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Catalog of rank-4 temperaments: Difference between revisions

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m Lériendil moved page Rank-4 temperament to Catalog of rank-4 temperaments: The article had no descriptive purpose, it is a list of temperaments and therefore should be considered a "catalog".
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A '''rank-4 temperament''' has a period and three additional independent generators. 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.  
{{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.  
 
== Cake (45/44) ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: [[45/44]]
 
{{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
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.4372{{c}}, ~3/2 = 693.2306{{c}}, ~5/4 = 374.6677{{c}}, ~7/4 = 963.8387{{c}}
: [[error map]]: {{val| +2.437 -6.287 -6.772 -0.113 +14.685 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.1966{{c}}, ~5/4 = 376.0821{{c}}, ~7/4 = 965.0600{{c}}
: error map: {{val| 0.000 -8.758 -10.232 -3.766 +11.157 }}
 
{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 26, 45 }}
 
[[Badness]] (Sintel): 0.400
 
== Mothwellsmic (99/98) ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: [[99/98]]
 
{{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
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3175{{c}}, ~3/2 = 700.0109{{c}}, ~5/4 = 385.6677{{c}}, ~7/4 = 973.2571{{c}}
: [[error map]]: {{val| +0.318 -1.627 -0.011 +5.066 -3.873 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.0599{{c}}, ~5/4 = 385.9307{{c}}, ~7/4 = 973.4880{{c}}
: error map: {{val| 0.000 -1.895 -0.383 +4.662 -4.462 }}
 
{{Optimal ET sequence|legend=1| 5, 8d, 9, 12, 17c, 19e, 22, 31, 53, 84e, 96, 127 }}
 
[[Badness]] (Sintel): 0.545


== Ptolemismic (100/99) ==
== Ptolemismic (100/99) ==
Line 7: Line 44:


{{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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.9532, ~5/4 = 384.0675, ~7/4 = 970.8803
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2633{{c}}, ~3/2 = 704.5205{{c}}, ~5/4 = 383.8317{{c}}, ~7/4 = 970.2843{{c}}
: [[error map]]: {{val| -0.737 +1.829 -3.955 -0.015 +4.358 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.4751{{c}}, ~5/4 = 383.0756{{c}}, ~7/4 = 969.8904{{c}}
: error map: {{val| 0.000 +2.520 -3.238 +1.065 +5.883 }}


{{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>
Line 14: Line 56:
<nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]]
<nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]]


[[Badness]]: 0.0225 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.411


== Biyatismic (121/120) ==
== Biyatismic (121/120) ==
Line 22: Line 64:


{{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:  
 
* [[WE]]: ~2 = 1200.6628{{c}}, ~3/2 = 701.8452{{c}}, ~11/10 = 157.8337{{c}}, ~7/4 = 967.4929{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589
: [[error map]]: {{val| +0.663 +0.553 +1.190 -0.007 -5.318 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9848{{c}}, ~11/10 = 157.6099{{c}}, ~7/4 = 967.8187{{c}}
: error map: {{val| 0.000 +0.030 +0.451 -1.007 -6.943 }}


{{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 }}


[[Badness]]: 0.0345 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.629


== Valinorsmic (176/175) ==
== Valinorsmic (176/175) ==
Line 37: Line 82:


{{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
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3113{{c}}, ~3/2 = 702.6414{{c}}, ~5/4 = 389.5404{{c}}, ~7/4 = 971.5534{{c}}
: [[error map]]: {{val| -0.689 -0.022 +1.849 +1.350 -2.061 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5869{{c}}, ~5/4 = 389.3935{{c}}, ~7/4 = 971.5196{{c}}
: error map: {{val| 0.000 +0.632 +3.080 +2.694 -1.011 }}
{{Optimal ET sequence|legend=1| 22, 27e, 31, 46, 53, 58, 80, 111, 237cd, 268cd }}


: mapping generators: ~2, ~3, ~5, ~7
[[Badness]] (Sintel): 0.339


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113
=== Tridecimal valinorsmic ===
Subgroup: 2.3.5.7.11.13


{{Optimal ET sequence|legend=1| 22, 31, 46, 53, 58, 80, 111 }}
Comma list: 176/175, 351/350


[[Badness]]: 0.0186 × 10<sup>-6</sup>
Mapping: {{Mapping| 1 0 0 0 -4 1 | 0 1 0 0 0 -3 | 0 0 1 0 2 2 | 0 0 0 1 1 1 }}
 
Optimal tunings:
* WE: ~2 = 1199.3397{{c}}, ~3/2 = 702.5360{{c}}, ~5/4 = 389.5451{{c}}, ~7/4 = 971.5415{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6943{{c}}, ~5/4 = 389.3823{{c}}, ~7/4 = 971.5305{{c}}
 
{{Optimal ET sequence|legend=0| 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 268cd }}
 
Badness (Sintel): 0.603


== Rastmic (243/242) ==
== Rastmic (243/242) ==
Rastmic halves [[3/2]] into two [[11/9]]'s, [[32/27]] into two [[12/11]]'s, and [[25/24]] into two [[45/44]][[~]][[55/54]]'s.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[243/242]]
[[Comma list]]: [[243/242]]


{{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
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0635{{c}}, ~11/9 = 350.5439{{c}}, ~5/4 = 386.1857{{c}}, ~7/4 = 968.6977{{c}}
: [[error map]]: {{val| +0.064 -0.804 -0.001 -0.001 +1.529 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5486{{c}}, ~5/4 = 386.2387{{c}}, ~7/4 = 968.7352{{c}}
: error map: {{val| 0.000 -0.858 -0.075 -0.091 +1.425 }}
 
{{Optimal ET sequence|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}
 
[[Badness]] (Sintel): 0.928
 
== Frostmic (245/242) ==
[[Subgroup]]: 2.3.5.7.11


: mapping generators: ~2, ~11/9, ~5, ~7
[[Comma list]]: [[245/242]]


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464
{{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


{{Optimal ET sequence|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2442{{c}}, ~3/2 = 701.7038{{c}}, ~11/7 = 792.3594{{c}}, ~7/4 = 964.4060{{c}}
: [[error map]]: {{val| +0.244 -0.007 -1.351 -3.931 +5.936 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7533{{c}}, ~11/7 = 792.3351{{c}}, ~7/4 = 964.5274{{c}}
: error map: {{val| 0.000 -0.202 -1.644 -4.298 +5.545 }}
 
{{Optimal ET sequence|legend=1| 9, 12, 15, 23de, 24, 26, 27e, 38d, 41, 91, 106d }}


[[Badness]]: 0.0509 × 10<sup>-6</sup>
[[Badness]] (Sintel): 1.64


== Akua (352/351, 847/845) ==
== Akua (352/351, 847/845) ==
Line 67: Line 153:


{{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:  
* [[WE]]: ~2 = 1199.6918{{c}}, ~3/2 = 702.7270{{c}}, ~5/4 = 386.9729{{c}}, ~11/8 = 551.3122{{c}}
: [[error map]]: {{val| -0.308 +0.464 +0.043 -0.064 -0.930 +1.062 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8960{{c}}, ~5/4 = 386.7835{{c}}, ~11/8 = 551.0679{{c}}
: error map: {{val| 0.000 +0.941 +0.470 +0.581 -0.025 +1.852 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538
{{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 }}
[[Badness]] (Sintel): 0.669


[[Badness]]: 2.550 × 10<sup>-6</sup>
== Keenanismic (385/384) ==
{{Main| Keenanismic family }}


== Werckismic (441/440) ==
[[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 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}}
: [[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 }}


{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 0 0 2 | 0 0 1 0 -1 | 0 0 0 1 2 }}
[[Minimax tuning]]:
* [[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 }}
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5


: mapping generators: ~2, ~3, ~5, ~7
{{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]] (Sintel): 0.277


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


Lattice basis:  
[[Optimal tuning]]s:  
: 3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657
* [[WE]]: ~2 = 1200.3494{{c}}, ~3/2 = 702.2131{{c}}, ~5/4 = 384.8666{{c}}, ~7/4 = 967.8487{{c}}
: [[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]]:  
[[Minimax tuning]]:  
* [[13-odd-limit]]
* [[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 }}
: [{{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 basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7.13/5
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/10, 8/7, 16/13
 
* [[15-odd-limit]]
{{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 }}
: [{{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]] (Sintel): 0.579


=== 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]]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 }}


{{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]] (Sintel): 0.675


=== 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]]s:
* [[WE]]: ~2 = 1200.6793{{c}}, ~3/2 = 700.7455{{c}}, ~5/4 = 385.5609{{c}}, ~7/4 = 967.5816{{c}}
: [[error map]]: {{val| +0.679 -0.530 +0.606 +0.114 -0.998 -1.343 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.5141{{c}}, ~5/4 = 385.6072{{c}}, ~7/4 = 967.1521{{c}}
: error map: {{val| 0.000 -1.441 -0.706 -1.674 -3.563 -3.703 }}


{{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]] (Sintel): 0.781


=== 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:
* [[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| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
 
[[Badness]] (Sintel): 0.880


{{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]]s:  
* [[WE]]: ~2 = 1200.4593{{c}}, ~3/2 = 701.9714{{c}}, ~5/4 = 384.8903{{c}}, ~7/4 = 966.4090{{c}}
: [[error map]]: {{val| +0.459 +0.476 -0.505 -1.498 +1.191 -1.325 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.9809{{c}}, ~5/4 = 384.8523{{c}}, ~7/4 = 966.1074{{c}}
: error map: {{val| 0.000 +0.026 -1.461 -2.719 -0.297 -2.612 }}


{{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]] (Sintel): 0.871


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


: mapping generators: ~2, ~3, ~5, ~13/11
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3530{{c}}, ~3/2 = 701.7283{{c}}, ~5/4 = 385.4633{{c}}, ~13/11 = 291.0352{{c}}
: [[error map]]: {{val| +0.353 +0.126 -0.144 -0.586 -1.175 +0.651 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7205{{c}}, ~5/4 = 385.3656{{c}}, ~13/11 = 290.9769{{c}}
: error map: {{val| 0.000 -0.235 -0.948 -1.507 -2.282 -0.515 }}


{{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]] (Sintel): 1.09
{{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
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.1734{{c}}, ~3/2 = 701.4907{{c}}, ~5/4 = 386.2784{{c}}, ~7/4 = 967.5667{{c}}
[[Transpose]]: [2 3 5 7 385/35]
: [[error map]]: {{val| +0.173 -0.291 +0.311 -0.912 +0.692 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4963{{c}}, ~5/4 = 386.4317{{c}}, ~7/4 = 967.5883{{c}}
: error map: {{val| 0.000 -0.459 +0.118 -1.238 +0.420 }}


[[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>
[[Badness]] (Sintel): 0.189


=== Martwin ===
=== Commas 364/363, 441/440 ===
[[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
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0145{{c}}, ~3/2 = 701.8745{{c}}, ~5/4 = 386.4853{{c}}, ~7/4 = 967.8244{{c}}
: [[error map]]: {{val| +0.014 -0.066 +0.201 -0.972 +1.609 -0.667 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8716{{c}}, ~5/4 = 386.4975{{c}}, ~7/4 = 967.8241{{c}}
: error map: {{val| 0.000 -0.083 +0.184 -1.002 +1.576 -0.692 }}


[[Minimax tuning]]: {{confirm}} <!-- contradiction with the data in Keenanismic family -->
[[Minimax tuning]]:  
* [[13-odd-limit|13-]] and [[15-odd-limit]]
* [[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 }}
: [{{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]]: 2.7.11/5.13
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.3.11/7.13/5
* [[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 basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3.11/9.13/7


{{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 }}
{{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>
[[Badness]] (Sintel): 0.452


=== 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]]s:  
* [[WE]]: ~2 = 1200.2006{{c}}, ~3/2 = 701.0354{{c}}, ~5/4 = 386.8579{{c}}, ~7/4 = 968.2226{{c}}
: [[error map]]: {{val| +0.201 -0.719 +0.945 -0.202 +0.541 -0.893 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.0371{{c}}, ~5/4 = 387.0417{{c}}, ~7/4 = 968.2546{{c}}
: error map: {{val| 0.000 -0.918 +0.728 -0.571 +0.224 -1.301 }}


{{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>
[[Badness]] (Sintel): 0.524


=== Commas 351/350, 385/384 ===
=== Commas 196/195, 352/351 ===
[[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 -3 2 | 0 1 0 0 2 -1 | 0 0 1 0 -1 -1 | 0 0 0 1 2 2 }}


{{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]]s:
* [[WE]]: ~2 = 1199.6830{{c}}, ~3/2 = 702.6689{{c}}, ~5/4 = 387.7765{{c}}, ~7/4 = 967.0260{{c}}
: [[error map]]: {{val| -0.317 +0.397 +0.829 -2.434 -0.022 +2.128 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.7536{{c}}, ~5/4 = 387.5834{{c}}, ~7/4 = 966.9373{{c}}
: error map: {{val| 0.000 +0.799 +1.270 -1.889 +0.480 +3.010 }}


{{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>
[[Badness]] (Sintel): 0.548


=== 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 -3 11 | 0 1 0 0 2 -4 | 0 0 1 0 -1 2 | 0 0 0 1 2 -2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7398{{c}}, ~3/2 = 700.6733{{c}}, ~5/4 = 385.4125{{c}}, ~7/4 = 966.9648{{c}}
: [[error map]]: {{val| +0.740 -0.542 +0.578 -0.381 -0.714 -1.147 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.3274{{c}}, ~5/4 = 385.8974{{c}}, ~7/4 = 966.8107{{c}}
: error map: {{val| 0.000 -1.628 -0.416 -2.015 -2.939 -3.664 }}


{{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 = 1200.000{{c}}, ~3/2 = 700.555{{c}}, ~5/4 = 387.816{{c}}, ~7/4 = 967.984{{c}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.657, ~5/4 = 385.632, ~7/4 = 967.829
{{Optimal ET sequence|legend=1| 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}


{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
[[Badness]] (Sintel): 1.23


[[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: {{mapping| 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 tunings:
* WE: ~2 = 1200.7320{{c}}, ~3/2 = 700.7374{{c}}, ~5/4 = 385.2303{{c}}, ~7/4 = 966.8369{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.3909{{c}}, ~5/4 = 385.7215{{c}}, ~7/4 = 966.6922{{c}}


{{Optimal ET sequence|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}
{{Optimal ET sequence|legend=0| 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef }}


[[Badness]]: 3.32 × 10<sup>-6</sup>
Badness (Sintel): 0.701


=== Commas 385/384, 847/845 ===
=== Commas 441/440, 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 tuning]]s:  
* [[WE]]: ~2 = 1200.1778{{c}}, ~3/2 = 701.4785{{c}}, ~5/4 = 385.4755{{c}}, ~13/11 = 290.3844{{c}}
: [[error map]]: {{val| +0.178 -0.299 +0.517 -1.226 -0.170 +1.005 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4843{{c}}, ~5/4 = 386.6332{{c}}, ~13/11 = 290.3162{{c}}
: error map: {{val| 0.000 -0.471 +0.319 -1.560 -0.451 +0.655 }}


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


[[Badness]]: 4.15 × 10<sup>-6</sup>
[[Badness]] (Sintel): 1.11


== Swetismic (540/539) ==
== Swetismic (540/539) ==
Line 266: Line 441:


{{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
* [[WE]]: ~2 = 1199.9149{{c}}, ~3/2 = 701.7196{{c}}, ~5/4 = 386.2545{{c}}, ~7/4 = 969.6659{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.6950, ~5/4 = 386.1796, ~7/4 = 969.6366
: [[error map]]: {{val| -0.085 -0.321 -0.229 +0.670 +0.508 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.6950{{c}}, ~5/4 = 386.1795{{c}}, ~7/4 = 969.6366{{c}}
: error map: {{val| 0.000 -0.260 -0.134 +0.811 +0.673 }}


{{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 }}


[[Badness]]: 0.0105 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.191


=== Commas 540/539, 729/728 ===
=== Commas 540/539, 729/728 ===
Line 283: Line 459:


{{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
* [[WE]]: ~2 = 1199.9170{{c}}, ~3/2 = 701.7128{{c}}, ~5/4 = 386.2521{{c}}, ~7/4 = 969.6598{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.7230, ~5/4 = 386.1818, ~7/4 = 969.6607
: [[error map]]: {{val| -0.083 -0.325 -0.228 +0.668 +0.504 +0.006 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7231{{c}}, ~5/4 = 386.1819{{c}}, ~7/4 = 969.6608{{c}}
: error map: {{val| 0.000 -0.232 -0.132 +0.835 +0.712 +0.150 }}


{{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 }}


[[Badness]]: 1.73 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.454
 
=== Commas 540/539, 847/845 ===
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 540/539, 847/845
 
{{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
 
{{Optimal ET sequence|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
 
[[Badness]]: 3.97 × 10<sup>-6</sup>


=== Commas 540/539, 625/624 ===
=== Commas 540/539, 625/624 ===
Line 313: Line 477:


{{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]]s:
* [[WE]]: ~2 = 1200.0035{{c}}, ~3/2 = 701.6973{{c}}, ~5/4 = 385.6264{{c}}, ~7/4 = 969.4616{{c}}
: [[error map]]: {{val| +0.004 -0.254 -0.680 +0.643 +0.488 +0.291 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6983{{c}}, ~5/4 = 385.6278{{c}}, ~7/4 = 969.4622{{c}}
: error map: {{val| 0.000 -0.257 -0.686 +0.636 +0.480 +0.285 }}


{{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 }}


[[Badness]]: 3.59 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.943


=== Commas 540/539, 676/675 ===
=== Commas 540/539, 676/675 ===
Line 324: Line 494:


{{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]]s:  
* [[WE]]: ~2 = 1199.7990{{c}}, ~26/15 = 950.9130{{c}}, ~5/4 = 386.8478{{c}}, ~7/4 = 970.1529{{c}}
: [[error map]]: {{val| -0.201 -0.129 +0.132 +0.925 +0.702 -1.142 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.9953{{c}}, ~5/4 = 386.7094{{c}}, ~7/4 = 970.1192{{c}}
: error map: {{val| 0.000 +0.036 +0.396 +1.293 +1.125 -0.832 }}


{{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 }}


[[Badness]]: 3.06 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.802
 
=== Commas 540/539, 847/845 ===
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 540/539, 847/845
 
{{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
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8339{{c}}, ~3/2 = 701.4961{{c}}, ~5/4 = 386.5624{{c}}, ~13/11 = 291.6840{{c}}
: [[error map]]: {{val| -0.166 -0.625 -0.084 +0.772 -0.627 +1.848 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4439{{c}}, ~5/4 = 386.4201{{c}}, ~13/11 = 291.7287{{c}}
: error map: {{val| 0.000 -0.511 +0.106 +1.052 -0.321 +2.198 }}
 
{{Optimal ET sequence|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
 
[[Badness]] (Sintel): 1.04


== Pentacircle (896/891) ==
== Pentacircle (896/891) ==
Line 337: Line 530:


{{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:  
 
* [[WE]]: ~2 = 1199.3774{{c}}, ~3/2 = 703.4693{{c}}, ~5/4 = 387.5573{{c}}, ~7/4 = 969.3690{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722
: [[error map]]: {{val| -0.623 +0.892 -0.002 -0.702 +1.061 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7426{{c}}, ~5/4 = 387.2445{{c}}, ~7/4 = 969.0476{{c}}
: error map: {{val| 0.000 +1.788 +0.931 +0.222 +2.759 }}


{{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 }}


[[Badness]]: 0.0658 × 10<sup>-6</sup>
[[Badness]] (Sintel): 1.20


=== Tridecimal pentacircle a.k.a. gentle ===
=== Tridecimal pentacircle (gentle) ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


Line 352: Line 548:


{{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]]s:
* [[WE]]: ~2 = 1199.3706{{c}}, ~3/2 = 703.4872{{c}}, ~5/4 = 387.5708{{c}}, ~7/4 = 969.3987{{c}}
: [[error map]]: {{val| -0.629 +0.903 -0.002 -0.686 +0.985 +0.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.8328{{c}}, ~5/4 = 387.2881{{c}}, ~7/4 = 969.1612{{c}}
: error map: {{val| 0.000 +1.878 +0.974 +0.335 +2.512 +1.804 }}


{{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 }}


[[Badness]]: 3.375 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.885


== Topsy (847/845, 1001/1000) ==
== Topsy (847/845, 1001/1000) ==
Line 362: Line 564:
[[Comma list]]: 847/845, 1001/1000
[[Comma list]]: 847/845, 1001/1000


{{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 3 2 | 0 1 0 0 0 0 | 0 0 1 1 1 1 | 0 0 0 4 -3 -1 }}
: Mapping generators: ~2, ~3, ~5, ~20/13
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0343{{c}}, ~3/2 = 701.9205{{c}}, ~5/4 = 386.6098{{c}}, ~20/13 = 745.4658{{c}}
: [[error map]]: {{val| +0.034 -0.000 +0.365 -0.353 -0.934 +0.753 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9261{{c}}, ~5/4 = 386.6458{{c}}, ~20/13 = 745.4431{{c}}
: error map: {{val| 0.000 -0.029 +0.332 -0.408 -1.001 +0.675 }}


: mapping generators: ~2, ~3, ~5, ~13/10
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.955{{c}}, ~5/4 = 386.699{{c}}, ~13/10 = 454.558{{c}}


{{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 }}
[[Badness]] (Sintel): 1.04


== Lehmerismic (3025/3024) ==
== Lehmerismic (3025/3024) ==
Line 374: Line 585:


{{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]]:  
* [[WE]]: ~2 = 1200.0197{{c}}, ~3/2 = 701.9725{{c}}, ~5/4 = 386.2210{{c}}, ~55/36 = 733.4363{{c}}
: [[error map]]: {{val| +0.020 +0.037 -0.053 +0.039 -0.118 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9782{{c}}, ~5/4 = 386.2319{{c}}, ~55/36 = 733.4299{{c}}
: error map: {{val| 0.000 +0.023 -0.082 +0.012 -0.164 }}


{{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| 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
 
[[Badness]] (Sintel): 0.0802
 
=== 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:
* [[WE]]: ~2 = 1199.9717{{c}}, ~3/2 = 702.0958{{c}}, ~5/4 = 386.2618{{c}}, ~55/36 = 733.4380{{c}}
: [[error map]]: {{val| -0.028 +0.113 -0.108 +0.118 -0.007 -0.134 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0990{{c}}, ~5/4 = 386.2477{{c}}, ~55/36 = 733.4490{{c}}
: error map: {{val| 0.000 +0.144 -0.066 +0.171 +0.081 -0.082 }}
 
{{Optimal ET sequence|legend=1| 15, 24d, 26, 31f, 39df, 41, 46, 65d, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}
 
[[Badness]] (Sintel): 0.272


== Trimitone (8019/8000) ==
== Trimitone (8019/8000) ==
Line 385: Line 620:


{{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
* [[WE]]: ~2 = 1200.1317{{c}}, ~3/2 = 701.4920{{c}}, ~5/4 = 386.4054{{c}}, ~7/4 = 968.5622{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.4729, ~5/4 = 386.5374, ~7/4 = 968.6210
: [[error map]]: {{val| +0.132 -0.331 +0.355 -0.000 -0.263 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4730{{c}}, ~5/4 = 386.5375{{c}}, ~7/4 = 968.6211{{c}}
: error map: {{val| 0.000 -0.482 +0.224 -0.205 -0.543 }}


{{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 }}


[[Badness]]: 0.0820 × 10<sup>-6</sup>
[[Badness]] (Sintel): 1.50


=== Commas 729/728, 1001/1000 ===
=== Commas 729/728, 1001/1000 ===
Line 403: Line 639:
{{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
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1318{{c}}, ~3/2 = 701.4926{{c}}, ~5/4 = 386.4063{{c}}, ~7/4 = 968.5613{{c}}
: [[error map]]: {{val| +0.132 -0.331 +0.356 -0.001 -0.264 -0.002 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4770{{c}}, ~5/4 = 386.5437{{c}}, ~7/4 = 968.6150{{c}}
: error map: {{val| 0.000 -0.478 +0.230 -0.211 -0.549 -0.280 }}
 
{{Optimal ET sequence|legend=1| 53, 58, 72, 111, 130, 183, 190, 243e, 248, 301, 373, 804, 1177be }}
 
[[Badness]] (Sintel): 0.875
 
=== Commas 351/350, 2080/2079 ===
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 351/350, 2080/2079
 
{{Mapping|legend=1| 1 0 0 0 6 1 | 0 1 0 0 -6 -3 | 0 0 1 0 3 2 | 0 0 0 1 0 1 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.5537, ~5/4 = 386.7680, ~7/4 = 968.8144
* [[WE]]: ~2 = 1199.9538{{c}}, ~3/2 = 701.5852{{c}}, ~5/4 = 387.1356{{c}}, ~7/4 = 969.7495{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.4770, ~5/4 = 386.5437, ~7/4 = 968.6150
: [[error map]]: {{val| -0.046 -0.416 +0.730 +0.831 +0.301 -1.447 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.5936{{c}}, ~5/4 = 387.0975{{c}}, ~7/4 = 969.7451{{c}}
: error map: {{val| 0.000 -0.361 +0.784 +0.919 +0.413 -1.369 }}


{{Optimal ET sequence|legend=1| 53, 58, 72, 111, 130, 183, 190, 243e, 248, 301, 373, 804, 1177be }}
{{Optimal ET sequence|legend=1| 19, 26, 39df, 46, 53, 58, 72, 111, 130, 183, 313 }}


[[Badness]]: 3.33 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.677


== Kalismic (9801/9800) ==
== Kalismic (9801/9800) ==
Line 419: Line 672:


{{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]]:  
* [[WE]]: ~99/70 = 600.0018{{c}}, ~3/2 = 701.9397{{c}}, ~5/4 = 386.3193{{c}}, ~7/4 = 968.8374{{c}}
: [[error map]]: {{val| +0.004 -0.012 +0.013 +0.019 -0.028 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9399{{c}}, ~5/4 = 386.3226{{c}}, ~7/4 = 968.8399{{c}}
: error map: {{val| 0.000 -0.015 +0.009 +0.014 -0.035 }}


{{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| 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 }}
 
[[Badness]] (Sintel): 0.0415


=== Commas 1716/1715, 2080/2079 ===
=== Commas 1716/1715, 2080/2079 ===
Line 433: Line 693:
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]]:
* [[WE]]: ~99/70 = 599.9732{{c}}, ~3/2 = 702.0465{{c}}, ~5/4 = 386.4214{{c}}, ~7/4 = 969.1757{{c}}
: [[error map]]: {{val| -0.054 +0.038 +0.001 +0.243 -0.001 -0.210 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0554{{c}}, ~5/4 = 386.3788{{c}}, ~7/4 = 969.1756{{c}}
: error map: {{val| 0.000 +0.100 +0.065 +0.350 +0.126 -0.121 }}


[[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]]: 2.5/3.9/7.13
: [[Eigenmonzo basis|Eigenmonzo (unchanged-intervals) basis]]: 2, 6/5, 16/13, 9/7
 
{{Optimal ET sequence|legend=1| 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d }}


{{Optimal ET sequence|legend=1| 10e, 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d }}
[[Badness]] (Sintel): 0.256


== Unisquary (12005/11979) ==
== Unisquary (12005/11979) ==
Line 446: Line 713:
[[Comma list]]: [[12005/11979]]
[[Comma list]]: [[12005/11979]]


{{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 -6 -8 | 0 1 0 2 2 | 0 0 1 2 3 | 0 0 0 3 4 }}
: Mapping generators: ~2, ~3, ~5, ~44/35


: Mapping generators: ~2, ~3, ~5, ~11/7
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0306{{c}}, ~5/4 = 386.2325{{c}}, ~44/35 = 397.2749{{c}}
: [[error map]]: {{val| -0.000 +0.076 -0.081 +0.243 -0.475 +0.541 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0306{{c}}, ~5/4 = 386.2325{{c}}, ~44/35 = 397.2750{{c}}
: error map: {{val| 0.000 +0.076 -0.081 -0.475 +0.541 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1＼1, ~3 = 1902.0307, ~5 = 2786.2325, ~11/7 = 783.5074
{{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, 46, 58, 72, 118, 121, 130, 190, 202, 239, 248, 311}}
[[Badness]] (Sintel): 2.43


=== Hensquary ===
=== Hensquary ===
Line 459: Line 731:
[[Comma list]]: 364/363, 1716/1715
[[Comma list]]: 364/363, 1716/1715


{{Mapping|legend=1| 1 0 0 0 0 -2 | 0 1 0 2 2 3 | 0 0 1 -1 -1 -1 | 0 0 0 3 4 5 }}
{{Mapping|legend=1| 1 0 0 -6 -8 -12 | 0 1 0 2 2 3 | 0 0 1 2 3 4 | 0 0 0 3 4 5 }}


: Mapping generators: ~2, ~3, ~5, ~11/7
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7826{{c}}, ~3/2 = 702.2831{{c}}, ~5/4 = 386.9208{{c}}, ~44/35 = 396.8581{{c}}
: [[error map]]: {{val| -0.217 +0.111 +0.172 +0.156 +1.443 -1.487 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2535{{c}}, ~5/4 = 386.7762{{c}}, ~44/35 = 397.0993{{c}}
: error map: {{val| 0.000 +0.298 +0.463 +0.531 +1.915 -1.166 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1＼1, ~3 = 1902.4103, ~5 = 2786.9909, ~11/7 = 783.9209
{{Optimal ET sequence|legend=1| 26, 37, 46, 58, 72, 121, 130, 193, 239, 251e, 323e, 369 }}


{{Optimal ET sequence|legend=1|9, 12f, 26, 37, 46, 49f, 58, 63, 72, 84, 118f, 121, 130}}
[[Badness]] (Sintel): 0.988


=== Ekasquary ===
=== Ekasquary ===
Line 472: Line 748:
[[Comma list]]: 1575/1573, 4459/4455
[[Comma list]]: 1575/1573, 4459/4455


{{Mapping|legend=1| 1 0 0 0 0 0 | 0 1 0 2 2 0 | 0 0 1 -1 -1 3 | 0 0 0 3 4 -5 }}
{{Mapping|legend=1| 1 0 0 -6 -8 10 | 0 1 0 2 2 0 | 0 0 1 2 3 -2 | 0 0 0 3 4 -5 }}


: Mapping generators: ~2, ~3, ~5, ~11/7
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9999{{c}}, ~3/2 = 701.9524{{c}}, ~5/4 = 386.1388{{c}}, ~44/35 = 397.4088{{c}}
: [[error map]]: {{val| -0.000 -0.003 -0.175 -0.417 +0.639 +0.150 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9524{{c}}, ~5/4 = 386.1388{{c}}, ~44/35 = 397.4089{{c}}
: error map: {{val| 0.000 -0.003 -0.175 -0.417 +0.639 +0.150 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1＼1, ~3 = 1901.9525, ~5 = 2786.1388, ~11/7 = 783.5477
{{Optimal ET sequence|legend=1| 46f, 49f, 58, 72, 118, 121, 130, 190, 193, 248, 311, 441, 752e, 1072 }}


{{Optimal ET sequence|legend=1|9, 12, 46f, 49f, 58, 60e, 63, 72, 118, 121, 130, 190, 193, 248, 311, 320, 383, 441}}
[[Badness]] (Sintel): 1.12


== Semicanousmic (14641/14580) ==
== Semicanousmic (14641/14580) ==
Line 485: Line 765:
[[Comma list]]: [[14641/14580]]
[[Comma list]]: [[14641/14580]]


{{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 0 | 0 1 2 0 2 | 0 0 4 0 1 | 0 0 0 1 0 }}
: Mapping generators: ~2, ~3, ~11/9, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0495{{c}}, ~3/2 = 702.2793{{c}}, ~11/9 = 345.4721{{c}}, ~7/4 = 968.7265{{c}}
: [[error map]]: {{val| +0.049 +0.374 +0.133 -0.000 -1.188 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2918{{c}}, ~11/9 = 345.4516{{c}}, ~7/4 = 968.7556{{c}}
: error map: {{val| 0.000 +0.337 +0.077 -0.070 -1.283 }}
 
{{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
 
[[Badness]] (Sintel): 6.40
 
== Semiporwellismic (16384/16335) ==
[[Subgroup]]: 2.3.5.7.11


: mapping generators: ~2, ~3, ~18/11, ~7
[[Comma list]]: 16384/16335


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866
{{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


{{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7329{{c}}, ~3/2 = 702.3657{{c}}, ~128/99 = 442.1589{{c}}, ~7/4 = 969.3599{{c}}
: [[error map]]: {{val| -0.267 +0.144 +0.103 -0.000 +0.456 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5107{{c}}, ~128/99 = 442.2243{{c}}, ~7/4 = 969.4120{{c}}
: error map: {{val| 0.000 +0.556 +0.646 +0.586 +1.436 }}
 
{{Optimal ET sequence|legend=1| 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee }}
 
[[Badness]] (Sintel): 4.00
 
=== Tridecimal semiporwellismic ===
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 2080/2079, 3584/3575
 
{{Mapping|legend=1| 1 0 0 0 7 2 | 0 1 1 0 -2 0 | 0 0 2 0 -1 -3 | 0 0 0 1 0 1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7102{{c}}, ~3/2 = 702.2488{{c}}, ~84/65 = 442.3476{{c}}, ~7/4 = 969.1583{{c}}
: [[error map]]: {{val| -0.290 +0.004 +0.341 -0.247 +0.388 +0.429 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3927{{c}}, ~84/65 = 442.4436{{c}}, ~7/4 = 969.1894{{c}}
: error map: {{val| 0.000 +0.438 +0.966 +0.364 +1.453 +1.331 }}
 
{{Optimal ET sequence|legend=1| 19, 22f, 27, 38e, 41, 46, 65d, 68, 84, 87, 111, 130, 198, 282, 328, 369f, 697cef, 1546bccdeeeffff, 1676bccdeeeffff }}


[[Badness]]: 0.351 × 10<sup>-6</sup>
[[Badness]] (Sintel): 1.13


=== Tridecimal semicanousmic ===
== Symbiotic (19712/19683) ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11


Comma list: 2080/2079, 14641/14580
[[Comma list]]: [[19712/19683]]


Mapping: {{mapping| 1 0 2 0 1 -6 | 0 1 2 0 2 3 | 0 0 -4 0 -1 3 | 0 0 0 1 0 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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.4931, ~18/11 = 854.6400, ~7/4 = 969.0099
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9290{{c}}, ~3/2 = 702.2265{{c}}, ~5/4 = 386.4556{{c}}, ~7/4 = 968.8979{{c}}
: [[error map]]: {{val| -0.071 +0.201 -0.000 -0.070 -0.106 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2440{{c}}, ~5/4 = 386.4064{{c}}, ~7/4 = 968.8593{{c}}
: error map: {{val| 0.000 +0.289 +0.093 +0.033 +0.019 }}


{{Optimal ET sequence|legend=1| 87, 111, 181, 198, 323, 410 }}
{{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: 17.1 × 10<sup>-6</sup>
[[Badness]] (Sintel): 2.19


==== 17-limit ====
=== Tridecimal symbiotic ===
Subgroup: 2.3.5.7.11.13.17
[[Subgroup]]: 2.3.5.7.11.13


Comma list: 715/714, 1089/1088, 14641/14580
[[Comma list]]: 2080/2079, 19712/19683


Mapping: {{mapping| 1 0 2 0 1 -6 -4 | 0 1 2 0 2 3 6 | 0 0 -4 0 -1 3 -2 | 0 0 0 1 0 1 0 }}
{{Mapping|legend=1| 1 0 0 0 -8 -13 | 0 1 0 0 9 12 | 0 0 1 0 0 -1 | 0 0 0 1 -1 0 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.4099, ~18/11 = 854.6338, ~7/4 = 969.0228
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9268{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.4554{{c}}, ~7/4 = 968.9114{{c}}
: [[error map]]: {{val| -0.073 +0.201 -0.005 -0.061 -0.093 -0.012 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2544{{c}}, ~5/4 = 386.4007{{c}}, ~7/4 = 968.8979{{c}}
: error map: {{val| 0.000 +0.299 +0.087 +0.072 +0.074 +0.125 }}


{{Optimal ET sequence|legend=1| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}
{{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: 34.0 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.868


==== 19-limit ====
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.19


Comma list: 715/714, 1089/1088, 1216/1215, 1445/1444
Comma list: 1216/1215, 1540/1539, 2080/2079


Mapping: {{mapping| 1 0 2 0 1 -6 -4 -4 | 0 1 2 0 2 3 6 7 | 0 0 -4 0 -1 3 -2 -4 | 0 0 0 1 0 1 0 0 }}
Mapping: {{mapping| 1 0 0 0 -8 -13 -6 | 0 1 0 0 9 12 5 | 0 0 1 0 0 -1 1 | 0 0 0 1 -1 0 0 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.3413, ~18/11 = 854.6472, ~7/4 = 968.9734
Optimal tunings:
* WE: ~2 = 1199.9273{{c}}, ~3/2 = 702.2289{{c}}, ~5/4 = 386.4508{{c}}, ~7/4 = 968.9092{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.2554{{c}}, ~5/4 = 386.4080{{c}}, ~7/4 = 968.9015{{c}}


{{Optimal ET sequence|legend=1| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}
{{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: 41.9 × 10<sup>-6</sup>
Badness (Sintel): 0.375


== Semiporwellismic (16384/16335) ==
== Argyric (41503/41472) ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 16384/16335
[[Comma list]]: [[41503/41472]]


{{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 1 3 | 0 1 0 0 2 | 0 0 1 0 0 | 0 0 0 2 -3 }}
: Mapping generators: ~2, ~3, ~5, ~144/77


: mapping generators: ~2, ~3, ~128/99, ~7
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0485{{c}}, ~3/2 = 701.9606{{c}}, ~5/4 = 386.2167{{c}}, ~144/77 = 1084.3250{{c}}
: [[error map]]: {{val| +0.049 +0.054 -0.000 -0.127 -0.129 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9739{{c}}, ~5/4 = 386.2460{{c}}, ~144/77 = 1084.2953{{c}}
: error map: {{val| 0.000 +0.019 -0.068 -0.235 -0.256 }}


{{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, 31, 41, 53, 72, 166, 167, 176, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}


[[Badness]]: 0.219 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.777


== Symbiotic (19712/19683) ==
== Chrysic (43923/43904) ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[19712/19683]]
[[Comma list]]: [[43923/43904]]
 
{{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|legend=1| 1 0 0 3 4 | 0 1 0 3 2 | 0 0 1 0 0 | 0 0 0 -4 -3 }}
: Mapping generators: ~2, ~3, ~5, ~33/14


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2681, ~5/4 = 386.4785, ~7/4 = 968.9552
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0167{{c}}, ~3/2 = 701.9323{{c}}, ~5/4 = 386.2803{{c}}, ~33/28 = 284.2370{{c}}
: [[error map]]: {{val| +0.017 -0.006 -0.000 +0.056 -0.114 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9341{{c}}, ~5/4 = 386.2927{{c}}, ~33/28 = 284.2352{{c}}
: error map: {{val| 0.000 -0.021 -0.021 +0.036 -0.155 }}


{{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, 26, 29, 34d, 43, 46, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 2715, 2985, 3327e, 3479, 3821, 4091e, 4433e, 5539e }}


[[Badness]]: 0.120 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.770


=== Tridecimal symbiotic ===
== Elysic (117649/117612) ==
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 2080/2079, 19712/19683
[[Comma list]]: [[117649/117612]]


{{Mapping|legend=1| 1 0 0 0 -8 -13 | 0 1 0 0 9 12 | 0 0 1 0 0 -1 | 0 0 0 1 -1 0 }}
{{Mapping|legend=1| 1 0 0 0 -1 | 0 2 0 0 -5 | 0 0 1 0 0 | 0 0 0 1 3 }}
: Mapping generators: ~2, ~343/198, ~5, ~7


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2721, ~5/4 = 386.4790, ~7/4 = 968.9705
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0027{{c}}, ~343/198 = 950.9861{{c}}, ~5/4 = 386.3082{{c}}, ~7/4 = 968.7558{{c}}
: [[error map]]: {{val| +0.003 +0.017 -0.000 -0.065 +0.033 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/198 = 950.9850{{c}}, ~5/4 = 386.3105{{c}}, ~7/4 = 968.7571{{c}}
: error map: {{val| 0.000 +0.015 -0.003 -0.069 +0.028 }}


{{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| 49, 58, 72, 130, 193, 198, 212, 270, 342, 612, 954, 1084, 1354, 1547, 1817, 2159, 3243, 3513, 5060, 5402, 5672, 8915d, 11074d }}


[[Badness]]: 3.31 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.901


== Olympic (131072/130977) ==
== Olympic (131072/130977) ==
Olympic is highly accurate yet easy to conceptualize and notate on the staff. It equates [[33/32]] with ([[64/63]])<sup>2</sup>, and [[36/35]] with [[1053/1024]], reducing the four [[formal comma]]s required for the [[13-limit]] to two. Therefore it can be notated in [[chain-of-fifths notation]] with two additional set of accidentals, one for the septimal comma step, and the other for the [[aberschisma]] step. It can be viewed as the [[detemperament]] of [[cassaschismic]] that decouples the Pythagorean and septimal commas.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 581: Line 925:


{{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]]s:  
* [[WE]]: ~2 = 1199.9460{{c}}, ~3/2 = 702.0489{{c}}, ~5/4 = 386.4217{{c}}, ~7/4 = 968.9839{{c}}
: [[error map]]: {{val| -0.054 +0.040 -0.000 +0.050 +0.038 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0853{{c}}, ~5/4 = 386.4125{{c}}, ~7/4 = 969.0242{{c}}
: error map: {{val| 0.000 +0.130 +0.099 +0.198 +0.207 }}


{{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 }}
[[Badness]] (Sintel): 2.05


=== Tridecimal olympic ===
=== Tridecimal olympic ===
Line 592: Line 943:


{{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]]s:
* [[WE]]: ~2 = 1199.9638{{c}}, ~3/2 = 702.0645{{c}}, ~5/4 = 386.3208{{c}}, ~7/4 = 968.9712{{c}}
: [[error map]]: {{val| -0.036 +0.073 -0.065 +0.073 +0.128 -0.166 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0884{{c}}, ~5/4 = 386.3269{{c}}, ~7/4 = 969.0015{{c}}
: error map: {{val| 0.000 +0.133 +0.013 +0.176 +0.237 -0.033 }}


{{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 }} *


<nowiki>*</nowiki> [[optimal patent val]]: [[3044edo|3044]]
<nowiki>*</nowiki> [[optimal patent val]]: [[3044edo|3044]]
[[Badness]] (Sintel): 0.292
== Alphaxenic (131769/131072) ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 131769/131072
{{Mapping|legend=1| 2 1 0 0 8 | 0 2 0 0 -1 | 0 0 1 0 0 | 0 0 0 1 0 }}
: mapping generators: ~363/256, ~16/11, ~5, ~7
[[Optimal tuning]]s:
* [[WE]]: ~363/256 = 600.1590{{c}}, ~16/11 = 650.8508{{c}}, ~5/4 = 385.6773{{c}}, ~7/4 = 968.1894{{c}}
: [[error map]]: {{val| +0.318 -0.094 -0.000 -0.000 -0.897 }}
* [[CWE]]: ~363/256 = 600.0000{{c}}, ~16/11 = 650.7321{{c}}, ~5/4 = 385.7900{{c}}, ~7/4 = 968.1927{{c}}
: error map: {{val| 0.000 -0.491 -0.524 -0.633 -2.050 }}
{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee }}
[[Badness]] (Sintel): 23.0
== Seascape (160083/160000) ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: [[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 generators: ~2, ~400/231, ~5, ~7
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0285{{c}}, ~400/231 = 950.9641{{c}}, ~5/4 = 386.3335{{c}}, ~7/4 = 968.7129{{c}}
: [[error map]]: {{val| +0.028 -0.027 +0.077 -0.056 -0.085 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 950.9492{{c}}, ~5/4 = 386.3638{{c}}, ~7/4 = 969.7159{{c}}
: error map: {{val| 0.000 -0.057 +0.050 -0.110 -0.154 }}
{{Optimal ET sequence|legend=1| 14c, 15, 19, 29, 39d, 43, 53, 58, 72, 130, 183, 198, 212, 255, 270, 342, 612, 954, 1084, 1354, 1696, 2833e, 3575, 3917e, 4187e, 4529de, 5883de }}
[[Badness]] (Sintel): 0.596
== Fifthchromismic (1610510/1594323) ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: [[1610510/1594323]]
{{Mapping|legend=1| 1 0 -6 0 1 | 0 1 3 0 2 | 0 0 5 0 -1 | 0 0 0 1 0 }}
: Mapping generators: ~2, ~3, ~18/11, ~7
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9757{{c}}, ~3/2 = 702.7610{{c}}, ~18/11 = 855.5656{{c}}, ~7/4 = 968.8735{{c}}
: [[error map]]: {{val| -0.024 +0.782 -0.130 -0.001 -1.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.7565{{c}}, ~18/11 = 855.5885{{c}}, ~7/4 = 968.8591{{c}}
: error map: {{val| 0.000 +0.801 -0.102 +0.033 -1.393 }}
{{Optimal ET sequence|legend=1| 70, 73, 77, 80, 87, 157, 167, 171e, 251, 258e, 338, 432, 519 }}
[[Badness]] (Sintel): 97.2
=== Fifthchroma ===
{{See also| Fifth-chroma temperaments }}
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: 352/351, 951665/944784
{{Mapping|legend=1| 1 0 -6 0 1 6 | 0 1 3 0 2 -1 | 0 0 5 0 -1 -1 | 0 0 0 1 0 0 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8431{{c}}, ~3/2 = 702.9005{{c}}, ~18/11 = 855.4090{{c}}, ~7/4 = 969.1386{{c}}
: [[error map]]: {{val| -0.157 +0.789 -0.097 -0.001 -1.396 +0.379 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.9326{{c}}, ~18/11 = 855.5400{{c}}, ~7/4 = 969.1437{{c}}
: error map: {{val| 0.000 +0.978 +0.184 +0.318 -0.993 +1.000 }}
{{Optimal ET sequence|legend=1| 70, 73, 77, 80, 87, 157, 167, 171e, 181, 244, 251, 261, 338, 599b }}
[[Badness]] (Sintel): 7.63
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 352/351, 715/714, 155771/155520
Mapping: {{mapping| 1 0 -6 0 1 6 0 | 0 1 3 0 2 -1 3 | 0 0 5 0 -1 -1 3 | 0 0 0 1 0 0 -1 }}
Optimal tunings:
* WE: ~2 = 1199.8126{{c}}, ~3/2 = 702.8981{{c}}, ~11/9 = 855.3764{{c}}, ~7/4 = 969.3543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9376{{c}}, ~11/9 = 855.5306{{c}}, ~7/4 = 969.4313{{c}}
{{Optimal ET sequence|legend=0| 70, 73g, 77, 80, 87, 94, 157, 167, 181, 251, 261, 338, 432, 599bg }}
Badness (Sintel): 3.65
==== 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 -6 0 1 6 0 0 | 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 tunings:
* WE: ~2 = 1199.8128{{c}}, ~3/2 = 702.9226{{c}}, ~11/9 = 855.3621{{c}}, ~7/4 = 969.3187{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9618{{c}}, ~11/9 = 855.5163{{c}}, ~7/4 = 969.3961{{c}}
{{Optimal ET sequence|legend=0| 70, 73gh, 77, 80, 87, 94, 157, 167h, 181, 251, 261, 338, 599bg }}
Badness (Sintel): 2.63
== Quartismic (117440512/117406179) ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 117440512/117406179
{{Mapping|legend=1| 1 0 0 1 5 | 0 1 0 1 -1 | 0 0 1 0 0 | 0 0 0 5 1 }}
: mapping generators: ~2, ~3, ~5, ~33/32
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9875{{c}}, ~3/2 = 701.9753{{c}}, ~5/4 = 386.3387{{c}}, ~33/32 = 53.3743{{c}}
: [[error map]]: {{val| -0.012 +0.008 -0.000 -0.004 +0.031 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9802{{c}}, ~5/4 = 386.3343{{c}}, ~33/32 = 53.3730{{c}}
: error map: {{val| 0.000 +0.025 +0.021 +0.019 +0.075 }}
{{Optimal ET sequence|legend=1| 21, 22, 43, 46, 65d, 68, 89, 111, 159, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2159, 2653, 3125, 3395, 7060, 7554, 10949e, 14614e, 15850ee, 22168bdee, 23404bcdee, 26799bcdeee, 34353bcdeeee }}
[[Badness]] (Sintel): 5.00
=== Tridecimal quartismic ===
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: 6656/6655, 123201/123200
{{Mapping|legend=1| 1 0 0 1 5 6 | 0 1 0 1 -1 -3 | 0 0 1 0 0 1 | 0 0 0 5 1 3 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9884{{c}}, ~3/2 = 701.9674{{c}}, ~5/4 = 386.3432{{c}}, ~33/32 = 53.3763{{c}}
: [[error map]]: {{val| -0.012 +0.001 +0.006 -0.000 +0.045 -0.016 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9728{{c}}, ~5/4 = 386.3385{{c}}, ~33/32 = 53.3748{{c}}
: error map: {{val| 0.000 +0.018 +0.025 +0.021 +0.084 +0.017 }}
{{Optimal ET sequence|legend=1| 22, 43f, 46, 65d, 89f, 111, 159, 224, 270, 494, 764, 1012, 1236, 1506, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816dee }}
[[Badness]] (Sintel): 0.456


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