Canopic clan: Difference between revisions

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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 =====
Line 172: Line 194:
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 ====
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: 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
Line 227: Line 257:
[[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 }}


Revision as of 10:54, 2 April 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.

The mirkwai clan of temperaments tempers out the mirkwai comma (monzo: [0 3 4 -5⟩, ratio: 16875/16807), a no-twos comma.

Canopus

Main article: Canopus

Subgroup: 3.5.7

Comma list: 16875/16807

Subgroup-val mapping: [⟨1 3 3], ⟨0 -5 -4]]

mapping generators: ~3, ~7/5

Optimal tuning (POTE): ~3 = 1901.9550 ¢, ~7/5 = 583.9584 ¢

Optimal ET sequence: b13, b62, b75, b88, b101, b114, b355, b469, b583, b697

Overview to extensions

The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid. These temperaments are distributed into different temperament collection pages.

The others are weak extensions. Mirkat tempers out 19683/19600, splitting the generator in two with a semitwelfth period. Sqrtphi tempers out 15625/15552, splitting the period in six. Miracle tempers out 225/224. Pluto tempers out 4000/3969. These split the generator in five. Quanharuk tempers out 32805/32768, splitting the generator in three with a 1/5-twelfth period. Semisept tempers out 1728/1715 and 3136/3125, splitting the generator in six. Kwai tempers out 5120/5103, splitting the generator in ten. Grendel tempers out 6144/6125, splitting the generator in eleven. Finally, eris tempers out 65625/65536, splitting the generator in sixteen.

Members of the clan discussed elsewhere are:

For no-twos extensions, see No-twos subgroup temperaments #Canopus.

Considered below are grendel, kwai, pluto, mirkat, eris, subsemifourth, septendesemi, gaster, subsedia, hemiseptisix, browser, and grazer.

Grendel

For the 5-limit version, see Syntonic–31 equivalence continuum #Counterwürschmidt.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 16875/16807

Mapping: [⟨1 -14 3 -6], ⟨0 23 -1 13]]

mapping generators: ~2, ~8/5

Optimal tunings:

  • WE: ~2 = 1199.7348 ¢, ~8/5 = 812.9574 ¢
error map: ⟨-0.265 -0.220 -0.067 +1.212]
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1311 ¢
error map: ⟨0.000 +0.059 +0.555 +1.878]

Optimal ET sequence: 31, 90, 121, 152, 335d, 822dd

Badness (Sintel): 1.31

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 5632/5625

Mapping: [⟨1 -14 3 -6 -25], ⟨0 23 -1 13 42]]

Optimal tunings:

  • WE: ~2 = 1199.7355 ¢, ~8/5 = 812.9622 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1353 ¢

Optimal ET sequence: 31, 90e, 121, 152, 335d, 487d

Badness (Sintel): 0.656

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 625/624, 1375/1372

Mapping: [⟨1 -14 3 -6 -25 22], ⟨0 23 -1 13 42 -27]]

Optimal tunings:

  • WE: ~2 = 1199.4412 ¢, ~8/5 = 812.7956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1209 ¢

Optimal ET sequence: 31, 90e, 121, 152f, 273def, 425deff

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 625/624, 715/714, 1275/1274

Mapping: [⟨1 -14 3 -6 -25 22 19], ⟨0 23 -1 13 42 -27 -22]]

Optimal tunings:

  • WE: ~2 = 1199.3029 ¢, ~8/5 = 812.7156 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1843 ¢

Optimal ET sequence: 31, 90e, 121, 152fg, 273defgg

Badness (Sintel): 1.09

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714

Mapping: [⟨1 -14 3 -6 -25 22 19 30], ⟨0 23 -1 13 42 -27 -22 -38]]

Optimal tunings:

  • WE: ~2 = 1199.3587 ¢, ~8/5 = 812.7462 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1796 ¢

Optimal ET sequence: 31, 90e, 121, 152fg, 273defgg

Badness (Sintel): 1.12

Kwai

For the 5-limit version, see Miscellaneous 5-limit temperaments #Kwai.

Named by Gene Ward Smith in 2004 for its "bridgeability"[1], kwai is generated by a perfect fifth, and can be described as 41 & 70.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 16875/16807

Mapping: [⟨1 0 -50 -40], ⟨0 1 33 27]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.7337 ¢, ~3/2 = 702.4600 ¢
error map: ⟨-0.266 +0.239 -0.607 +1.055]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6085 ¢
error map: ⟨0.000 +0.653 -0.234 +1.603]

Optimal ET sequence: 41, 111, 152, 345, 497d

Badness (Sintel): 1.38

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 5120/5103

Mapping: [⟨1 0 -50 -40 32], ⟨0 1 33 27 -18]]

Optimal tunings:

  • WE: ~2 = 1199.6672 ¢, ~3/2 = 702.4282 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6189 ¢

Optimal ET sequence: 41, 111, 152, 497de, 649dde

Badness (Sintel): 0.867

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1375/1372

Mapping: [⟨1 0 -50 -40 32 27], ⟨0 1 33 27 -18 -21]]

Optimal tunings:

  • WE: ~2 = 1199.4772 ¢, ~3/2 = 702.3379 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6409 ¢

Optimal ET sequence: 41, 111, 152f, 415dff

Badness (Sintel): 1.01

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088

Mapping: [⟨1 0 -50 -40 32 27 58], ⟨0 1 33 27 -18 -21 -34]]

Optimal tunings:

  • WE: ~2 = 1199.3537 ¢, ~3/2 = 702.2850 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6589 ¢

Optimal ET sequence: 41, 70, 111, 152fg, 263dfg

Badness (Sintel): 1.12

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845

Mapping: [⟨1 0 -50 -40 32 27 58 -56], ⟨0 1 33 27 -18 -21 -34 38]]

Optimal tunings:

  • WE: ~2 = 1199.3401 ¢, ~3/2 = 702.2705 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6548 ¢

Optimal ET sequence: 41, 70h, 111, 152fg, 263dfgh

Badness (Sintel): 1.03

Hemikwai

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 676/675, 1375/1372, 5120/5103

Mapping: [⟨1 0 -50 -40 32 -51], ⟨0 2 66 54 -36 69]]

mapping generators: ~2, ~26/15

Optimal tunings:

  • WE: ~2 = 1199.6968 ¢, ~26/15 = 951.0740 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3123 ¢

Optimal ET sequence: 82, 111, 193, 304d

Badness (Sintel): 1.82

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103

Mapping: [⟨1 0 -50 -40 32 -51 -30], ⟨0 2 66 54 -36 69 43]]

Optimal tunings:

  • WE: ~2 = 1199.6861 ¢, ~26/15 = 951.0654 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3120 ¢

Optimal ET sequence: 82, 111, 193, 304d

Badness (Sintel): 1.31

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444

Mapping: [⟨1 0 -50 -40 32 -51 -30 -56], ⟨0 2 66 54 -36 69 43 76]]

Optimal tunings:

  • WE: ~2 = 1199.6718 ¢, ~26/15 = 951.0526 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3103 ¢

Optimal ET sequence: 82, 111, 193, 304dh

Badness (Sintel): 1.16

Pluto

Not to be confused with Plutus.

Pluto, named by Gene Ward Smith in 2010[2], can be described as the 41 & 80 temperament. It is generated by a flattened 10/7, and 59\121 is about perfect as a tuning.

Subgroup: 2.3.5.7

Comma list: 4000/3969, 10976/10935

Mapping: [⟨1 -2 -11 -10], ⟨0 7 26 25]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1199.4720 ¢, ~10/7 = 614.5826 ¢
error map: ⟨-0.528 +1.179 -1.358 +1.020]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8282 ¢
error map: ⟨0.000 +1.843 -0.780 +1.880]

Optimal ET sequence: 39d, 41, 80, 121, 283d, 404bd

Badness (Sintel): 1.46

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 1375/1372

Mapping: [⟨1 -2 -11 -10 5], ⟨0 7 26 25 -3]]

Optimal tunings:

  • WE: ~2 = 1200.1835 ¢, ~10/7 = 614.4677 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8616 ¢

Optimal ET sequence: 39d, 41, 80, 121

Badness (Sintel): 0.987

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 540/539

Mapping: [⟨1 -2 -11 -10 5 16], ⟨0 7 26 25 -3 -24]]

Optimal tunings:

  • WE: ~2 = 1199.2469 ¢, ~10/7 = 614.4908 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8684 ¢

Optimal ET sequence: 39d, 41, 80, 121

Badness (Sintel): 1.06

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363, 540/539

Mapping: [⟨1 -2 -11 -10 5 16 21], ⟨0 7 26 25 -3 -24 -33]]

Optimal tunings:

  • WE: ~2 = 1199.2009 ¢, ~10/7 = 614.4746 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8813 ¢

Optimal ET sequence: 39d, 41, 80, 121

Badness (Sintel): 1.09

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594

Mapping: [⟨1 -2 -11 -10 5 16 21 -6], ⟨0 7 26 25 -3 -24 -33 20]]

Optimal tunings:

  • WE: ~2 = 1199.2833 ¢, ~10/7 = 614.5237 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8876 ¢

Optimal ET sequence: 39d, 41, 80, 121

Badness (Sintel): 1.07

Orcus

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 275/273, 896/891

Mapping: [⟨1 -2 -11 -10 5 -5], ⟨0 7 26 25 -3 17]]

Optimal tunings:

  • WE: ~2 = 1198.7372 ¢, ~10/7 = 614.2418 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8502 ¢

Optimal ET sequence: 39df, 41, 80f, 121ff

Badness (Sintel): 1.38

Plutino

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 10976/10935

Mapping: [⟨1 -2 -11 -10 -16], ⟨0 7 26 25 38]]

Optimal tunings:

  • WE: ~2 = 1199.7793 ¢, ~10/7 = 614.6040 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.7074 ¢

Optimal ET sequence: 39dee, 41

Badness (Sintel): 1.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 245/242, 729/728

Mapping: [⟨1 5 15 15 22 12], ⟨0 7 26 25 38 17]]

Optimal tunings:

  • WE: ~2 = 1199.0328 ¢, ~10/7 = 614.2729 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.7277 ¢

Optimal ET sequence: 39deef, 41

Badness (Sintel): 1.66

Mirkat

Subgroup: 2.3.5.7

Comma list: 16875/16807, 19683/19600

Mapping: [⟨3 2 1 2], ⟨0 6 13 14]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~10/9 = 183.539 ¢

Optimal ET sequence: 39d, 72, 111, 183, 255

Badness (Smith): 0.059376

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 8019/8000

Mapping: [⟨3 2 1 2 9], ⟨0 6 13 14 3]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~10/9 = 183.528 ¢

Optimal ET sequence: 39d, 72, 111, 183, 255

Badness (Smith): 0.022126

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 1375/1372

Mapping: [⟨3 2 1 2 9 1], ⟨0 6 13 14 3 22]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~10/9 = 183.577 ¢

Optimal ET sequence: 39df, 72, 111, 183

Badness (Smith): 0.018632

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 540/539, 561/560, 715/714

Mapping: [⟨3 2 1 2 9 1 4], ⟨0 6 13 14 3 22 18]]

Optimal tuning (POTE): ~34/27 = 400.000 ¢, ~10/9 = 183.579 ¢

Optimal ET sequence: 39dfg, 72, 111, 183

Badness (Smith): 0.011775

Eris

The 2.5.7 subgroup restriction of this temperament is exodia.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 65625/65536

Mapping: [⟨1 10 0 6], ⟨0 -29 8 -11]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~60/49 = 348.216 ¢

Optimal ET sequence: 31, 131, 162, 193, 224, 1823cd, 2271cd

Badness (Smith): 0.074719

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 65625/65536

Mapping: [⟨1 10 0 6 20], ⟨0 -29 8 -11 -57]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/9 = 348.219 ¢

Optimal ET sequence: 31, 193, 224, 703, 927d, 1151cd

Badness (Smith): 0.027621

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 1375/1372, 4096/4095

Mapping: [⟨1 10 0 6 20 -14], ⟨0 -29 8 -11 -57 61]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/9 = 348.213 ¢

Optimal ET sequence: 31, 193, 224

Badness (Smith): 0.025137

Subsemifourth

Subgroup: 2.3.5.7

Comma list: 16875/16807, 26873856/26796875

Mapping: [⟨1 39 27 45], ⟨0 -47 -31 -53]]

mapping generators: ~2, ~125/72

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~144/125 = 244.719 ¢

Optimal ET sequence: 49, 103, 152, 255, 407

Badness (Smith): 0.135173

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 234375/234256

Mapping: [⟨1 39 27 45 56], ⟨0 -47 -31 -53 -66]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~121/105 = 244.719 ¢

Optimal ET sequence: 49, 103, 152, 255, 407, 966d

Badness (Smith): 0.034276

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 847/845, 1375/1372, 1575/1573

Mapping: [⟨1 39 27 45 56 65], ⟨0 -47 -31 -53 -66 -77]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/13 = 244.714 ¢

Optimal ET sequence: 49f, 103, 152f, 255, 407f, 662df

Badness (Smith): 0.028387

Septendesemi

The name septendesemi means a septendecimal semitone (17/16). The septendesemi temperament (80 & 103) tempers out the mirkwai comma and 1959552/1953125 (parkleiness comma, zotritrigu) in the 7-limit. 183edo provides an excellent tuning for 7-, 11-, 13-, and 17-limit septendesemi.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1959552/1953125

Mapping: [⟨1 39 37 53], ⟨0 -41 -38 -55]]

mapping generators: ~2, ~648/343

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~343/324 = 104.916 ¢

Optimal ET sequence: 80, 103, 183

Badness (Smith): 0.146795

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 43923/43750

Mapping: [⟨1 39 37 53 50], ⟨0 -41 -38 -55 -51]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~35/33 = 104.916 ¢

Optimal ET sequence: 80, 103, 183

Badness (Smith): 0.041554

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 1375/1372, 4225/4224

Mapping: [⟨1 39 37 53 50 11], ⟨0 -41 -38 -55 -51 -8]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~35/33 = 104.908 ¢

Optimal ET sequence: 80, 103, 183, 469f, 652def

Badness (Smith): 0.027908

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224

Mapping: [⟨1 39 37 53 50 11 5], ⟨0 -41 -38 -55 -51 -8 -1]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~17/16 = 104.909 ¢

Optimal ET sequence: 80, 103, 183, 469f, 652def

Badness (Smith): 0.020128

Gaster

For the 5-limit version, see Very high accuracy temperaments #Gaster.
Main article: Gaster temperament

The gaster temperament (111 & 113) tempers out [-70 72 -19⟩ (quadbila-negu) in the 5-limit; mirkwai comma (16875/16807) and scheme comma (14348907/14336000) in the 7-limit. The word "gaster" means abdomen or stomach, but also a restructuring of the words "gassormic tritone", which is a generator of this temperament. This temperament is sufficient to obtain high prime limit harmonics like a stomach, so that patent vals 111, 113 and 224 support it even in the 41-limit.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 14348907/14336000

Mapping: [⟨1 11 38 37], ⟨0 -19 -72 -69]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~800/567 = 594.641 ¢

Optimal ET sequence: 111, 224

Badness (Smith): 0.154521

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 14348907/14336000

Mapping: [⟨1 11 38 37 -1], ⟨0 -19 -72 -69 9]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~512/363 = 594.639 ¢

Optimal ET sequence: 111, 224, 783d, 1007d, 1231dd

Badness (Smith): 0.054060

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372, 2200/2197

Mapping: [⟨1 11 38 37 -1 26], ⟨0 -19 -72 -69 9 -45]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~55/39 = 594.639 ¢

Optimal ET sequence: 111, 224, 783df, 1007df, 1231ddf

Badness (Smith): 0.024882

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197

Mapping: [⟨1 11 38 37 -1 26 14], ⟨0 -19 -72 -69 9 -45 -20]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.636 ¢

Optimal ET sequence: 111, 224, 559dgg

Badness (Smith): 0.021436

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444

Mapping: [⟨1 11 38 37 -1 26 14 32], ⟨0 -19 -72 -69 9 -45 -20 -56]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.636 ¢

Optimal ET sequence: 111, 224

Badness (Smith): 0.018370

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714

Mapping: [⟨1 11 38 37 -1 26 14 32 7], ⟨0 -19 -72 -69 9 -45 -20 -56 -5]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.641 ¢

Optimal ET sequence: 111, 224

Badness (Smith): 0.017619

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714

Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.646 ¢

Optimal ET sequence: 111, 113, 224

Badness (Smith): 0.016815

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714

Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11 0], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32 10]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.644 ¢

Optimal ET sequence: 111, 113, 224

Badness (Smith): 0.014790

37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714

Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11 0 -27], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.644 ¢

Optimal ET sequence: 111, 113, 224

Badness (Smith): 0.014377

41-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 533/532, 540/539, 575/574, 667/666

Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11 0 -27 45], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 -80]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.643 ¢

Optimal ET sequence: 111, 113, 224

Badness (Smith): 0.012858

Subsedia

The generator for subsedia (10 & 111) is 0.5 cents flat of 15/14-wide semitone and tempers out the mirkwai comma and buzzardsma. In this temperament, three generators makes ~16/13, five of them equals ~24/17, twelve of them equals ~16/7, sixteen of them equals ~3/1, and 45 of them equals ~22/1.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 65536/64827

Mapping: [⟨1 0 5 4], ⟨0 16 -27 -12]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 118.965 ¢

Optimal ET sequence: 10, 101, 111, 121, 232d

Badness (Smith): 0.157658

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 65536/64827

Mapping: [⟨1 0 5 4 -1], ⟨0 16 -27 -12 45]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 118.968 ¢

Optimal ET sequence: 10, 101, 111, 121, 232d

Badness (Smith): 0.066838

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 676/675, 1375/1372

Mapping: [⟨1 0 5 4 -1 4], ⟨0 16 -27 -12 45 -3]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 118.968 ¢

Optimal ET sequence: 10, 101, 111, 121, 232d

Badness (Smith): 0.031635

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 442/441, 540/539, 715/714

Mapping: [⟨1 0 5 4 -1 4 3], ⟨0 16 -27 -12 45 -3 11]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 118.968 ¢

Optimal ET sequence: 10, 101, 111, 121, 232dg

Badness (Smith): 0.019707

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 400/399, 442/441, 456/455, 715/714

Mapping: [⟨1 0 5 4 -1 4 3 10], ⟨0 16 -27 -12 45 -3 11 -58]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 118.964 ¢

Optimal ET sequence: 10, 101h, 111, 121, 232dg

Badness (Smith): 0.017935

Hemiseptisix

The name hemiseptisix means a half of septimal major sixth (12/7). The hemiseptisix temperament (103 & 121) tempers out the mirkwai comma and 95703125/95551488 (pontiqak comma, lazozotritriyo) in the 7-limit. 224edo provides an excellent tuning for 7-, 11-, and 13-limit hemiseptisix.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 95703125/95551488

Mapping: [⟨1 34 17 34], ⟨0 -53 -24 -51]]

mapping generators: ~2, ~75/49

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~98/75 = 466.071 ¢

Optimal ET sequence: 103, 121, 224

Badness (Smith): 0.162826

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 2734375/2725888

Mapping: [⟨1 34 17 34 53], ⟨0 -53 -24 -51 -81]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~55/42 = 466.070 ¢

Optimal ET sequence: 103, 121, 224

Badness (Smith): 0.043381

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 1375/1372, 2200/2197

Mapping: [⟨1 34 17 34 53 30], ⟨0 -53 -24 -51 -81 -43]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~55/42 = 466.071 ¢

Optimal ET sequence: 103, 121, 224

Badness (Smith): 0.021127

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197

Mapping: [⟨1 34 17 34 53 30 31], ⟨0 -53 -24 -51 -81 -43 -44]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~17/13 = 466.074 ¢

Optimal ET sequence: 103, 121, 224

Badness (Smith): 0.018611

Browser

See also: Sensipent family

This can also be considered a non-over-1 temperament, with considerable scope for harmony in the 2.5/3.7/3.11/3.13/3.17/3 subgroup with MOSes of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8et, with a generator very slightly sharp of 1\8.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 78732/78125

Mapping: [⟨1 6 8 10], ⟨0 -35 -45 -57]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/45 = 151.399 ¢

Optimal ET sequence: 103, 111, 214

Badness (Smith): 0.180645

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 78732/78125

Mapping: [⟨1 6 8 10 8], ⟨0 -35 -45 -57 -36]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 151.405 ¢

Optimal ET sequence: 103, 214

Badness (Smith): 0.057634

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845, 1375/1372

Mapping: [⟨1 6 8 10 8 9], ⟨0 -35 -45 -57 -36 -42]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 151.403 ¢

Optimal ET sequence: 103, 111, 214

Badness (Smith): 0.028822

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 715/714, 847/845

Mapping: [⟨1 6 8 10 8 9 8], ⟨0 -35 -45 -57 -36 -42 -31]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 151.397 ¢

Optimal ET sequence: 103, 111, 214

Badness (Smith): 0.020384

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714

Mapping: [⟨1 6 8 10 8 9 8 18], ⟨0 -35 -45 -57 -36 -42 -31 -109]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 151.396 ¢

Optimal ET sequence: 103h, 111, 214

Badness (Smith): 0.017570

Grazer

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1071875/1062882

Mapping: [⟨1 34 47 58], ⟨0 -37 -51 -63]]

mapping generators: ~2, ~90/49

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/45 = 148.719 ¢

Optimal ET sequence: 113, 121, 234

Badness (Smith): 0.217166

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 218750/216513

Mapping: [⟨1 34 47 58 35], ⟨0 -37 -51 -63 -36]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 148.729 ¢

Optimal ET sequence: 113, 121, 234, 355e, 589cee

Badness (Smith): 0.076062

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 540/539, 2200/2197

Mapping: [⟨1 34 47 58 35 44], ⟨0 -37 -51 -63 -36 -46]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 148.729 ¢

Optimal ET sequence: 113, 121, 234, 355e, 589cee

Badness (Smith): 0.036248

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989

Mapping: [⟨1 34 47 58 35 44 33], ⟨0 -37 -51 -63 -36 -46 -33]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 148.735 ¢

Optimal ET sequence: 113, 121, 234g, 355eg

Badness (Smith): 0.025410

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663

Mapping: [⟨1 34 47 58 35 44 33 6], ⟨0 -37 -51 -63 -36 -46 -33 -2]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 148.727 ¢

Optimal ET sequence: 113, 121, 234g, 355eg, 589ceegg

Badness (Smith): 0.022574

References

  1. ↑ Yahoo! Tuning Group | Kwai
  2. ↑ Yahoo! Tuning Group | 19-limit Pluto temperament
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