Xenharmonic Wiki

Tetracot family: Difference between revisions

← Older editNewer edit →
VisualWikitext
Switch to Sintel's badness, WE & CWE tunings, per community consensus (2/)
Switch to Sintel's badness, WE & CWE tunings, per community consensus (3/3)
Line 806: Line 806:


{{Mapping|legend=1| 1 1 1 1 | 0 12 27 37 }}
{{Mapping|legend=1| 1 1 1 1 | 0 12 27 37 }}
: mapping generators: ~2, ~28/27


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~28/27 = 58.648{{c}}
* [[WE]]: ~2 = 1199.6912{{c}}, ~28/27 = 58.6600{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~28/27 = 58.675{{c}}
: [[error map]]: {{val| -0.309 +1.657 -2.802 +1.287 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 58.6624{{c}}
: error map: {{val| 0.000 +1.993 -2.430 +1.681 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~28/27 = 58.648{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~28/27 = 58.675{{c}} -->


{{Optimal ET sequence|legend=1| 41, 143d, 184, 225, 266c }}
{{Optimal ET sequence|legend=1| 20cd, 41, 143d, 184, 225 }}


[[Badness]] (Smith): 0.119761
[[Badness]] (Sintel): 3.03


=== 11-limit ===
=== 11-limit ===
Line 822: Line 827:
Mapping: {{mapping| 1 1 1 1 2 | 0 12 27 37 30 }}
Mapping: {{mapping| 1 1 1 1 2 | 0 12 27 37 30 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~28/27 = 58.602{{c}}
* WE: ~2 = 1199.3125{{c}}, ~28/27 = 58.6317{{c}}
* POTE: ~2 = 1200.000{{c}}, ~28/27 = 58.665{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.6360{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~28/27 = 58.602{{c}}
* POTE: ~2 = 1200.000{{c}}, ~28/27 = 58.665{{c}} -->


{{Optimal ET sequence|legend=0| 20cde, 41 }}
{{Optimal ET sequence|legend=0| 20cde, 41 }}


Badness (Smith): 0.059528
Badness (Sintel): 1.97


=== 13-limit ===
=== 13-limit ===
Line 837: Line 844:
Mapping: {{mapping| 1 1 1 1 2 2 | 0 12 27 37 30 35 }}
Mapping: {{mapping| 1 1 1 1 2 2 | 0 12 27 37 30 35 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~27/26 = 58.551{{c}}
* WE: ~2 = 1199.0713{{c}}, ~28/27 = 58.5932{{c}}
* POTE: ~2 = 1200.000{{c}}, ~27/26 = 58.639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.5982{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~28/27 = 58.551{{c}}
* POTE: ~2 = 1200.000{{c}}, ~28/27 = 58.639{{c}} -->


{{Optimal ET sequence|legend=0| 20cdef, 21cdef, 41 }}
{{Optimal ET sequence|legend=0| 20cdef, 41 }}


Badness: 0.043645
Badness (Sintel): 1.80


== Byhearted ==
== Byhearted ==
Line 855: Line 864:
: mapping generators: ~7/5, ~10/9
: mapping generators: ~7/5, ~10/9


[[Optimal tuning]] ([[CTE]]): ~7/5 = 600.000{{c}}, ~10/9 = 175.472{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 599.6934{{c}}, ~10/9 = 175.5626{{c}}
: [[error map]]: {{val| -0.613 -0.318 -6.864 +10.318 }}
* [[CWE]]: ~7/5 = 1200.0000{{c}}, ~10/9 = 175.5632{{c}}
: error map: {{val| 0.000 +0.298 -6.245 +11.243 }}
<!-- * [[CTE]]: ~7/5 = 600.000{{c}}, ~10/9 = 175.472{{c}} -->


{{Optimal ET sequence|legend=1| 14c, 34d, 48, 82d, 130cdd }}
{{Optimal ET sequence|legend=1| 14c, 34d, 48 }}


[[Badness]] (Smith): 0.111574
[[Badness]] (Sintel): 2.82


=== 11-limit ===
=== 11-limit ===
Line 868: Line 882:
Mapping: {{mapping| 2 2 2 3 4 | 0 4 9 9 10 }}
Mapping: {{mapping| 2 2 2 3 4 | 0 4 9 9 10 }}


Optimal tuning (CTE): ~7/5 = 600.000{{c}}, ~10/9 = 175.401{{c}}
Optimal tunings:
* WE: ~7/5 = 599.6525{{c}}, ~10/9 = 175.5103{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.5086{{c}}
<!-- * CTE: ~7/5 = 600.000{{c}}, ~10/9 = 175.401{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 34d, 48, 82d, 130cdd }}
{{Optimal ET sequence|legend=0| 14c, 34d, 48 }}


Badness (Smith): 0.043923
Badness (Sintel): 1.45


=== 13-limit ===
=== 13-limit ===
Line 881: Line 898:
Mapping: {{mapping| 2 2 2 3 4 3 | 0 4 9 9 10 15 }}
Mapping: {{mapping| 2 2 2 3 4 3 | 0 4 9 9 10 15 }}


Optimal tuning (CTE): ~7/5 = 600.000{{c}}, ~10/9 = 175.586{{c}}
Optimal tunings:
* WE: ~7/5 = 599.4539{{c}}, ~10/9 = 175.7393{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.7502{{c}}
<!-- * CTE: ~7/5 = 600.000{{c}}, ~10/9 = 175.586{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 34d, 48f, 82d }}
{{Optimal ET sequence|legend=0| 14cf, 20cdef, 34d }}


Badness (Smith): 0.031948
Badness (Sintel): 1.32


=== 17-limit ===
=== 17-limit ===
Line 894: Line 914:
Mapping: {{mapping| 2 2 2 3 4 3 7 | 0 4 9 9 10 15 4 }}
Mapping: {{mapping| 2 2 2 3 4 3 7 | 0 4 9 9 10 15 4 }}


Optimal tuning (CTE): ~7/5 = 600.000{{c}}, ~10/9 = 175.596{{c}}
Optimal tunings:
* WE: ~7/5 = 599.7509{{c}}, ~10/9 = 175.6684{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.6839{{c}}
<!-- * CTE: ~7/5 = 600.000{{c}}, ~10/9 = 175.596{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 34d, 48f, 82d }}
{{Optimal ET sequence|legend=0| 14cf, 20cdef, 34d }}


Badness (Smith): 0.026119
Badness (Sintel): 1.33


=== 19-limit ===
=== 19-limit ===
Line 907: Line 930:
Mapping: {{mapping| 2 2 2 3 4 3 7 5 | 0 4 9 9 10 15 4 12 }}
Mapping: {{mapping| 2 2 2 3 4 3 7 5 | 0 4 9 9 10 15 4 12 }}


Optimal tuning (CTE): ~7/5 = 600.000{{c}}, ~10/9 = 175.498{{c}}
Optimal tunings:
* WE: ~7/5 = 599.6682{{c}}, ~10/9 = 175.5994{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.6190{{c}}
<!-- * CTE: ~7/5 = 600.000{{c}}, ~10/9 = 175.498{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 34dh, 48f, 82dh }}
{{Optimal ET sequence|legend=0| 14cf, 20cdefhh, 34dh, 48f }}


Badness (Smith): 0.021029
Badness (Sintel): 1.28


== Other subgroup temperaments ==
== Other subgroup extensions ==
=== Devisemi (2.3.5.7.19) ===
=== Devisemi (2.3.5.19) ===
==== 2.3.5.19 subgroup ====
[[Subgroup]]: 2.3.5.19
Subgroup: 2.3.5.19


[[Comma list]]: 361/360, 20000/19683
[[Comma list]]: 361/360, 20000/19683


[[Gencom]]: [2 19/18; 361/360 20000/19683]
{{Mapping|legend=2| 1 1 1 3 | 0 8 18 17 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 0 0 0 3}}, {{val|0 8 18 0 0 0 0 17}}]
{{Mapping|legend=3| 1 1 1 0 0 0 0 3 | 0 8 18 0 0 0 0 17 }}


[[Mapping|Sval mapping]]: [{{val|1 1 1 3}}, {{val|0 8 18 17}}]
: mapping generators: ~2, ~20/19


[[Tp tuning|POL2 generator]]: ~19/18 = 88.077
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6900{{c}}, ~20/19 = 88.0541{{c}}
: [[error map]]: {{val| -0.310 +2.168 -1.649 -1.523 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~20/19 = 88.0538{{c}}
: error map: {{val| 0.000 +2.475 -1.345 -0.598 }}


{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5701 cents
[[Badness]] (Sintel): 1.30


==== 2.3.5.7.19 subgroup ====
==== 2.3.5.7.19 subgroup ====
Subgroup: 2.3.5.7.19
Subgroup: 2.3.5.7.19


[[Comma list]]: 190/189, 245/243, 361/360
Comma list: 190/189, 245/243, 361/360


[[Gencom]]: [2 19/18; 190/189 245/243 361/360]
Subgroup-val mapping: {{mapping| 1 1 1 2 3 | 0 8 18 11 17 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 1 2 0 0 0 3}}, {{val|0 8 18 11 0 0 0 17}}]
Gencom mapping: {{mapping| 1 1 1 2 0 0 0 3 | 0 8 18 11 0 0 0 17 }}


[[Mapping|Sval mapping]]: [{{val|1 1 1 2 3}}, {{val|0 8 18 11 17}}]
Optimal tunings:  
* WE: ~2 = 1199.7591{{c}}, ~20/19 = 88.0570{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.0564{{c}}


[[Tp tuning|POL2 generator]]: ~19/18 = 88.075
{{Optimal ET sequence|legend=0| 14c, 27, 41, 68, 109 }}


{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
Badness (Sintel): 0.508
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5780 cents


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