961edo - 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"

2026-03-13T13:32:43Z

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 65:		Line 65:
	\| [[Pontiac]]		\| [[Pontiac]]
	\|}		\|}
	<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

	== Scales ==		== Scales ==

ArrowHead294 at 12:43, 21 February 2025

2025-02-21T12:43:08Z

← Older revision		Revision as of 12:43, 21 February 2025
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	== Theory ==		== Theory ==
	961edo has a reasonable 7-limit interpretation. The equal temperament [[Tempering out\|tempers out]] the [[schisma]] in the 5-limit; [[4375/4374]], [[65625/65536]], and [[14348907/14336000]] in the 7-limit, supporting [[pontiac]], the 395 & 566 temperament.		961edo has a reasonable 7-limit interpretation. The equal temperament [[Tempering out\|tempers out]] the [[schisma]] in the 5-limit; [[4375/4374]], [[65625/65536]], and [[14348907/14336000]] in the 7-limit, supporting [[pontiac]], the {{nowrap\|395 & 566}} temperament.

	In the 11-limit, the 961e [[val]] {{val\| 961 1523 2231 '''2698''' '''3324''' }} scores the best, which tempers out 102487/102400 and 234375/234256. It prompts us to consider the 961de val {{val\| 961 1523 2231 '''2697''' '''3324''' }}, which tempers out [[3025/3024]] and 184877/184320. The [[patent val]] {{val\| 961 1523 2231 '''2698''' '''3325''' }} tempers out [[4000/3993]] and 46656/46585.		In the 11-limit, the 961e [[val]] {{val\| 961 1523 2231 '''2698''' '''3324''' }} scores the best, which tempers out 102487/102400 and 234375/234256. It prompts us to consider the 961de val {{val\| 961 1523 2231 '''2697''' '''3324''' }}, which tempers out [[3025/3024]] and 184877/184320. The [[patent val]] {{val\| 961 1523 2231 '''2698''' '''3325''' }} tempers out [[4000/3993]] and 46656/46585.

Francium: changed EDO intro to ED intro

2025-02-21T06:04:02Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:04, 21 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|961~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 17:44, 15 January 2025

2025-01-15T17:44:19Z

← Older revision		Revision as of 17:44, 15 January 2025
Line 30:		Line 30:
	\| {{monzo\|-1523 961}}		\| {{monzo\|-1523 961}}
	\| {{mapping\| 961 1523 }}		\| {{mapping\| 961 1523 }}
	\| 0.0587		\| +0.0587
	\| 0.0587		\| 0.0587
	\| 4.70		\| 4.70
Line 37:		Line 37:
	\| 32805/32768, {{monzo\| -22 -137 103 }}		\| 32805/32768, {{monzo\| -22 -137 103 }}
	\| {{mapping\| 961 1523 2231 }}		\| {{mapping\| 961 1523 2231 }}
	\| 0.1060		\| +0.1060
	\| 0.0823		\| 0.0823
	\| 6.59		\| 6.59
Line 44:		Line 44:
	\| 4375/4374, 32805/32768, {{monzo\| 15 9 14 -22 }}		\| 4375/4374, 32805/32768, {{monzo\| 15 9 14 -22 }}
	\| {{mapping\| 961 1523 2231 2698 }}		\| {{mapping\| 961 1523 2231 2698 }}
	\| 0.0648		\| +0.0648
	\| 0.1008		\| 0.1008
	\| 8.01		\| 8.01
Line 65:		Line 65:
	\| [[Pontiac]]		\| [[Pontiac]]
	\|}		\|}
	<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

	== Scales ==		== Scales ==

ArrowHead294: Partial undo

2024-11-16T13:09:34Z

Partial undo

← Older revision		Revision as of 13:09, 16 November 2024
Line 16:		Line 16:

	== 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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	\| 0.1008		\| 0.1008
	\| 8.01		\| 8.01
	~~{{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 48:		Line 64:
	\| 4/3		\| 4/3
	\| [[Pontiac]]		\| [[Pontiac]]
	~~{{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

	== Scales ==		== Scales ==

ArrowHead294 at 02:13, 16 November 2024

2024-11-16T02:13:43Z

← Older revision		Revision as of 02:13, 16 November 2024
Line 16:		Line 16:

	== 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 46:		Line 38:
	\| 0.1008		\| 0.1008
	\| 8.01		\| 8.01
	\|}		{{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 62:		Line 48:
	\| 4/3		\| 4/3
	\| [[Pontiac]]		\| [[Pontiac]]
	\|}		{{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}}

	== Scales ==		== Scales ==

FloraC: Adopt template: Factorization; misc. cleanup

2023-11-02T11:20:57Z

Adopt template: Factorization; misc. cleanup

← Older revision		Revision as of 11:20, 2 November 2023
Line 13:		Line 13:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 961 factors into ~~31<sup>2</sup>~~, 961edo has [[31edo]] as its subset edo.		Since 961 factors into {{factorization\|961}}, 961edo has [[31edo]] as its subset edo.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 54:		Line 54:
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-

FloraC: /* Theory */ expand

2023-10-20T08:34:49Z

Theory: expand

← Older revision		Revision as of 08:34, 20 October 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	The equal temperament [[Tempering out\|tempers out]] [[~~32805/32768~~]] in the 5-limit; [[4375/4374]], [[65625/65536]], and [[14348907/14336000]] in the 7-limit. In the 11-limit, the 961e [[val]] {{val\| 961 1523 2231 '''2698''' '''3324''' }} scores the best, which tempers out 102487/102400 and 234375/234256. It prompts us to consider the 961de val {{val\| 961 1523 2231 '''2697''' '''3324''' }}, which tempers out [[3025/3024]] and 184877/184320. The [[patent val]] {{val\| 961 1523 2231 '''2698''' '''3325''' }} tempers out [[4000/3993]] and 46656/46585.		961edo has a reasonable 7-limit interpretation. The equal temperament [[Tempering out\|tempers out]] the [[schisma]] in the 5-limit; [[4375/4374]], [[65625/65536]], and [[14348907/14336000]] in the 7-limit, supporting [[pontiac]], the 395 & 566 temperament.

			In the 11-limit, the 961e [[val]] {{val\| 961 1523 2231 '''2698''' '''3324''' }} scores the best, which tempers out 102487/102400 and 234375/234256. It prompts us to consider the 961de val {{val\| 961 1523 2231 '''2697''' '''3324''' }}, which tempers out [[3025/3024]] and 184877/184320. The [[patent val]] {{val\| 961 1523 2231 '''2698''' '''3325''' }} tempers out [[4000/3993]] and 46656/46585.

			It works much better as a 2.3.5.7.13.17 [[subgroup temperament]], in which case it tempers out [[10985/10976]], [[1275/1274]], [[2025/2023]] and [[4914/4913]].

	=== Odd harmonics ===		=== Odd harmonics ===

FloraC: Rework theory; comma bases; formatting; clarify the title row of the rank-2 temp table

2023-10-20T08:25:02Z

Rework theory; comma bases; formatting; clarify the title row of the rank-2 temp table

← Older revision		Revision as of 08:25, 20 October 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|961}}		{{EDO intro\|961}}
	==Theory==
	~~961et~~ tempers out [[32805/32768]] in the 5-limit; [[~~14348907~~/~~14336000~~]], [[~~4375~~/~~4374~~]] and [[~~65625~~/~~65536~~]] in the 7-limit~~; 1019215872~~/~~1019046875~~, ~~2097152~~/~~2096325, 26214400~~/~~26198073, 5767168~~/~~5764801~~ and ~~3294225~~/~~3294172 in the 11-limit~~.		== Theory ==
	===Odd harmonics===		The equal temperament [[Tempering out\|tempers out]] [[32805/32768]] in the 5-limit; [[4375/4374]], [[65625/65536]], and [[14348907/14336000]] in the 7-limit. In the 11-limit, the 961e [[val]] {{val\| 961 1523 2231 '''2698''' '''3324''' }} scores the best, which tempers out 102487/102400 and 234375/234256. It prompts us to consider the 961de val {{val\| 961 1523 2231 '''2697''' '''3324''' }}, which tempers out [[3025/3024]] and 184877/184320. The [[patent val]] {{val\| 961 1523 2231 '''2698''' '''3325''' }} tempers out [[4000/3993]] and 46656/46585.

			=== Odd harmonics ===
	{{Harmonics in equal\|961}}		{{Harmonics in equal\|961}}
	===Subsets and supersets===
	961 factors into 31<sup>2</sup> ~~with~~ [[31edo]] as subset edo.		=== Subsets and supersets ===
	==Regular temperament properties==		Since 961 factors into 31<sup>2</sup>, 961edo has [[31edo]] as its subset edo.

			== 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\|-1523 961}}		\| {{monzo\|-1523 961}}
	\|{{~~val~~\|961 1523}}		\| {{mapping\| 961 1523 }}
	\| 0.0587		\| 0.0587
	\| 0.0587		\| 0.0587
	\| 4.70		\| 4.70
	\|-		\|-
	\|2.3.5		\| 2.3.5
	\|32805/32768, {{monzo\|-22 -137 103}}		\| 32805/32768, {{monzo\| -22 -137 103 }}
	\|{{~~val~~\|961 1523 2231}}		\| {{mapping\| 961 1523 2231 }}
	\| 0.1060		\| 0.1060
	\| 0.0823		\| 0.0823
	\| 6.59		\| 6.59
	\|-		\|-
	\|2.3.5.7		\| 2.3.5.7
	\|4375/4374, 32805/32768, ~~65625/65536~~		\| 4375/4374, 32805/32768, {{monzo\| 15 9 14 -22 }}
	\|{{~~val~~\|961 1523 2231 2698}}		\| {{mapping\| 961 1523 2231 2698 }}
	\| 0.0648		\| 0.0648
	\| 0.1008		\| 0.1008
	\| 8.01		\| 8.01
	\|}		\|}

	=== 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		\|+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
	\|399\961		\| 399\961
	\|498.231		\| 498.231
	\|4/3		\| 4/3
	\|~~[[Helmholtz]] /~~ [[Pontiac]]		\| [[Pontiac]]
	\|}		\|}
	==Scales==		<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

			== Scales ==
	* [[Haumea5]]		* [[Haumea5]]
	* [[Haumea9]]		* [[Haumea9]]
	* [[Haumea14]]		* [[Haumea14]]
	* [[Haumea19]]		* [[Haumea19]]

Francium: Created page with "{{Infobox ET}} {{EDO intro|961}} ==Theory== 961et tempers out 32805/32768 in the 5-limit; 14348907/14336000, 4375/4374 and 65625/65536 in the 7-limit; 10192158..."

2023-10-02T15:03:26Z

Created page with "{{Infobox ET}} {{EDO intro|961}} ==Theory== 961et tempers out 32805/32768 in the 5-limit; 14348907/14336000, 4375/4374 and 65625/65536 in the 7-limit; 10192158..."

New page

{{Infobox ET}}
{{EDO intro|961}}
==Theory==
961et tempers out [[32805/32768]] in the 5-limit; [[14348907/14336000]], [[4375/4374]] and [[65625/65536]] in the 7-limit; 1019215872/1019046875, 2097152/2096325, 26214400/26198073, 5767168/5764801 and 3294225/3294172 in the 11-limit.
===Odd harmonics===
{{Harmonics in equal|961}}
===Subsets and supersets===
961 factors into 312 with [[31edo]] as subset edo.
==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|-1523 961}}
|{{val|961 1523}}
| 0.0587
| 0.0587
| 4.70
|-
|2.3.5
|32805/32768, {{monzo|-22 -137 103}}
|{{val|961 1523 2231}}
| 0.1060
| 0.0823
| 6.59
|-
|2.3.5.7
|4375/4374, 32805/32768, 65625/65536
|{{val|961 1523 2231 2698}}
| 0.0648
| 0.1008
| 8.01
|}
=== 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
|399\961
|498.231
|4/3
|[[Helmholtz]] / [[Pontiac]]
|}
==Scales==
* [[Haumea5]]
* [[Haumea9]]
* [[Haumea14]]
* [[Haumea19]]