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Biyatismic clan: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''biyatismic clan''' of [[Rank-3 temperament|rank-3]] [[Temperament|temperaments]] [[Tempering out|tempers out]] the [[biyatisma]], 121/120 = {{monzo| -3 -1 -1 0 2 }}.
The '''biyatismic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the biyatisma, [[121/120]].


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
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{{Mapping|legend=2| 1 0 1 2 | 0 1 1 1 | 0 0 -2 -1 }}
{{Mapping|legend=2| 1 0 1 2 | 0 1 1 1 | 0 0 -2 -1 }}
: mapping generators: ~2, ~3, ~11/10


: Mapping generators: ~2, ~3, ~11/10
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.4578{{c}}, ~11/10 = 157.7466{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466


{{Optimal ET sequence|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }}
{{Optimal ET sequence|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }}


[[Badness]]: 0.0297 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.0297 × 10<sup>-3</sup>


== Zeus ==
== Zeus ==
Line 41: Line 40:
: Angle (11/10, 11/8) = 87.464 degrees
: Angle (11/10, 11/8) = 87.464 degrees


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.1530, ~11/10 = 157.0881
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.1530{{c}}, ~11/10 = 157.0881{{c}}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.9/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.9/7


{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }}
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }}


[[Badness]]: 0.400 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.400 × 10<sup>-3</sup>


[[Projection pair]]s: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5
[[Projection pair]]s: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5
Line 77: Line 76:
: Angle (11/10, 11/8) = 106.7439 degrees
: Angle (11/10, 11/8) = 106.7439 degrees


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8679, ~11/10 = 156.9582
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.8679{{c}}, ~11/10 = 156.9582{{c}}


Minimax tuning:  
Minimax tuning:  
Line 89: Line 88:
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 77, 99, 130e }}
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 77, 99, 130e }}


Badness: 0.934 × 10<sup>-3</sup>
Badness (Smith): 0.934 × 10<sup>-3</sup>


Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5
Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5
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Mapping: {{mapping| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}
Mapping: {{mapping| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.3420, ~11/10 = 155.3666
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 699.3420{{c}}, ~11/10 = 155.3666{{c}}


{{Optimal ET sequence|legend=1| 7, 9, 15, 22f, 24, 31 }}
{{Optimal ET sequence|legend=0| 7, 9, 15, 22f, 24, 31 }}


Badness: 0.808 × 10<sup>-3</sup>
Badness (Smith): 0.808 × 10<sup>-3</sup>


== Artemis ==
== Artemis ==
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{{Mapping|legend=1| 1 0 1 -3 2 | 0 1 1 4 1 | 0 0 -2 -4 -1 }}
{{Mapping|legend=1| 1 0 1 -3 2 | 0 1 1 4 1 | 0 0 -2 -4 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 699.8719, ~11/10 = 158.3232
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 699.8719{{c}}, ~11/10 = 158.3232{{c}}


{{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}
{{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}
Line 129: Line 128:
Mapping: {{mapping| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}
Mapping: {{mapping| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7090, ~11/10 = 158.7117
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 698.7090{{c}}, ~11/10 = 158.7117{{c}}


{{Optimal ET sequence|legend=1| 9, 20, 22f, 29, 31 }}
{{Optimal ET sequence|legend=0| 9, 20, 22f, 29, 31 }}


=== Diana ===
=== Diana ===
Line 140: Line 139:
Mapping: {{mapping| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}
Mapping: {{mapping| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9789, ~11/10 = 159.0048
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.9789{{c}}, ~11/10 = 159.0048{{c}}


{{Optimal ET sequence|legend=1| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
{{Optimal ET sequence|legend=0| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}


== Oxpecker ==
== Oxpecker ==
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{{Mapping|legend=1| 1 0 1 2 2 | 0 1 1 1 1 | 0 0 -2 -6 -1 }}
{{Mapping|legend=1| 1 0 1 2 2 | 0 1 1 1 1 | 0 0 -2 -6 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.8882, ~11/10 = 155.7756
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 700.8882{{c}}, ~11/10 = 155.7756{{c}}


{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}


[[Badness]]: 0.699 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.699 × 10<sup>-3</sup>


=== Woodpecker ===
=== Woodpecker ===
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Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}
Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5946, ~11/10 = 154.8652
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.5946{{c}}, ~11/10 = 154.8652{{c}}


{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31 }}
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 23de, 24d, 31 }}


Badness: 1.093 × 10<sup>-3</sup>
Badness (Smith): 1.093 × 10<sup>-3</sup>


== Aphrodite ==
== Aphrodite ==
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{{Mapping|legend=1| 1 0 1 3 | 0 1 1 0 | 0 0 -4 -3 }}
{{Mapping|legend=1| 1 0 1 3 | 0 1 1 0 | 0 0 -4 -3 }}
: mapping generators: ~2, ~3, ~21/20


: Mapping generators: ~2, ~3, ~21/20
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 700.2144{{c}}, ~21/20 = 78.5694{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.2144, ~21/20 = 78.5694


{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}


[[Badness]]: 0.943 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.943 × 10<sup>-3</sup>


[[Projection pair]]s: 5 320000/64827 7 64000/9261 to 2.3.7/5
[[Projection pair]]s: 5 320000/64827 7 64000/9261 to 2.3.7/5
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{{Mapping|legend=1| 1 0 1 3 2 | 0 1 1 0 1 | 0 0 -4 -3 -2 }}
{{Mapping|legend=1| 1 0 1 3 2 | 0 1 1 0 1 | 0 0 -4 -3 -2 }}
: mapping generators: ~2, ~3, ~22/21


: Mapping generators: ~2, ~3, ~22/21
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.3200{{c}}, ~21/20 = 78.6421{{c}}
 
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.3200, ~21/20 = 78.6421


{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}


[[Badness]]: 0.583 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.583 × 10<sup>-3</sup>


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }}
Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.1158, ~21/20 = 78.5211
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.1158{{c}}, ~21/20 = 78.5211{{c}}


{{Optimal ET sequence|legend=1| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}


Badness: 1.456 × 10<sup>-3</sup>
Badness (Smith): 1.456 × 10<sup>-3</sup>


==== Eros ====
==== Eros ====
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Mapping: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}
Mapping: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5014, ~21/20 = 78.6143
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.5014{{c}}, ~21/20 = 78.6143{{c}}


{{Optimal ET sequence|legend=1| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
{{Optimal ET sequence|legend=0| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}


Badness: 1.150 × 10<sup>-3</sup>
Badness (Smith): 1.150 × 10<sup>-3</sup>


===== 17-limit =====
===== 17-limit =====
Line 243: Line 240:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.9299, ~22/21 = 78.2539
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9299{{c}}, ~22/21 = 78.2539{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.7925, ~22/21 = 78.6203
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7925{{c}}, ~22/21 = 78.6203{{c}}


Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 77, 106de }}
{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 77, 106de }}


Badness:  
Badness:  
* Smith: 0.979 × 10<sup>-3</sup>
* Smith: 0.979 × 10<sup>-3</sup>
* Dirichlet: 0.931
* Sintel: 0.931


===== 19-limit =====
===== 19-limit =====
Line 260: Line 257:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.5642, ~22/21 = 78.2353
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5642{{c}}, ~22/21 = 78.2353{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.6963, ~22/21 = 78.6479
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6963{{c}}, ~22/21 = 78.6479{{c}}


Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}
{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}


Badness:  
Badness:  
* Smith: 1.13 × 10<sup>-3</sup>
* Smith: 1.13 × 10<sup>-3</sup>
* Dirichlet: 1.159
* Sintel: 1.159


===== 23-limit =====
===== 23-limit =====
Line 277: Line 274:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3 = 1901.7115, ~23/22 = 78.2054
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7115{{c}}, ~23/22 = 78.2054{{c}}
* CWE: ~2 = 1\1, ~3 = 1901.8010, ~23/22 = 78.7188
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8010{{c}}, ~23/22 = 78.7188{{c}}


Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}
{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}


Badness:  
Badness:  
* Smith: 0.939 × 10<sup>-3</sup>
* Smith: 0.939 × 10<sup>-3</sup>
* Dirichlet: 1.084
* Sintel: 1.084


==== Inanna ====
==== Inanna ====
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Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}
Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 698.7754{{c}}, ~21/20 = 79.6096{{c}}


{{Optimal ET sequence|legend=1| 14cf, 15, 29, 31, 45ef, 60e }}
{{Optimal ET sequence|legend=0| 14cf, 15, 29, 31, 45ef, 60e }}


Badness: 1.077 × 10<sup>-3</sup>
Badness (Sintel): 1.077 × 10<sup>-3</sup>


==== Ishtar ====
==== Ishtar ====
Line 306: Line 303:
Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}
Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.3952, ~21/20 = 78.9578
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 703.3952{{c}}, ~21/20 = 78.9578{{c}}


{{Optimal ET sequence|legend=1| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
{{Optimal ET sequence|legend=0| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}


Badness: 1.151 × 10<sup>-3</sup>
Badness (Sintel): 1.151 × 10<sup>-3</sup>


== Notes ==
== Notes ==
Retrieved from "https://en.xen.wiki/w/Biyatismic_clan"