Keenanismic family: Difference between revisions

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The '''keenanismic family''' of rank-4 temperaments tempers out the [[keenanisma]], the 11-limit comma 385/384 = {{monzo| -7 -1 1 1 1 }} of 4.503 [[cent]]s.
{{Technical data page}}
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 ==
== Keenanismic ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 385/384
[[Comma list]]: 385/384


[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 1 }}, {{val| 0 0 1 0 -1 }}, {{val| 0 0 0 1 -1 }}]
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 0 0 1 | 0 0 1 0 -1 | 0 0 0 1 -1 }}


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


[[Minimax tuning]]:  
[[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 }}]
: [{{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 }}]
: Eigenmonzos: 2, 3, 7/5, 11/5
: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5


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


[[Badness]]: 15.159 × 10<sup>-9</sup>
[[Badness]]: 15.2 × 10<sup>-9</sup>


== Martwin ==
== Martwin ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


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


[[Mapping]]: [{{val| 1 0 0 0 7 2 }}, {{val| 0 1 0 0 1 4 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 0 1 -1 0 }}]
{{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]
[[Transpose]]: [2 3 5 7 385/35 324/25]


[[Minimax tuning]]:  
[[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 }}]
: [{{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 }}]
: Eigenmonzos: 2, 3, 11/5, 13/5
: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.11/5.13/5


{{Val list|legend=1| 7, 15, 19, 26, 34, 41, 46, 53, 72, 87, 140, 159, 212, 299 }}
{{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 }}


[[Badness]]: 2.206 × 10<sup>-6</sup>
[[Badness]]: 2.21 × 10<sup>-6</sup>


== Ancient ==
== Ancient ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


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


[[Mapping]]: [{{val| 1 0 0 0 7 -4 }}, {{val| 0 1 0 0 1 -1 }}, {{val| 0 0 1 0 -1 4 }}, {{val| 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 7 385/35 625/48]
 
{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}
 
[[Badness]]: 2.57 × 10<sup>-6</sup>
 
== Zaxa ==
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 352/351, 385/384
 
{{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


{{Val list|legend=1| 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 }}


[[Badness]]: 2.573 × 10<sup>-6</sup>
[[Badness]]: 3.35 × 10<sup>-6</sup>


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Keenanismic family| ]] <!-- main article -->
[[Category:Keenanismic family| ]] <!-- main article -->
[[Category:Keenanismic| ]] <!-- key article -->
[[Category:Rank 4]]
[[Category:Rank 4]]
[[Category:Keenanismic]]

Latest revision as of 00:37, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The keenanismic family of rank-4 temperaments tempers out the keenanisma, the 11-limit comma 385/384 = [-7 -1 1 1 1⟩ of 4.503 cents.

Keenanismic

Subgroup: 2.3.5.7.11

Comma list: 385/384

Mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 1], ⟨0 0 1 0 -1], ⟨0 0 0 1 -1]]

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

Transpose: [2 3 5 7 385/35]

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [7/3 1/3 2/3 -1/3 -1/3⟩, [7/3 1/3 -1/3 2/3 -1/3⟩, [7/3 1/3 -1/3 -1/3 2/3⟩]
Unchanged-interval (eigenmonzo) basis: 2.3.7/5.11/5

Optimal ET sequence: 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de

Badness: 15.2 × 10-9

Martwin

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384

Mapping: [⟨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: [to be confirmed]

[[1 0 0 0 0 0⟩, [0 1 0 0 0 0⟩, [2/3 4/3 1/3 0 0 -1/3⟩, [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⟩]
Unchanged-interval (eigenmonzo) basis: 2.3.11/5.13/5

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

Badness: 2.21 × 10-6

Ancient

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624

Mapping: [⟨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 7 385/35 625/48]

Optimal ET sequence: 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def

Badness: 2.57 × 10-6

Zaxa

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 1 -2], ⟨0 0 1 0 -1 -1], ⟨0 0 0 1 -1 -1]]

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

Optimal ET sequence: 7, 10, 15f, 17, 19f, 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e

Badness: 3.35 × 10-6

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