Keenanismic family: Difference between revisions

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~~__FORCETOC__~~

The '''keenanismic family''' of rank ~~four~~ temperaments tempers out the [[keenanisma~~|keenanisma (385/384)~~]], the 11-limit comma 385/384 = {{~~Monzo~~| -7 -1 1 1 1 }} of 4.503 ~~cents~~.

The '''keenanismic family''' of [[Catalog of rank-4 temperaments|rank-4]] temperaments [[Tempering out|tempers out]] the [[keenanisma]], the 11-limit comma 385/384 = {{monzo| -7 -1 1 1 1 }} of 4.503 [[cent]]s.

== Keenanismic ==

[[Subgroup]]: 2.3.5.7.11

Comma: 385/384

[[Comma list]]: 385/384

~~Minimax tuning~~

{{Mapping|legend=1| 1 0 0 0 7 | 0 1 0 0 1 | 0 0 1 0 -1 | 0 0 0 1 -1 }}

~~[|1 0 0 0 0>~~, ~~|0 1 0 0 0>, |7/3 1/~~3 ~~2/3 -1/3 -1/3>~~,

: mapping generators: ~2, ~3, ~5, ~7

~~|7/3 1/3 -1/3 2/3 -1/3>~~, |7~~/3 1/3 -1/3 -1/3 2/3>]~~

~~Eigenmonzos~~: 2, 3, 7/~~5, 11/5~~

[[Transpose]]: [2 3 5 7 385/35]

~~Map~~: [~~<~~1 0 0 0 7|~~, <~~0 1 0 0 1|~~, <0 0~~ 1 0 -1|~~, <0 0 0~~ 1 -1|]

[[Minimax tuning]]:

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 7/3 1/3 2/3 -1/3 -1/3 }}, {{monzo| 7/3 1/3 -1/3 2/3 -1/3 }}, {{monzo| 7/3 1/3 -1/3 -1/3 2/3 }}]

: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5

~~Transpose: [2 3 5 7 385/35]~~

{{Optimal ET sequence|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}

~~EDOs~~: ~~{{EDOs| 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212 and 284 }}~~

[[Badness]]: 15.2 × 10-9

~~10^9*Badness~~: 15.~~159~~

== Martwin ==

[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: 325/324, 385/384

{{Mapping|legend=1| 1 0 0 0 7 2 | 0 1 0 0 1 4 | 0 0 1 0 -1 -2 | 0 0 0 1 -1 0 }}

[[Transpose]]: [2 3 5 7 385/35 324/25]

[[Minimax tuning]]: {{confirm}}

: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 2/3 4/3 1/3 0 0 -1/3 }}, {{monzo| 17/3 -5/3 1/3 0 -1 2/3 }}, {{monzo| 2/3 4/3 -2/3 0 1 -1/3 }}, {{monzo| 2/3 4/3 -2/3 0 0 2/3 }}]

: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.11/5.13/5

=~~= Martwin ==~~

{{Optimal ET sequence|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}

~~Commas~~: ~~385~~/~~384, 325/324~~

[[Badness]]: 2.21 × 10-6

~~Minimax tuning~~

== Ancient ==

~~||1 0 0 0 0 0>, |0 1 0 0 0 0>, |~~2~~/3 4/~~3 ~~1/3 0 0 -1/3>,~~

[[Subgroup]]: 2.3.5.7.11.13

~~|17/3 -~~5~~/3 1/3 0 -1 2/3>, |2/3 4/3 -2/3 0 1 -1/3>, |2/3 4/3 -2/3 0 0 2/3>]~~

~~Eigenmonzos~~: ~~2, 3, 11~~/5, 13/5

[[Comma list]]: 385/384, 625/624

~~Map: [<~~1 0 0 0 7 2|~~, <~~0 1 0 0 1 4|~~, <~~0 0 1 0 -1 -2|~~, <~~0 0 0 1 -1 0|]

{{Mapping|legend=1| 1 0 0 0 7 -4 | 0 1 0 0 1 -1 | 0 0 1 0 -1 4 | 0 0 0 1 -1 0 }}

Transpose: [2 3 5 7 385/35 ~~324~~/25]

[[Transpose]]: [2 3 5 7 385/35 625/48]

~~EDOs:~~ {{~~EDOs~~| 7, 15, 19, ~~26, 34~~, 41, 46, 53, 72, 87, 140, 159, ~~212 and 299~~ }}

{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}

10^6*Badness: 2.~~206~~

[[Badness]]: 2.57 × 10-6

== ~~Ancient~~ ==

== Zaxa ==

[[Subgroup]]: 2.3.5.7.11.13

~~Commas~~: 385/384~~, 625/624~~

[[Comma list]]: 352/351, 385/384

~~Map: [<~~1 0 0 0 7 -4|~~, <~~0 1 0 0 1 -1|~~, <~~0 0 1 0 -1 4|~~, <~~0 0 0 1 -1 ~~0|]~~

{{Mapping|legend=1| 1 0 0 0 7 12 |0 1 0 0 1 -2 | 0 0 1 0 -1 -1 | 0 0 0 1 -1 -1 }}

~~Transpose~~: [2 3 5 7 ~~385/35 625~~/~~48]~~

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.657, ~5/4 = 385.632, ~7/4 = 967.829

~~EDOs:~~ {{~~EDOs~~| 6, 15, 19, 22, 31, 50, 53, 72, 87, ~~103~~, 140, ~~159~~, ~~190~~ }}

{{Optimal ET sequence|legend=1| 7, 10, 15f, 17, 19f, 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}

10^6*Badness: 2.~~573~~

[[Badness]]: 3.35 × 10-6

[[Category:Temperament]]

[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Keenanismic family| ]]

[[Category:Keenanismic| ]]

[[Category:Rank 4]]

~~[[Category:Keenanismic]]~~

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-__FORCETOC__
+{{Technical data page}}
-The '''keenanismic family''' of rank four temperaments tempers out the [[keenanisma|keenanisma (385/384)]], the 11-limit comma 385/384 = {{Monzo| -7 -1 1 1 1 }} of 4.503 cents.
+The '''keenanismic family''' of [[Catalog of rank-4 temperaments|rank-4]] temperaments [[Tempering out|tempers out]] the [[keenanisma]], the 11-limit comma 385/384 = {{monzo| -7 -1 1 1 1 }} of 4.503 [[cent]]s.
 == Keenanismic ==
+[[Subgroup]]: 2.3.5.7.11
-Comma: 385/384
+[[Comma list]]: 385/384
-Minimax tuning
+{{Mapping|legend=1| 1 0 0 0 7 | 0 1 0 0 1 | 0 0 1 0 -1 | 0 0 0 1 -1 }}
-[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |7/3 1/3 2/3 -1/3 -1/3&gt;,
+: mapping generators: ~2, ~3, ~5, ~7
-|7/3 1/3 -1/3 2/3 -1/3&gt;, |7/3 1/3 -1/3 -1/3 2/3&gt;]
-Eigenmonzos: 2, 3, 7/5, 11/5
+[[Transpose]]: [2 3 5 7 385/35]
-Map: [&lt;1 0 0 0 7|, &lt;0 1 0 0 1|, &lt;0 0 1 0 -1|, &lt;0 0 0 1 -1|]
+[[Minimax tuning]]:
+* [[11-odd-limit]]
+: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 7/3 1/3 2/3 -1/3 -1/3 }}, {{monzo| 7/3 1/3 -1/3 2/3 -1/3 }}, {{monzo| 7/3 1/3 -1/3 -1/3 2/3 }}]
+: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5
-Transpose: [2 3 5 7 385/35]
+{{Optimal ET sequence|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}
-EDOs: {{EDOs| 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212 and 284 }}
+[[Badness]]: 15.2 × 10<sup>-9</sup>
-^9*Badness: 15.159
+== Martwin ==
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 325/324, 385/384
+{{Mapping|legend=1| 1 0 0 0 7 2 | 0 1 0 0 1 4 | 0 0 1 0 -1 -2 | 0 0 0 1 -1 0 }}
+[[Transpose]]: [2 3 5 7 385/35 324/25]
+[[Minimax tuning]]: {{confirm}} <!-- contradiction with the data in Rank-4 temperament -->
+: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 2/3 4/3 1/3 0 0 -1/3 }}, {{monzo| 17/3 -5/3 1/3 0 -1 2/3 }}, {{monzo| 2/3 4/3 -2/3 0 1 -1/3 }}, {{monzo| 2/3 4/3 -2/3 0 0 2/3 }}]
+: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.11/5.13/5
-== Martwin ==
+{{Optimal ET sequence|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}
-Commas: 385/384, 325/324
+[[Badness]]: 2.21 × 10<sup>-6</sup>
-Minimax tuning
+== Ancient ==
-||1 0 0 0 0 0&gt;, |0 1 0 0 0 0&gt;, |2/3 4/3 1/3 0 0 -1/3&gt;,
+[[Subgroup]]: 2.3.5.7.11.13
-|17/3 -5/3 1/3 0 -1 2/3&gt;, |2/3 4/3 -2/3 0 1 -1/3&gt;, |2/3 4/3 -2/3 0 0 2/3&gt;]
-Eigenmonzos: 2, 3, 11/5, 13/5
+[[Comma list]]: 385/384, 625/624
-Map: [&lt;1 0 0 0 7 2|, &lt;0 1 0 0 1 4|, &lt;0 0 1 0 -1 -2|, &lt;0 0 0 1 -1 0|]
+{{Mapping|legend=1| 1 0 0 0 7 -4 | 0 1 0 0 1 -1 | 0 0 1 0 -1 4 | 0 0 0 1 -1 0 }}
-Transpose: [2 3 5 7 385/35 324/25]
+[[Transpose]]: [2 3 5 7 385/35 625/48]
-EDOs: {{EDOs| 7, 15, 19, 26, 34, 41, 46, 53, 72, 87, 140, 159, 212 and 299 }}
+{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}
-^6*Badness: 2.206
+[[Badness]]: 2.57 × 10<sup>-6</sup>
-== Ancient ==
+== Zaxa ==
+[[Subgroup]]: 2.3.5.7.11.13
-Commas: 385/384, 625/624
+[[Comma list]]: 352/351, 385/384
-Map: [&lt;1 0 0 0 7 -4|, &lt;0 1 0 0 1 -1|, &lt;0 0 1 0 -1 4|, &lt;0 0 0 1 -1 0|]
+{{Mapping|legend=1| 1 0 0 0 7 12 |0 1 0 0 1 -2 | 0 0 1 0 -1 -1 | 0 0 0 1 -1 -1 }}
-Transpose: [2 3 5 7 385/35 625/48]
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.657, ~5/4 = 385.632, ~7/4 = 967.829
-EDOs: {{EDOs| 6, 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190 }}
+{{Optimal ET sequence|legend=1| 7, 10, 15f, 17, 19f, 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
-^6*Badness: 2.573
+[[Badness]]: 3.35 × 10<sup>-6</sup>
-[[Category:Temperament]]
+[[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
+[[Category:Keenanismic family| ]] <!-- main article -->
+[[Category:Keenanismic| ]] <!-- key article -->
 [[Category:Rank 4]]
-[[Category:Keenanismic]]