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Keemic temperaments: 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 (2/2)
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{{Mapping|legend=1| 1 -9 -6 -4 | 0 14 11 9 }}
{{Mapping|legend=1| 1 -9 -6 -4 | 0 14 11 9 }}
: mapping generators: ~2, ~42/25
: mapping generators: ~2, ~42/25


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{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }}
{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }}
: mapping generators: ~2, ~36/35


: Mapping generators: ~2, ~25/24
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1201.4060{{c}}, ~36/35 = 53.8791{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25/24 = 53.816{{c}}
: [[error map]]: {{val| +1.406 -0.121 -6.348 +3.810 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 53.9055{{c}}
: error map: {{val| 0.000 -1.183 -8.975 +1.474 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~36/35 = 53.816{{c}} -->


{{Optimal ET sequence|legend=1| 22, 45, 67c }}
{{Optimal ET sequence|legend=1| 22, 45, 67c }}


[[Badness]] (Smith): 0.090769
[[Badness]] (Sintel): 2.30


== Barbad ==
== Barbad ==
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[[Comma list]]: 875/864, 16875/16807
[[Comma list]]: 875/864, 16875/16807


{{Mapping|legend=1| 1 9 7 11 | 0 -19 -12 -21 }}
{{Mapping|legend=1| 1 -10 -5 -10 | 0 19 12 21 }}
: mapping generators: ~2, ~98/75


: Mapping generators: ~2, ~98/75
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1201.0462{{c}}, ~75/49 = 732.3071{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~98/75 = 468.331{{c}}
: [[error map]]: {{val| +1.046 +1.418 -3.859 -0.838 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/49 = 731.7183{{c}}
: error map: {{val| 0.000 +0.692 -5.694 -2.742 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~75/49 = 731.669{{c}} -->


{{Optimal ET sequence|legend=0| 18, 23d, 41 }}
{{Optimal ET sequence|legend=0| 18, 23d, 41 }}


[[Badness]] (Smith): 0.110448
[[Badness]] (Sintel): 2.80


=== 11-limit ===
=== 11-limit ===
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Comma list: 245/242, 540/539, 625/616
Comma list: 245/242, 540/539, 625/616


Mapping: {{mapping| 1 9 7 11 14 | 0 -19 -12 -21 -27 }}
Mapping: {{mapping| 1 -10 -5 -10 -13 | 0 19 12 21 27 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~98/75 = 468.367{{c}}
Optimal tunings:
* WE: ~2 = 1200.8513{{c}}, ~75/49 = 732.1519{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 731.6740{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~75/49 = 731.633{{c}} -->


{{Optimal ET sequence|legend=0| 18e, 23de, 41, 228ccdd }}
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}


Badness (Smith): 0.050105
Badness (Sintel): 1.66


=== 13-limit ===
=== 13-limit ===
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Comma list: 144/143, 196/195, 245/242, 275/273
Comma list: 144/143, 196/195, 245/242, 275/273


Mapping: {{mapping| 1 9 7 11 14 8 | 0 -19 -12 -21 -27 -11 }}
Mapping: {{mapping| 1 -10 -5 -10 -13 -3 | 0 19 12 21 27 11 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/10 = 468.270{{c}}
Optimal tunings:
* WE: ~2 = 1199.7960{{c}}, ~20/13 = 731.6053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 731.7208{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~20/13 = 731.730{{c}} -->


{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}


Badness (Smith): 0.039183
Badness (Sintel): 1.62


== Hyperkleismic ==
== Hyperkleismic ==
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{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}
{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}
: mapping generators: ~2, ~6/5


: Mapping generators: ~2, ~6/5
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.0290{{c}}, ~6/5 = 323.7882{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 323.780{{c}}
: [[error map]]: {{val| +0.029 +2.358 -5.759 +2.597 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 323.7816{{c}}
: error map: {{val| 0.000 +2.332 -5.808 +2.519 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 323.780{{c}} -->


{{Optimal ET sequence|legend=1| 26, 37, 63 }}
{{Optimal ET sequence|legend=1| 26, 37, 63 }}


[[Badness]] (Smith): 0.157830
[[Badness]] (Sintel): 3.99


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}
Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 323.796{{c}}
Optimal tunings:
* WE: ~2 = 1199.9010{{c}}, ~6/5 = 323.7691{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7931{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 323.796{{c}} -->


{{Optimal ET sequence|legend=0| 26, 37, 63 }}
{{Optimal ET sequence|legend=0| 26, 37, 63 }}


Badness (Smith): 0.065356
Badness (Sintel): 2.16


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}
Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 323.790{{c}}
Optimal tunings:
* WE: ~2 = 1200.0524{{c}}, ~6/5 = 323.8039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7912{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 323.790{{c}} -->


{{Optimal ET sequence|legend=0| 26, 37, 63 }}
{{Optimal ET sequence|legend=0| 26, 37, 63 }}


Badness (Smith): 0.035724
Badness (Sintel): 1.48


== Sevond ==
== Sevond ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sevond (5-limit)]].''  
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sevond (5-limit)]].''  


10/9 is tempered to be exactly 1\7 of an octave. Therefore 3/2 is 1 generator sharp of a 7edo step and 5/4 is 2 generators sharp.
10/9 is tempered to be exactly 1\7. Therefore 3/2 is 1 generator sharp of a 7edo step and 5/4 is 2 generators sharp.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }}
{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }}
: mapping generators: ~10/9, ~3


: Mapping generators: ~10/9, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~10/9 = 171.4007{{c}}, ~3/2 = 705.4982{{c}}
[[Optimal tuning]] ([[POTE]]): ~10/9 = 171.429{{c}}, ~3/2 = 705.613{{c}}
: [[error map]]: {{val| -0.195 +3.348 -4.112 -0.499 }}
* [[CWE]]: ~10/9 = 171.4286{{c}}, ~3/2 = 705.6057{{c}}
: error map: {{val| 0.000 +3.651 -3.674 +0.071 }}
<!-- * [[POTE]]: ~10/9 = 171.429{{c}}, ~3/2 = 705.613{{c}} -->


{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=1| 7, …, 56, 63, 119 }}


[[Badness]] (Smith): 0.206592
[[Badness]] (Sintel): 5.23


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}
Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}


Optimal tuning (POTE): ~10/9 = 171.429{{c}}, ~3/2 = 705.518{{c}}
Optimal tunings:
* WE: ~11/10 = 171.3859{{c}}, ~3/2 = 705.3421{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.4973{{c}}
<!-- * POTE: ~11/10 = 171.429{{c}}, ~3/2 = 705.518{{c}} -->


{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}


Badness (Smith): 0.070437
Badness (Sintel): 2.33


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}
Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}


Optimal tuning (POTE): ~10/9 = 171.429{{c}}, ~3/2 = 705.344{{c}}
Optimal tunings:
* WE: ~11/10 = 171.4163{{c}}, ~3/2 = 705.2930{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.3402{{c}}
<!-- * POTE: ~10/9 = 171.429{{c}}, ~3/2 = 705.344{{c}} -->


{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}


Badness (Smith): 0.041238
Badness (Sintel): 1.70


[[Category:Temperament collections]]
[[Category:Temperament collections]]
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