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Tetracot family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus (1/)
Switch to Sintel's badness, WE & CWE tunings, per community consensus (2/)
Line 536: Line 536:
{{Mapping|legend=1| 1 1 1 2 | 0 8 18 11 }}
{{Mapping|legend=1| 1 1 1 2 | 0 8 18 11 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 88.023{{c}}
* [[WE]]: ~2 = 1199.6782{{c}}, ~21/20 = 88.0528{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 88.076{{c}}
: [[error map]]: {{val| -0.322 +2.145 -1.686 -0.889 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 88.0525{{c}}
: error map: {{val| 0.000 +2.465 -1.369 -0.248 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 88.023{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 88.076{{c}} -->


{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109, 150, 259, 409bc, 668bbcc }}
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}


[[Badness]] (Smith): 0.033845
[[Badness]] (Sintel): 0.857


=== 11-limit ===
=== 11-limit ===
Line 551: Line 555:
Mapping: {{mapping| 1 1 1 2 2 | 0 8 18 11 20 }}
Mapping: {{mapping| 1 1 1 2 2 | 0 8 18 11 20 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.910{{c}}
* WE: ~2 = 1199.6025{{c}}, ~21/20 = 87.9460{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 87.975{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.9453{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.910{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 87.975{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 27e, 41, 150ee, 191ee, 232cee }}
{{Optimal ET sequence|legend=0| 14c, 27e, 41, 109e }}


Badness (Smith): 0.024078
Badness (Sintel): 0.796


==== 13-limit ====
==== 13-limit ====
Line 566: Line 572:
Mapping: {{mapping| 1 1 1 2 2 4 | 0 8 18 11 20 -4 }}
Mapping: {{mapping| 1 1 1 2 2 4 | 0 8 18 11 20 -4 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.926{{c}}
* WE: ~2 = 1198.8609{{c}}, ~21/20 = 87.0219{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.106{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0557{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.926{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.106{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 27e, 41, 150eef, 191eeff, 232ceeff }}
{{Optimal ET sequence|legend=0| 14c, 27e, 41, 68e, 109ef }}


Badness (Smith): 0.023276
Badness (Sintel): 0.962


===== 17-limit =====
===== 17-limit =====
Line 581: Line 589:
Mapping: {{mapping| 1 1 1 2 2 4 3 | 0 8 18 11 20 -4 15 }}
Mapping: {{mapping| 1 1 1 2 2 4 3 | 0 8 18 11 20 -4 15 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~18/17 = 87.842{{c}}
* WE: ~2 = 1198.4494{{c}}, ~21/20 = 87.9878{{c}}
* POTE: ~2 = 1200.000{{c}}, ~18/17 = 88.102{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0324{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.842{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.102{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 27eg, 41 }}
{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}


Badness (Smith): 0.021088
Badness (Sintel): 1.07


===== 19-limit =====
===== 19-limit =====
Line 596: Line 606:
Mapping: {{mapping| 1 1 1 2 2 4 3 3 | 0 8 18 11 20 -4 15 17 }}
Mapping: {{mapping| 1 1 1 2 2 4 3 3 | 0 8 18 11 20 -4 15 17 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~18/17 = 87.866{{c}}
* WE: ~2 = 1198.5995{{c}}, ~20/19 = 88.0081{{c}}
* POTE: ~2 = 1200.000{{c}}, ~18/17 = 88.111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.0471{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~20/19 = 87.866{{c}}
* POTE: ~2 = 1200.000{{c}}, ~20/19 = 88.111{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 27eg, 41 }}
{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}


Badness (Smith): 0.016652
Badness (Sintel): 1.01


==== Octocat ====
==== Octocat ====
Line 611: Line 623:
Mapping: {{mapping| 1 1 1 2 2 2 | 0 8 18 11 20 23 }}
Mapping: {{mapping| 1 1 1 2 2 2 | 0 8 18 11 20 23 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.090{{c}}
* WE: ~2 = 1199.4441{{c}}, ~21/20 = 88.1380{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.179{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1375{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.090{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.179{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27e, 41f, 68ef, 109eff }}
{{Optimal ET sequence|legend=0| 14cf, 27e, 41f }}


Badness (Smith): 0.027601
Badness (Sintel): 1.14


===== 17-limit =====
===== 17-limit =====
Line 626: Line 640:
Mapping: {{mapping| 1 1 1 2 2 2 3 | 0 8 18 11 20 23 15 }}
Mapping: {{mapping| 1 1 1 2 2 2 3 | 0 8 18 11 20 23 15 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~18/17 = 88.011{{c}}
Optimal tunings:
* WE: ~2 = 1198.4257{{c}}, ~21/20 = 88.1636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1642{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.011{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27eg, 41f, 109effgg }}
{{Optimal ET sequence|legend=0| 14cf, 27eg }}


Badness (Smith): 0.023345
Badness (Sintel): 1.19


===== 19-limit =====
===== 19-limit =====
Line 639: Line 656:
Mapping: {{mapping| 1 1 1 2 2 2 3 3 | 0 8 18 11 20 23 15 17 }}
Mapping: {{mapping| 1 1 1 2 2 2 3 3 | 0 8 18 11 20 23 15 17 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~18/17 = 88.017{{c}}
Optimal tunings:
* WE: ~2 = 1198.5748{{c}}, ~20/19 = 88.1631{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1637{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~20/19 = 88.017{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27eg, 41f, 109effgg }}
{{Optimal ET sequence|legend=0| 14cf, 27eg }}


Badness (Smith): 0.017916
Badness (Sintel): 1.09


==== Octopod ====
==== Octopod ====
Line 652: Line 672:
Mapping: {{mapping| 1 1 1 2 2 1 | 0 8 18 11 20 37 }}
Mapping: {{mapping| 1 1 1 2 2 1 | 0 8 18 11 20 37 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.770{{c}}
* WE: ~2 = 1200.5116{{c}}, ~21/20 = 87.7346{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 87.697{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7257{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.770{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 87.697{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27eff, 41 }}
{{Optimal ET sequence|legend=0| 14cf, 27eff, 41 }}


Badness (Smith): 0.028326
Badness (Sintel): 1.17


===== 17-limit =====
===== 17-limit =====
Line 667: Line 689:
Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 8 18 11 20 37 15 }}
Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 8 18 11 20 37 15 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~18/17 = 87.728{{c}}
Optimal tunings:
* WE: ~2 = 1199.6667{{c}}, ~21/20 = 87.7494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7559{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.728{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}


Badness (Smith): 0.024660
Badness (Sintel): 1.26


===== 19-limit =====
===== 19-limit =====
Line 680: Line 705:
Mapping: {{mapping| 1 1 1 2 2 1 3 3 | 0 8 18 11 20 37 15 17 }}
Mapping: {{mapping| 1 1 1 2 2 1 3 3 | 0 8 18 11 20 37 15 17 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~18/17 = 87.750{{c}}
Optimal tunings:
* WE: ~2 = 1199.9909{{c}}, ~20/19 = 87.7474{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 87.7476{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~20/19 = 87.750{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}


Badness (Smith): 0.019504
Badness (Sintel): 1.19


==== Dificot ====
==== Dificot ====
Line 691: Line 719:
Comma list: 100/99, 243/242, 245/242, 343/338
Comma list: 100/99, 243/242, 245/242, 343/338


Mapping: {{mapping| 1 9 19 13 22 19 | 0 -16 -36 -22 -40 -33 }}
Mapping: {{mapping| 1 -7 -17 -9 -18 -14 | 0 16 36 22 40 33 }}
: mapping generators: ~2, ~13/9


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~13/9 = 643.916{{c}}
* WE: ~2 = 1199.1496{{c}}, ~13/9 = 643.5328{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/9 = 643.989{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 643.9567{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~13/9 = 643.916{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/9 = 643.989{{c}} -->


{{Optimal ET sequence|legend=0| 13cdeef, 28ccdef, 41 }}
{{Optimal ET sequence|legend=0| 13cdeef, 28ccdef, 41 }}


Badness (Smith): 0.051876
Badness (Sintel): 2.14


=== October ===
=== October ===
Line 708: Line 739:
Mapping: {{mapping| 1 1 1 2 5 | 0 8 18 11 -21 }}
Mapping: {{mapping| 1 1 1 2 5 | 0 8 18 11 -21 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.026{{c}}
* WE: ~2 = 1199.8843{{c}}, ~21/20 = 88.0261{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0329{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.026{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.035{{c}} -->


{{Optimal ET sequence|legend=0| 27, 41, 68, 109, 150, 259 }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109, 150, 259 }}


Badness (Smith): 0.039643
Badness (Sintel): 1.31


==== 13-limit ====
==== 13-limit ====
Line 723: Line 756:
Mapping: {{mapping| 1 1 1 2 5 4 | 0 8 18 11 -21 -4 }}
Mapping: {{mapping| 1 1 1 2 5 4 | 0 8 18 11 -21 -4 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.041{{c}}
* WE: ~2 = 1199.5060{{c}}, ~21/20 = 88.0388{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.075{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0697{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.041{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.075{{c}} -->


{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}


Badness (Smith): 0.031136
Badness (Sintel): 1.29


==== 17-limit ====
==== 17-limit ====
Line 738: Line 773:
Mapping: {{mapping| 1 1 1 2 5 4 6 | 0 8 18 11 -21 -4 -26 }}
Mapping: {{mapping| 1 1 1 2 5 4 6 | 0 8 18 11 -21 -4 -26 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.093{{c}}
* WE: ~2 = 1199.3845{{c}}, ~21/20 = 88.0589{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.104{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1027{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.093{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.104{{c}} -->


{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}


Badness (Smith): 0.026833
Badness (Sintel): 1.37


==== 19-limit ====
==== 19-limit ====
Line 753: Line 790:
Mapping: {{mapping| 1 1 1 2 5 4 6 3 | 0 8 18 11 -21 -4 -26 17 }}
Mapping: {{mapping| 1 1 1 2 5 4 6 3 | 0 8 18 11 -21 -4 -26 17 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~19/18 = 88.093{{c}}
* WE: ~2 = 1199.4449{{c}}, ~20/19 = 88.0723{{c}}
* POTE: ~2 = 1200.000{{c}}, ~19/18 = 88.113{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1107{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~20/19 = 88.093{{c}}
* POTE: ~2 = 1200.000{{c}}, ~20/19 = 88.113{{c}} -->


{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}


Badness (Smith): 0.020511
Badness (Sintel): 1.25
 
=== Devisemi (2.3.5.7.19) ===
==== 2.3.5.19 subgroup ====
Subgroup: 2.3.5.19
 
[[Comma list]]: 361/360, 20000/19683
 
[[Gencom]]: [2 19/18; 361/360 20000/19683]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 0 0 0 3}}, {{val|0 8 18 0 0 0 0 17}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 1 3}}, {{val|0 8 18 17}}]
 
[[Tp tuning|POL2 generator]]: ~19/18 = 88.077
 
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5701 cents
 
==== 2.3.5.7.19 subgroup ====
Subgroup: 2.3.5.7.19
 
[[Comma list]]: 190/189, 245/243, 361/360
 
[[Gencom]]: [2 19/18; 190/189 245/243 361/360]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 1 2 0 0 0 3}}, {{val|0 8 18 11 0 0 0 17}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 1 2 3}}, {{val|0 8 18 11 17}}]
 
[[Tp tuning|POL2 generator]]: ~19/18 = 88.075
 
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5780 cents


== Dodecacot ==
== Dodecacot ==
Line 908: Line 912:


Badness (Smith): 0.021029
Badness (Smith): 0.021029
== Other subgroup temperaments ==
=== Devisemi (2.3.5.7.19) ===
==== 2.3.5.19 subgroup ====
Subgroup: 2.3.5.19
[[Comma list]]: 361/360, 20000/19683
[[Gencom]]: [2 19/18; 361/360 20000/19683]
[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 0 0 0 3}}, {{val|0 8 18 0 0 0 0 17}}]
[[Mapping|Sval mapping]]: [{{val|1 1 1 3}}, {{val|0 8 18 17}}]
[[Tp tuning|POL2 generator]]: ~19/18 = 88.077
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.5701 cents
==== 2.3.5.7.19 subgroup ====
Subgroup: 2.3.5.7.19
[[Comma list]]: 190/189, 245/243, 361/360
[[Gencom]]: [2 19/18; 190/189 245/243 361/360]
[[Gencom|Gencom mapping]]: [{{val|1 1 1 2 0 0 0 3}}, {{val|0 8 18 11 0 0 0 17}}]
[[Mapping|Sval mapping]]: [{{val|1 1 1 2 3}}, {{val|0 8 18 11 17}}]
[[Tp tuning|POL2 generator]]: ~19/18 = 88.075
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.5780 cents


== References ==
== References ==
Retrieved from "https://en.xen.wiki/w/Tetracot_family"