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Syntonic–chromatic equivalence continuum: Difference between revisions

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| {{Monzo| -17 21 -7 }}
| {{Monzo| -17 21 -7 }}
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{{Mapping|legend=1| 1 0 -15 | 0 1 11 }}
{{Mapping|legend=1| 1 0 -15 | 0 1 11 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 689.879{{c}}
* [[WE]]: ~2 = 1203.7664{{c}}, ~3/2 = 691.1525{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 689.312{{c}}
: [[error map]]: {{val| +3.766 -7.036 +1.298 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 689.3122{{c}}
: error map: {{val| 0.000 -12.643 -3.879 }}


{{Optimal ET sequence|legend=1| 7, 33, 40, 47, 87b }}
{{Optimal ET sequence|legend=1| 7, 33, 40, 47, 54b }}


[[Badness]] (Smith): 0.403
[[Badness]] (Sintel): 9.44


== Shallowtone (5-limit) ==
== Shallowtone (5-limit) ==
Line 207: Line 209:


{{Mapping|legend=1| 1 0 18 | 0 1 -10 }}
{{Mapping|legend=1| 1 0 18 | 0 1 -10 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 681.801{{c}}
* [[WE]]: ~2 = 1206.3211{{c}}, ~3/2 = 686.6308{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 682.662{{c}}
: [[error map]]: {{val| +6.321 -9.003 -2.053 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 682.6617{{c}}
: error map: {{val| 0.000 -19.293 -12.931 }}


{{Optimal ET sequence|legend=1| 7, 30b, 37b, 44b, 51b, 58bc, 65bbc }}
{{Optimal ET sequence|legend=1| 7, 30b, 37b, 44b, 51b, 58bc, 65bbc }}


[[Badness]] (Smith): 0.666
[[Badness]] (Sintel): 15.6


== Nethertone ==
== Nethertone ==
Line 224: Line 227:


{{Mapping|legend=1| 1 1 -1 | 0 2 15 }}
{{Mapping|legend=1| 1 1 -1 | 0 2 15 }}
: mapping generators: ~2, ~2560/2187
: mapping generators: ~2, ~2560/2187


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~2560/2187 = 345.946{{c}}
* [[WE]]: ~2 = 1203.1196{{c}}, ~2560/2187 = 346.2857{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~2560/2187 = 345.599{{c}}
: [[error map]]: {{val| +3.120 -6.264 +1.733 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2560/2187 = 345.5992{{c}}
: error map: {{val| 0.000 -10.757 -2.326 }}


{{Optimal ET sequence|legend=1| 7, 38c, 45c, 52, 59b, 66b }}
{{Optimal ET sequence|legend=1| 7, 38c, 45c, 52, 59b, 66b }}


[[Badness]] (Smith): 0.828
[[Badness]] (Sintel): 19.4


== Enipucrop ==
== Enipucrop ==
Line 243: Line 247:


{{Mapping|legend=1| 1 2 2 | 0 -3 2 }}
{{Mapping|legend=1| 1 2 2 | 0 -3 2 }}
: mapping generators: ~2, ~16/15
: mapping generators: ~2, ~16/15


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~16/15 = 170.671{{c}}
* [[WE]]: ~2 = 1210.1294{{c}}, ~16/15 = 174.5613{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~16/15 = 173.101{{c}}
: [[error map]]: {{val| +10.129 -5.380 -16.932 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 172.4620{{c}}
: error map: {{val| 0.000 -19.341 -41.390 }}


{{Optimal ET sequence|legend=1| 6b, 7 }}
{{Optimal ET sequence|legend=1| 6b, 7 }}


[[Badness]] (Smith): 0.1439
[[Badness]] (Sintel): 3.38


== Nadir ==
== Nadir ==
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{{Mapping|legend=1| 1 2 5 | 0 -3 -19 }}
{{Mapping|legend=1| 1 2 5 | 0 -3 -19 }}
: mapping generators: ~2, ~729/640
: mapping generators: ~2, ~729/640


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~729/640 = 168.983{{c}}
* [[WE]]: ~2 = 1202.743{{c}}, ~729/640 = 169.7633{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~729/640 = 169.223{{c}}
: [[error map]]: {{val| +2.743 -5.758 +1.900 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~729/640 = 169.2234{{c}}
: error map: {{val| 0.000 -9.625 -1.559 }}


{{Optimal ET sequence|legend=1| 7, 57c, 64, 71b, 78b, 85b }}
{{Optimal ET sequence|legend=1| 7, 57c, 64, 71b, 78b, 85b }}


[[Badness]] (Smith): 1.47
[[Badness]] (Sintel): 34.6


== Sixix (5-limit) ==
== Sixix (5-limit) ==
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{{Mapping|legend=1| 1 3 4 | 0 -5 -6 }}
{{Mapping|legend=1| 1 3 4 | 0 -5 -6 }}
: mapping generators: ~2, ~6/5
: mapping generators: ~2, ~6/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.005{{c}}
* [[WE]]: ~2 = 1200.9934{{c}}, ~6/5 = 338.6456{{c}}
: [[error map]]: {{val| 0.000 +8.020 -14.344 }}
: [[error map]]: {{val| +0.993 +7.797 -14.214 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.365{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 338.1959{{c}}
: error map: {{val| 0.000 +6.217 -16.507 }}
: error map: {{val| 0.000 +7.065 -15.489 }}


{{Optimal ET sequence|legend=1| 7, 25, 32, 39c }}
{{Optimal ET sequence|legend=1| 7, 25, 32, 39c }}


[[Badness]] (Smith): 0.153088
[[Badness]] (Sintel): 3.59


== Absurdity (5-limit) ==
== Absurdity (5-limit) ==
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{{Mapping|legend=1| 7 0 -17 | 0 1 3 }}
{{Mapping|legend=1| 7 0 -17 | 0 1 3 }}
: mapping generators: ~800/729, ~3
: mapping generators: ~800/729, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~800/729 = 171.429{{c}}, ~3/2 = 700.538{{c}} (~81/80 = 14.824{{c}})
* [[WE]]: ~800/729 = 171.4824{{c}}, ~3/2 = 700.4067{{c}} (~81/80 = 14.4772{{c}})
* [[POTE]]: ~800/729 = 171.429{{c}}, ~3/2 = 700.187{{c}} (~81/80 = 14.473{{c}})
: [[error map]]: {{val| +0.376 -1.172 +0.836 }}
* [[CWE]]: ~800/729 = 171.4286{{c}}, ~3/2 = 700.3453{{c}} (~81/80 = 14.6310{{c}})
: error map: {{val| 0.000 +7.065 -15.489 }}


{{Optimal ET sequence|legend=1| 7, …, 70, 77, 84, 329, 413b, 497b }}
{{Optimal ET sequence|legend=1| 7, …, 70, 77, 84, 329, 413b, 497b }}


[[Badness]] (Smith): 0.341202
[[Badness]] (Sintel): 8.00


== Sevond (5-limit) ==
== Sevond (5-limit) ==
Line 323: Line 329:


{{Mapping|legend=1| 7 0 -6 | 0 1 2 }}
{{Mapping|legend=1| 7 0 -6 | 0 1 2 }}
: mapping generators: ~10/9, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~10/9 = 171.429{{c}}, ~3/2 = 705.526{{c}} (~250/243 = 19.812{{c}})
* [[WE]]: ~10/9 = 171.3446{{c}}, ~3/2 = 705.9421{{c}} (~250/243 = 20.5637{{c}})
* [[POTE]]: ~10/9 = 171.429{{c}}, ~3/2 = 706.288{{c}} (~250/243 = 20.574{{c}})
: [[error map]]: {{val| -0.588 +3.399 -3.673 }}
* [[CWE]]: ~10/9 = 171.4286{{c}}, ~3/2 = 705.9119{{c}} (~250/243 = 20.1977{{c}})
: error map: {{val| 0.000 +3.957 -3.061 }}


{{Optimal ET sequence|legend=1| 7, 42, 49, 56, 119 }}
{{Optimal ET sequence|legend=1| 7, …, 42, 49, 56, 119 }}


[[Badness]] (Smith): 0.339335
[[Badness]] (Sintel): 7.96


== Seville ==
== Seville ==
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{{Mapping|legend=1| 7 0 5 | 0 1 1 }}
{{Mapping|legend=1| 7 0 5 | 0 1 1 }}
: mapping generators: ~125/108, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~125/108 = 171.429{{c}}, ~3/2 = 710.606{{c}} (~25/24 = 24.891{{c}})
* [[WE]]: ~125/108 = 171.7481{{c}}, ~3/2 = 707.7272{{c}} (~25/24 = 20.7346{{c}})
* [[POTE]]: ~125/108 = 171.429{{c}}, ~3/2 = 706.410{{c}} (~25/24 = 20.696{{c}})
: [[error map]]: {{val| +2.237 +8.009 -17.609 }}
* [[CWE]]: ~125/108 = 171.4286{{c}}, ~3/2 = 708.3739{{c}} (~25/24 = 22.6596{{c}})
: error map: {{val| 0.000 +6.419 -20.797 }}


{{Optimal ET sequence|legend=1| 7, 35b, 42c }}
{{Optimal ET sequence|legend=1| 7, 35b, 42c }}


[[Badness]] (Smith): 0.4377
[[Badness]] (Sintel): 10.3


== Artoneutral (5-limit) ==
== Artoneutral (5-limit) ==
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[[Comma list]]: {{monzo| 14 -22 9 }}
[[Comma list]]: {{monzo| 14 -22 9 }}


{{Mapping|legend=1| 1 8 18 | 0 -9 -22 }}
{{Mapping|legend=1| 1 -1 -4 | 0 9 22 }}
: mapping generators: ~2, ~243/200
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7434{{c}}, ~243/200 = 344.7454{{c}}
: [[error map]]: {{val| -0.257 +1.010 -0.889 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/200 = 344.8041{{c}}
: error map: {{val| 0.000 +1.282 -0.624 }}


: mapping generators: ~2, ~400/243
{{Optimal ET sequence|legend=1| 7, … 73, 80, 87 }}
 
[[Badness]] (Sintel): 8.17
 
== Geb (5-limit) ==
: ''For extensions, see [[Metric microtemperaments #Geb]] and [[Breedsmic temperaments #Osiris]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -31 43 -16 }}
 
{{Mapping|legend=1| 1 -3 -10 | 0 16 43 }}
: mapping generators: ~2, ~8000/6561


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~400/243 = 855.213{{c}}
* [[WE]]: ~2 = 1200.0135{{c}}, ~8000/6561 = 343.8718{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~400/243 = 855.196{{c}}
: [[error map]]: {{val| +0.014 -0.047 +0.038 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8000/6561 = 343.8683{{c}}
: error map: {{val| 0.000 -0.062 +0.024 }}


{{Optimal ET sequence|legend=1| 7, … 73, 80, 87 }}
{{Optimal ET sequence|legend=1| 7, …, 157, 164, 171, 506, 677, 848 }}
 
[[Badness]] (Sintel): 2.78
 
== Undetrita (5-limit) ==
: ''For extensions, see [[Hemimean clan #Undetrita]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -22 30 -11 }}
 
{{Mapping|legend=1| 1 0 -2 | 0 11 30 }}
: mapping generators: ~2, ~177147/160000
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0596{{c}}, ~177147/160000 = 172.8864{{c}}
: [[error map]]: {{val| +0.060 -0.204 +0.161 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~177147/160000 = 172.8804{{c}}
: error map: {{val| 0.000 -0.271 +0.098 }}
 
{{Optimal ET sequence|legend=1| 7, …, 111, 118, 1541, 1659, 1777, 1895b, …, 2839b, 2957b }}


[[Badness]] (Smith): 0.348
[[Badness]] (Sintel): 4.09


[[Category:7edo]]
[[Category:7edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
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