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11th-octave temperaments: Difference between revisions

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{{Technical data page}}
{{Infobox fractional-octave|11}}
{{Infobox fractional-octave|11}}
Although [[11edo]] itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 11 are.
An 11th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 11. Although [[11edo]] itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 11 are.


== 5-limit temperaments ==
== 5-limit temperaments ==
=== Hendecapent ===
=== Hendecapent ===
This temperament has a period of 1/11 octave, which represents [[16/15]] in the 5-limit. There are some 7-limit extensions include [[#Hendecatonic|hendecatonic]] (22 & 77) and [[#Hendeca|hendeca]] (22 & 33).
This temperament has a period of 1/11 octave, which represents [[16/15]] in the 5-limit. There are some 7-limit extensions include [[Porwell temperaments #Hendecatonic|hendecatonic]] ({{nowrap| 22 & 77 }}) and [[#Hendeca|hendeca]] ({{nowrap| 22 & 33 }}).


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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: Mapping generators: ~16/15, ~3
: Mapping generators: ~16/15, ~3


[[Optimal tuning]] ([[POTE]]): ~16/15 = 1＼11, ~3/2 = 702.947
[[Optimal tuning]]s:
* [[CTE]]: ~16/15 = 109.091¢, ~3/2 = 702.794¢
* [[CWE]]: ~16/15 = 109.091¢, ~3/2 = 702.908¢


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


[[Badness]]: 0.454083
[[Badness]] (Sintel): 10.652


=== Undekapent ===
=== Undekapent ===
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: Mapping generators: ~648/625, ~3
: Mapping generators: ~648/625, ~3


[[Optimal tuning]] ([[POTE]]): ~648/625 = 1＼11, ~3/2 = 703.829
[[Optimal tuning]]s:
* [[CTE]]: ~648/625 = 109.091¢, ~3/2 = 705.652¢
* [[CWE]]: ~648/625 = 109.091¢, ~3/2 = 704.698¢


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


[[Badness]]: 0.727539
[[Badness]] (Sintel): 17.067


== Hendecatonic ==
=== Elven ===
: <Small>''For 11-limit extensions, see [[Porwell temperaments #Hendecatonic]].''</small>
{{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, four of them represent [[9/7]] in the 7-limit.
[[Subgroup]]: 2.3.5


[[Subgroup]]: 2.3.5.7
[[Comma list]]: {{monzo|-23 -66 55}}


[[Comma list]]: 6144/6125, 10976/10935
{{Mapping|legend=1| 11 2 7 | 0 5 6 }}


{{Mapping|legend=1| 11 0 43 -4 | 0 1 -1 2 }}
: Mapping generators: ~3125/2916, ~{{monzo| 10 29 -24 }}


: Mapping generators: ~16/15, ~3
[[Optimal tuning]]s:  
* [[CTE]]: ~3125/2916 = 109.091¢, ~5/3 = 882.353¢
* [[CWE]]: ~3125/2916 = 109.091¢, ~5/3 = 882.336¢


{{Multival|legend=1| 11 -11 22 -43 4 82 }}
{{Optimal ET sequence|legend=1| 121, 253, 374 }}


[[Optimal tuning]] ([[POTE]]): ~16/15 = 1＼11, ~3/2 = 703.054
[[Badness]] (Sintel): 429.4


{{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}
== Hendeca ==
{{Distinguish|Hendec}}


[[Badness]]: 0.041081
== Hendeca ==
{{distinguish|Hendec}}
This temperament has a period of 1/11 octave, nine of them represent [[7/4]] in the 7-limit, as well as [[#undeka|undeka]].
This temperament has a period of 1/11 octave, nine of them represent [[7/4]] in the 7-limit, as well as [[#undeka|undeka]].


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: Mapping generators: ~16/15, ~3
: Mapping generators: ~16/15, ~3


{{Multival|legend=1| 11 -11 0 -43 -31 31 }}
[[Optimal tuning]] ([[POTE]]): ~16/15 = 109.091, ~3/2 = 703.313
 
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1＼11, ~3/2 = 703.313


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


[[Badness]]: 0.172714
[[Badness]] (Sintel): 4.371


=== 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 = 1＼11, ~3/2 = 703.090
Optimal tuning (POTE): ~16/15 = 109.091, ~3/2 = 703.090


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


Badness: 0.074543
Badness (Sintel): 2.464


== Undeka ==
== Undeka ==
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: Mapping generators: ~21/20, ~3
: Mapping generators: ~21/20, ~3


{{Multival|legend=1| 11 11 0 -8 -31 -31 }}
[[Optimal tuning]] ([[POTE]]): ~21/20 = 109.091, ~3/2 = 708.792
 
[[Optimal tuning]] ([[POTE]]): ~21/20 = 1＼11, ~3/2 = 708.792


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


[[Badness]]: 0.141782
[[Badness]] (Sintel): 3.588


=== 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 = 1＼11, ~3/2 = 706.768
Optimal tuning (POTE): ~21/20 = 109.091, ~3/2 = 706.768


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


Badness: 0.068672
Badness (Sintel): 2.270


=== 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 = 1＼11, ~3/2 = 707.764
Optimal tuning (POTE): ~13/12 = 109.091, ~3/2 = 707.764


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


Badness: 0.056528
Badness (Sintel): 2.336


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}
Retrieved from "https://en.xen.wiki/w/11th-octave_temperaments"