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Kleismic family: Difference between revisions

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Badness (Sintel): 0.690
Badness (Sintel): 0.690
== Quadritikleismic ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 15625/15552
{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
: mapping generators: ~25/21, ~6/5
[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 300.0520{{c}}, ~6/5 = 317.0548{{c}} (~126/125 = 17.0029{{c}})
: [[error map]]: {{val| +0.208 +0.374 -0.832 -0.243 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0301{{c}} (~126/125 = 17.0301{{c}})
: error map: {{val| 0.000 +0.225 -1.163 -0.706 }}
{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
[[Badness]] (Sintel): 0.993
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 6250/6237
Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
Optimal tunings:
* WE: ~25/21 = 300.0995{{c}}, ~6/5 = 317.0298{{c}} (~100/99 = 16.9303{{c}})
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9540{{c}} (~100/99 = 16.9540{{c}})
{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}
Badness (Sintel): 0.774
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 625/624, 1375/1372
Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
Optimal tunings:
* WE: ~25/21 = 300.0985{{c}}, ~6/5 = 317.0899{{c}} (~100/99 = 16.9941{{c}})
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}
Badness (Sintel): 0.774
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 325/324, 385/384, 442/441, 625/624
Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}
Optimal tunings:
* WE: ~25/21 = 300.1102{{c}}, ~6/5 = 317.1011{{c}} (~100/99 = 16.9909{{c}})
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}
Badness (Sintel): 0.651
== Kleiboh ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 1728/1715, 3125/3087
{{Mapping|legend=1| 1 -12 -9 -7 | 0 18 15 13 }}
: mapping generators: ~2, ~42/25
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5290{{c}}, ~42/25 = 905.3417{{c}}
: [[error map]]: {{val| -0.471 -0.152 -1.949 +3.914 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 905.6741{{c}}
: error map: {{val| 0.000 +0.178 -1.203 +4.937 }}
{{Optimal ET sequence|legend=1| 49, 53 }}
[[Badness]] (Sintel): 1.93
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 176/175, 540/539, 3125/3087
Mapping: {{mapping| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}
Optimal tunings:
* WE: ~2 = 1199.1389{{c}}, ~42/25 = 905.1688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 905.7840{{c}}
{{Optimal ET sequence|legend=0| 49, 53, 102d }}
Badness (Sintel): 1.75
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 275/273, 325/324, 540/539
Mapping: {{mapping| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}
Optimal tunings:
* WE: ~2 = 1199.1517{{c}}, ~22/13 = 905.1727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 905.7801{{c}}
{{Optimal ET sequence|legend=0| 49f, 53, 102df }}
Badness (Sintel): 1.28


== Marfifths ==
== Marfifths ==
Line 995: Line 884:
Badness (Sintel): 1.58
Badness (Sintel): 1.58


== Marthirds ==
== Kleiboh ==
The ''marthirds'' temperament (19 & 193) tempers out the breeze comma (laquadru-atruyo comma), [[2460375/2458624]]. It splits the interval of minor tenth (~12/5) into four marvelous major third ([[56/45]]) intervals, and uses it for a generator.
 
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15625/15552, 2460375/2458624
[[Comma list]]: 1728/1715, 3125/3087


{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}
{{Mapping|legend=1| 1 -12 -9 -7 | 0 18 15 13 }}
: mapping generators: ~2, ~56/45
: mapping generators: ~2, ~42/25


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1662{{c}}, ~56/45 = 379.3041{{c}}
* [[WE]]: ~2 = 1199.5290{{c}}, ~42/25 = 905.3417{{c}}
: [[error map]]: {{val| +0.166 +0.347 -0.896 +0.000 }}
: [[error map]]: {{val| -0.471 -0.152 -1.949 +3.914 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 379.2552{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 905.6741{{c}}
: error map: {{val| 0.000 +0.171 -1.209 -0.214 }}
: error map: {{val| 0.000 +0.178 -1.203 +4.937 }}


{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}
{{Optimal ET sequence|legend=1| 49, 53 }}


[[Badness]] (Sintel): 2.64
[[Badness]] (Sintel): 1.93


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 15625/15552, 19712/19683
Comma list: 176/175, 540/539, 3125/3087


Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}
Mapping: {{mapping| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1189{{c}}, ~56/45 = 379.2942{{c}}
* WE: ~2 = 1199.1389{{c}}, ~42/25 = 905.1688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 905.7840{{c}}


{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c }}
{{Optimal ET sequence|legend=0| 49, 53, 102d }}


Badness (Sintel): 2.50
Badness (Sintel): 1.75


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 625/624, 1375/1372, 19712/19683
Comma list: 176/175, 275/273, 325/324, 540/539


Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
Mapping: {{mapping| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2154{{c}}, ~56/45 = 379.3236{{c}}
* WE: ~2 = 1199.1517{{c}}, ~22/13 = 905.1727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 905.7801{{c}}


{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405f, 617cff }}
{{Optimal ET sequence|legend=0| 49f, 53, 102df }}


Badness (Sintel): 1.81
Badness (Sintel): 1.28
 
== Quartkeenlig ==
Quartkeenlig uses a generator in the 11-limit that is 33/32~36/35 tempered together, and is called so because it tempers out the [[quartisma]] by virtue of five 33/32's being with 7/6, keenanisma, 385/384, tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). It can also be viewed as a regular temperament interpretation of [[23edo and octave stretching|stretched 23edo]].


== Quadritikleismic ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15625/15552, 117649/116640
[[Comma list]]: 2401/2400, 15625/15552


{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
: mapping generator: ~2, ~36/35
: mapping generators: ~25/21, ~6/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.2825{{c}}, ~36/35 = 52.8528{{c}}
* [[WE]]: ~25/21 = 300.0520{{c}}, ~6/5 = 317.0548{{c}} (~126/125 = 17.0029{{c}})
: [[error map]]: {{val| +0.282 +0.745 -0.448 -1.579 }}
: [[error map]]: {{val| +0.208 +0.374 -0.832 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 52.8476{{c}}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0301{{c}} (~126/125 = 17.0301{{c}})
: error map: {{val| 0.000 +0.558 -0.886 -2.074 }}
: error map: {{val| 0.000 +0.225 -1.163 -0.706 }}


{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}


[[Badness]] (Sintel): 3.69
[[Badness]] (Sintel): 0.993


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 6250/6237, 67228/66825
Comma list: 385/384, 1375/1372, 6250/6237


Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2526{{c}}, ~36/35 = 52.8534{{c}}
* WE: ~25/21 = 300.0995{{c}}, ~6/5 = 317.0298{{c}} (~100/99 = 16.9303{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8446{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9540{{c}} (~100/99 = 16.9540{{c}})


{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}
{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}


Badness (Sintel): 2.86
Badness (Sintel): 0.774


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 385/384, 625/624, 16807/16731
Comma list: 325/324, 385/384, 625/624, 1375/1372
 
Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
 
Optimal tunings:
* WE: ~25/21 = 300.0985{{c}}, ~6/5 = 317.0899{{c}} (~100/99 = 16.9941{{c}})
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
 
{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}
 
Badness (Sintel): 0.774
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 325/324, 385/384, 442/441, 625/624


Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2564{{c}}, ~36/35 = 52.8568{{c}}
* WE: ~25/21 = 300.1102{{c}}, ~6/5 = 317.1011{{c}} (~100/99 = 16.9909{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8479{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
 
{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}


{{Optimal ET sequence|legend=0| 68, 159, 386d, 545ddf }}
Badness (Sintel): 0.651


Badness (Sintel): 1.97
== Marthirds ==
The ''marthirds'' temperament (19 & 193) tempers out the breeze comma (laquadru-atruyo comma), [[2460375/2458624]]. It splits the interval of minor tenth (~12/5) into four marvelous major third ([[56/45]]) intervals, and uses it for a generator.


== Novemkleismic ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15625/15552, 40353607/40310784
[[Comma list]]: 15625/15552, 2460375/2458624


{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}
: mapping generators: ~2592/2401, ~6/5
: mapping generators: ~2, ~56/45


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2592/2401 = 133.3488{{c}}, ~6/5 = 317.0413{{c}} (~36/35 = 50.3437{{c}})
* [[WE]]: ~2 = 1200.1662{{c}}, ~56/45 = 379.3041{{c}}
: [[error map]]: {{val| +0.139 +0.293 -0.968 +0.259 }}
: [[error map]]: {{val| +0.166 +0.347 -0.896 +0.000 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~6/5 = 317.0260{{c}} (~36/35 = 50.3593{{c}})
* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 379.2552{{c}}
: error map: {{val| 0.000 +0.201 -1.184 -0.003 }}
: error map: {{val| 0.000 +0.171 -1.209 -0.214 }}


{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}
{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}


[[Badness]] (Sintel): 4.90
[[Badness]] (Sintel): 2.64


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 4000/3993, 15625/15552
Comma list: 1375/1372, 15625/15552, 19712/19683


Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~250/231 = 133.3465{{c}}, ~6/5 = 317.0416{{c}} (~36/35 = 50.3486{{c}})
* WE: ~2 = 1200.1189{{c}}, ~56/45 = 379.2942{{c}}
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0264{{c}} (~36/35 = 50.3597{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}


{{Optimal ET sequence|legend=0| 72, 261, 333, 405, 882c }}
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c }}


Badness (Sintel): 1.71
Badness (Sintel): 2.50


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 625/624, 1375/1372, 4000/3993
Comma list: 325/324, 625/624, 1375/1372, 19712/19683


Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~250/231 = 133.3385{{c}}, ~6/5 = 317.0978{{c}} (~36/35 = 50.4208{{c}})
* WE: ~2 = 1200.2154{{c}}, ~56/45 = 379.3236{{c}}
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0910{{c}} (~36/35 = 50.4243{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}


{{Optimal ET sequence|legend=0| 72, 189f, 261, 333, 738cf }}
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405f, 617cff }}


Badness (Sintel): 1.61
Badness (Sintel): 1.81


== Sqrtphi ==
== Sqrtphi ==
Line 1,224: Line 1,126:


Badness (Sintel): 0.897
Badness (Sintel): 0.897
== Quartkeenlig ==
Quartkeenlig uses a generator in the 11-limit that is 33/32~36/35 tempered together, and is called so because it tempers out the [[quartisma]] by virtue of five 33/32's being with 7/6, keenanisma, 385/384, tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). It can also be viewed as a regular temperament interpretation of [[23edo and octave stretching|stretched 23edo]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 15625/15552, 117649/116640
{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
: mapping generator: ~2, ~36/35
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2825{{c}}, ~36/35 = 52.8528{{c}}
: [[error map]]: {{val| +0.282 +0.745 -0.448 -1.579 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 52.8476{{c}}
: error map: {{val| 0.000 +0.558 -0.886 -2.074 }}
{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
[[Badness]] (Sintel): 3.69
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 6250/6237, 67228/66825
Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
Optimal tunings:
* WE: ~2 = 1200.2526{{c}}, ~36/35 = 52.8534{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8446{{c}}
{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}
Badness (Sintel): 2.86
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 625/624, 16807/16731
Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
Optimal tunings:
* WE: ~2 = 1200.2564{{c}}, ~36/35 = 52.8568{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8479{{c}}
{{Optimal ET sequence|legend=0| 68, 159, 386d, 545ddf }}
Badness (Sintel): 1.97
== Novemkleismic ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 15625/15552, 40353607/40310784
{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
: mapping generators: ~2592/2401, ~6/5
[[Optimal tuning]]s:
* [[WE]]: ~2592/2401 = 133.3488{{c}}, ~6/5 = 317.0413{{c}} (~36/35 = 50.3437{{c}})
: [[error map]]: {{val| +0.139 +0.293 -0.968 +0.259 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~6/5 = 317.0260{{c}} (~36/35 = 50.3593{{c}})
: error map: {{val| 0.000 +0.201 -1.184 -0.003 }}
{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}
[[Badness]] (Sintel): 4.90
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 4000/3993, 15625/15552
Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
Optimal tunings:
* WE: ~250/231 = 133.3465{{c}}, ~6/5 = 317.0416{{c}} (~36/35 = 50.3486{{c}})
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0264{{c}} (~36/35 = 50.3597{{c}})
{{Optimal ET sequence|legend=0| 72, 261, 333, 405, 882c }}
Badness (Sintel): 1.71
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 625/624, 1375/1372, 4000/3993
Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
Optimal tunings:
* WE: ~250/231 = 133.3385{{c}}, ~6/5 = 317.0978{{c}} (~36/35 = 50.4208{{c}})
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0910{{c}} (~36/35 = 50.4243{{c}})
{{Optimal ET sequence|legend=0| 72, 189f, 261, 333, 738cf }}
Badness (Sintel): 1.61


== Subgroup extensions ==
== Subgroup extensions ==
Retrieved from "https://en.xen.wiki/w/Kleismic_family"