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Switch to Sintel's badness, WE & CWE tunings (1/)
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== Grendel ==
== Grendel ==
: ''For the 5-limit version of this temperament, see [[Syntonic–31 equivalence continuum #Counterwürschmidt]].''
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Counterwürschmidt]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 47: Line 47:
[[Comma list]]: 6144/6125, 16875/16807
[[Comma list]]: 6144/6125, 16875/16807


{{Mapping|legend=1| 1 9 2 7 | 0 -23 1 -13 }}
{{Mapping|legend=1| 1 -14 3 -6 | 0 23 -1 13 }}
: mapping generators: ~2, ~5/4
: mapping generators: ~2, ~8/5


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~5/4 = 386.863{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7348{{c}}, ~8/5 = 812.9574{{c}}
: [[error map]]: {{val| -0.265 -0.220 -0.067 +1.212 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 813.1311{{c}}
: error map: {{val| 0.000 +0.059 +0.555 +1.878 }}


{{Optimal ET sequence|legend=1| 31, 90, 121, 152, 335d }}
{{Optimal ET sequence|legend=1| 31, 90, 121, 152, 335d, 822dd }}


[[Badness]] (Smith): 0.051834
[[Badness]] (Sintel): 1.31


=== 11-limit ===
=== 11-limit ===
Line 61: Line 65:
Comma list: 540/539, 1375/1372, 5632/5625
Comma list: 540/539, 1375/1372, 5632/5625


Mapping: {{mapping| 1 9 2 7 17 | 0 -23 1 -13 -42 }}
Mapping: {{mapping| 1 -14 3 -6 -25 | 0 23 -1 13 42 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 386.856{{c}}
Optimal tunings:
* WE: ~2 = 1199.7355{{c}}, ~8/5 = 812.9622{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1353{{c}}


{{Optimal ET sequence|legend=0| 31, 90e, 121, 152, 335d, 487d }}
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152, 335d, 487d }}


Badness (Smith): 0.019845
Badness (Sintel): 0.656


=== 13-limit ===
=== 13-limit ===
Line 74: Line 80:
Comma list: 352/351, 540/539, 625/624, 1375/1372
Comma list: 352/351, 540/539, 625/624, 1375/1372


Mapping: {{mapping| 1 9 2 7 17 -5 | 0 -23 1 -13 -42 27 }}
Mapping: {{mapping| 1 -14 3 -6 -25 22 | 0 23 -1 13 42 -27 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 386.826{{c}}
Optimal tunings:
* WE: ~2 = 1199.4412{{c}}, ~8/5 = 812.7956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1209{{c}}


{{Optimal ET sequence|legend=0| 31, 121, 152f, 425deff }}
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152f, 273def, 425deff }}


Badness (Smith): 0.024839
Badness (Sintel): 1.03


=== 17-limit ===
=== 17-limit ===
Line 87: Line 95:
Comma list: 256/255, 352/351, 625/624, 715/714, 1275/1274
Comma list: 256/255, 352/351, 625/624, 715/714, 1275/1274


Mapping: {{mapping| 1 9 2 7 17 -5 -3 | 0 -23 1 -13 -42 27 22 }}
Mapping: {{mapping| 1 -14 3 -6 -25 22 19 | 0 23 -1 13 42 -27 -22 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 386.812{{c}}
Optimal tunings:
* WE: ~2 = 1199.3029{{c}}, ~8/5 = 812.7156{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1843{{c}}


{{Optimal ET sequence|legend=0| 31, 121, 273defgg }}
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152fg, 273defgg }}


Badness (Smith): 0.021400
Badness (Sintel): 1.09


=== 19-limit ===
=== 19-limit ===
Line 100: Line 110:
Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714
Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714


Mapping: {{mapping| 1 9 2 7 17 -5 -3 -8 | 0 -23 1 -13 -42 27 22 38 }}
Mapping: {{mapping| 1 -14 3 -6 -25 22 19 30 | 0 23 -1 13 42 -27 -22 -38 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 386.819{{c}}
Optimal tunings:
* WE: ~2 = 1199.3587{{c}}, ~8/5 = 812.7462{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1796{{c}}


{{Optimal ET sequence|legend=0| 31, 121, 152fg, 273defgg }}
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152fg, 273defgg }}


Badness (Smith): 0.018413
Badness (Sintel): 1.12


== Kwai ==
== Kwai ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kwai]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''


Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a fifth, and can be described as 41 & 70.  
Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a [[3/2|perfect fifth]], and can be described as {{nowrap| 41 & 70 }}.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 120: Line 132:
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.616{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7337{{c}}, ~3/2 = 702.4600{{c}}
: [[error map]]: {{val| -0.266 +0.239 -0.607 +1.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6085{{c}}
: error map: {{val| 0.000 +0.653 -0.234 +1.603 }}


{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}


[[Badness]] (Smith): 0.054476
[[Badness]] (Sintel): 1.38


=== 11-limit ===
=== 11-limit ===
Line 133: Line 149:
Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}
Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.623{{c}}
Optimal tunings:
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


{{Optimal ET sequence|legend=0| 29cd, 41, 111, 152 }}
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}


Badness (Smith): 0.026219
Badness (Sintel): 0.867


==== 13-limit ====
==== 13-limit ====
Line 146: Line 164:
Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}
Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.644{{c}}
Optimal tunings:
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}


{{Optimal ET sequence|legend=0| 29cd, 41, 111, 152f }}
{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}


Badness (Smith): 0.024555
Badness (Sintel): 1.01


===== 17-limit =====
===== 17-limit =====
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Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}
Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.660{{c}}
Optimal tunings:
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}


{{Optimal ET sequence|legend=0| 29cdg, 41, 111, 152fg, 263dfg }}
{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}


Badness (Smith): 0.021950
Badness (Sintel): 1.12


===== 19-limit =====
===== 19-limit =====
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Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}
Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.657{{c}}
Optimal tunings:
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}


{{Optimal ET sequence|legend=0| 29cdgh, 41, 111, 152fg, 263dfgh }}
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}


Badness (Smith): 0.016957
Badness (Sintel): 1.03


==== Hemikwai ====
==== Hemikwai ====
Line 186: Line 210:
: mapping generators: ~2, ~26/15
: mapping generators: ~2, ~26/15


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 951.314{{c}}
Optimal tunings:
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}


{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


Badness (Smith): 0.044108
Badness (Sintel): 1.82


===== 17-limit =====
===== 17-limit =====
Line 199: Line 225:
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 951.314{{c}}
Optimal tunings:
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}


{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


Badness (Smith): 0.025806
Badness (Sintel): 1.31


===== 19-limit =====
===== 19-limit =====
Line 212: Line 240:
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 951.313{{c}}
Optimal tunings:
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}


{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}


Badness (Smith): 0.019146
Badness (Sintel): 1.16


== Pluto ==
== Pluto ==
{{Distinguish| Plutus }}
{{Distinguish| Plutus }}


Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group | ''19-limit Pluto temperament'']</ref>, can be described as the {{nowrap| 41 & 80 }} temperament. It is generated by a sharpened [[7/5]], and [[121edo|59\121]] is about perfect as a tuning.  
Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group | ''19-limit Pluto temperament'']</ref>, can be described as the {{nowrap| 41 & 80 }} temperament. It is generated by a flattened [[10/7]], and [[121edo|59\121]] is about perfect as a tuning.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 4000/3969, 10976/10935
[[Comma list]]: 4000/3969, 10976/10935


{{Mapping|legend=1| 1 5 15 15 | 0 -7 -26 -25 }}
{{Mapping|legend=1| 1 -2 -11 -10 | 0 7 26 25 }}
: mapping generators: ~2, ~10/7


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 585.147{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4720{{c}}, ~10/7 = 614.5826{{c}}
: [[error map]]: {{val| -0.528 +1.179 -1.358 +1.020 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.8282{{c}}
: error map: {{val| 0.000 +1.843 -0.780 +1.880 }}


{{Optimal ET sequence|legend=1| 39d, 41, 80, 121, 404bd }}
{{Optimal ET sequence|legend=1| 39d, 41, 80, 121, 283d, 404bd }}


[[Badness]] (Smith): 0.057514
[[Badness]] (Sintel): 1.46


=== 11-limit ===
=== 11-limit ===
Line 240: Line 275:
Comma list: 540/539, 896/891, 1375/1372
Comma list: 540/539, 896/891, 1375/1372


Mapping: {{mapping| 1 5 15 15 2 | 0 -7 -26 -25 3 }}
Mapping: {{mapping| 1 -2 -11 -10 5 | 0 7 26 25 -3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.114{{c}}
Optimal tunings:
* WE: ~2 = 1200.1835{{c}}, ~10/7 = 614.4677{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8616{{c}}


{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}


Badness (Smith): 0.029844
Badness (Sintel): 0.987


==== 13-limit ====
==== 13-limit ====
Line 253: Line 290:
Comma list: 325/324, 352/351, 364/363, 540/539
Comma list: 325/324, 352/351, 364/363, 540/539


Mapping: {{mapping| 1 5 15 15 2 -8 | 0 -7 -26 -25 3 24 }}
Mapping: {{mapping| 1 -2 -11 -10 5 16 | 0 7 26 25 -3 -24 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.123{{c}}
Optimal tunings:
* WE: ~2 = 1199.2469{{c}}, ~10/7 = 614.4908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8684{{c}}


{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}


Badness (Smith): 0.025717
Badness (Sintel): 1.06


===== 17-limit =====
===== 17-limit =====
Line 266: Line 305:
Comma list: 256/255, 325/324, 352/351, 364/363, 540/539
Comma list: 256/255, 325/324, 352/351, 364/363, 540/539


Mapping: {{mapping| 1 5 15 15 2 -8 -12 | 0 -7 -26 -25 3 24 33 }}
Mapping: {{mapping| 1 -2 -11 -10 5 16 21 | 0 7 26 25 -3 -24 -33 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.116{{c}}
Optimal tunings:
* WE: ~2 = 1199.2009{{c}}, ~10/7 = 614.4746{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8813{{c}}


{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}


Badness (Smith): 0.021463
Badness (Sintel): 1.09


===== 19-limit =====
===== 19-limit =====
Line 279: Line 320:
Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594
Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594


Mapping: {{mapping| 1 5 15 15 2 -8 -12 14 | 0 -7 -26 -25 3 24 33 -20 }}
Mapping: {{mapping| 1 -2 -11 -10 5 16 21 -6 | 0 7 26 25 -3 -24 -33 20 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.109{{c}}
Optimal tunings:
* WE: ~2 = 1199.2833{{c}}, ~10/7 = 614.5237{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8876{{c}}


{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}


Badness (Smith): 0.017650
Badness (Sintel): 1.07


==== Orcus ====
==== Orcus ====
Line 292: Line 335:
Comma list: 144/143, 196/195, 275/273, 896/891
Comma list: 144/143, 196/195, 275/273, 896/891


Mapping: {{mapping| 1 5 15 15 2 12 | 0 -7 -26 -25 3 -17 }}
Mapping: {{mapping| 1 -2 -11 -10 5 -5 | 0 7 26 25 -3 17 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.111{{c}}
Optimal tunings:
* WE: ~2 = 1198.7372{{c}}, ~10/7 = 614.2418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8502{{c}}


{{Optimal ET sequence|legend=0| 39df, 41, 80f, 121ff }}
{{Optimal ET sequence|legend=0| 39df, 41, 80f, 121ff }}


Badness (Smith): 0.033441
Badness (Sintel): 1.38


=== Plutino ===
=== Plutino ===
Line 305: Line 350:
Comma list: 100/99, 245/242, 10976/10935
Comma list: 100/99, 245/242, 10976/10935


Mapping: {{mapping| 1 5 15 15 22 | 0 -7 -26 -25 -38 }}
Mapping: {{mapping| 1 -2 -11 -10 -16 | 0 7 26 25 38 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.283{{c}}
Optimal tunings:
* WE: ~2 = 1199.7793{{c}}, ~10/7 = 614.6040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7074{{c}}


{{Optimal ET sequence|legend=0| 39dee, 41 }}
{{Optimal ET sequence|legend=0| 39dee, 41 }}


Badness (Smith): 0.057966
Badness (Sintel): 1.92


==== 13-limit ====
==== 13-limit ====
Line 318: Line 365:
Comma list: 100/99, 196/195, 245/242, 729/728
Comma list: 100/99, 196/195, 245/242, 729/728


Mapping: {{mapping| 1 5 15 15 22 12 | 0 -7 -26 -25 -38 -17 }}
Mapping: {{mapping| 1 5 15 15 22 12 | 0 7 26 25 38 17 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.232{{c}}
Optimal tunings:
* WE: ~2 = 1199.0328{{c}}, ~10/7 = 614.2729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7277{{c}}


{{Optimal ET sequence|legend=0| 39deef, 41 }}
{{Optimal ET sequence|legend=0| 39deef, 41 }}


Badness (Smith): 0.040182
Badness (Sintel): 1.66


== Mirkat ==
== Mirkat ==
Line 515: Line 564:


== Gaster ==
== Gaster ==
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Gaster]].''
: ''For the 5-limit version, see [[Very high accuracy temperaments #Gaster]].''
{{Main| Gaster temperament }}
{{Main| Gaster temperament }}


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