Catalog of rank-4 temperaments: Difference between revisions

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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 }}
[[Badness]] (Sintel): 0.339
=== Tridecimal valinorsmic ===
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350


: mapping generators: ~2, ~3, ~5, ~7
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 tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113
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=1| 22, 31, 46, 53, 58, 80, 111 }}
{{Optimal ET sequence|legend=0| 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 268cd }}


[[Badness]]: 0.0186 × 10<sup>-6</sup>
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 }}


: mapping generators: ~2, ~11/9, ~5, ~7
[[Badness]] (Sintel): 0.928


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464
== Frostmic (245/242) ==
[[Subgroup]]: 2.3.5.7.11


{{Optimal ET sequence|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}
[[Comma list]]: [[245/242]]
 
{{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 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}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538
: [[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 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]]: 2.550 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.669
 
== 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


{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 0 0 2 | 0 0 1 0 -1 | 0 0 0 1 2 }}
[[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 generators: ~2, ~3, ~5, ~7
[[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


{{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]] (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]]
: [{{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 }}
{{Optimal ET sequence|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}
 
[[Badness]] (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 }}


{{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 }}
[[Badness]] (Sintel): 0.880


{{Optimal ET sequence|legend=1| 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}
=== Commas 364/363, 385/384 ===
[[Subgroup]]: 2.3.5.7.11.13


==== 17-limit ====
[[Comma list]]: 364/363, 385/384
[[Subgroup]]: 2.3.5.7.11.13.17


[[Comma list]]: 273/272, 441/440, 561/560
{{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 }}


{{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 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 }}


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


=== Commas 441/440, 847/845 ===
[[Badness]] (Sintel): 0.871
 
=== 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:  
* [[WE]]: ~2 = 1199.9149{{c}}, ~3/2 = 701.7196{{c}}, ~5/4 = 386.2545{{c}}, ~7/4 = 969.6659{{c}}
: [[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 }}


=== Commas 540/539, 847/845 ===
[[Badness]] (Sintel): 0.191
 
=== Commas 540/539, 729/728 ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 540/539, 847/845
[[Comma list]]: 540/539, 729/728
 
{{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|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 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9170{{c}}, ~3/2 = 701.7128{{c}}, ~5/4 = 386.2521{{c}}, ~7/4 = 969.6598{{c}}
: [[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 }}


: mapping generators: ~2, ~3, ~5, ~13/11
{{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| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
[[Badness]] (Sintel): 0.454


=== Commas 540/539, 625/624 ===
=== Commas 540/539, 625/624 ===
Line 288: 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]] (Sintel): 0.943


=== Commas 540/539, 676/675 ===
=== Commas 540/539, 676/675 ===
Line 297: 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]] (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 308: 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 323: 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 333: 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 345: 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 356: 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:  
* [[WE]]: ~2 = 1200.1317{{c}}, ~3/2 = 701.4920{{c}}, ~5/4 = 386.4054{{c}}, ~7/4 = 968.5622{{c}}
: [[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]] (Sintel): 1.50
=== Commas 729/728, 1001/1000 ===
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: 729/728, 1001/1000
{{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 }}
[[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:
* [[WE]]: ~2 = 1199.9538{{c}}, ~3/2 = 701.5852{{c}}, ~5/4 = 387.1356{{c}}, ~7/4 = 969.7495{{c}}
: [[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| 19, 26, 39df, 46, 53, 58, 72, 111, 130, 183, 313 }}
[[Badness]] (Sintel): 0.677


== Kalismic (9801/9800) ==
== Kalismic (9801/9800) ==
Line 367: 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 381: 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| 10e, 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d }}
{{Optimal ET sequence|legend=1| 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d }}


== Semicanousmic (14641/14580) ==
[[Badness]] (Sintel): 0.256
 
== Unisquary (12005/11979) ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[14641/14580]]
[[Comma list]]: [[12005/11979]]


{{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 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, ~18/11, ~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, ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866
{{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| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
[[Badness]] (Sintel): 2.43


[[Badness]]: 0.351 × 10<sup>-6</sup>
=== Hensquary ===
[[Subgroup]]: 2.3.5.7.11.13


=== Tridecimal semicanousmic ===
[[Comma list]]: 364/363, 1716/1715
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, 14641/14580
{{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: {{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 }}
[[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, ~3/2 = 702.4931, ~18/11 = 854.6400, ~7/4 = 969.0099
{{Optimal ET sequence|legend=1| 26, 37, 46, 58, 72, 121, 130, 193, 239, 251e, 323e, 369 }}


{{Optimal ET sequence|legend=1| 87, 111, 181, 198, 323, 410 }}
[[Badness]] (Sintel): 0.988


Badness: 17.1 × 10<sup>-6</sup>
=== Ekasquary ===
[[Subgroup]]: 2.3.5.7.11.13


==== 17-limit ====
[[Comma list]]: 1575/1573, 4459/4455
Subgroup: 2.3.5.7.11.13.17


Comma list: 715/714, 1089/1088, 14641/14580
{{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: {{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 }}
[[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, ~3/2 = 702.4099, ~18/11 = 854.6338, ~7/4 = 969.0228
{{Optimal ET sequence|legend=1| 46f, 49f, 58, 72, 118, 121, 130, 190, 193, 248, 311, 441, 752e, 1072 }}


{{Optimal ET sequence|legend=1| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}
[[Badness]] (Sintel): 1.12


Badness: 34.0 × 10<sup>-6</sup>
== Semicanousmic (14641/14580) ==
[[Subgroup]]: 2.3.5.7.11


==== 19-limit ====
[[Comma list]]: [[14641/14580]]
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 715/714, 1089/1088, 1216/1215, 1445/1444
{{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


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 }}
[[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 tuning (POTE): ~2 = 1\1, ~3/2 = 702.3413, ~18/11 = 854.6472, ~7/4 = 968.9734
{{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| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}
[[Badness]] (Sintel): 6.40
 
Badness: 41.9 × 10<sup>-6</sup>


== Semiporwellismic (16384/16335) ==
== Semiporwellismic (16384/16335) ==
Line 449: Line 784:


{{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]]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 }}
{{Optimal ET sequence|legend=1| 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee }}


[[Badness]]: 0.219 × 10<sup>-6</sup>
[[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]] (Sintel): 1.13


== Symbiotic (19712/19683) ==
== Symbiotic (19712/19683) ==
Line 462: Line 819:


{{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:  
 
* [[WE]]: ~2 = 1199.9290{{c}}, ~3/2 = 702.2265{{c}}, ~5/4 = 386.4556{{c}}, ~7/4 = 968.8979{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2681, ~5/4 = 386.4785, ~7/4 = 968.9552
: [[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| 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]]: 0.120 × 10<sup>-6</sup>
[[Badness]] (Sintel): 2.19


=== Tridecimal symbiotic ===
=== Tridecimal symbiotic ===
Line 478: Line 838:
{{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 -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 = 702.2721, ~5/4 = 386.4790, ~7/4 = 968.9705
[[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| 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]]: 3.31 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.868
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1216/1215, 1540/1539, 2080/2079
 
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 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=0| 17ch, 24f, 34dffh, 36ceh, 41, 53, 58h, 94, 111, 152f, 212h, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bfh }}
 
Badness (Sintel): 0.375
 
== Argyric (41503/41472) ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: [[41503/41472]]
 
{{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
 
[[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, 31, 41, 53, 72, 166, 167, 176, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}
 
[[Badness]] (Sintel): 0.777
 
== Chrysic (43923/43904) ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: [[43923/43904]]
 
{{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]]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, 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]] (Sintel): 0.770
 
== Elysic (117649/117612) ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: [[117649/117612]]
 
{{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]]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| 49, 58, 72, 130, 193, 198, 212, 270, 342, 612, 954, 1084, 1354, 1547, 1817, 2159, 3243, 3513, 5060, 5402, 5672, 8915d, 11074d }}
 
[[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 490: 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 501: 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 -->

Latest revision as of 05:05, 5 June 2026

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

A rank-4 temperament has a period and three additional independent generators, for a total of four 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: [⟨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 tunings:

  • WE: ~2 = 1202.4372 ¢, ~3/2 = 693.2306 ¢, ~5/4 = 374.6677 ¢, ~7/4 = 963.8387 ¢
error map: ⟨+2.437 -6.287 -6.772 -0.113 +14.685]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.1966 ¢, ~5/4 = 376.0821 ¢, ~7/4 = 965.0600 ¢
error map: ⟨0.000 -8.758 -10.232 -3.766 +11.157]

Optimal ET sequence: 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: [⟨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 tunings:

  • WE: ~2 = 1200.3175 ¢, ~3/2 = 700.0109 ¢, ~5/4 = 385.6677 ¢, ~7/4 = 973.2571 ¢
error map: ⟨+0.318 -1.627 -0.011 +5.066 -3.873]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.0599 ¢, ~5/4 = 385.9307 ¢, ~7/4 = 973.4880 ¢
error map: ⟨0.000 -1.895 -0.383 +4.662 -4.462]

Optimal ET sequence: 5, 8d, 9, 12, 17c, 19e, 22, 31, 53, 84e, 96, 127

Badness (Sintel): 0.545

Ptolemismic (100/99)

Subgroup: 2.3.5.7.11

Comma list: 100/99

Mapping: [⟨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 tunings:

  • WE: ~2 = 1199.2633 ¢, ~3/2 = 704.5205 ¢, ~5/4 = 383.8317 ¢, ~7/4 = 970.2843 ¢
error map: ⟨-0.737 +1.829 -3.955 -0.015 +4.358]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.4751 ¢, ~5/4 = 383.0756 ¢, ~7/4 = 969.8904 ¢
error map: ⟨0.000 +2.520 -3.238 +1.065 +5.883]

Optimal ET sequence: 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e *

* optimal patent val: 104

Badness (Sintel): 0.411

Biyatismic (121/120)

Subgroup: 2.3.5.7.11

Comma list: 121/120

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1200.6628 ¢, ~3/2 = 701.8452 ¢, ~11/10 = 157.8337 ¢, ~7/4 = 967.4929 ¢
error map: ⟨+0.663 +0.553 +1.190 -0.007 -5.318]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9848 ¢, ~11/10 = 157.6099 ¢, ~7/4 = 967.8187 ¢
error map: ⟨0.000 +0.030 +0.451 -1.007 -6.943]

Optimal ET sequence: 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee

Badness (Sintel): 0.629

Valinorsmic (176/175)

Subgroup: 2.3.5.7.11

Comma list: 176/175

Mapping: [⟨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 tunings:

  • WE: ~2 = 1199.3113 ¢, ~3/2 = 702.6414 ¢, ~5/4 = 389.5404 ¢, ~7/4 = 971.5534 ¢
error map: ⟨-0.689 -0.022 +1.849 +1.350 -2.061]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5869 ¢, ~5/4 = 389.3935 ¢, ~7/4 = 971.5196 ¢
error map: ⟨0.000 +0.632 +3.080 +2.694 -1.011]

Optimal ET sequence: 22, 27e, 31, 46, 53, 58, 80, 111, 237cd, 268cd

Badness (Sintel): 0.339

Tridecimal valinorsmic

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350

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 ¢, ~3/2 = 702.5360 ¢, ~5/4 = 389.5451 ¢, ~7/4 = 971.5415 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6943 ¢, ~5/4 = 389.3823 ¢, ~7/4 = 971.5305 ¢

Optimal ET sequence: 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 268cd

Badness (Sintel): 0.603

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

Comma list: 243/242

Mapping: [⟨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 tunings:

  • WE: ~2 = 1200.0635 ¢, ~11/9 = 350.5439 ¢, ~5/4 = 386.1857 ¢, ~7/4 = 968.6977 ¢
error map: ⟨+0.064 -0.804 -0.001 -0.001 +1.529]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.5486 ¢, ~5/4 = 386.2387 ¢, ~7/4 = 968.7352 ¢
error map: ⟨0.000 -0.858 -0.075 -0.091 +1.425]

Optimal ET sequence: 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

Comma list: 245/242

Mapping: [⟨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 tunings:

  • WE: ~2 = 1200.2442 ¢, ~3/2 = 701.7038 ¢, ~11/7 = 792.3594 ¢, ~7/4 = 964.4060 ¢
error map: ⟨+0.244 -0.007 -1.351 -3.931 +5.936]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7533 ¢, ~11/7 = 792.3351 ¢, ~7/4 = 964.5274 ¢
error map: ⟨0.000 -0.202 -1.644 -4.298 +5.545]

Optimal ET sequence: 9, 12, 15, 23de, 24, 26, 27e, 38d, 41, 91, 106d

Badness (Sintel): 1.64

Akua (352/351, 847/845)

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1199.6918 ¢, ~3/2 = 702.7270 ¢, ~5/4 = 386.9729 ¢, ~11/8 = 551.3122 ¢
error map: ⟨-0.308 +0.464 +0.043 -0.064 -0.930 +1.062]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8960 ¢, ~5/4 = 386.7835 ¢, ~11/8 = 551.0679 ¢
error map: ⟨0.000 +0.941 +0.470 +0.581 -0.025 +1.852]

Optimal ET sequence: 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f

Badness (Sintel): 0.669

Keenanismic (385/384)

Main article: Keenanismic family

Subgroup: 2.3.5.7.11

Comma list: 385/384

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

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

Optimal tunings:

  • WE: ~2 = 1200.4104 ¢, ~3/2 = 701.6916 ¢, ~5/4 = 385.1761 ¢, ~7/4 = 967.5421 ¢
error map: ⟨+0.410 +0.147 -0.317 -0.463 -0.703]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6933 ¢, ~5/4 = 385.1491 ¢, ~7/4 = 967.3004 ¢
error map: ⟨0.000 -0.262 -1.165 -1.525 -2.074]

Minimax tuning:

  • 11-odd-limit: ~2 = [1 0 0 0 0⟩, ~3 = [0 1 0 0 0⟩, ~5 = [7/3 1/3 2/3 -1/3 -1/3⟩, ~7 = [7/3 1/3 -1/3 2/3 -1/3⟩
Unchanged-interval (eigenmonzo) basis: 2.3.7/5.11/5

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

Badness (Sintel): 0.277

Martwin

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384

Mapping: [⟨1 0 0 0 7 2], ⟨0 1 0 0 1 4], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 -1 0]]

Lattice basis:

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

Optimal tunings:

  • WE: ~2 = 1200.3494 ¢, ~3/2 = 702.2131 ¢, ~5/4 = 384.8666 ¢, ~7/4 = 967.8487 ¢
error map: ⟨+0.349 +0.608 -0.748 -0.278 -0.422 -0.710]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1659 ¢, ~5/4 = 384.8721 ¢, ~7/4 = 967.6109 ¢
error map: ⟨0.000 +0.211 -1.442 -1.215 -1.635 -1.608]

Minimax tuning:

  • 13- and 15-odd-limit: ~2 = ⟨1 0 0 0 0 0], ~3 = ⟨0 1 0 0 0 0], ~5 = ⟨2/3 4/3 1/3 0 0 -1/3], ~7 = ⟨19/6 -1/6 -1/6 1/2 -1/2 1/6]
unchanged-interval (eigenmonzo) basis: 2.3.11/7.13/5

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

Badness (Sintel): 0.579

Ancient

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624

Mapping: [⟨1 0 0 0 7 -4], ⟨0 1 0 0 1 -1], ⟨0 0 1 0 -1 4], ⟨0 0 0 1 -1 0]]

Optimal tunings:

  • WE: ~2 = 1200.4130 ¢, ~3/2 = 701.6863 ¢, ~5/4 = 385.2319 ¢, ~7/4 = 967.5188 ¢
error map: ⟨+0.413 +0.144 -0.256 -0.481 -0.730 -0.047]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6838 ¢, ~5/4 = 385.2497 ¢, ~7/4 = 967.2556 ¢
error map: ⟨0.000 -0.271 -1.064 -1.570 -2.139 -1.213]

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

Badness (Sintel): 0.675

Commas 351/350, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384

Mapping: [⟨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 tunings:

  • WE: ~2 = 1200.6793 ¢, ~3/2 = 700.7455 ¢, ~5/4 = 385.5609 ¢, ~7/4 = 967.5816 ¢
error map: ⟨+0.679 -0.530 +0.606 +0.114 -0.998 -1.343]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.5141 ¢, ~5/4 = 385.6072 ¢, ~7/4 = 967.1521 ¢
error map: ⟨0.000 -1.441 -0.706 -1.674 -3.563 -3.703]

Optimal ET sequence: 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff

Badness (Sintel): 0.781

Zaxa

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384

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

Optimal tunings:

  • WE: ~2 = 1200.1117 ¢, ~3/2 = 702.6588 ¢, ~5/4 = 385.6821 ¢, ~7/4 = 968.0061 ¢
error map: ⟨+0.112 +0.815 -0.408 -0.596 -1.901 +1.137]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.6029 ¢, ~5/4 = 385.6441 ¢, ~7/4 = 967.9030 ¢
error map: ⟨0.000 +0.648 -0.670 -0.923 -2.262 +0.720]

Optimal ET sequence: 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e

Badness (Sintel): 0.880

Commas 364/363, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 385/384

Mapping: [⟨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 = 1200.4593 ¢, ~3/2 = 701.9714 ¢, ~5/4 = 384.8903 ¢, ~7/4 = 966.4090 ¢
error map: ⟨+0.459 +0.476 -0.505 -1.498 +1.191 -1.325]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.9809 ¢, ~5/4 = 384.8523 ¢, ~7/4 = 966.1074 ¢
error map: ⟨0.000 +0.026 -1.461 -2.719 -0.297 -2.612]

Optimal ET sequence: 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159

Badness (Sintel): 0.871

Commas 385/384, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 847/845

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1200.3530 ¢, ~3/2 = 701.7283 ¢, ~5/4 = 385.4633 ¢, ~13/11 = 291.0352 ¢
error map: ⟨+0.353 +0.126 -0.144 -0.586 -1.175 +0.651]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7205 ¢, ~5/4 = 385.3656 ¢, ~13/11 = 290.9769 ¢
error map: ⟨0.000 -0.235 -0.948 -1.507 -2.282 -0.515]

Optimal ET sequence: 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee

Badness (Sintel): 1.09

Werckismic (441/440)

Subgroup: 2.3.5.7.11

Comma list: 441/440

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1200.1734 ¢, ~3/2 = 701.4907 ¢, ~5/4 = 386.2784 ¢, ~7/4 = 967.5667 ¢
error map: ⟨+0.173 -0.291 +0.311 -0.912 +0.692]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4963 ¢, ~5/4 = 386.4317 ¢, ~7/4 = 967.5883 ¢
error map: ⟨0.000 -0.459 +0.118 -1.238 +0.420]

Minimax tuning:

  • 11-odd-limit: ~2 = [1 0 0 0 0⟩, ~3 = [1/2 2/3 1/6 -1/3 1/6⟩, ~5 = [0 0 1 0 0⟩, ~7 = [1 -2/3 1/3 1/3 1/3⟩
unchanged-interval (eigenmonzo) basis: 2.5.9/7.11

Optimal ET sequence: 10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d

Badness (Sintel): 0.189

Commas 364/363, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440

Mapping: [⟨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 to lattice: [⟨0 1 1 -1 -1 0], ⟨0 0 1 0 -1 -2], ⟨0 0 1 1 1 1]]

Lattice basis:

3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657

Optimal tunings:

  • WE: ~2 = 1200.0145 ¢, ~3/2 = 701.8745 ¢, ~5/4 = 386.4853 ¢, ~7/4 = 967.8244 ¢
error map: ⟨+0.014 -0.066 +0.201 -0.972 +1.609 -0.667]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8716 ¢, ~5/4 = 386.4975 ¢, ~7/4 = 967.8241 ¢
error map: ⟨0.000 -0.083 +0.184 -1.002 +1.576 -0.692]

Minimax tuning:

  • 13-odd-limit: ~2 = [1 0 0 0 0 0⟩, ~3 = [5/3 0 1/3 -1/3 -1/3 1/3⟩, ~5 = [1/6 0 5/6 2/3 -5/6 1/3⟩, ~7 = [0 0 0 1 0 0⟩
eigenmonzo (unchanged-interval) basis: 2.7.11/5.13
  • 15-odd-limit: ~2 = [1 0 0 0 0 0⟩, ~3 = [5/4 1/4 1/4 -1/4 -1/4 1/4⟩, ~5 = [5/4 -3/4 5/4 -1/4 -1/4 1/4⟩, ~7 = [17/8 -11/8 5/8 -1/8 3/8 1/8⟩
eigenmonzo (unchanged-interval) basis: 2.5/3.11/9.13/7

Optimal ET sequence: 12f, 14cf, 15, 17c, 26, 29, 31f, 41, 46, 58, 72, 87, 130, 217, 289

Badness (Sintel): 0.452

Commas 351/350, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440

Mapping: [⟨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 tunings:

  • WE: ~2 = 1200.2006 ¢, ~3/2 = 701.0354 ¢, ~5/4 = 386.8579 ¢, ~7/4 = 968.2226 ¢
error map: ⟨+0.201 -0.719 +0.945 -0.202 +0.541 -0.893]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.0371 ¢, ~5/4 = 387.0417 ¢, ~7/4 = 968.2546 ¢
error map: ⟨0.000 -0.918 +0.728 -0.571 +0.224 -1.301]

Optimal ET sequence: 12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf

Badness (Sintel): 0.524

Commas 196/195, 352/351

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351

Mapping: [⟨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 tunings:

  • WE: ~2 = 1199.6830 ¢, ~3/2 = 702.6689 ¢, ~5/4 = 387.7765 ¢, ~7/4 = 967.0260 ¢
error map: ⟨-0.317 +0.397 +0.829 -2.434 -0.022 +2.128]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.7536 ¢, ~5/4 = 387.5834 ¢, ~7/4 = 966.9373 ¢
error map: ⟨0.000 +0.799 +1.270 -1.889 +0.480 +3.010]

Optimal ET sequence: 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232

Badness (Sintel): 0.548

Tannic

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1287/1280

Mapping: [⟨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 tunings:

  • WE: ~2 = 1200.7398 ¢, ~3/2 = 700.6733 ¢, ~5/4 = 385.4125 ¢, ~7/4 = 966.9648 ¢
error map: ⟨+0.740 -0.542 +0.578 -0.381 -0.714 -1.147]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.3274 ¢, ~5/4 = 385.8974 ¢, ~7/4 = 966.8107 ¢
error map: ⟨0.000 -1.628 -0.416 -2.015 -2.939 -3.664]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 700.555 ¢, ~5/4 = 387.816 ¢, ~7/4 = 967.984 ¢

Optimal ET sequence: 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff

Badness (Sintel): 1.23

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 441/440, 561/560

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]]

Optimal tunings:

  • WE: ~2 = 1200.7320 ¢, ~3/2 = 700.7374 ¢, ~5/4 = 385.2303 ¢, ~7/4 = 966.8369 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.3909 ¢, ~5/4 = 385.7215 ¢, ~7/4 = 966.6922 ¢

Optimal ET sequence: 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef

Badness (Sintel): 0.701

Commas 441/440, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 847/845

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1200.1778 ¢, ~3/2 = 701.4785 ¢, ~5/4 = 385.4755 ¢, ~13/11 = 290.3844 ¢
error map: ⟨+0.178 -0.299 +0.517 -1.226 -0.170 +1.005]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4843 ¢, ~5/4 = 386.6332 ¢, ~13/11 = 290.3162 ¢
error map: ⟨0.000 -0.471 +0.319 -1.560 -0.451 +0.655]

Optimal ET sequence: 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d

Badness (Sintel): 1.11

Swetismic (540/539)

Subgroup: 2.3.5.7.11

Comma list: 540/539

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1199.9149 ¢, ~3/2 = 701.7196 ¢, ~5/4 = 386.2545 ¢, ~7/4 = 969.6659 ¢
error map: ⟨-0.085 -0.321 -0.229 +0.670 +0.508]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.6950 ¢, ~5/4 = 386.1795 ¢, ~7/4 = 969.6366 ¢
error map: ⟨0.000 -0.260 -0.134 +0.811 +0.673]

Optimal ET sequence: 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 (Sintel): 0.191

Commas 540/539, 729/728

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1199.9170 ¢, ~3/2 = 701.7128 ¢, ~5/4 = 386.2521 ¢, ~7/4 = 969.6598 ¢
error map: ⟨-0.083 -0.325 -0.228 +0.668 +0.504 +0.006]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7231 ¢, ~5/4 = 386.1819 ¢, ~7/4 = 969.6608 ¢
error map: ⟨0.000 -0.232 -0.132 +0.835 +0.712 +0.150]

Optimal ET sequence: 12e, 14cf, 17c, 19, 22f, 31f, 39df, 41, 53, 58, 72, 111, 130, 183, 224, 354, 578, 985d, 1267ddef

Badness (Sintel): 0.454

Commas 540/539, 625/624

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624

Mapping: [⟨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 tunings:

  • WE: ~2 = 1200.0035 ¢, ~3/2 = 701.6973 ¢, ~5/4 = 385.6264 ¢, ~7/4 = 969.4616 ¢
error map: ⟨+0.004 -0.254 -0.680 +0.643 +0.488 +0.291]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6983 ¢, ~5/4 = 385.6278 ¢, ~7/4 = 969.4622 ¢
error map: ⟨0.000 -0.257 -0.686 +0.636 +0.480 +0.285]

Optimal ET sequence: 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224

Badness (Sintel): 0.943

Commas 540/539, 676/675

Subgroup: 2.3.5.7.11

Comma list: 540/539, 676/675

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1199.7990 ¢, ~26/15 = 950.9130 ¢, ~5/4 = 386.8478 ¢, ~7/4 = 970.1529 ¢
error map: ⟨-0.201 -0.129 +0.132 +0.925 +0.702 -1.142]
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.9953 ¢, ~5/4 = 386.7094 ¢, ~7/4 = 970.1192 ¢
error map: ⟨0.000 +0.036 +0.396 +1.293 +1.125 -0.832]

Optimal ET sequence: 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de

Badness (Sintel): 0.802

Commas 540/539, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 847/845

Mapping: [⟨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 tunings:

  • WE: ~2 = 1199.8339 ¢, ~3/2 = 701.4961 ¢, ~5/4 = 386.5624 ¢, ~13/11 = 291.6840 ¢
error map: ⟨-0.166 -0.625 -0.084 +0.772 -0.627 +1.848]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4439 ¢, ~5/4 = 386.4201 ¢, ~13/11 = 291.7287 ¢
error map: ⟨0.000 -0.511 +0.106 +1.052 -0.321 +2.198]

Optimal ET sequence: 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f

Badness (Sintel): 1.04

Pentacircle (896/891)

Subgroup: 2.3.5.7.11

Comma list: 896/891

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1199.3774 ¢, ~3/2 = 703.4693 ¢, ~5/4 = 387.5573 ¢, ~7/4 = 969.3690 ¢
error map: ⟨-0.623 +0.892 -0.002 -0.702 +1.061]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7426 ¢, ~5/4 = 387.2445 ¢, ~7/4 = 969.0476 ¢
error map: ⟨0.000 +1.788 +0.931 +0.222 +2.759]

Optimal ET sequence: 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee

Badness (Sintel): 1.20

Tridecimal pentacircle (gentle)

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363

Mapping: [⟨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 tunings:

  • WE: ~2 = 1199.3706 ¢, ~3/2 = 703.4872 ¢, ~5/4 = 387.5708 ¢, ~7/4 = 969.3987 ¢
error map: ⟨-0.629 +0.903 -0.002 -0.686 +0.985 +0.055]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.8328 ¢, ~5/4 = 387.2881 ¢, ~7/4 = 969.1612 ¢
error map: ⟨0.000 +1.878 +0.974 +0.335 +2.512 +1.804]

Optimal ET sequence: 12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff

Badness (Sintel): 0.885

Topsy (847/845, 1001/1000)

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000

Mapping: [⟨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 tunings:

  • WE: ~2 = 1200.0343 ¢, ~3/2 = 701.9205 ¢, ~5/4 = 386.6098 ¢, ~20/13 = 745.4658 ¢
error map: ⟨+0.034 -0.000 +0.365 -0.353 -0.934 +0.753]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9261 ¢, ~5/4 = 386.6458 ¢, ~20/13 = 745.4431 ¢
error map: ⟨0.000 -0.029 +0.332 -0.408 -1.001 +0.675]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 701.955 ¢, ~5/4 = 386.699 ¢, ~13/10 = 454.558 ¢

Optimal ET sequence: 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)

Subgroup: 2.3.5.7.11

Comma list: 3025/3024

Mapping: [⟨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

Optimal tuning:

  • WE: ~2 = 1200.0197 ¢, ~3/2 = 701.9725 ¢, ~5/4 = 386.2210 ¢, ~55/36 = 733.4363 ¢
error map: ⟨+0.020 +0.037 -0.053 +0.039 -0.118]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9782 ¢, ~5/4 = 386.2319 ¢, ~55/36 = 733.4299 ¢
error map: ⟨0.000 +0.023 -0.082 +0.012 -0.164]

Optimal ET sequence: 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: [⟨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 tunings:

  • WE: ~2 = 1199.9717 ¢, ~3/2 = 702.0958 ¢, ~5/4 = 386.2618 ¢, ~55/36 = 733.4380 ¢
error map: ⟨-0.028 +0.113 -0.108 +0.118 -0.007 -0.134]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0990 ¢, ~5/4 = 386.2477 ¢, ~55/36 = 733.4490 ¢
error map: ⟨0.000 +0.144 -0.066 +0.171 +0.081 -0.082]

Optimal ET sequence: 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)

Subgroup: 2.3.5.7.11

Comma list: 8019/8000

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1200.1317 ¢, ~3/2 = 701.4920 ¢, ~5/4 = 386.4054 ¢, ~7/4 = 968.5622 ¢
error map: ⟨+0.132 -0.331 +0.355 -0.000 -0.263]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4730 ¢, ~5/4 = 386.5375 ¢, ~7/4 = 968.6211 ¢
error map: ⟨0.000 -0.482 +0.224 -0.205 -0.543]

Optimal ET sequence: 12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be

Badness (Sintel): 1.50

Commas 729/728, 1001/1000

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1001/1000

Mapping: [⟨1 0 0 0 6 -3], ⟨0 1 0 0 -6 6], ⟨0 0 1 0 3 0], ⟨0 0 0 1 0 -1]]

Optimal tunings:

  • WE: ~2 = 1200.1318 ¢, ~3/2 = 701.4926 ¢, ~5/4 = 386.4063 ¢, ~7/4 = 968.5613 ¢
error map: ⟨+0.132 -0.331 +0.356 -0.001 -0.264 -0.002]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4770 ¢, ~5/4 = 386.5437 ¢, ~7/4 = 968.6150 ¢
error map: ⟨0.000 -0.478 +0.230 -0.211 -0.549 -0.280]

Optimal ET sequence: 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: [⟨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 tunings:

  • WE: ~2 = 1199.9538 ¢, ~3/2 = 701.5852 ¢, ~5/4 = 387.1356 ¢, ~7/4 = 969.7495 ¢
error map: ⟨-0.046 -0.416 +0.730 +0.831 +0.301 -1.447]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.5936 ¢, ~5/4 = 387.0975 ¢, ~7/4 = 969.7451 ¢
error map: ⟨0.000 -0.361 +0.784 +0.919 +0.413 -1.369]

Optimal ET sequence: 19, 26, 39df, 46, 53, 58, 72, 111, 130, 183, 313

Badness (Sintel): 0.677

Kalismic (9801/9800)

Subgroup: 2.3.5.7.11

Comma list: 9801/9800

Mapping: [⟨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

Optimal tuning:

  • WE: ~99/70 = 600.0018 ¢, ~3/2 = 701.9397 ¢, ~5/4 = 386.3193 ¢, ~7/4 = 968.8374 ¢
error map: ⟨+0.004 -0.012 +0.013 +0.019 -0.028]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.9399 ¢, ~5/4 = 386.3226 ¢, ~7/4 = 968.8399 ¢
error map: ⟨0.000 -0.015 +0.009 +0.014 -0.035]

Optimal ET sequence: 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

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079

Mapping: [⟨2 0 0 0 3 -7], ⟨0 1 0 0 -2 1], ⟨0 0 1 0 1 0], ⟨0 0 0 1 1 2]]

Lattice basis:

3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299

Optimal tuning:

  • WE: ~99/70 = 599.9732 ¢, ~3/2 = 702.0465 ¢, ~5/4 = 386.4214 ¢, ~7/4 = 969.1757 ¢
error map: ⟨-0.054 +0.038 +0.001 +0.243 -0.001 -0.210]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.0554 ¢, ~5/4 = 386.3788 ¢, ~7/4 = 969.1756 ¢
error map: ⟨0.000 +0.100 +0.065 +0.350 +0.126 -0.121]

Minimax tuning:

  • 13- and 15-odd-limit: ~99/70 = [1/2 0 0 0 0 0⟩, ~3 = [7/10 4/5 0 -2/5 0 1/5⟩, ~5 = [7/10 -1/5 1 -2/5 0 1/5⟩, ~7 = [7/5 -2/5 0 1/5 0 2/5⟩
eigenmonzo (unchanged-intervals) basis: 2.5/3.9/7.13

Optimal ET sequence: 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)

Subgroup: 2.3.5.7.11

Comma list: 12005/11979

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~3/2 = 702.0306 ¢, ~5/4 = 386.2325 ¢, ~44/35 = 397.2749 ¢
error map: ⟨-0.000 +0.076 -0.081 +0.243 -0.475 +0.541]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0306 ¢, ~5/4 = 386.2325 ¢, ~44/35 = 397.2750 ¢
error map: ⟨0.000 +0.076 -0.081 -0.475 +0.541]

Optimal ET sequence: 12, 21, 23de, 26, 37, 46, 58, 72, 118, 130, 190, 239, 311, 441, 559, 752e, 870

Badness (Sintel): 2.43

Hensquary

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 1716/1715

Mapping: [⟨1 0 0 -6 -8 -12], ⟨0 1 0 2 2 3], ⟨0 0 1 2 3 4], ⟨0 0 0 3 4 5]]

Optimal tunings:

  • WE: ~2 = 1199.7826 ¢, ~3/2 = 702.2831 ¢, ~5/4 = 386.9208 ¢, ~44/35 = 396.8581 ¢
error map: ⟨-0.217 +0.111 +0.172 +0.156 +1.443 -1.487]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2535 ¢, ~5/4 = 386.7762 ¢, ~44/35 = 397.0993 ¢
error map: ⟨0.000 +0.298 +0.463 +0.531 +1.915 -1.166]

Optimal ET sequence: 26, 37, 46, 58, 72, 121, 130, 193, 239, 251e, 323e, 369

Badness (Sintel): 0.988

Ekasquary

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 4459/4455

Mapping: [⟨1 0 0 -6 -8 10], ⟨0 1 0 2 2 0], ⟨0 0 1 2 3 -2], ⟨0 0 0 3 4 -5]]

Optimal tunings:

  • WE: ~2 = 1199.9999 ¢, ~3/2 = 701.9524 ¢, ~5/4 = 386.1388 ¢, ~44/35 = 397.4088 ¢
error map: ⟨-0.000 -0.003 -0.175 -0.417 +0.639 +0.150]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9524 ¢, ~5/4 = 386.1388 ¢, ~44/35 = 397.4089 ¢
error map: ⟨0.000 -0.003 -0.175 -0.417 +0.639 +0.150]

Optimal ET sequence: 46f, 49f, 58, 72, 118, 121, 130, 190, 193, 248, 311, 441, 752e, 1072

Badness (Sintel): 1.12

Semicanousmic (14641/14580)

Subgroup: 2.3.5.7.11

Comma list: 14641/14580

Mapping: [⟨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 tunings:

  • WE: ~2 = 1200.0495 ¢, ~3/2 = 702.2793 ¢, ~11/9 = 345.4721 ¢, ~7/4 = 968.7265 ¢
error map: ⟨+0.049 +0.374 +0.133 -0.000 -1.188]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2918 ¢, ~11/9 = 345.4516 ¢, ~7/4 = 968.7556 ¢
error map: ⟨0.000 +0.337 +0.077 -0.070 -1.283]

Optimal ET sequence: 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

Comma list: 16384/16335

Mapping: [⟨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 tunings:

  • WE: ~2 = 1199.7329 ¢, ~3/2 = 702.3657 ¢, ~128/99 = 442.1589 ¢, ~7/4 = 969.3599 ¢
error map: ⟨-0.267 +0.144 +0.103 -0.000 +0.456]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5107 ¢, ~128/99 = 442.2243 ¢, ~7/4 = 969.4120 ¢
error map: ⟨0.000 +0.556 +0.646 +0.586 +1.436]

Optimal ET sequence: 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: [⟨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 tunings:

  • WE: ~2 = 1199.7102 ¢, ~3/2 = 702.2488 ¢, ~84/65 = 442.3476 ¢, ~7/4 = 969.1583 ¢
error map: ⟨-0.290 +0.004 +0.341 -0.247 +0.388 +0.429]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3927 ¢, ~84/65 = 442.4436 ¢, ~7/4 = 969.1894 ¢
error map: ⟨0.000 +0.438 +0.966 +0.364 +1.453 +1.331]

Optimal ET sequence: 19, 22f, 27, 38e, 41, 46, 65d, 68, 84, 87, 111, 130, 198, 282, 328, 369f, 697cef, 1546bccdeeeffff, 1676bccdeeeffff

Badness (Sintel): 1.13

Symbiotic (19712/19683)

Subgroup: 2.3.5.7.11

Comma list: 19712/19683

Mapping: [⟨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 tunings:

  • WE: ~2 = 1199.9290 ¢, ~3/2 = 702.2265 ¢, ~5/4 = 386.4556 ¢, ~7/4 = 968.8979 ¢
error map: ⟨-0.071 +0.201 -0.000 -0.070 -0.106]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2440 ¢, ~5/4 = 386.4064 ¢, ~7/4 = 968.8593 ¢
error map: ⟨0.000 +0.289 +0.093 +0.033 +0.019]

Optimal ET sequence: 17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b

Badness (Sintel): 2.19

Tridecimal symbiotic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 19712/19683

Mapping: [⟨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 tunings:

  • WE: ~2 = 1199.9268 ¢, ~3/2 = 702.2293 ¢, ~5/4 = 386.4554 ¢, ~7/4 = 968.9114 ¢
error map: ⟨-0.073 +0.201 -0.005 -0.061 -0.093 -0.012]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2544 ¢, ~5/4 = 386.4007 ¢, ~7/4 = 968.8979 ¢
error map: ⟨0.000 +0.299 +0.087 +0.072 +0.074 +0.125]

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

Badness (Sintel): 0.868

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 2080/2079

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 tunings:

  • WE: ~2 = 1199.9273 ¢, ~3/2 = 702.2289 ¢, ~5/4 = 386.4508 ¢, ~7/4 = 968.9092 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.2554 ¢, ~5/4 = 386.4080 ¢, ~7/4 = 968.9015 ¢

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

Badness (Sintel): 0.375

Argyric (41503/41472)

Subgroup: 2.3.5.7.11

Comma list: 41503/41472

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1200.0485 ¢, ~3/2 = 701.9606 ¢, ~5/4 = 386.2167 ¢, ~144/77 = 1084.3250 ¢
error map: ⟨+0.049 +0.054 -0.000 -0.127 -0.129]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9739 ¢, ~5/4 = 386.2460 ¢, ~144/77 = 1084.2953 ¢
error map: ⟨0.000 +0.019 -0.068 -0.235 -0.256]

Optimal ET sequence: 19, 22, 31, 41, 53, 72, 166, 167, 176, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee

Badness (Sintel): 0.777

Chrysic (43923/43904)

Subgroup: 2.3.5.7.11

Comma list: 43923/43904

Mapping: [⟨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 tunings:

  • WE: ~2 = 1200.0167 ¢, ~3/2 = 701.9323 ¢, ~5/4 = 386.2803 ¢, ~33/28 = 284.2370 ¢
error map: ⟨+0.017 -0.006 -0.000 +0.056 -0.114]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9341 ¢, ~5/4 = 386.2927 ¢, ~33/28 = 284.2352 ¢
error map: ⟨0.000 -0.021 -0.021 +0.036 -0.155]

Optimal ET sequence: 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 (Sintel): 0.770

Elysic (117649/117612)

Subgroup: 2.3.5.7.11

Comma list: 117649/117612

Mapping: [⟨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 tunings:

  • WE: ~2 = 1200.0027 ¢, ~343/198 = 950.9861 ¢, ~5/4 = 386.3082 ¢, ~7/4 = 968.7558 ¢
error map: ⟨+0.003 +0.017 -0.000 -0.065 +0.033]
  • CWE: ~2 = 1200.0000 ¢, ~343/198 = 950.9850 ¢, ~5/4 = 386.3105 ¢, ~7/4 = 968.7571 ¢
error map: ⟨0.000 +0.015 -0.003 -0.069 +0.028]

Optimal ET sequence: 49, 58, 72, 130, 193, 198, 212, 270, 342, 612, 954, 1084, 1354, 1547, 1817, 2159, 3243, 3513, 5060, 5402, 5672, 8915d, 11074d

Badness (Sintel): 0.901

Olympic (131072/130977)

Olympic is highly accurate yet easy to conceptualize and notate on the staff. It equates 33/32 with (64/63)2, and 36/35 with 1053/1024, reducing the four formal commas 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

Comma list: 131072/130977

Mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1199.9460 ¢, ~3/2 = 702.0489 ¢, ~5/4 = 386.4217 ¢, ~7/4 = 968.9839 ¢
error map: ⟨-0.054 +0.040 -0.000 +0.050 +0.038]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0853 ¢, ~5/4 = 386.4125 ¢, ~7/4 = 969.0242 ¢
error map: ⟨0.000 +0.130 +0.099 +0.198 +0.207]

Optimal ET sequence: 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

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095

Mapping: [⟨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 tunings:

  • WE: ~2 = 1199.9638 ¢, ~3/2 = 702.0645 ¢, ~5/4 = 386.3208 ¢, ~7/4 = 968.9712 ¢
error map: ⟨-0.036 +0.073 -0.065 +0.073 +0.128 -0.166]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0884 ¢, ~5/4 = 386.3269 ¢, ~7/4 = 969.0015 ¢
error map: ⟨0.000 +0.133 +0.013 +0.176 +0.237 -0.033]

Optimal ET sequence: 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e *

* optimal patent val: 3044

Badness (Sintel): 0.292

Alphaxenic (131769/131072)

Subgroup: 2.3.5.7.11

Comma list: 131769/131072

Mapping: [⟨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 tunings:

  • WE: ~363/256 = 600.1590 ¢, ~16/11 = 650.8508 ¢, ~5/4 = 385.6773 ¢, ~7/4 = 968.1894 ¢
error map: ⟨+0.318 -0.094 -0.000 -0.000 -0.897]
  • CWE: ~363/256 = 600.0000 ¢, ~16/11 = 650.7321 ¢, ~5/4 = 385.7900 ¢, ~7/4 = 968.1927 ¢
error map: ⟨0.000 -0.491 -0.524 -0.633 -2.050]

Optimal ET sequence: 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: [⟨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 tunings:

  • WE: ~2 = 1200.0285 ¢, ~400/231 = 950.9641 ¢, ~5/4 = 386.3335 ¢, ~7/4 = 968.7129 ¢
error map: ⟨+0.028 -0.027 +0.077 -0.056 -0.085]
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 950.9492 ¢, ~5/4 = 386.3638 ¢, ~7/4 = 969.7159 ¢
error map: ⟨0.000 -0.057 +0.050 -0.110 -0.154]

Optimal ET sequence: 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: [⟨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 tunings:

  • WE: ~2 = 1199.9757 ¢, ~3/2 = 702.7610 ¢, ~18/11 = 855.5656 ¢, ~7/4 = 968.8735 ¢
error map: ⟨-0.024 +0.782 -0.130 -0.001 -1.434]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.7565 ¢, ~18/11 = 855.5885 ¢, ~7/4 = 968.8591 ¢
error map: ⟨0.000 +0.801 -0.102 +0.033 -1.393]

Optimal ET sequence: 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: [⟨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 tunings:

  • WE: ~2 = 1199.8431 ¢, ~3/2 = 702.9005 ¢, ~18/11 = 855.4090 ¢, ~7/4 = 969.1386 ¢
error map: ⟨-0.157 +0.789 -0.097 -0.001 -1.396 +0.379]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.9326 ¢, ~18/11 = 855.5400 ¢, ~7/4 = 969.1437 ¢
error map: ⟨0.000 +0.978 +0.184 +0.318 -0.993 +1.000]

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 702.8981 ¢, ~11/9 = 855.3764 ¢, ~7/4 = 969.3543 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.9376 ¢, ~11/9 = 855.5306 ¢, ~7/4 = 969.4313 ¢

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 702.9226 ¢, ~11/9 = 855.3621 ¢, ~7/4 = 969.3187 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.9618 ¢, ~11/9 = 855.5163 ¢, ~7/4 = 969.3961 ¢

Optimal ET sequence: 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: [⟨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 tunings:

  • WE: ~2 = 1199.9875 ¢, ~3/2 = 701.9753 ¢, ~5/4 = 386.3387 ¢, ~33/32 = 53.3743 ¢
error map: ⟨-0.012 +0.008 -0.000 -0.004 +0.031]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9802 ¢, ~5/4 = 386.3343 ¢, ~33/32 = 53.3730 ¢
error map: ⟨0.000 +0.025 +0.021 +0.019 +0.075]

Optimal ET sequence: 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: [⟨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 tunings:

  • WE: ~2 = 1199.9884 ¢, ~3/2 = 701.9674 ¢, ~5/4 = 386.3432 ¢, ~33/32 = 53.3763 ¢
error map: ⟨-0.012 +0.001 +0.006 -0.000 +0.045 -0.016]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9728 ¢, ~5/4 = 386.3385 ¢, ~33/32 = 53.3748 ¢
error map: ⟨0.000 +0.018 +0.025 +0.021 +0.084 +0.017]

Optimal ET sequence: 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

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