Keenanismic family: Difference between revisions

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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.

The '''keenanismic family''' of [[rank-4 temperament|rank-4]] [[regular temperament|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 list]]: 385/384

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

{{Mapping|legend=1| 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

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.4104{{c}}, ~3/2 = 701.6916{{c}}, ~5/4 = 385.1761{{c}}, ~7/4 = 967.5421{{c}}

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

: [[error map]]: {{val| +0.410 +0.147 -0.317 -0.463 -0.703 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6933{{c}}, ~5/4 = 385.1491{{c}}, ~7/4 = 967.3004{{c}}

: error map: {{val| 0.000 -0.262 -1.165 -1.525 -2.074 }}

[[Minimax tuning]]:

* [[11-odd-limit]]

* [[11-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 }}, ~3 = {{monzo| 0 1 0 0 0 }}, ~5 = {{monzo| 7/3 1/3 2/3 -1/3 -1/3 }}, ~7 = {{monzo| 7/3 1/3 -1/3 2/3 -1/3 }}

: [{{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 }}]~~

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

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

{{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.~~2 × 10-9~~

[[Badness]] (Sintel): 0.277

== Martwin ==

=== Martwin ===

[[Subgroup]]: 2.3.5.7.11.13

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{{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]~~

Lattice basis:

: 4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247

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

[[Optimal tuning]]s:

: [{{~~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 }}]

* [[WE]]: ~2 = 1200.3494{{c}}, ~3/2 = 702.2131{{c}}, ~5/4 = 384.8666{{c}}, ~7/4 = 967.8487{{c}}

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

: [[error map]]: {{val| +0.349 +0.608 -0.748 -0.278 -0.422 -0.710 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1659{{c}}, ~5/4 = 384.8721{{c}}, ~7/4 = 967.6109{{c}}

: error map: {{val| 0.000 +0.211 -1.442 -1.215 -1.635 -1.608 }}

[[Minimax tuning]]:

* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~2 = {{val| 1 0 0 0 0 0 }}, ~3 = {{val| 0 1 0 0 0 0 }}, ~5 = {{val| 2/3 4/3 1/3 0 0 -1/3 }}, ~7 = {{val| 19/6 -1/6 -1/6 1/2 -1/2 1/6 }}

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7.13/5

{{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.~~21 × 10-6~~

[[Badness]] (Sintel): 0.579

== Ancient ==

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{{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 tuning]]s:

* [[WE]]: ~2 = 1200.4130{{c}}, ~3/2 = 701.6863{{c}}, ~5/4 = 385.2319{{c}}, ~7/4 = 967.5188{{c}}

: [[error map]]: {{val| +0.413 +0.144 -0.256 -0.481 -0.730 -0.047 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6838{{c}}, ~5/4 = 385.2497{{c}}, ~7/4 = 967.2556{{c}}

: error map: {{val| 0.000 -0.271 -1.064 -1.570 -2.139 -1.213 }}

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

[[Badness]]: 2.~~57 × 10-6~~

[[Badness]] (Sintel): 0.675

== Zaxa ==

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[[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 }}

{{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 }}

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1\1, ~3/2 = 702.~~657~~, ~5/4 = 385.~~632~~, ~7/4 = 967.~~829~~

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1117{{c}}, ~3/2 = 702.6588{{c}}, ~5/4 = 385.6821{{c}}, ~7/4 = 968.0061{{c}}

: [[error map]]: {{val| +0.112 +0.815 -0.408 -0.596 -1.901 +1.137 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.6029{{c}}, ~5/4 = 385.6441{{c}}, ~7/4 = 967.9030{{c}}

: error map: {{val| 0.000 +0.648 -0.670 -0.923 -2.262 +0.720 }}

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

{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}

[[Badness]]: 3.~~35 × 10-6~~

[[Badness]] (Sintel): 0.880

[[Category:Temperament families]]

@@ Line 1: / Line 1: @@
 {{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.
+The '''keenanismic family''' of [[rank-4 temperament|rank-4]] [[regular temperament|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 list]]: 385/384
+[[Comma list]]: [[385/384]]
 {{Mapping|legend=1| 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
-: mapping generators: ~2, ~3, ~5, ~7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.4104{{c}}, ~3/2 = 701.6916{{c}}, ~5/4 = 385.1761{{c}}, ~7/4 = 967.5421{{c}}
-[[Transpose]]: [2 3 5 7 385/35]
+: [[error map]]: {{val| +0.410 +0.147 -0.317 -0.463 -0.703 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6933{{c}}, ~5/4 = 385.1491{{c}}, ~7/4 = 967.3004{{c}}
+: error map: {{val| 0.000 -0.262 -1.165 -1.525 -2.074 }}
 [[Minimax tuning]]:
-* [[11-odd-limit]]
+* [[11-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 }}, ~3 = {{monzo| 0 1 0 0 0 }}, ~5 = {{monzo| 7/3 1/3 2/3 -1/3 -1/3 }}, ~7 = {{monzo| 7/3 1/3 -1/3 2/3 -1/3 }}
-: [{{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 }}]
+: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5
-: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5
 {{Optimal ET sequence|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}
-[[Badness]]: 15.2 × 10<sup>-9</sup>
+[[Badness]] (Sintel): 0.277
 == Martwin ==
+=== Martwin ===
 [[Subgroup]]: 2.3.5.7.11.13
@@ Line 29: / Line 34: @@
 {{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]
+Lattice basis:
+: 4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247
-[[Minimax tuning]]: {{confirm}} <!-- contradiction with the data in Rank-4 temperament -->
+[[Optimal tuning]]s:
-: [{{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 }}]
+* [[WE]]: ~2 = 1200.3494{{c}}, ~3/2 = 702.2131{{c}}, ~5/4 = 384.8666{{c}}, ~7/4 = 967.8487{{c}}
-: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.11/5.13/5
+: [[error map]]: {{val| +0.349 +0.608 -0.748 -0.278 -0.422 -0.710 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1659{{c}}, ~5/4 = 384.8721{{c}}, ~7/4 = 967.6109{{c}}
+: error map: {{val| 0.000 +0.211 -1.442 -1.215 -1.635 -1.608 }}
+[[Minimax tuning]]:
+* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~2 = {{val| 1 0 0 0 0 0 }}, ~3 = {{val| 0 1 0 0 0 0 }}, ~5 = {{val| 2/3 4/3 1/3 0 0 -1/3 }}, ~7 = {{val| 19/6 -1/6 -1/6 1/2 -1/2 1/6 }}
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7.13/5
 {{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]]: 2.21 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 0.579
 == Ancient ==
@@ Line 46: / Line 58: @@
 {{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 tuning]]s:
+* [[WE]]: ~2 = 1200.4130{{c}}, ~3/2 = 701.6863{{c}}, ~5/4 = 385.2319{{c}}, ~7/4 = 967.5188{{c}}
+: [[error map]]: {{val| +0.413 +0.144 -0.256 -0.481 -0.730 -0.047 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6838{{c}}, ~5/4 = 385.2497{{c}}, ~7/4 = 967.2556{{c}}
+: error map: {{val| 0.000 -0.271 -1.064 -1.570 -2.139 -1.213 }}
 {{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>
+[[Badness]] (Sintel): 0.675
 == Zaxa ==
@@ Line 57: / Line 73: @@
 [[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 }}
+{{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 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.657, ~5/4 = 385.632, ~7/4 = 967.829
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1117{{c}}, ~3/2 = 702.6588{{c}}, ~5/4 = 385.6821{{c}}, ~7/4 = 968.0061{{c}}
+: [[error map]]: {{val| +0.112 +0.815 -0.408 -0.596 -1.901 +1.137 }}
+* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.6029{{c}}, ~5/4 = 385.6441{{c}}, ~7/4 = 967.9030{{c}}
+: error map: {{val| 0.000 +0.648 -0.670 -0.923 -2.262 +0.720 }}
-{{Optimal ET sequence|legend=1| 7, 10, 15f, 17, 19f, 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
+{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
-[[Badness]]: 3.35 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 0.880
 [[Category:Temperament families]]