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11th-octave temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings
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{{Mapping|legend=1| 11 0 43 | 0 1 -1 }}
{{Mapping|legend=1| 11 0 43 | 0 1 -1 }}
: Mapping generators: ~16/15, ~3
: Mapping generators: ~16/15, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~16/15 = 109.091¢, ~3/2 = 702.794¢
* [[WE]]: ~16/15 = 109.0904{{c}}, ~3/2 = 702.6861{{c}}
* [[CWE]]: ~16/15 = 109.091¢, ~3/2 = 702.908¢
: [[error map]]: {{val| -0.445 +0.286 +0.613 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 702.9082{{c}}
: error map: {{val| 0.000 +0.953 +1.687 }}


{{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}
{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 869bcc, 968bcc, 1067bccc, 1166bccc }}


[[Badness]] (Sintel): 10.652
[[Badness]] (Sintel): 10.7


=== Undekapent ===
=== Undekapent ===
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{{Mapping|legend=1| 11 0 8 | 0 1 1 }}
{{Mapping|legend=1| 11 0 8 | 0 1 1 }}
: Mapping generators: ~648/625, ~3
: Mapping generators: ~648/625, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~648/625 = 109.091¢, ~3/2 = 705.652¢
* [[WE]]: ~648/625 = 109.1809{{c}}, ~3/2 = 704.4091{{c}}
* [[CWE]]: ~648/625 = 109.091¢, ~3/2 = 704.698¢
: [[error map]]: {{val| +0.990 +3.444 -7.468 }}
* [[CWE]]: ~648/625 = 109.0909{{c}}, ~3/2 = 704.6983{{c}}
: error map: {{val| 0.000 +2.743 -8.888 }}


{{Optimal ET sequence|legend=1| 22, 77c, 99c, 121c }}
{{Optimal ET sequence|legend=1| 22, 77c, 99c }}


[[Badness]] (Sintel): 17.067
[[Badness]] (Sintel): 17.1


=== Elven ===
=== Elven ===
{{see also| Sensibeta temperaments #Elven }}
{{See also| Sensibeta temperaments #Elven }}
 
This temperament has a period of 1/11 octave, which represents [[3125/2916]] in the 5-limit.
This temperament has a period of 1/11 octave, which represents [[3125/2916]] in the 5-limit.


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


[[Comma list]]: {{monzo|-23 -66 55}}
[[Comma list]]: {{monzo| -23 -66 55 }}


{{Mapping|legend=1| 11 2 7 | 0 5 6 }}
{{Mapping|legend=1| 11 2 7 | 0 5 6 }}
 
: mapping generators: ~3125/2916, ~{{monzo| 10 29 -24 }}
: Mapping generators: ~3125/2916, ~{{monzo| 10 29 -24 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~3125/2916 = 109.091¢, ~5/3 = 882.353¢
* [[WE]]: ~3125/2916 = 109.0998{{c}}, ~{{monzo| 10 29 -24 }} = 336.8921{{c}}
* [[CWE]]: ~3125/2916 = 109.091¢, ~5/3 = 882.336¢
: [[error map]]: {{val| +0.097 +0.705 -1.263 }}
* [[CWE]]: ~3125/2916 = 109.0909{{c}}, ~{{monzo| 10 29 -24 }} = 336.8814{{c}}
: error map: {{val| 0.000 +0.634 -1.389 }}


{{Optimal ET sequence|legend=1| 121, 253, 374 }}
{{Optimal ET sequence|legend=1| 121, 253, 374, 627c }}


[[Badness]] (Sintel): 429.4
[[Badness]] (Sintel): 429


== Hendeca ==
== Hendeca ==
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{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


: Mapping generators: ~16/15, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
[[Optimal tuning]] ([[POTE]]): ~16/15 = 109.091, ~3/2 = 703.313
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}


{{Optimal ET sequence|legend=1| 11, 22, 55d, 77d, 99dd }}
{{Optimal ET sequence|legend=1| 11, 22, 55d, 77d, 99dd }}


[[Badness]] (Sintel): 4.371
[[Badness]] (Sintel): 4.37


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}


Optimal tuning (POTE): ~16/15 = 109.091, ~3/2 = 703.090
Optimal tunings:
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


{{Optimal ET sequence|legend=0| 11, 22, 55d, 77d, 99dd }}
{{Optimal ET sequence|legend=0| 11, 22, 55d }}


Badness (Sintel): 2.464
Badness (Sintel): 2.46


== Undeka ==
== Undeka ==
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{{Mapping|legend=1| 11 0 8 31 | 0 1 1 0 }}
{{Mapping|legend=1| 11 0 8 31 | 0 1 1 0 }}
: mapping generators: ~21/20, ~3


: Mapping generators: ~21/20, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~21/20 = 108.9318{{c}}, ~3/2 = 707.7579{{c}}
[[Optimal tuning]] ([[POTE]]): ~21/20 = 109.091, ~3/2 = 708.792
: [[error map]]: {{val| -1.750 +4.053 -8.852 +8.059 }}
* [[CWE]]: ~21/20 = 109.0909{{c}}, ~3/2 = 707.7526{{c}}
: error map: {{val| 0.000 +5.798 -5.834 +12.992 }}


{{Optimal ET sequence|legend=1| 11c, 22 }}
{{Optimal ET sequence|legend=1| 11b, 22 }}


[[Badness]] (Sintel): 3.588
[[Badness]] (Sintel): 3.59


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 11 0 8 31 38 | 0 1 1 0 0 }}
Mapping: {{mapping| 11 0 8 31 38 | 0 1 1 0 0 }}


Optimal tuning (POTE): ~21/20 = 109.091, ~3/2 = 706.768
Optimal tunings:
* WE: ~21/20 = 109.0321{{c}}, ~3/2 = 706.3870{{c}}
* CWE: ~21/20 = 109.0909{{c}}, ~3/2 = 706.4785{{c}}


{{Optimal ET sequence|legend=0| 11c, 22 }}
{{Optimal ET sequence|legend=0| 11c, 22 }}


Badness (Sintel): 2.270
Badness (Sintel): 2.27


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 11 0 8 31 38 23 | 0 1 1 0 0 1 }}
Mapping: {{mapping| 11 0 8 31 38 23 | 0 1 1 0 0 1 }}


Optimal tuning (POTE): ~13/12 = 109.091, ~3/2 = 707.764
Optimal tunings:
* WE: ~13/12 = 109.1184{{c}}, ~3/2 = 707.9422{{c}}
* CWE: ~13/12 = 109.0909{{c}}, ~3/2 = 707.9065{{c}}


{{Optimal ET sequence|legend=0| 11cf, 22 }}
{{Optimal ET sequence|legend=0| 11cf, 22 }}


Badness (Sintel): 2.336
Badness (Sintel): 2.34


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}
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