407edo - 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"

2026-03-13T13:32:54Z

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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	\| [[Helmholtz (temperament)\|Helmholtz]]		\| [[Helmholtz (temperament)\|Helmholtz]]
	\|}		\|}
	<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:Subsemifourth]]		[[Category:Subsemifourth]]

Lériendil: Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"

2025-04-17T02:29:23Z

Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"

← Older revision		Revision as of 02:29, 17 April 2025
Line 64:		Line 64:
	\| 498.28		\| 498.28
	\| 4/3		\| 4/3
	\| [[Helmholtz temperament\|Helmholtz]]		\| [[Helmholtz (temperament)\|Helmholtz]]
	\|}		\|}
	<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:Subsemifourth]]		[[Category:Subsemifourth]]

Sintel: fix helmholtz link

2025-04-16T15:33:30Z

fix helmholtz link

← Older revision		Revision as of 15:33, 16 April 2025
Line 64:		Line 64:
	\| 498.28		\| 498.28
	\| 4/3		\| 4/3
	\| [[Helmholtz]]		\| [[Helmholtz temperament\|Helmholtz]]
	\|}		\|}
	<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:Subsemifourth]]		[[Category:Subsemifourth]]

Francium: changed EDO intro to ED intro

2025-02-21T06:22:38Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:22, 21 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|407~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 16:55, 15 January 2025

2025-01-15T16:55:39Z

← Older revision		Revision as of 16:55, 15 January 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	407 factors into 11 × 37, with [[11edo]] and [[37edo]] as its subset edos. [[814edo]], which doubles it, gives a good correction to harmonics 7 and 17, and is a notable full 23-limit temperament.		407 factors into {{nowrap\|11 × 37}}, with [[11edo]] and [[37edo]] as its subset edos. [[814edo]], which doubles it, gives a good correction to harmonics 7 and 17, and is a notable full 23-limit temperament.

	== Regular temperament properties ==		== Regular temperament properties ==
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	\| [[Helmholtz]]		\| [[Helmholtz]]
	\|}		\|}
	<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:Subsemifourth]]		[[Category:Subsemifourth]]

ArrowHead294: Partial undo

2024-11-16T13:13:42Z

Partial undo

← Older revision		Revision as of 13:13, 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		\| 2.3
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	\| 0.0638		\| 0.0638
	\| 2.16		\| 2.16
	~~{{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		\| 4/3
	\| [[Helmholtz]]		\| [[Helmholtz]]
	~~{{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:Subsemifourth]]		[[Category:Subsemifourth]]

ArrowHead294 at 01:51, 16 November 2024

2024-11-16T01:51:51Z

← Older revision		Revision as of 01:51, 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
Line 36:		Line 27:
	\| 0.0638		\| 0.0638
	\| 2.16		\| 2.16
	\|}		{{comma basis end}}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+ 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		\| 4/3
	\| [[Helmholtz]]		\| [[Helmholtz]]
	\|}		{{rank-2 end}}
	{{orf}}		{{orf}}

	[[Category:Subsemifourth]]		[[Category:Subsemifourth]]

ArrowHead294 at 01:14, 16 November 2024

2024-11-16T01:14:29Z

← Older revision		Revision as of 01:14, 16 November 2024
Line 13:		Line 13:
	== 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
	\|-		\|-
Line 39:		Line 40:
	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+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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	\| [[Helmholtz]]		\| [[Helmholtz]]
	\|}		\|}
	~~<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:Subsemifourth]]		[[Category:Subsemifourth]]

FloraC: Review

2024-01-01T11:12:18Z

Review

← Older revision		Revision as of 11:12, 1 January 2024
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	== Theory ==		== Theory ==
	~~407et~~ tempers out [[32805/32768]] in the 5-limit; [[~~4096000/4084101~~]], ~~134217728~~/~~133984375, 26873856/26796875~~, [[~~78125000~~/~~78121827~~]] and ~~48828125~~/~~48771072~~ in the 7-limit. It ~~supports~~ the ~~pinkanberry chords and~~ the temperament [[~~subsemifourth~~]].		407edo is a strong 5-limit system and 2.3.5.11.13.19.23 [[subgroup]] system. The equal temperament [[tempering out\|tempers out]] [[32805/32768]] in the 5-limit; using the [[patent val]], [[16875/16807]], [[4096000/4084101]], and 26873856/26796875 in the 7-limit. It [[support]]s and provides the [[optimal patent val]] for the [[subsemifourth]] temperament in the 7- and 11-limit. [[Essentially tempered chord]]s available in 407et include [[pinkanberry chords]].

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

	=== Subsets and supersets ===		=== Subsets and supersets ===
	407 factors into 11 × 37, with [[11edo]] and [[37edo]] as its subset edos.		407 factors into 11 × 37, with [[11edo]] and [[37edo]] as its subset edos. [[814edo]], which doubles it, gives a good correction to harmonics 7 and 17, and is a notable full 23-limit temperament.

	== 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\|-645 407}}		\| {{monzo\| -645 407 }}
	\|{{mapping\|407 645}}		\| {{mapping\| 407 645 }}
	\| 0.0742		\| +0.0742
	\| 0.0742		\| 0.0742
	\| 2.52		\| 2.52
	\|-		\|-
	\|2.3.5		\| 2.3.5
	\|32805/32768, {{monzo\|30 47 -45}}		\| 32805/32768, {{monzo\| 30 47 -45 }}
	\|{{mapping\|407 645 945}}		\| {{mapping\| 407 645 945 }}
	\| 0.0599		\| +0.0599
	\| 0.0638		\| 0.0638
	\| 2.16		\| 2.16
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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
	\|-		\|-
	\|1		\| 1
	\|63\407		\| 63\407
	\|185.75		\| 185.75
	\|{{monzo\|24 4 -13}}		\| {{monzo\| 24 4 -13 }}
	\|[[Pirate]]		\| [[Pirate]]
			\|-
			\| 1
			\| 83\407
			\| 244.72
			\| 15/13
			\| [[Subsemifourth]] (407f)
	\|-		\|-
	\|1		\| 1
	\|169\407		\| 169\407
	\|498.28		\| 498.28
	\|4/3		\| 4/3
	\|[[Helmholtz]]		\| [[Helmholtz]]
	\|}		\|}
			<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:Subsemifourth]]

Francium: Created page with "{{Infobox ET}} {{EDO intro|407}} == Theory == 407et tempers out 32805/32768 in the 5-limit; 4096000/4084101, 134217728/133984375, 26873856/26796875, 78125000/781218..."

2023-12-22T16:24:48Z

Created page with "{{Infobox ET}} {{EDO intro|407}} == Theory == 407et tempers out 32805/32768 in the 5-limit; 4096000/4084101, 134217728/133984375, 26873856/26796875, 78125000/781218..."

New page

{{Infobox ET}}
{{EDO intro|407}}

== Theory ==
407et tempers out [[32805/32768]] in the 5-limit; [[4096000/4084101]], 134217728/133984375, 26873856/26796875, [[78125000/78121827]] and 48828125/48771072 in the 7-limit. It supports the pinkanberry chords and the temperament [[subsemifourth]].

=== Prime harmonics ===
{{Harmonics in equal|407}}

=== Subsets and supersets ===
407 factors into 11 × 37, with [[11edo]] and [[37edo]] 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 8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo|-645 407}}
|{{mapping|407 645}}
| 0.0742
| 0.0742
| 2.52
|-
|2.3.5
|32805/32768, {{monzo|30 47 -45}}
|{{mapping|407 645 945}}
| 0.0599
| 0.0638
| 2.16
|}

=== 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
|63\407
|185.75
|{{monzo|24 4 -13}}
|[[Pirate]]
|-
|1
|169\407
|498.28
|4/3
|[[Helmholtz]]
|}

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