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Syntonic–31 equivalence continuum: Difference between revisions

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== Quadlayo (31 & 12c) ==
== Quadlayo (31 & 12c) ==
In the [[chain-of-fifths notation]], 5/4 is mapped to the quadruple-diminished fifth (C-Gbbbb).
In the [[chain-of-fifths notation]], 5/4 is mapped to the quadruple-diminished fifth (C-Gbbbb).


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -45 27 1 }
[[Comma list]]: {{monzo| -45 27 1 }}


{{Mapping|legend=1| 1 0 45 | 0 1 -27 }}
{{Mapping|legend=1| 1 0 45 | 0 1 -27 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 696.950{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.6167{{c}}, ~3/2 = 697.8886{{c}}
: [[error map]]: {{val| +1.617 -2.450 -0.204 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 696.9075{{c}}
: error map: {{val| 0.000 -5.048 -2.815 }}


{{Optimal ET sequence|legend=1| 12c, 19c, 31, 43c, 50c }}
{{Optimal ET sequence|legend=1| 12c, 19c, 31, 136bc, 167bc, 198bc, 229bc }}


[[Badness]] (Smith): 2.993628
[[Badness]] (Sintel): 70.2


[http://x31eq.com/cgi-bin/rt.cgi?ets=31_12c&limit=5 The temperament finder - 5-limit 31 & 12c]
[http://x31eq.com/cgi-bin/rt.cgi?ets=31_12c&limit=5 The temperament finder - 5-limit 31 & 12c]
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{{Mapping|legend=1| 1 1 3 | 0 6 -7 }}
{{Mapping|legend=1| 1 1 3 | 0 6 -7 }}
: mapping generators: ~2, ~16/15


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~16/15 = 116.701{{c}}
* [[WE]]: ~2 = 1200.8367{{c}}, ~16/15 = 116.7546{{c}}
: [[error map]]: {{val| 0.000 -1.750 -3.219 }}
: [[error map]]: {{val| +0.837 -0.591 -1.086 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~16/15 = 116.680{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~16/15 = 116.6802{{c}}
: error map: {{val| 0.000 -1.874 -3.075 }}
: error map: {{val| 0.000 -1.874 -3.075 }}


{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}
{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}


[[Badness]]:
[[Badness]] (Sintel): 3.89
* Smith: 0.165755
* Dirichlet: 3.89


== Valentine (5-limit) ==
== Valentine (5-limit) ==
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{{Mapping|legend=1| 1 1 2 | 0 9 5 }}
{{Mapping|legend=1| 1 1 2 | 0 9 5 }}
: mapping generators: ~2, ~25/24


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~25/24 = 78.039{{c}}
* [[WE]]: ~2 = 1199.3579{{c}}, ~25/24 = 77.9973{{c}}
: [[error map]]: {{val| 0.000 +0.397 +3.882 }}
: [[error map]]: {{val| -0.642 -0.621 +2.389 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~25/24 = 78.039{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/24 = 77.9807{{c}}
: error map: {{val| 0.000 -0.829 +3.201 }}
: error map: {{val| 0.000 -0.129 +3.590 }}


{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 123 }}
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 123 }}


[[Badness]] (Smith): 0.122765
[[Badness]] (Sintel): 2.88


== Quadlaleyo (31 & 70c) ==
== Quadlaleyo (31 & 70c) ==
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[[Comma list]]: {{monzo| -54 18 11 }}
[[Comma list]]: {{monzo| -54 18 11 }}


{{Mapping|legend=1| 1 3 0 | 0 -11 18 }}
{{Mapping|legend=1| 1 -8 18 | 0 11 -18 }}
: mapping generators: ~2, ~30375/16384


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~32768/30375 = 154.597{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0416{{c}}, ~32768/30375 = 1046.3102{{c}}
: [[error map]]: {{val| +1.042 -0.876 -1.149 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32768/30375 = 1045.4008{{c}}
: error map: {{val| 0.000 -2.546 -3.529 }}


{{Optimal ET sequence|legend=1| 8c, 23c, 31, 39c, 132, 163 }}
{{Optimal ET sequence|legend=1| 8c, 31, 101c, 132, 163 }}


[[Badness]] (Smith): 2.067160
[[Badness]] (Sintel): 48.5


[http://x31eq.com/cgi-bin/rt.cgi?ets=31_70c&limit=5 The temperament finder - 5-limit 31 & 70c]
[http://x31eq.com/cgi-bin/rt.cgi?ets=31_70c&limit=5 The temperament finder - 5-limit 31 & 70c]
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[[Comma list]]: 847288609443/781250000000
[[Comma list]]: 847288609443/781250000000


{{Mapping|legend=1| 1 7 12 | 0 -14 -25 }}
{{Mapping|legend=1| 1 -7 -13 | 0 14 25 }}
: mapping generators: ~2, ~19683/12500
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3614{{c}}, ~19683/12500 = 735.7984{{c}}
: [[error map]]: {{val| +0.361 -3.307 +3.498 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~19683/12500 = 735.5950{{c}}
: error map: {{val| 0.000 -3.625 -3.560 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25000/19683 = 464.423{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25000/19683 = 464.423{{c}}


{{Optimal ET sequence|legend=1| 13c, 18bc, 31, 44c, 49bc, 75c, 80bc }}
{{Optimal ET sequence|legend=1| 13c, 18bc, 31 }}


[[Badness]] (Smith): 2.094918
[[Badness]] (Sintel): 49.1


[http://x31eq.com/cgi-bin/rt.cgi?ets=31_13c&limit=5 The temperament finder - 5-limit 31 & 13c]
[http://x31eq.com/cgi-bin/rt.cgi?ets=31_13c&limit=5 The temperament finder - 5-limit 31 & 13c]
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[[Comma list]]: {{monzo| 55 -1 -23 }}
[[Comma list]]: {{monzo| 55 -1 -23 }}


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


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~5/4 = 386.8710{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000{{c}}, ~8/5 = 813.0556{{c}}
: [[error map]]: {{val| -0.120 +0.005 +0.271 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 813.1344{{c}}
: error map: {{val| 0.000 +0.135 +0.552 }}


{{Optimal ET sequence|legend=1| 28b, 31, 90, 121, 152, 335, 822, 1157c, 1492c }}
{{Optimal ET sequence|legend=1| 28b, 31, 90, 121, 152, 335, 822, 1157c, 1492c, 2649cc }}


[[Badness]] (Smith): 0.420
[[Badness]] (Sintel): 9.86


[[Category:31edo]]
[[Category:31edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
Retrieved from "https://en.xen.wiki/w/Syntonic–31_equivalence_continuum"