405edo - 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"

2026-03-13T13:33:06Z

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
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	\| [[Novemkleismic]]		\| [[Novemkleismic]]
	\|}		\|}
	<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

2025-02-21T06:25:56Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:25, 21 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|405~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 17:03, 15 January 2025

2025-01-15T17:03:03Z

← Older revision		Revision as of 17:03, 15 January 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 405 factors into ~~3<sup>4</sup> × 5~~, 405edo has subset edos {{EDOs\| 3, 5, 9, 15, 27, 45, 81, and 135 }}.		Since 405 factors into {{factorisation\|405}}, 405edo has subset edos {{EDOs\| 3, 5, 9, 15, 27, 45, 81, and 135 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 73:		Line 73:
	\| [[Novemkleismic]]		\| [[Novemkleismic]]
	\|}		\|}
	<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

2024-11-16T13:08:30Z

Partial undo

← Older revision		Revision as of 13:08, 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
Line 34:		Line 43:
	\| 0.2207		\| 0.2207
	\| 7.45		\| 7.45
	~~{{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 56:		Line 72:
	\| 6/5<br />(36/35)		\| 6/5<br />(36/35)
	\| [[Novemkleismic]]		\| [[Novemkleismic]]
	~~{{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 02:27, 16 November 2024

2024-11-16T02:27:28Z

← Older revision		Revision as of 02:27, 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
Line 42:		Line 34:
	\| 0.2207		\| 0.2207
	\| 7.45		\| 7.45
	\|}		{{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
Line 66:		Line 52:
	\|-		\|-
	\| 9		\| 9
	\| 107\405<br>(17\405)		\| 107\405<br />(17\405)
	\| 317.04<br>(50.37)		\| 317.04<br />(50.37)
	\| 6/5<br>(36/35)		\| 6/5<br />(36/35)
	\| [[Novemkleismic]]		\| [[Novemkleismic]]
	\|}		{{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

2024-01-19T14:48:46Z

Review

← Older revision		Revision as of 14:48, 19 January 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	~~405et~~ is consistent to the [[7-odd-limit]]~~. Using~~ the ~~patent val, it tempers out~~ [[~~4096000~~/~~4084101~~]]~~, 2460375/2458624,~~ [[~~33554432/33480783~~]], [[~~2100875/2097152~~]] ~~and~~ [[~~40353607~~/~~40310784~~]] in the 7-limit; ~~56723625/56689952, 55296000/55240493,~~ [[~~117440512~~/~~117406179~~]], ~~820125~~/~~819896~~, [[4000/3993]]~~, 78675968/78594219, 7168000/7144929, 496125/495616, 117649/117612, 1296000/1294139, [[131072/130977]], 1265625/1261568, 42875/42768, 5767168/5764801, 1375/1372~~, [[19712/19683]], [[41503/41472]]~~, 199297406/199290375, 1362944/1361367, 166375/165888 and 67110351/67108864~~ in the 11-limit. It ~~supports~~ [[marthirds]], [[novemkleismic]] and [[kleirtismic]]. It provides the [[optimal patent val]] for novemkleismic.		405edo is [[enfactoring\|enfactored]] in the 3-limit, with the same tuning as [[135edo]]. Like 135edo, it is [[consistent]] to the [[7-odd-limit]] with a poor approximation to the [[harmonic]] [[5/1\|5]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] [[15625/15552]] in the 5-limit; [[2100875/2097152]], and 2460375/2458624 in the 7-limit; 1375/1372, [[4000/3993]], [[19712/19683]], and [[41503/41472]] in the 11-limit. It [[support]]s [[marthirds]], [[novemkleismic]] and [[kleirtismic]]. It provides the [[optimal patent val]] for 7- and 11-limit novemkleismic.

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

	=== Subsets and supersets ===		=== Subsets and supersets ===
	405 factors into 3<sup>4</sup> × 5, ~~with~~ subset edos {{EDOs\|3, 5, 9, 15, 27, 45, 81, and 135}}.		Since 405 factors into 3<sup>4</sup> × 5, 405edo has subset edos {{EDOs\| 3, 5, 9, 15, 27, 45, 81, and 135 }}.

	== 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\|214 -135}}~~		\| 15625/15552, {{monzo\| 110 -65 -3 }}
	~~\|{{mapping\|405 642}}~~		\| {{mapping\| 405 642 940 }}
	~~\| -0.0843~~
	~~\| 0.0843~~
	~~\| 2.85~~
	\|-
	\|2.3.5
	\|15625/15552, {{monzo\|110 -65 -3}}
	\|{{mapping\|405 642 940}}
	\| +0.1058		\| +0.1058
	\| 0.2776		\| 0.2776
	\| 9.37		\| 9.37
	\|-		\|-
	\|2.3.5.7		\| 2.3.5.7
	\|15625/15552, 2460375/2458624~~, 4096000/4084101~~		\| 15625/15552, 2100875/2097152, 2460375/2458624
	\|{{mapping\|405 642 940 1137}}		\| {{mapping\| 405 642 940 1137 }}
	\| +0.0737		\| +0.0737
	\| 0.2467		\| 0.2467
	\| 8.33		\| 8.33
	\|-		\|-
	\|2.3.5.7.11		\| 2.3.5.7.11
	\|1375/1372, ~~42875~~/~~42768~~, 19712/19683, 41503/41472		\| 1375/1372, 4000/3993, 19712/19683, 41503/41472
	\|{{mapping\|405 642 940 1137 1401}}		\| {{mapping\| 405 642 940 1137 1401 }}
	\| +0.0709		\| +0.0709
	\| 0.2207		\| 0.2207
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	! Temperaments		! Temperaments
	\|-		\|-
	\|1		\| 1
	\|56\405		\| 56\405
	\|165.93		\| 165.93
	\|~~8575~~/~~7776~~		\| 11/10
	\|[[Satin]]		\| [[Satin]]
	\|-		\|-
	\|1		\| 1
	\|107\405		\| 107\405
	\|317.04		\| 317.04
	\|6/5		\| 6/5
	\|[[Hanson]]		\| [[Hanson]]
	\|-		\|-
	\|9		\| 9
	\|107\405<br>(17\405)		\| 107\405<br>(17\405)
	\|317.04<br>(50.37)		\| 317.04<br>(50.37)
	\|6/5<br>(36/35)		\| 6/5<br>(36/35)
	\|[[Novemkleismic]]		\| [[Novemkleismic]]
	\|}		\|}

	<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|405}} == Theory == 405et is consistent to the 7-odd-limit. Using the patent val, it tempers out 4096000/4084101, 2460375/2458624, 3355443..."

2024-01-10T21:13:16Z

Created page with "{{Infobox ET}} {{EDO intro|405}} == Theory == 405et is consistent to the 7-odd-limit. Using the patent val, it tempers out 4096000/4084101, 2460375/2458624, 3355443..."

New page

{{Infobox ET}}
{{EDO intro|405}}

== Theory ==
405et is consistent to the [[7-odd-limit]]. Using the patent val, it tempers out [[4096000/4084101]], 2460375/2458624, [[33554432/33480783]], [[2100875/2097152]] and [[40353607/40310784]] in the 7-limit; 56723625/56689952, 55296000/55240493, [[117440512/117406179]], 820125/819896, [[4000/3993]], 78675968/78594219, 7168000/7144929, 496125/495616, 117649/117612, 1296000/1294139, [[131072/130977]], 1265625/1261568, 42875/42768, 5767168/5764801, 1375/1372, [[19712/19683]], [[41503/41472]], 199297406/199290375, 1362944/1361367, 166375/165888 and 67110351/67108864 in the 11-limit. It supports [[marthirds]], [[novemkleismic]] and [[kleirtismic]]. It provides the [[optimal patent val]] for novemkleismic.

=== Prime harmonics ===
{{Harmonics in equal|405}}

=== Subsets and supersets ===
405 factors into 34 × 5, with subset edos {{EDOs|3, 5, 9, 15, 27, 45, 81, and 135}}.

== 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|214 -135}}
|{{mapping|405 642}}
| -0.0843
| 0.0843
| 2.85
|-
|2.3.5
|15625/15552, {{monzo|110 -65 -3}}
|{{mapping|405 642 940}}
| +0.1058
| 0.2776
| 9.37
|-
|2.3.5.7
|15625/15552, 2460375/2458624, 4096000/4084101
|{{mapping|405 642 940 1137}}
| +0.0737
| 0.2467
| 8.33
|-
|2.3.5.7.11
|1375/1372, 42875/42768, 19712/19683, 41503/41472
|{{mapping|405 642 940 1137 1401}}
| +0.0709
| 0.2207
| 7.45
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods per 8ve
! Generator*
! Cents*
! Associated Ratio*
! Temperaments
|-
|1
|56\405
|165.93
|8575/7776
|[[Satin]]
|-
|1
|107\405
|317.04
|6/5
|[[Hanson]]
|-
|9
|107\405 (17\405)
|317.04 (50.37)
|6/5 (36/35)
|[[Novemkleismic]]
|}

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