961edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|961}}
{{ED intro}}


== 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 {{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.
 
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 ===  
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=== 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 ==
{| 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]]
! 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]] (%)
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| {{monzo|-1523 961}}
| {{monzo|-1523 961}}
| {{mapping| 961 1523 }}
| {{mapping| 961 1523 }}
| 0.0587
| +0.0587
| 0.0587
| 0.0587
| 4.70
| 4.70
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| 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
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| 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
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=== 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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| [[Pontiac]]
| [[Pontiac]]
|}
|}
<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 />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==

Latest revision as of 12:43, 21 February 2025

← 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

961 equal divisions of the octave (abbreviated 961edo or 961ed2), also called 961-tone equal temperament (961tet) or 961 equal temperament (961et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 961 equal parts of about 1.25 ¢ each. Each step represents a frequency ratio of 21/961, or the 961st root of 2.

Theory

961edo has a reasonable 7-limit interpretation. The equal temperament 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 ⟨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.

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

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 distinct

Scales

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