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

Revision as of 08:25, 20 October 2023

← 960edo 961edo 962edo →
Prime factorization 312
Step size 1.2487 ¢ 
Fifth 562\961 (701.769 ¢)
Semitones (A1:m2) 90:73 (112.4 ¢ : 91.16 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

The equal temperament 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 ⟨961 1523 2231 2698 3324] scores the best, which tempers out 102487/102400 and 234375/234256. It prompts us to consider the 961de val ⟨961 1523 2231 2697 3324], which tempers out 3025/3024 and 184877/184320. The patent val ⟨961 1523 2231 2698 3325] tempers out 4000/3993 and 46656/46585.

Odd harmonics

Approximation of odd harmonics in 961edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.186 -0.466 +0.165 -0.372 +0.607 -0.153 +0.597 -0.065 -0.323 -0.021 -0.179
Relative (%) -14.9 -37.3 +13.2 -29.8 +48.6 -12.3 +47.8 -5.2 -25.8 -1.7 -14.3
Steps
(reduced) 		1523
(562) 		2231
(309) 		2698
(776) 		3046
(163) 		3325
(442) 		3556
(673) 		3755
(872) 		3928
(84) 		4082
(238) 		4221
(377) 		4347
(503)

Subsets and supersets

Since 961 factors into 312, 961edo has 31edo as its subset edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-1523 961⟩ [⟨961 1523]] 0.0587 0.0587 4.70
2.3.5 32805/32768, [-22 -137 103⟩ [⟨961 1523 2231]] 0.1060 0.0823 6.59
2.3.5.7 4375/4374, 32805/32768, [15 9 14 -22⟩ [⟨961 1523 2231 2698]] 0.0648 0.1008 8.01

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio 		Temperaments
1 399\961 498.231 4/3 Pontiac

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Scales

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