444edo - 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"

2026-03-13T13:33:02Z

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 13:33, 13 March 2026
Line 53:		Line 53:
	\| [[Undim]] (444d)		\| [[Undim]] (444d)
	\|}		\|}
	<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

Francium: changed EDO intro to ED intro

2025-02-21T06:23:02Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:23, 21 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|444~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 16:57, 15 January 2025

2025-01-15T16:57:27Z

← Older revision		Revision as of 16:57, 15 January 2025
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	== Theory ==		== Theory ==
	444edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1\|7]] is about halfway between its steps. 444 = 4 × 111, ~~and~~ its harmonic [[3/1\|3]] derives from [[111edo]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] [[250047/250000]], 29360128/29296875, 67108864/66976875 and in the 7-limit; [[3025/3024]], [[5632/5625]], 42592/42525, 102487/102400, [[131072/130977]], 160083/160000, 172032/171875, 322102/321489, 391314/390625 and [[1771561/1769472]] in the 11-limit. It [[support]]s the [[magnesium]] temperament.		444edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1\|7]] is about halfway between its steps. Since {{nowrap\|444 {{=}} 4 × 111}}, its harmonic [[3/1\|3]] derives from [[111edo]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] [[250047/250000]], 29360128/29296875, 67108864/66976875 and in the 7-limit; [[3025/3024]], [[5632/5625]], 42592/42525, 102487/102400, [[131072/130977]], 160083/160000, 172032/171875, 322102/321489, 391314/390625 and [[1771561/1769472]] in the 11-limit. It [[support]]s the [[magnesium]] temperament.

	=== Odd harmonics ===		=== Odd harmonics ===
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	\| {{monzo\| 41 -20 -4 }}, {{monzo\| -29 -11 20 }}		\| {{monzo\| 41 -20 -4 }}, {{monzo\| -29 -11 20 }}
	\| {{mapping\| 444 704 1031 }}		\| {{mapping\| 444 704 1031 }}
	\| ~~−0~~.1821		\| −0.1821
	\| 0.2071		\| 0.2071
	\| 7.66		\| 7.66
Line 53:		Line 53:
	\| [[Undim]] (444d)		\| [[Undim]] (444d)
	\|}		\|}
	<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

2024-11-16T13:20:56Z

Partial undo

← Older revision		Revision as of 13:20, 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]] (%)
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{monzo\| 41 -20 -4 }}, {{monzo\| -29 -11 20 }}		\| {{monzo\| 41 -20 -4 }}, {{monzo\| -29 -11 20 }}
	\| {{mapping\| 444 704 1031 }}		\| {{mapping\| 444 704 1031 }}
	\| -0.1821		\| −0.1821
	\| 0.2071		\| 0.2071
	\| 7.66		\| 7.66
	~~{{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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	\| 4/3<br>(35/33)		\| 4/3<br>(35/33)
	\| [[Undim]] (444d)		\| [[Undim]] (444d)
	~~{{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 01:32, 16 November 2024

2024-11-16T01:32:37Z

← Older revision		Revision as of 01:32, 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
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	\| 0.2071		\| 0.2071
	\| 7.66		\| 7.66
	\|}		{{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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	\| 4/3<br>(35/33)		\| 4/3<br>(35/33)
	\| [[Undim]] (444d)		\| [[Undim]] (444d)
	\|}		{{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: It's enfactored in the 3-limit

2023-12-15T08:49:20Z

It's enfactored in the 3-limit

← Older revision		Revision as of 08:49, 15 December 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	~~444et~~ is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1\|7]] is about halfway between its steps. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] [[250047/250000]], 29360128/29296875, 67108864/66976875 and in the 7-limit; [[3025/3024]], [[5632/5625]], 42592/42525, 102487/102400, [[131072/130977]], 160083/160000, 172032/171875, 322102/321489, 391314/390625 and [[1771561/1769472]] in the 11-limit. It [[support]]s the [[magnesium]] temperament.		444edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1\|7]] is about halfway between its steps. 444 = 4 × 111, and its harmonic [[3/1\|3]] derives from [[111edo]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] [[250047/250000]], 29360128/29296875, 67108864/66976875 and in the 7-limit; [[3025/3024]], [[5632/5625]], 42592/42525, 102487/102400, [[131072/130977]], 160083/160000, 172032/171875, 322102/321489, 391314/390625 and [[1771561/1769472]] in the 11-limit. It [[support]]s the [[magnesium]] temperament.

	=== Odd harmonics ===		=== Odd harmonics ===
Line 21:		Line 21:
	! [[TE error\|Absolute]] (¢)		! [[TE error\|Absolute]] (¢)
	! [[TE simple badness\|Relative]] (%)		! [[TE simple badness\|Relative]] (%)
	\|-
	~~\| 2.3~~
	~~\| {{monzo\| 176 -111 }}~~
	~~\| {{mapping\| 444 704 }}~~
	~~\| -0.2359~~
	~~\| 0.2358~~
	~~\| 8.72~~
	\|-		\|-
	\| 2.3.5		\| 2.3.5
Line 56:		Line 49:
	\| 497.30<br>(102.70)		\| 497.30<br>(102.70)
	\| 4/3<br>(35/33)		\| 4/3<br>(35/33)
	\| [[Undim]]		\| [[Undim]] (444d)
	\|}		\|}
	<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

FloraC: Review

2023-12-13T15:22:24Z

Review

← Older revision		Revision as of 15:22, 13 December 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	444et is only consistent to the 5-limit. Using the patent val, it tempers out ~~67108864/66976875, 29360128/29296875 and~~ [[250047/250000]] in the 7-limit; ~~100663296/100656875, 2097152/2096325,~~ [[~~131072~~/~~130977~~]]~~, 172032/171875~~, [[5632/5625]], ~~47265625~~/~~47258883~~, [[~~3025~~/~~3024~~]], 160083/160000, ~~42592~~/~~42525~~, 391314/390625~~, 102487/102400, 322102/321489~~ and [[1771561/1769472]] in the 11-limit. It ~~provides the~~ [[~~optimal patent val~~]] ~~for~~ the [[magnesium]] temperament.		444et is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1\|7]] is about halfway between its steps. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] [[250047/250000]], 29360128/29296875, 67108864/66976875 and in the 7-limit; [[3025/3024]], [[5632/5625]], 42592/42525, 102487/102400, [[131072/130977]], 160083/160000, 172032/171875, 322102/321489, 391314/390625 and [[1771561/1769472]] in the 11-limit. It [[support]]s the [[magnesium]] temperament.

	=== Odd harmonics ===		=== Odd harmonics ===
Line 9:		Line 9:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	444 factors into ~~2<sub>2</sub> × 3 × 37~~, ~~with~~ subset edos {{EDOs\|2, 3, 4, 6, 12, 37, 74, 111, 148, and 222}}. [[1332edo]], which triples it, gives a good correction to the harmonic 7.		Since 444 factors into {{factorization\|444}}, 444edo has subset edos {{EDOs\| 2, 3, 4, 6, 12, 37, 74, 111, 148, and 222 }}. [[1332edo]], which triples it, gives a good correction to the harmonic 7.

	== 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
	\|{{monzo\|176 -111}}		\| {{monzo\| 176 -111 }}
	\|{{mapping\|444 704}}		\| {{mapping\| 444 704 }}
	\| -0.2359		\| -0.2359
	\| 0.2358		\| 0.2358
	\| 8.72		\| 8.72
	\|-		\|-
	\|2.3.5		\| 2.3.5
	\|{{monzo\|41 -20 -4}}, {{monzo\|-29 -11 20}}		\| {{monzo\| 41 -20 -4 }}, {{monzo\| -29 -11 20 }}
	\|{{mapping\|444 704 1031}}		\| {{mapping\| 444 704 1031 }}
	\| -0.1821		\| -0.1821
	\| 0.2071		\| 0.2071
Line 41:		Line 41:
	\|+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
	\|-		\|-
	\|1		\| 1
	\|13\444		\| 13\444
	\|35.14		\| 35.14
	\|1990656/1953125		\| 1990656/1953125
	\|[[Gammic]]		\| [[Gammic]] (5-limit)
	\|-		\|-
	\|4		\| 4
	\|184\444<br>(38\444)		\| 184\444<br>(38\444)
	\|497.30<br>(102.70)		\| 497.30<br>(102.70)
	\|4/3<br>(35/33)		\| 4/3<br>(35/33)
	\|[[Undim]]		\| [[Undim]]
	\|}		\|}

	<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|444}} == Theory == 444et is only consistent to the 5-limit. Using the patent val, it tempers out 67108864/66976875, 29360128/29296875 and 250047/..."

2023-12-13T08:04:00Z

Created page with "{{Infobox ET}} {{EDO intro|444}} == Theory == 444et is only consistent to the 5-limit. Using the patent val, it tempers out 67108864/66976875, 29360128/29296875 and 250047/..."

New page

{{Infobox ET}}
{{EDO intro|444}}

== Theory ==
444et is only consistent to the 5-limit. Using the patent val, it tempers out 67108864/66976875, 29360128/29296875 and [[250047/250000]] in the 7-limit; 100663296/100656875, 2097152/2096325, [[131072/130977]], 172032/171875, [[5632/5625]], 47265625/47258883, [[3025/3024]], 160083/160000, 42592/42525, 391314/390625, 102487/102400, 322102/321489 and [[1771561/1769472]] in the 11-limit. It provides the [[optimal patent val]] for the [[magnesium]] temperament.

=== Odd harmonics ===
{{Harmonics in equal|444}}

=== Subsets and supersets ===
444 factors into 22 × 3 × 37, with subset edos {{EDOs|2, 3, 4, 6, 12, 37, 74, 111, 148, and 222}}. [[1332edo]], which triples it, gives a good correction to the harmonic 7.

== 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 8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo|176 -111}}
|{{mapping|444 704}}
| -0.2359
| 0.2358
| 8.72
|-
|2.3.5
|{{monzo|41 -20 -4}}, {{monzo|-29 -11 20}}
|{{mapping|444 704 1031}}
| -0.1821
| 0.2071
| 7.66
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods per 8ve
! Generator (reduced)*
! Cents (reduced)*
! Associated Ratio*
! Temperaments
|-
|1
|13\444
|35.14
|1990656/1953125
|[[Gammic]]
|-
|4
|184\444 (38\444)
|497.30 (102.70)
|4/3 (35/33)
|[[Undim]]
|}

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