412edo - 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"

2026-03-13T13:31:45Z

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 13:31, 13 March 2026
Line 79:		Line 79:
	\| [[Septisuperfourth]] (7-limit)		\| [[Septisuperfourth]] (7-limit)
	\|}		\|}
	<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

2025-02-21T06:23:21Z

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~~\|412~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 16:59, 15 January 2025

2025-01-15T16:59:59Z

← Older revision		Revision as of 16:59, 15 January 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	412 factors into ~~2<sup>2</sup> × 103~~, with subset edos {{EDOs\|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to harmonics 5, 7, and 11.		412 factors into {{factorisation\|412}}, with subset edos {{EDOs\|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to harmonics 5, 7, and 11.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 33:		Line 33:
	\| {{monzo\| 32 -7 -9 }}, {{monzo\| -5 31 -19 }}		\| {{monzo\| 32 -7 -9 }}, {{monzo\| -5 31 -19 }}
	\| {{mapping\| 412 653 957 }}		\| {{mapping\| 412 653 957 }}
	\| ~~−0~~.1501		\| −0.1501
	\| 0.2182		\| 0.2182
	\| 7.49		\| 7.49
Line 40:		Line 40:
	\| 6144/6125, 2460375/2458624, 49009212/48828125		\| 6144/6125, 2460375/2458624, 49009212/48828125
	\| {{mapping\| 412 653 957 1157 }}		\| {{mapping\| 412 653 957 1157 }}
	\| ~~−0~~.2085		\| −0.2085
	\| 0.2143		\| 0.2143
	\| 7.36		\| 7.36
Line 79:		Line 79:
	\| [[Septisuperfourth]] (7-limit)		\| [[Septisuperfourth]] (7-limit)
	\|}		\|}
	<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

2024-11-16T13:12:41Z

Partial undo

← Older revision		Revision as of 13:12, 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		\| 2.3
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	\| 0.2143		\| 0.2143
	\| 7.36		\| 7.36
	~~{{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
Line 62:		Line 78:
	\| 16875/16384		\| 16875/16384
	\| [[Septisuperfourth]] (7-limit)		\| [[Septisuperfourth]] (7-limit)
	~~{{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 01:57, 16 November 2024

2024-11-16T01:57:33Z

← Older revision		Revision as of 01:57, 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 32:		Line 24:
	\| {{monzo\| 32 -7 -9 }}, {{monzo\| -5 31 -19 }}		\| {{monzo\| 32 -7 -9 }}, {{monzo\| -5 31 -19 }}
	\| {{mapping\| 412 653 957 }}		\| {{mapping\| 412 653 957 }}
	\| -0.1501		\| −0.1501
	\| 0.2182		\| 0.2182
	\| 7.49		\| 7.49
Line 39:		Line 31:
	\| 6144/6125, 2460375/2458624, 49009212/48828125		\| 6144/6125, 2460375/2458624, 49009212/48828125
	\| {{mapping\| 412 653 957 1157 }}		\| {{mapping\| 412 653 957 1157 }}
	\| -0.2085		\| −0.2085
	\| 0.2143		\| 0.2143
	\| 7.36		\| 7.36
	\|}		{{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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	\| 16875/16384		\| 16875/16384
	\| [[Septisuperfourth]] (7-limit)		\| [[Septisuperfourth]] (7-limit)
	\|}		{{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: Review

2024-01-01T10:34:09Z

Review

← Older revision		Revision as of 10:34, 1 January 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	~~412et~~ tempers out ~~2460375/2458624,~~ [[6144/6125]], ~~102760448~~/~~102515625~~, [[~~1640558367/1638400000~~]] ~~and 200120949/200000000~~ in the 7-limit. It supports [[nanic]] and [[counterschismic]].		412edo has a very accurate [[3/2\|perfect fifth]], but it is not quite accurate beyond that. The equal temperament [[tempering out\|tempers out]] {{monzo\| 32 -7 -9 }} ([[escapade comma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the 5-limit; [[6144/6125]], 118098/117649, 2460375/2458624, 49009212/48828125, and notably the [[nanisma]] in the 7-limit. It supports [[nanic]] and [[counterschismic]].

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

	=== Subsets and supersets ===		=== Subsets and supersets ===
	412 factors into 2<sup>2</sup> × 103, with subset edos {{EDOs\|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to ~~the harmonic~~ 11.		412 factors into 2<sup>2</sup> × 103, with subset edos {{EDOs\|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to harmonics 5, 7, and 11.

	== 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\|-653 412}}		\| {{monzo\|-653 412}}
	\|{{mapping\|412 653}}		\| {{mapping\| 412 653 }}
	\| +0.0042		\| +0.0042
	\| 0.0042		\| 0.0042
	\| 0.14		\| 0.14
	\|-		\|-
	\|2.3.5		\| 2.3.5
	\|{{monzo\|32 -7 -9}}, {{monzo\|-5 31 -19}}		\| {{monzo\| 32 -7 -9 }}, {{monzo\| -5 31 -19 }}
	\|{{mapping\|412 653 957}}		\| {{mapping\| 412 653 957 }}
	\| -0.1501		\| -0.1501
	\| 0.2182		\| 0.2182
	\| 7.49		\| 7.49
	\|-		\|-
	\|2.3.5.7		\| 2.3.5.7
	\|6144/6125, 2460375/2458624, ~~100442349~~/~~100000000~~		\| 6144/6125, 2460375/2458624, 49009212/48828125
	\|{{mapping\|412 653 957 1157}}		\| {{mapping\| 412 653 957 1157 }}
	\| -0.2085		\| -0.2085
	\| 0.2143		\| 0.2143
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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
	\|9\412		\| 9\412
	\|26.21		\| 26.21
	\|49/48		\| 49/48
	\|[[Sfourth]]		\| [[Sfourth]] (5-limit)
	\|-		\|-
	\|1		\| 1
	\|19\412		\| 19\412
	\|55.34		\| 55.34
	\|16875/16384		\| 16875/16384
	\|[[Escapade]]		\| [[Escapade]] (5-limit)
	\|-		\|-
	\|1		\| 1
	\|171\412		\| 171\412
	\|498.06		\| 498.06
	\|4/3		\| 4/3
	\|[[Counterschismic]]		\| [[Counterschismic]]<br>[[Nanic]]
	\|-		\|-
	\|2		\| 2
	\|19\412		\| 19\412
	\|55.34		\| 55.34
	\|16875/16384		\| 16875/16384
	\|[[Septisuperfourth]]		\| [[Septisuperfourth]] (7-limit)
	\|}		\|}

	<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: /* Rank-2 temperaments */ typo

2023-12-22T13:36:30Z

Rank-2 temperaments: typo

← Older revision		Revision as of 13:36, 22 December 2023
Line 75:		Line 75:
	\|55.34		\|55.34
	\|16875/16384		\|16875/16384
	\|[[~~Semisuperfourth~~]]		\|[[Septisuperfourth]]
	\|}		\|}

	<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|412}} == Theory == 412et tempers out 2460375/2458624, 6144/6125, 102760448/102515625, 1640558367/1638400000 and 200120949/200000000 in the..."

2023-12-22T13:35:14Z

Created page with "{{Infobox ET}} {{EDO intro|412}} == Theory == 412et tempers out 2460375/2458624, 6144/6125, 102760448/102515625, 1640558367/1638400000 and 200120949/200000000 in the..."

New page

{{Infobox ET}}
{{EDO intro|412}}

== Theory ==
412et tempers out 2460375/2458624, [[6144/6125]], 102760448/102515625, [[1640558367/1638400000]] and 200120949/200000000 in the 7-limit. It supports [[nanic]] and [[counterschismic]].

=== Prime harmonics ===
{{Harmonics in equal|412}}

=== Subsets and supersets ===
412 factors into 22 × 103, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to the harmonic 11.

== 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|-653 412}}
|{{mapping|412 653}}
| +0.0042
| 0.0042
| 0.14
|-
|2.3.5
|{{monzo|32 -7 -9}}, {{monzo|-5 31 -19}}
|{{mapping|412 653 957}}
| -0.1501
| 0.2182
| 7.49
|-
|2.3.5.7
|6144/6125, 2460375/2458624, 100442349/100000000
|{{mapping|412 653 957 1157}}
| -0.2085
| 0.2143
| 7.36
|}

=== 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
|9\412
|26.21
|49/48
|[[Sfourth]]
|-
|1
|19\412
|55.34
|16875/16384
|[[Escapade]]
|-
|1
|171\412
|498.06
|4/3
|[[Counterschismic]]
|-
|2
|19\412
|55.34
|16875/16384
|[[Semisuperfourth]]
|}

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