {{mapping\| 451 715 1047 }}		\| {{mapping\| 451 715 1047 }}
	\| -0.0294		\| -0.0294
Line 59:		Line 59:
	! Associated<br>Ratio*		! Associated<br>Ratio*
	! Temperaments		! Temperaments
			\|-
			\| 1
			\| 17\451
			\| 45.23
			\| 250/243
			\| [[Quartonic]] (5-limit)
	\|-		\|-
	\| 1		\| 1
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	\|}		\|}
	<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct		<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:Quartonic]]

FloraC: Review

2024-01-21T10:39:08Z

Review

← Older revision		Revision as of 10:39, 21 January 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	~~451et~~ is consistent to the [[7-odd-limit]]~~. Using~~ the patent val~~, it~~ tempers out ~~52734375/52706752,~~ [[~~703125~~/~~702464~~]], [[65625/65536]], [[~~2401~~/~~2400~~]], [[2100875/2097152]] and ~~201768035~~/~~201326592~~ in the 7-limit; [[~~117440512~~/~~117406179~~]], ~~35156250~~/~~35153041~~, ~~234375~~/~~234256~~, [[~~536870912~~/~~535869675~~]], ~~104857600~~/~~104825259~~, [[~~131072~~/~~130977~~]], [[~~6250~~/~~6237~~]], ~~200704~~/~~200475, 42875/42768, 1879453125/1879048192, 1362944/1361367, 42592/42525, 166375/165888, 456533/455625~~, ~~3294225~~/~~3294172~~, ~~43923/43904~~ and ~~102487~~/~~102400~~ in the 11-limit. It [[support]]s [[~~hemermacomp~~]], [[~~quartonic~~]] and [[~~tertiseptisix~~]].		451 = 11 × 41, and 451edo shares its [[3/2\|fifth]] with [[41edo]]. Unlike 41, however, 451 is only [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation up to the [[13-limit]] using the [[patent val]]. The equal temperament [[tempering out\|tempers out]] [[2401/2400]], [[65625/65536]], [[703125/702464]], [[2100875/2097152]], and 390625000/387420489 in the 7-limit; [[6250/6237]], 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and [[625/624]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]], 4459/4455, and 17303/17280 in the 13-limit. It [[support]]s [[tertiaseptal]], [[tertiseptisix]], and [[hemermacomp]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 9:		Line 9:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	451 factors into 11 × 41, ~~with~~ [[11edo]] and [[41edo]] as its ~~subset edos~~.		Since 451 factors into 11 × 41, 451edo has [[11edo]] and [[41edo]] as its subsets.

	== 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\|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]] (¢)
	![[TE simple badness\|Relative]] (%)		! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	~~\|2.3~~		\| 2.3.5
	~~\|{{monzo\|65 -41}}~~		\| {{monzo\| 3 -18 11 }}, {{monzo\| -59 5 22 }}
	~~\|{{mapping\|451 715}}~~		\| {{mapping\| 451 715 1047 }}
	~~\| -0.1527~~
	~~\| 0.1527~~
	~~\| 5.74~~
	\|-
	\|2.3.5
	\|{{monzo\|3 -18 11}}, {{monzo\|-59 5 22}}
	\|{{mapping\|451 715 1047}}
	\| -0.0294		\| -0.0294
	\| 0.2144		\| 0.2144
	\| 8.06		\| 8.06
	\|-		\|-
	\|2.3.5.7		\| 2.3.5.7
	\|2401/2400, ~~703125~~/~~702464~~, 390625000/387420489		\| 2401/2400, 65625/65536, 390625000/387420489
	\|{{mapping\|451 715 1047 ~~1266~~}}		\| {{mapping\| 451 715 1047 126 6}}
	\| +0.0057		\| +0.0057
	\| 0.1953		\| 0.1953
	\| 7.34		\| 7.34
	\|-		\|-
	\|2.3.5.7.11		\| 2.3.5.7.11
	\|2401/2400, 6250/6237, 42592/42525, 43923/43904		\| 2401/2400, 6250/6237, 42592/42525, 43923/43904
	\|{{mapping\|451 715 1047 1266 1560}}		\| {{mapping\| 451 715 1047 1266 1560 }}
	\| +0.0359		\| +0.0359
	\| 0.1849		\| 0.1849
	\| 6.95		\| 6.95
	\|-		\|-
	\|2.3.5.7.11.13		\| 2.3.5.7.11.13
	\|~~2080/2079,~~ 625/624, ~~4459~~/~~4455~~, 2200/2197, ~~20449~~/~~20412~~		\| 625/624, 2080/2079, 2200/2197, 2401/2400, 17303/17280
	\|{{mapping\|451 715 1047 1266 1560 1669}}		\| {{mapping\| 451 715 1047 1266 1560 1669 }}
	\| +0.0177		\| +0.0177
	\| 0.1736		\| 0.1736
Line 67:		Line 60:
	! Temperaments		! Temperaments
	\|-		\|-
	\|1		\| 1
	\|29\451		\| 29\451
	\|77.16		\| 77.16
	\|256/245		\| 256/245
	\|[[Tertiaseptal]]		\| [[Tertiaseptal]]
	\|}		\|}

	<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct		<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

Francium: Created page with "{{Infobox ET}} {{EDO intro|451}} == Theory == 451et is consistent to the 7-odd-limit. Using the patent val, it tempers out 52734375/52706752, 703125/702464, 65625/6..."

2024-01-18T11:57:23Z

Created page with "{{Infobox ET}} {{EDO intro|451}} == Theory == 451et is consistent to the 7-odd-limit. Using the patent val, it tempers out 52734375/52706752, 703125/702464, 65625/6..."

New page

{{Infobox ET}}
{{EDO intro|451}}

== Theory ==
451et is consistent to the [[7-odd-limit]]. Using the patent val, it tempers out 52734375/52706752, [[703125/702464]], [[65625/65536]], [[2401/2400]], [[2100875/2097152]] and 201768035/201326592 in the 7-limit; [[117440512/117406179]], 35156250/35153041, 234375/234256, [[536870912/535869675]], 104857600/104825259, [[131072/130977]], [[6250/6237]], 200704/200475, 42875/42768, 1879453125/1879048192, 1362944/1361367, 42592/42525, 166375/165888, 456533/455625, 3294225/3294172, 43923/43904 and 102487/102400 in the 11-limit. It [[support]]s [[hemermacomp]], [[quartonic]] and [[tertiseptisix]].

=== Prime harmonics ===
{{Harmonics in equal|451}}

=== Subsets and supersets ===
451 factors into 11 × 41, with [[11edo]] and [[41edo]] as its subset edos.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! 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
|{{monzo|65 -41}}
|{{mapping|451 715}}
| -0.1527
| 0.1527
| 5.74
|-
|2.3.5
|{{monzo|3 -18 11}}, {{monzo|-59 5 22}}
|{{mapping|451 715 1047}}
| -0.0294
| 0.2144
| 8.06
|-
|2.3.5.7
|2401/2400, 703125/702464, 390625000/387420489
|{{mapping|451 715 1047 1266}}
| +0.0057
| 0.1953
| 7.34
|-
|2.3.5.7.11
|2401/2400, 6250/6237, 42592/42525, 43923/43904
|{{mapping|451 715 1047 1266 1560}}
| +0.0359
| 0.1849
| 6.95
|-
|2.3.5.7.11.13
|2080/2079, 625/624, 4459/4455, 2200/2197, 20449/20412
|{{mapping|451 715 1047 1266 1560 1669}}
| +0.0177
| 0.1736
| 6.52
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|1
|29\451
|77.16
|256/245
|[[Tertiaseptal]]
|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct