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Catalog of rank-4 temperaments: Difference between revisions

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{{Technical data page}}
A '''rank-4 temperament''' has a [[period]] and three additional independent [[generator]]s, for a total of four [[rank and codimension|dimensions]]. Typical examples include [[7-limit]] [[JI]], full 11-limit temperament with a one-dimensional [[comma basis]], and full 13-limit temperament with a two-dimensional comma basis.  
A '''rank-4 temperament''' has a [[period]] and three additional independent [[generator]]s, for a total of four [[rank and codimension|dimensions]]. Typical examples include [[7-limit]] [[JI]], full 11-limit temperament with a one-dimensional [[comma basis]], and full 13-limit temperament with a two-dimensional comma basis.  


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


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 694.823, ~5/4 = 378.660, ~7/4 = 968.826
* [[CTE]]: ~2 = 1＼1, ~3/2 = 694.823, ~5/4 = 378.660, ~7/4 = 968.826
* [[CWE]]: ~2 = 1\1, ~3/2 = 693.197, ~5/4 = 376.082, ~7/4 = 965.060
* [[CWE]]: ~2 = 1＼1, ~3/2 = 693.197, ~5/4 = 376.082, ~7/4 = 965.060


{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 26, 45 }}
{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 26, 45 }}
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{{Mapping|legend=1| 1 0 0 0 1 | 0 1 0 0 -2 | 0 0 1 0 0 | 0 0 0 1 2 }}
{{Mapping|legend=1| 1 0 0 0 1 | 0 1 0 0 -2 | 0 0 1 0 0 | 0 0 0 1 2 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 700.306, ~5/4 = 386.314, ~7/4 = 974.000
* [[CTE]]: ~2 = 1＼1, ~3/2 = 700.306, ~5/4 = 386.314, ~7/4 = 974.000
* [[CWE]]: ~2 = 1\1, ~3/2 = 700.060, ~5/4 = 385.931, ~7/4 = 973.388
* [[CWE]]: ~2 = 1＼1, ~3/2 = 700.060, ~5/4 = 385.931, ~7/4 = 973.388


{{Optimal ET sequence|legend=1| 5, 8d, 9, 12, 17c, 19e, 22, 31, 53, 84e, 96, 127 }}
{{Optimal ET sequence|legend=1| 5, 8d, 9, 12, 17c, 19e, 22, 31, 53, 84e, 96, 127 }}
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{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 0 -2 | 0 0 1 0 2 | 0 0 0 1 0 }}
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 0 -2 | 0 0 1 0 2 | 0 0 0 1 0 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 703.961, ~5/4 = 382.009, ~7/4 = 968.826
* [[CTE]]: ~2 = 1＼1, ~3/2 = 703.961, ~5/4 = 382.009, ~7/4 = 968.826
* [[CWE]]: ~2 = 1\1, ~3/2 = 704.475, ~5/4 = 383.076, ~7/4 = 969.890
* [[CWE]]: ~2 = 1＼1, ~3/2 = 704.475, ~5/4 = 383.076, ~7/4 = 969.890


{{Optimal ET sequence|legend=1| 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=1| 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e }} <nowiki>*</nowiki>
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{{Mapping|legend=1| 1 0 1 0 2 | 0 1 1 0 1 | 0 0 -2 0 -1 | 0 0 0 1 0 }}
{{Mapping|legend=1| 1 0 1 0 2 | 0 1 1 0 1 | 0 0 -2 0 -1 | 0 0 0 1 0 }}


: mapping generators: ~2, ~3, ~11/10, ~7
: Mapping generators: ~2, ~3, ~11/10, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.602, ~11/10 = 157.450, ~7/4 = 968.826
* [[CTE]]: ~2 = 1＼1, ~3/2 = 702.602, ~11/10 = 157.450, ~7/4 = 968.826
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.985, ~11/10 = 157.610, ~7/4 = 967.819
* [[CWE]]: ~2 = 1＼1, ~3/2 = 701.985, ~11/10 = 157.610, ~7/4 = 967.819


{{Optimal ET sequence|legend=1| 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee }}
{{Optimal ET sequence|legend=1| 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee }}
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{{Mapping|legend=1| 1 0 0 0 -4 | 0 1 0 0 0 | 0 0 1 0 2 | 0 0 0 1 1 }}
{{Mapping|legend=1| 1 0 0 0 -4 | 0 1 0 0 0 | 0 0 1 0 2 | 0 0 0 1 1 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.955, ~5/4 = 388.882, ~7/4 = 970.703
* [[CTE]]: ~2 = 1＼1, ~3/2 = 701.955, ~5/4 = 388.882, ~7/4 = 970.703
* [[CWE]]: ~2 = 1\1, ~3/2 = 702.587, ~5/4 = 389.393, ~7/4 = 971.520
* [[CWE]]: ~2 = 1＼1, ~3/2 = 702.587, ~5/4 = 389.393, ~7/4 = 971.520


{{Optimal ET sequence|legend=1| 22, 31, 46, 53, 58, 80, 111 }}
{{Optimal ET sequence|legend=1| 22, 31, 46, 53, 58, 80, 111 }}
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{{Mapping|legend=1| 1 1 0 0 2 | 0 2 0 0 5 | 0 0 1 0 0 | 0 0 0 1 0 }}
{{Mapping|legend=1| 1 1 0 0 2 | 0 2 0 0 5 | 0 0 1 0 0 | 0 0 0 1 0 }}


: mapping generators: ~2, ~11/9, ~5, ~7
: Mapping generators: ~2, ~11/9, ~5, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~11/9 = 350.572, ~5/4 = 386.314, ~7/4 = 968.826
* [[CTE]]: ~2 = 1＼1, ~11/9 = 350.572, ~5/4 = 386.314, ~7/4 = 968.826
* [[CWE]]: ~2 = 1\1, ~11/9 = 350.549, ~5/4 = 386.239, ~7/4 = 968.735
* [[CWE]]: ~2 = 1＼1, ~11/9 = 350.549, ~5/4 = 386.239, ~7/4 = 968.735


{{Optimal ET sequence|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}
{{Optimal ET sequence|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}
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{{Mapping|legend=1| 1 0 1 0 0 | 0 1 0 0 0 | 0 0 2 0 1 | 0 0 0 1 1 }}
{{Mapping|legend=1| 1 0 1 0 0 | 0 1 0 0 0 | 0 0 2 0 1 | 0 0 0 1 1 }}


: mapping generators: ~2, ~3, ~11/7, ~7
: Mapping generators: ~2, ~3, ~11/7, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.955, ~11/7 = 792.479, ~7/4 = 964.861
* [[CTE]]: ~2 = 1＼1, ~3/2 = 701.955, ~11/7 = 792.479, ~7/4 = 964.861
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.753, ~5/4 = 792.335, ~7/4 = 964.527
* [[CWE]]: ~2 = 1＼1, ~3/2 = 701.753, ~5/4 = 792.335, ~7/4 = 964.527


{{Optimal ET sequence|legend=1| 9, 12, 15, 23de, 24, 26, 27e, 38d, 41, 91, 106d }}
{{Optimal ET sequence|legend=1| 9, 12, 15, 23de, 24, 26, 27e, 38d, 41, 91, 106d }}
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{{Mapping|legend=1| 1 0 0 10 0 5 | 0 1 0 -6 0 -3 | 0 0 1 1 0 0 | 0 0 0 0 1 1 }}
{{Mapping|legend=1| 1 0 0 10 0 5 | 0 1 0 -6 0 -3 | 0 0 1 1 0 0 | 0 0 0 0 1 1 }}


: mapping generators: ~2, ~3, ~5, ~11
: Mapping generators: ~2, ~3, ~5, ~11


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.873, ~5/4 = 386.213, ~11/8 = 550.306
* [[CTE]]: ~2 = 1＼1, ~3/2 = 702.873, ~5/4 = 386.213, ~11/8 = 550.306
* [[CWE]]: ~2 = 1\1, ~3/2 = 702.896, ~5/4 = 386.784, ~11/8 = 551.068
* [[CWE]]: ~2 = 1＼1, ~3/2 = 702.896, ~5/4 = 386.784, ~11/8 = 551.068


{{Optimal ET sequence|legend=1| 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f }}
{{Optimal ET sequence|legend=1| 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f }}
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[[Badness]]: 2.550 × 10<sup>-6</sup>
[[Badness]]: 2.550 × 10<sup>-6</sup>


== Werckismic (441/440) ==
== Keenanismic (385/384) ==
{{Main| Keenanismic family }}
 
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[441/440]]
[[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|legend=1| 1 0 0 0 -3 | 0 1 0 0 2 | 0 0 1 0 -1 | 0 0 0 1 2 }}
: Mapping generators: ~2, ~3, ~5, ~7


: mapping generators: ~2, ~3, ~5, ~7
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1＼1, ~3/2 = 702.363, ~5/4 = 385.439, ~7/4 = 967.547
* [[CWE]]: ~2 = 1＼1, ~3/2 = 701.693, ~5/4 = 385.149, ~7/4 = 967.300


[[Minimax tuning]]:  
[[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| 1/2 2/3 1/6 -1/3 1/6 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 1 -2/3 1/3 1/3 1/3 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.9/7.11


{{Optimal ET sequence|legend=1| 10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d }}
{{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>


=== Commas 364/363, 441/440 ===
=== Martwin ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 364/363, 441/440
[[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 }}


{{Mapping|legend=1| 1 0 0 0 -3 -8 | 0 1 0 0 2 5 | 0 0 1 0 -1 -2 | 0 0 0 1 2 3 }}
Lattice basis:
: 4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247


[[Mapping to lattice]]: [{{val| 0 1 1 -1 -1 0 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 1 1 1 1 }}]
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 702.629, ~5/4 = 385.219, ~7/4 = 967.741


Lattice basis:  
[[Minimax tuning]]: {{confirm}} <!-- contradiction with the data in Keenanismic family -->
: 3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657
* [[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


[[Minimax tuning]]:
{{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 }}
* [[13-odd-limit]]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/3 0 1/3 -1/3 -1/3 1/3 }}, {{monzo| 1/6 0 5/6 2/3 -5/6 1/3 }}, {{monzo| 0 0 0 1 0 0 }}, {{monzo| 1/6 0 -1/6 2/3 1/6 1/3 }}, {{monzo| 0 0 0 0 0 1 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/10, 8/7, 16/13
* [[15-odd-limit]]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/4 1/4 1/4 -1/4 -1/4 1/4 }}, {{monzo| 5/4 -3/4 5/4 -1/4 -1/4 1/4 }}, {{monzo| 17/8 -11/8 5/8 -1/8 3/8 1/8 }}, {{monzo| 5/2 -3/2 1/2 -1/2 1/2 1/2 }}, {{monzo| 17/8 -11/8 5/8 -9/8 3/8 9/8 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 14/13, 6/5, 11/9


{{Optimal ET sequence|legend=1| 12f, 14cf, 15, 17c, 26, 29, 31f, 41, 46, 58, 72, 87, 130, 217, 289 }}
[[Badness]]: 2.21 × 10<sup>-6</sup>


=== Commas 351/350, 441/440 ===
=== Ancient ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 351/350, 441/440
[[Comma list]]: 385/384, 625/624
 
{{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 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 702.358, ~5/4 = 385.674, ~7/4 = 967.452


{{Mapping|legend=1| 1 0 0 0 -3 1 | 0 1 0 0 2 -3 | 0 0 1 0 -1 2 | 0 0 0 1 2 1 }}
{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}


{{Optimal ET sequence|legend=1| 12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf }}
[[Badness]]: 2.57 × 10<sup>-6</sup>


=== Commas 196/195, 352/351 ===
=== Commas 351/350, 385/384 ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 196/195, 352/351
[[Comma list]]: 351/350, 385/384
 
{{Mapping|legend=1| 1 0 0 0 7 1 | 0 1 0 0 1 -3 | 0 0 1 0 -1 2 | 0 0 0 1 -1 1 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 701.205, ~5/4 = 386.715, ~7/4 = 967.646


{{Mapping|legend=1| 1 0 0 0 -3 2 | 0 1 0 0 2 -1 | 0 0 1 0 -1 -1 | 0 0 0 1 2 2 }}
{{Optimal ET sequence|legend=1| 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff }}


{{Optimal ET sequence|legend=1| 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232 }}
[[Badness]]: 2.98 × 10<sup>-6</sup>


=== Tannic ===
=== Zaxa ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 441/440, 1287/1280
[[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 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1＼1, ~3/2 = 702.657, ~5/4 = 385.632, ~7/4 = 967.829
* [[CWE]]: ~2 = 1＼1, ~3/2 = 702.603, ~5/4 = 385.644, ~7/4 = 967.903
 
{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
 
[[Badness]]: 3.35 × 10<sup>-6</sup>


{{Mapping|legend=1| 1 0 0 0 -3 11 | 0 1 0 0 2 -4 | 0 0 1 0 -1 2 | 0 0 0 1 2 -2 }}
=== Commas 364/363, 385/384 ===
[[Subgroup]]: 2.3.5.7.11.13


{{Optimal ET sequence|legend=1| 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}
[[Comma list]]: 364/363, 385/384


==== 17-limit ====
{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 1 3 | 0 0 1 0 -1 -2 | 0 0 0 1 -1 -3 }}
[[Subgroup]]: 2.3.5.7.11.13.17


[[Comma list]]: 273/272, 441/440, 561/560
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 702.761, ~5/4 = 385.159, ~7/4 = 966.298


{{Mapping|legend=1| 1 0 0 0 -3 11 7 | 0 1 0 0 2 -4 -3 | 0 0 1 0 -1 2 2 | 0 0 0 1 2 -2 -1 }}
{{Optimal ET sequence|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}


{{Optimal ET sequence|legend=1| 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef }}
[[Badness]]: 3.32 × 10<sup>-6</sup>


=== Commas 441/440, 847/845 ===
=== Commas 385/384, 847/845 ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 441/440, 847/845
[[Comma list]]: 385/384, 847/845
 
{{Mapping|legend=1| 1 0 0 0 7 7 | 0 1 0 0 1 1 | 0 0 1 1 -2 -2 | 0 0 0 2 -2 -1 }}


{{Mapping|legend=1| 1 0 0 0 -3 -3 | 0 1 0 0 2 2 | 0 0 1 1 1 1 | 0 0 0 2 4 5 }}
: Mapping generators: ~2, ~3, ~5, ~13/11


: mapping generators: ~2, ~3, ~5, ~13/11
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 702.368, ~5/4 = 385.416, ~13/11 = 291.066


{{Optimal ET sequence|legend=1| 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d }}
{{Optimal ET sequence|legend=1| 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee }}


== Keenanismic (385/384) ==
[[Badness]]: 4.15 × 10<sup>-6</sup>
{{Main| Keenanismic family }}


== Werckismic (441/440) ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[385/384]]
[[Comma list]]: [[441/440]]


{{Mapping|legend=1| 1 0 0 0 7 | 0 1 0 0 1 | 0 0 1 0 -1 | 0 0 0 1 -1 }}
{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 0 0 2 | 0 0 1 0 -1 | 0 0 0 1 2 }}


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


[[Transpose]]: [2 3 5 7 385/35]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1＼1, ~3/2 = 701.620, ~5/4 = 386.673, ~7/4 = 967.775
* [[CWE]]: ~2 = 1＼1, ~3/2 = 701.496, ~5/4 = 386.432, ~7/4 = 967.588


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 }}, ~3 = {{monzo| 1/2 2/3 1/6 -1/3 1/6 }}, ~5 = {{monzo| 0 0 1 0 0 }}, ~7 = {{monzo| 1 -2/3 1/3 1/3 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]]: 2.5.9/7.11
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) 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 }}
{{Optimal ET sequence|legend=1| 10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d }}


[[Badness]]: 15.2 × 10<sup>-9</sup>
=== Commas 364/363, 441/440 ===
 
=== Martwin ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 325/324, 385/384
[[Comma list]]: 364/363, 441/440


{{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 }}
{{Mapping|legend=1| 1 0 0 0 -3 -8 | 0 1 0 0 2 5 | 0 0 1 0 -1 -2 | 0 0 0 1 2 3 }}


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


Lattice basis:  
Lattice basis:  
: 4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247
: 3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657


[[Minimax tuning]]: {{confirm}} <!-- contradiction with the data in Keenanismic family -->
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 701.880, ~5/4 = 386.523, ~7/4 = 967.841
* [[13-odd-limit|13-]] and [[15-odd-limit]]
: [{{val| 1 0 0 0 0 0 }}, {{val| 0 1 0 0 0 0 }}, {{val| 2/3 4/3 1/3 0 0 -1/3 }}, {{val| 19/6 -1/6 -1/6 1/2 -1/2 1/6 }}, {{val| 19/6 -1/6 -1/6 -1/2 1/2 1/6 }}, {{val| 2/3 4/3 -2/3 0 0 2/3 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) 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 }}
[[Minimax tuning]]:
* [[13-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 0 }}, ~3 = {{monzo| 5/3 0 1/3 -1/3 -1/3 1/3 }}, ~5 = {{monzo| 1/6 0 5/6 2/3 -5/6 1/3 }}, ~7 = {{monzo| 0 0 0 1 0 0 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/10, 8/7, 16/13
* [[15-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 0 }}, ~3 = {{monzo| 5/4 1/4 1/4 -1/4 -1/4 1/4 }}, ~5 = {{monzo| 5/4 -3/4 5/4 -1/4 -1/4 1/4 }}, ~7 = {{monzo| 17/8 -11/8 5/8 -1/8 3/8 1/8 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 14/13, 6/5, 11/9


[[Badness]]: 2.21 × 10<sup>-6</sup>
{{Optimal ET sequence|legend=1| 12f, 14cf, 15, 17c, 26, 29, 31f, 41, 46, 58, 72, 87, 130, 217, 289 }}


=== Ancient ===
=== Commas 351/350, 441/440 ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 385/384, 625/624
[[Comma list]]: 351/350, 441/440


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


[[Transpose]]: [2 3 5 7 385/35 625/48]
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 701.175, ~5/4 = 387.330, ~7/4 = 968.480


{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}
{{Optimal ET sequence|legend=1| 12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf }}


[[Badness]]: 2.57 × 10<sup>-6</sup>
=== Commas 196/195, 352/351 ===
 
=== Commas 351/350, 385/384 ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 351/350, 385/384
[[Comma list]]: 196/195, 352/351


{{Mapping|legend=1| 1 0 0 0 7 1 | 0 1 0 0 1 -3 | 0 0 1 0 -1 2 | 0 0 0 1 -1 1 }}
{{Mapping|legend=1| 1 0 0 0 -3 2 | 0 1 0 0 2 -1 | 0 0 1 0 -1 -1 | 0 0 0 1 2 2 }}


{{Optimal ET sequence|legend=1| 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 702.601, ~5/4 = 387.137, ~7/4 = 966.418


[[Badness]]: 2.98 × 10<sup>-6</sup>
{{Optimal ET sequence|legend=1| 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232 }}


=== Zaxa ===
=== Tannic ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 352/351, 385/384
[[Comma list]]: 441/440, 1287/1280


{{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 -3 11 | 0 1 0 0 2 -4 | 0 0 1 0 -1 2 | 0 0 0 1 2 -2 }}


[[Optimal tuning]]s:
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 700.555, ~5/4 = 387.816, ~7/4 = 967.984
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.657, ~5/4 = 385.632, ~7/4 = 967.829
* [[CWE]]: ~2 = 1\1, ~3/2 = 702.603, ~5/4 = 385.644, ~7/4 = 967.903


{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
{{Optimal ET sequence|legend=1| 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}


[[Badness]]: 3.35 × 10<sup>-6</sup>
==== 17-limit ====
[[Subgroup]]: 2.3.5.7.11.13.17


=== Commas 364/363, 385/384 ===
[[Comma list]]: 273/272, 441/440, 561/560
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 364/363, 385/384
{{Mapping|legend=1| 1 0 0 0 -3 11 7 | 0 1 0 0 2 -4 -3 | 0 0 1 0 -1 2 2 | 0 0 0 1 2 -2 -1 }}


{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 1 3 | 0 0 1 0 -1 -2 | 0 0 0 1 -1 -3 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 700.606, ~5/4 = 387.655, ~7/4 = 967.876


{{Optimal ET sequence|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}
{{Optimal ET sequence|legend=1| 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef }}


[[Badness]]: 3.32 × 10<sup>-6</sup>
=== Commas 441/440, 847/845 ===
 
=== Commas 385/384, 847/845 ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 385/384, 847/845
[[Comma list]]: 441/440, 847/845


{{Mapping|legend=1| 1 0 0 0 7 7 | 0 1 0 0 1 1 | 0 0 1 1 -2 -2 | 0 0 0 2 -2 -1 }}
{{Mapping|legend=1| 1 0 0 0 -3 -3 | 0 1 0 0 2 2 | 0 0 1 1 1 1 | 0 0 0 2 4 5 }}


: mapping generators: ~2, ~3, ~5, ~13/11
: Mapping generators: ~2, ~3, ~5, ~13/11


{{Optimal ET sequence|legend=1| 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 701.611, ~5/4 = 386.882, ~13/11 = 290.287


[[Badness]]: 4.15 × 10<sup>-6</sup>
{{Optimal ET sequence|legend=1| 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d }}


== Swetismic (540/539) ==
== Swetismic (540/539) ==
Line 337: Line 357:
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 0 3 | 0 0 1 0 1 | 0 0 0 1 -2 }}
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 0 3 | 0 0 1 0 1 | 0 0 0 1 -2 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.6167, ~5/4 = 386.0717, ~7/4 = 969.5334
* [[CTE]]: ~2 = 1＼1, ~3/2 = 701.6167, ~5/4 = 386.0717, ~7/4 = 969.5334
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.6950, ~5/4 = 386.1796, ~7/4 = 969.6366
* [[CWE]]: ~2 = 1＼1, ~3/2 = 701.6950, ~5/4 = 386.1796, ~7/4 = 969.6366


{{Optimal ET sequence|legend=1| 8d, 9, 10, 12e, 14c, 17c, 19, 22, 27e, 31, 41, 53, 58, 72, 130, 152, 224, 354, 506e, 578, 730de, 761d, 985d, 1115de, 1267dde }}
{{Optimal ET sequence|legend=1| 8d, 9, 10, 12e, 14c, 17c, 19, 22, 27e, 31, 41, 53, 58, 72, 130, 152, 224, 354, 506e, 578, 730de, 761d, 985d, 1115de, 1267dde }}
Line 354: Line 374:
{{Mapping|legend=1| 1 0 0 0 2 -3 | 0 1 0 0 3 6 | 0 0 1 0 1 0 | 0 0 0 1 -2 -1 }}
{{Mapping|legend=1| 1 0 0 0 2 -3 | 0 1 0 0 3 6 | 0 0 1 0 1 0 | 0 0 0 1 -2 -1 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.6687, ~5/4 = 386.0441, ~7/4 = 969.5668
* [[CTE]]: ~2 = 1＼1, ~3/2 = 701.6687, ~5/4 = 386.0441, ~7/4 = 969.5668
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.7230, ~5/4 = 386.1818, ~7/4 = 969.6607
* [[CWE]]: ~2 = 1＼1, ~3/2 = 701.7230, ~5/4 = 386.1818, ~7/4 = 969.6607


{{Optimal ET sequence|legend=1| 12e, 14cf, 17c, 19, 22f, 31f, 39df, 41, 53, 58, 72, 111, 130, 183, 224, 354, 578, 985d, 1267ddef }}
{{Optimal ET sequence|legend=1| 12e, 14cf, 17c, 19, 22f, 31f, 39df, 41, 53, 58, 72, 111, 130, 183, 224, 354, 578, 985d, 1267ddef }}
Line 371: Line 391:
{{Mapping|legend=1| 1 0 0 0 2 2 | 0 1 0 0 3 3 | 0 0 1 1 -1 -1 | 0 0 0 2 -4 -3 }}
{{Mapping|legend=1| 1 0 0 0 2 2 | 0 1 0 0 3 3 | 0 0 1 1 -1 -1 | 0 0 0 2 -4 -3 }}


: mapping generators: ~2, ~3, ~5, ~13/11
: Mapping generators: ~2, ~3, ~5, ~13/11
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 701.284, ~5/4 = 386.211, ~13/11 = 291.733


{{Optimal ET sequence|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
{{Optimal ET sequence|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
Line 383: Line 405:


{{Mapping|legend=1| 1 0 0 0 2 -4 | 0 1 0 0 3 -1 | 0 0 1 0 1 4 | 0 0 0 1 -2 0 }}
{{Mapping|legend=1| 1 0 0 0 2 -4 | 0 1 0 0 3 -1 | 0 0 1 0 1 4 | 0 0 0 1 -2 0 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 701.702, ~5/4 = 385.631, ~7/4 = 969.467


{{Optimal ET sequence|legend=1| 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224 }}
{{Optimal ET sequence|legend=1| 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224 }}
Line 395: Line 419:
{{Mapping|legend=1| 1 0 0 0 2 -1 | 0 2 0 0 6 3 | 0 0 1 0 1 1 | 0 0 0 1 -2 0 }}
{{Mapping|legend=1| 1 0 0 0 2 -1 | 0 2 0 0 6 3 | 0 0 1 0 1 1 | 0 0 0 1 -2 0 }}


: mapping generators: ~2, ~26/15, ~5, ~7
: Mapping generators: ~2, ~26/15, ~5, ~7
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~26/15 = 950.906, ~5/4 = 386.474, ~7/4 = 969.892


{{Optimal ET sequence|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}
{{Optimal ET sequence|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}
Line 408: Line 434:
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 0 0 -4 | 0 0 1 0 0 | 0 0 0 1 1 }}
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 0 0 -4 | 0 0 1 0 0 | 0 0 0 1 1 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 703.576, ~5/4 = 386.314, ~7/4 = 968.126
* [[CTE]]: ~2 = 1＼1, ~3/2 = 703.576, ~5/4 = 386.314, ~7/4 = 968.126
* [[CWE]]: ~2 = 1\1, ~3/2 = 703.743, ~5/4 = 387.245, ~7/4 = 969.048
* [[CWE]]: ~2 = 1＼1, ~3/2 = 703.743, ~5/4 = 387.245, ~7/4 = 969.048


{{Optimal ET sequence|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}
{{Optimal ET sequence|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}
Line 424: Line 450:


{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 -4 -7 | 0 0 1 0 0 0 | 0 0 0 1 1 1 }}
{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 -4 -7 | 0 0 1 0 0 0 | 0 0 0 1 1 1 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 703.786, ~5/4 = 386.314, ~7/4 = 967.665


{{Optimal ET sequence|legend=1| 12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff }}
{{Optimal ET sequence|legend=1| 12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff }}
Line 436: Line 464:
{{Mapping|legend=1| 1 0 0 2 0 1 | 0 1 0 0 0 0 | 0 0 1 1 1 1 | 0 0 0 4 -3 1 }}
{{Mapping|legend=1| 1 0 0 2 0 1 | 0 1 0 0 0 0 | 0 0 1 1 1 1 | 0 0 0 4 -3 1 }}


: mapping generators: ~2, ~3, ~5, ~13/10
: Mapping generators: ~2, ~3, ~5, ~13/10
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 701.955, ~5/4 = 386.699, ~13/10 = 454.558


{{Optimal ET sequence|legend=1| 16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e }}
{{Optimal ET sequence|legend=1| 16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e }}
Line 447: Line 477:
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 1 2 | 0 0 1 0 -1 | 0 0 0 2 1 }}
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 1 2 | 0 0 1 0 -1 | 0 0 0 2 1 }}


: mapping generators: ~2, ~3, ~5, ~55/36
: Mapping generators: ~2, ~3, ~5, ~55/36
 
[[Optimal tuning]]:
* [[CTE]]: ~2 = 1＼1, ~3/2 = 701.998, ~5/4 = 386.252, ~55/36 = 733.436
* [[CWE]]: ~2 = 1＼1, ~3/2 = 701.978, ~5/4 = 386.232, ~55/36 = 733.430


{{Optimal ET sequence|legend=1| 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
{{Optimal ET sequence|legend=1| 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
=== Commas 2080/2079, 3025/3024 ===
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: 2080/2079, 3025/3024
{{Mapping|legend=1| 1 0 0 0 2 -3 | 0 1 0 1 2 6 | 0 0 1 0 -1 -2 | 0 0 0 2 1 3 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1＼1, ~3/2 = 702.076, ~5/4 = 386.208, ~55/36 = 733.438
* [[CWE]]: ~2 = 1＼1, ~3/2 = 702.099, ~5/4 = 386.248, ~55/36 = 733.449
{{Optimal ET sequence|legend=1| 15, 26, 41, 46, 72, 87, 111, 183, 224, 270, 311, 494 }}


== Trimitone (8019/8000) ==
== Trimitone (8019/8000) ==
Line 458: Line 505:
{{Mapping|legend=1| 1 0 0 0 6 | 0 1 0 0 -6 | 0 0 1 0 3 | 0 0 0 1 0 }}
{{Mapping|legend=1| 1 0 0 0 6 | 0 1 0 0 -6 | 0 0 1 0 3 | 0 0 0 1 0 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.5449, ~5/4 = 386.7538, ~7/4 = 968.8259
* [[CTE]]: ~2 = 1＼1, ~3/2 = 701.5449, ~5/4 = 386.7538, ~7/4 = 968.8259
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.4729, ~5/4 = 386.5374, ~7/4 = 968.6210
* [[CWE]]: ~2 = 1＼1, ~3/2 = 701.4729, ~5/4 = 386.5374, ~7/4 = 968.6210


{{Optimal ET sequence|legend=1| 12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be }}
{{Optimal ET sequence|legend=1| 12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be }}
Line 475: Line 522:
{{Mapping|legend=1| 1 0 0 0 6 -3 | 0 1 0 0 -6 6 | 0 0 1 0 3 0 | 0 0 0 1 0 -1 }}
{{Mapping|legend=1| 1 0 0 0 6 -3 | 0 1 0 0 -6 6 | 0 0 1 0 3 0 | 0 0 0 1 0 -1 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.5537, ~5/4 = 386.7680, ~7/4 = 968.8144
* [[CTE]]: ~2 = 1＼1, ~3/2 = 701.5537, ~5/4 = 386.7680, ~7/4 = 968.8144
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.4770, ~5/4 = 386.5437, ~7/4 = 968.6150
* [[CWE]]: ~2 = 1＼1, ~3/2 = 701.4770, ~5/4 = 386.5437, ~7/4 = 968.6150


{{Optimal ET sequence|legend=1| 53, 58, 72, 111, 130, 183, 190, 243e, 248, 301, 373, 804, 1177be }}
{{Optimal ET sequence|legend=1| 53, 58, 72, 111, 130, 183, 190, 243e, 248, 301, 373, 804, 1177be }}
Line 492: Line 539:
{{Mapping|legend=1| 2 0 0 0 3 | 0 1 0 0 -2 | 0 0 1 0 1 | 0 0 0 1 1 }}
{{Mapping|legend=1| 2 0 0 0 3 | 0 1 0 0 -2 | 0 0 1 0 1 | 0 0 0 1 1 }}


: mapping generators: ~99/70, ~3, ~5, ~7
: Mapping generators: ~99/70, ~3, ~5, ~7
 
[[Optimal tuning]]:
* [[CTE]]: ~99/70 = 1＼2, ~3/2 = 701.942, ~5/4 = 386.327, ~7/4 = 968.846
* [[CWE]]: ~99/70 = 1＼2, ~3/2 = 701.940, ~5/4 = 386.323, ~7/4 = 968.840


{{Optimal ET sequence|legend=1| 8d, 10, 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e }}
{{Optimal ET sequence|legend=1| 8d, 10, 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e }}
Line 505: Line 556:
Lattice basis:  
Lattice basis:  
: 3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299
: 3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299
[[Optimal tuning]] ([[CTE]]): ~99/70 = 1＼1, ~3/2 = 702.024, ~5/4 = 386.294, ~7/4 = 969.114


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[13-odd-limit|13-]] and [[15-odd-limit]]
* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~99/70 = {{monzo| 1/2 0 0 0 0 0 }}, ~3 = {{monzo| 7/10 4/5 0 -2/5 0 1/5 }}, ~5 = {{monzo| 7/10 -1/5 1 -2/5 0 1/5 }}, ~7 = {{monzo| 7/5 -2/5 0 1/5 0 2/5 }}
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 7/10 4/5 0 -2/5 0 1/5 }}, {{monzo| 7/10 -1/5 1 -2/5 0 1/5 }}, {{monzo| 7/5 -2/5 0 1/5 0 2/5 }}, {{monzo| 11/5 -11/5 1 3/5 0 1/5 }}, {{monzo| 0 0 0 0 0 1 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-intervals) basis]]: 2, 6/5, 16/13, 9/7
: [[Eigenmonzo basis|Eigenmonzo (unchanged-intervals) basis]]: 2, 6/5, 16/13, 9/7


Line 520: Line 572:
{{Mapping|legend=1| 1 0 0 0 0 | 0 1 0 2 2 | 0 0 1 -1 -1 | 0 0 0 3 4 }}
{{Mapping|legend=1| 1 0 0 0 0 | 0 1 0 2 2 | 0 0 1 -1 -1 | 0 0 0 3 4 }}


: mapping generators: ~2, ~3, ~5, ~11/7
: Mapping generators: ~2, ~3, ~5, ~11/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.031, ~5/4 = 386.233, ~11/7 = 783.508
* [[CTE]] and [[CWE]]: ~2 = 1＼1, ~3/2 = 702.031, ~5/4 = 386.233, ~11/7 = 783.508
* [[CWE]]: ~2 = 1\1, ~3/2 = 702.031, ~5/4 = 386.232, ~11/7 = 783.507


{{Optimal ET sequence|legend=1| 12, 21, 23de, 26, 37, 46, 58, 72, 118, 130, 190, 239, 311, 441, 559, 752e, 870 }}
{{Optimal ET sequence|legend=1| 12, 21, 23de, 26, 37, 46, 58, 72, 118, 130, 190, 239, 311, 441, 559, 752e, 870 }}
Line 538: Line 589:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.072, ~5/4 = 386.528, ~11/7 = 783.823
* [[CTE]]: ~2 = 1＼1, ~3/2 = 702.072, ~5/4 = 386.528, ~11/7 = 783.823
* [[CWE]]: ~2 = 1\1, ~3/2 = 702.253, ~5/4 = 386.776, ~11/7 = 783.875
* [[CWE]]: ~2 = 1＼1, ~3/2 = 702.253, ~5/4 = 386.776, ~11/7 = 783.875


{{Optimal ET sequence|legend=1| 26, 37, 46, 58, 72, 121, 130, 193, 239, 251e, 323e, 369 }}
{{Optimal ET sequence|legend=1| 26, 37, 46, 58, 72, 121, 130, 193, 239, 251e, 323e, 369 }}
Line 553: Line 604:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.952, ~5/4 = 386.139, ~11/7 = 783.188
* [[CTE]] and [[CWE]]: ~2 = 1＼1, ~3/2 = 701.952, ~5/4 = 386.139, ~11/7 = 783.548
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.952, ~5/4 = 386.139, ~11/7 = 783.188


{{Optimal ET sequence|legend=1| 46f, 49f, 58, 72, 118, 121, 130, 190, 193, 248, 311, 441, 752e, 1072 }}
{{Optimal ET sequence|legend=1| 46f, 49f, 58, 72, 118, 121, 130, 190, 193, 248, 311, 441, 752e, 1072 }}
Line 565: Line 615:
{{Mapping|legend=1| 1 0 2 0 1 | 0 1 2 0 2 | 0 0 -4 0 -1 | 0 0 0 1 0 }}
{{Mapping|legend=1| 1 0 2 0 1 | 0 1 2 0 2 | 0 0 -4 0 -1 | 0 0 0 1 0 }}


: mapping generators: ~2, ~3, ~18/11, ~7
: Mapping generators: ~2, ~3, ~18/11, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.334, ~18/11 = 854.555, ~7/4 = 968.826
* [[CTE]]: ~2 = 1＼1, ~3/2 = 702.334, ~18/11 = 854.555, ~7/4 = 968.826
* [[CWE]]: ~2 = 1\1, ~3/2 = 702.292, ~18/11 = 854.548, ~7/4 = 968.756
* [[CWE]]: ~2 = 1＼1, ~3/2 = 702.292, ~18/11 = 854.548, ~7/4 = 968.756


{{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
{{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
Line 583: Line 633:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 702.469, ~18/11 = 854.644, ~7/4 = 968.959
* CTE: ~2 = 1＼1, ~3/2 = 702.469, ~18/11 = 854.644, ~7/4 = 968.959
* CWE: ~2 = 1\1, ~3/2 = 702.483, ~18/11 = 854.642, ~7/4 = 968.989
* CWE: ~2 = 1＼1, ~3/2 = 702.483, ~18/11 = 854.642, ~7/4 = 968.989


{{Optimal ET sequence|legend=1| 87, 111, 181, 198, 323, 410 }}
{{Optimal ET sequence|legend=0| 87, 111, 181, 198, 323, 410 }}


Badness: 17.1 × 10<sup>-6</sup>
Badness: 17.1 × 10<sup>-6</sup>
Line 598: Line 648:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 702.413, ~18/11 = 854.633, ~7/4 = 969.032
* CTE: ~2 = 1＼1, ~3/2 = 702.413, ~18/11 = 854.633, ~7/4 = 969.032
* CWE: ~2 = 1\1, ~3/2 = 702.411, ~18/11 = 854.634, ~7/4 = 969.026
* CWE: ~2 = 1＼1, ~3/2 = 702.411, ~18/11 = 854.634, ~7/4 = 969.026


{{Optimal ET sequence|legend=1| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}
{{Optimal ET sequence|legend=0| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}


Badness: 34.0 × 10<sup>-6</sup>
Badness: 34.0 × 10<sup>-6</sup>
Line 613: Line 663:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 702.368, ~18/11 = 854.639, ~7/4 = 969.075
* CTE: ~2 = 1＼1, ~3/2 = 702.368, ~18/11 = 854.639, ~7/4 = 969.075
* CWE: ~2 = 1\1, ~3/2 = 702.351, ~18/11 = 854.644, ~7/4 = 969.010
* CWE: ~2 = 1＼1, ~3/2 = 702.351, ~18/11 = 854.644, ~7/4 = 969.010


{{Optimal ET sequence|legend=1| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}
{{Optimal ET sequence|legend=0| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}


Badness: 41.9 × 10<sup>-6</sup>
Badness: 41.9 × 10<sup>-6</sup>
Line 627: Line 677:
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 0 -2 | 0 0 2 0 -1 | 0 0 0 1 0 }}
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 0 -2 | 0 0 2 0 -1 | 0 0 0 1 0 }}


: mapping generators: ~2, ~3, ~128/99, ~7
: Mapping generators: ~2, ~3, ~128/99, ~7
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 702.470, ~128/99 = 442.106, ~7/4 = 968.826


{{Optimal ET sequence|legend=1| 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee }}
{{Optimal ET sequence|legend=1| 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee }}
Line 640: Line 692:
{{Mapping|legend=1| 1 0 0 0 -8 | 0 1 0 0 9 | 0 0 1 0 0 | 0 0 0 1 -1 }}
{{Mapping|legend=1| 1 0 0 0 -8 | 0 1 0 0 9 | 0 0 1 0 0 | 0 0 0 1 -1 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 702.213, ~5/4 = 386.314, ~7/4 = 968.736
* [[CTE]]: ~2 = 1＼1, ~3/2 = 702.213, ~5/4 = 386.314, ~7/4 = 968.736
* [[CWE]]: ~2 = 1200.000, ~3/2 = 702.244, ~5/4 = 386.406, ~7/4 = 968.859
* [[CWE]]: ~2 = 1＼1, ~3/2 = 702.244, ~5/4 = 386.406, ~7/4 = 968.859


{{Optimal ET sequence|legend=1| 17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b }}
{{Optimal ET sequence|legend=1| 17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b }}


[[Badness]] (Smith): 0.120 × 10<sup>-6</sup>
[[Badness]]: 0.120 × 10<sup>-6</sup>


=== Tridecimal symbiotic ===
=== Tridecimal symbiotic ===
Line 658: Line 710:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 702.228, ~5/4 = 386.284, ~7/4 = 968.789
* [[CTE]]: ~2 = 1＼1, ~3/2 = 702.228, ~5/4 = 386.284, ~7/4 = 968.789
* [[CWE]]: ~2 = 1200.000, ~3/2 = 702.254, ~5/4 = 386.401, ~7/4 = 968.898
* [[CWE]]: ~2 = 1＼1, ~3/2 = 702.254, ~5/4 = 386.401, ~7/4 = 968.898


{{Optimal ET sequence|legend=1| 17c, 34dff, 36ce, 41, 53, 58, 94, 111, 152f, 212, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bf }}
{{Optimal ET sequence|legend=1| 17c, 34dff, 36ce, 41, 53, 58, 94, 111, 152f, 212, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bf }}


[[Badness]] (Smith): 3.31 × 10<sup>-6</sup>
[[Badness]]: 3.31 × 10<sup>-6</sup>


=== 2.3.5.7.11.13.19 subgroup ===
=== 2.3.5.7.11.13.19 subgroup ===
Line 673: Line 725:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 702.230, ~5/4 = 386.304, ~7/4 = 968.797
* CTE: ~2 = 1＼1, ~3/2 = 702.230, ~5/4 = 386.304, ~7/4 = 968.797
* CWE: ~2 = 1200.000, ~3/2 = 702.255, ~5/4 = 386.408, ~7/4 = 968.902
* CWE: ~2 = 1＼1, ~3/2 = 702.255, ~5/4 = 386.408, ~7/4 = 968.902


{{Optimal ET sequence|legend=0| 17ch, 24f, 34dffh, 36ceh, 41, 53, 58h, 94, 111, 152f, 212h, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bfh }}
{{Optimal ET sequence|legend=0| 17ch, 24f, 34dffh, 36ceh, 41, 53, 58h, 94, 111, 152f, 212h, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bfh }}


Badness (Smith): 4.37 × 10<sup>-6</sup>
Badness: 4.37 × 10<sup>-6</sup>


== Olympic (131072/130977) ==
== Olympic (131072/130977) ==
Line 687: Line 739:
{{Mapping|legend=1| 1 0 0 0 17 | 0 1 0 0 -5 | 0 0 1 0 0 | 0 0 0 1 -2 }}
{{Mapping|legend=1| 1 0 0 0 17 | 0 1 0 0 -5 | 0 0 1 0 0 | 0 0 0 1 -2 }}


: mapping generators: ~2, ~3, ~5, ~7
: Mapping generators: ~2, ~3, ~5, ~7
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 702.103, ~5/4 = 386.314, ~7/4 = 969.012


{{Optimal ET sequence|legend=1| 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee }}
{{Optimal ET sequence|legend=1| 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee }}
Line 697: Line 751:


{{Mapping|legend=1| 1 0 0 0 17 12 | 0 1 0 0 -5 -2 | 0 0 1 0 0 -1 | 0 0 0 1 -2 -1 }}
{{Mapping|legend=1| 1 0 0 0 17 12 | 0 1 0 0 -5 -2 | 0 0 1 0 0 -1 | 0 0 0 1 -2 -1 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 702.103, ~5/4 = 386.298, ~7/4 = 969.007


{{Optimal ET sequence|legend=1| 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e }} *
{{Optimal ET sequence|legend=1| 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e }} *
Line 705: Line 761:
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]:
[[Comma list]]: [[Seascape comma|160083/160000]]
[[Seascape comma|160083/160000]]


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


: mapping generators: ~2, ~400/231, ~5, ~7
: Mapping generators: ~2, ~400/231, ~5, ~7
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~400/231 = 950.960, ~5/4 = 386.417, ~7/4 = 968.751
 
{{Optimal ET sequence|legend=1| 15, 19, 29, 43, 53, 58, 68, 72 }}
 
== Fifthchromismic (1610510/1594323) ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: [[1610510/1594323]]
 
{{Mapping|legend=1| 1 0 4 0 -1 | 0 1 3 0 2 | 0 0 -5 0 1 | 0 0 0 1 0 }}
 
: Mapping generators: ~2, ~3, ~22/9, ~7
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 702.738, ~11/9 = 344.406, ~7/4 = 968.826
 
{{Optimal ET sequence|legend=1| 77, 80, 87, 94, 167 }}
 
=== Fifthchroma ===
{{see also| Fifth-chroma temperaments }}
 
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 352/351, 951665/944784
 
{{Mapping|legend=1| 1 0 4 0 -1 4 | 0 1 3 0 2 -1 | 0 0 -5 0 1 1 | 0 0 0 1 0 0 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~3/2 = 702.776, ~11/9 = 344.410, ~7/4 = 968.826
 
{{Optimal ET sequence|legend=1| 77, 80, 87, 94, 167 }}
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 352/351, 715/714, 155771/155520
 
Mapping: {{mapping| 1 0 4 0 -1 4 6 | 0 1 3 0 2 -1 3 | 0 0 -5 0 1 1 -3 | 0 0 0 1 0 0 -1 }}
 
Optimal tuning (CTE): ~2 = 1＼1, ~3/2 = 702.713, ~11/9 = 344.403, ~7/4 = 969.193
 
{{Optimal ET sequence|legend=0| 77, 80, 87, 94, 167 }}
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 352/351, 715/714, 1463/1458, 2057/2052
 
Mapping: {{mapping| 1 0 4 0 -1 4 6 2 | 0 1 3 0 2 -1 3 4 | 0 0 -5 0 1 1 -3 -1 | 0 0 0 1 0 0 -1 -1 }}
 
Optimal tuning (CTE): ~2 = 1＼1, ~3/2 = 702.736, ~11/9 = 344.417, ~7/4 = 969.159
 
{{Optimal ET sequence|legend=0| 77, 80, 87, 94, 167h }}


[[Category:Temperaments]]
[[Category:Temperaments]]
[[Category:Lists of temperaments]]
[[Category:Lists of temperaments]]
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]
[[Category:Rank 4| ]] <!-- main article -->
[[Category:Rank 4| ]] <!-- main article -->
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