961edo: Difference between revisions

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== 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 ===

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=== Subsets and supersets ===

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

Since 961 factors into {{factorization|961}}, 961edo has [[31edo]] as its subset edo.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

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| {{monzo|-1523 961}}

| {{mapping| 961 1523 }}

| 0.0587

| +0.0587

| 0.0587

| 4.70

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| 32805/32768, {{monzo| -22 -137 103 }}

| {{mapping| 961 1523 2231 }}

| 0.1060

| +0.1060

| 0.0823

| 6.59

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| 4375/4374, 32805/32768, {{monzo| 15 9 14 -22 }}

| {{mapping| 961 1523 2231 2698 }}

| 0.0648

| +0.0648

| 0.1008

| 8.01

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=== Rank-2 temperaments ===

{| 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 per 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

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| [[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 ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|961}}
+{{ED intro}}
 == 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.
+edo 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 &amp; 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 ===
@@ Line 9: / Line 13: @@
 === 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 ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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 simple badness|Relative]] (%)
@@ Line 25: / Line 30: @@
 | {{monzo|-1523 961}}
 | {{mapping| 961 1523 }}
-| 0.0587
+| +0.0587
 | 0.0587
 | 4.70
@@ Line 32: / Line 37: @@
 | 32805/32768, {{monzo| -22 -137 103 }}
 | {{mapping| 961 1523 2231 }}
-| 0.1060
+| +0.1060
 | 0.0823
 | 6.59
@@ Line 39: / Line 44: @@
 | 4375/4374, 32805/32768, {{monzo| 15 9 14 -22 }}
 | {{mapping| 961 1523 2231 2698 }}
-| 0.0648
+| +0.0648
 | 0.1008
 | 8.01
@@ Line 46: / Line 51: @@
 === Rank-2 temperaments ===
 {| 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*
 ! Cents*
-! Associated<br>Ratio
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 59: / Line 65: @@
 | [[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 ==