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Greenwoodmic temperaments: Difference between revisions

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This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[greenwoodma]], 405/392 ({{monzo| -3 4 1 -2 }}).
{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] '''greenwoodmic temperaments''', which [[tempering out|temper out]] [[405/392]], the [[greenwoodma]].  


Temperaments discussed elsewhere are
Temperaments discussed elsewhere are
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* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[Schism]]'' (+64/63) → [[Archytas clan #Schism|Archytas clan]]
* ''[[Schism]]'' (+64/63) → [[Archytas clan #Schism|Archytas clan]]
* ''[[Greenwood]]'' (+1323/1280) → [[Whitewood family #Greenwood|Apotome family]]
* ''[[Greenwood]]'' (+1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]
 
Considered below are secund and semishallowtone.


== Secund ==
== Secund ==
Secund tempers out the greendwoodma, the [[avicennma]], and the [[orwellisma]]. It may be described as the {{nowrap| 9 & 26 }} temperament, with a [[ploidacot]] signature of pentacot. It divides the [[3/2|perfect fifth]] into five [[~]][[16/15]] generators, two for [[7/6]] and three for [[9/7]].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}
{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}
 
: mapping generators: ~2, ~16/15
{{Multival|legend=1| 5 -6 7 -21 -3 33 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000, ~16/15 = 137.958
* [[WE]]: ~2 = 1203.6786{{c}}, ~16/15 = 138.3805{{c}}
: [[error map]]: {{val| +3.679 -6.374 -5.561 +7.195 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 138.0648{{c}}
: error map: {{val| 0.000 -11.631 -14.702 -2.372 }}


{{Optimal ET sequence|legend=1| 9, 17, 26, 61bc, 87bcc }}
{{Optimal ET sequence|legend=1| 9, 17, 26, 61bc, 87bcc }}


[[Badness]] (Smith): 0.089840
[[Badness]] (Sintel): 2.27


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000, ~12/11 = 137.900
* WE: ~2 = 1202.9415{{c}}, ~12/11 = 138.2377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.9970{{c}}


{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc, 87bcc }}
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}


Badness (Smith): 0.042562
Badness (Sintel): 1.41


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000, ~12/11 = 136.937
* WE: ~2 = 1202.4465{{c}}, ~12/11 = 137.2165{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.0309{{c}}


{{Optimal ET sequence|legend=0| 9, 26f, 35 }}
{{Optimal ET sequence|legend=0| 9, 35 }}


Badness (Smith): 0.049684
Badness (Sintel): 2.05


==== Secundly ====
==== Secundly ====
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Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000, ~12/11 = 137.917
* WE: ~2 = 1203.0195{{c}}, ~13/12 = 138.2642{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 138.0317{{c}}


{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf, 87bccf }}
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf }}


Badness (Smith): 0.026212
Badness (Sintel): 1.08


== Semishallowtone ==
== Semishallowtone ==
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[[Comma list]]: 405/392, 35721/32768
[[Comma list]]: 405/392, 35721/32768


{{Mapping|legend=1| 2 2 16 9 | 0 1 -10 -3 }}
{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}
: mapping generators: ~189/128, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~189/128 = 600.000, ~3/2 = 681.532
* [[WE]]: ~189/128 = 603.1653{{c}}, ~3/2 = 686.6365{{c}} (~64/63 = 83.4712{{c}})
: [[error map]]: {{val| +6.331 -8.988 -2.034 -0.248 }}
* [[CWE]]: ~189/128 = 600.0000{{c}}, ~3/2 = 682.6498{{c}} (~64/63 = 82.6498{{c}})
: error map: {{val| 0.000 -19.305 -12.811 -16.775 }}


[[Support]]ing [[ET]]s: {{EDOs| 14c, 30b, 44bd, 58bcd, 72bbccdd }}
{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}


[[Badness]] (Dirichlet): 8.420
[[Badness]] (Sintel): 8.42


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 99/98, 43659/40960
Comma list: 45/44, 99/98, 35721/32768


Mapping: {{mapping| 2 2 16 9 16 | 0 1 -10 -3 -8}}
Mapping: {{mapping| 2 0 36 15 32 | 0 1 -10 -3 -8 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~189/128 = 600.000, ~3/2 = 681.438
* WE: ~189/128 = 603.0536{{c}}, ~3/2 = 686.6175{{c}} (~64/63 = 83.5640{{c}})
* CWE: ~189/128 = 600.0000{{c}}, ~3/2 = 682.7238{{c}} (~64/63 = 82.7238{{c}})


Supporting ETs: {{EDOs| 14c, 30b, 44bd, 58bcde, 72bbccddee }}
{{Optimal ET sequence|legend=0| 14c, 44bd, 58bcde, 72bbccddee }}


Badness (Dirichlet): 4.148
Badness (Sintel): 4.15


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 27/26, 99/98, 35721/32768
Comma list: 27/26, 45/44, 99/98, 35721/32768


Mapping: {{mapping| 2 2 16 9 16 4 | 0 1 -10 -3 -8 3 }}
Mapping: {{mapping| 2 0 36 15 32 -2 | 0 1 -10 -3 -8 3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~189/128 = 600.000, ~3/2 = 681.406
* WE: ~91/64 = 601.8872{{c}}, ~3/2 = 684.4774{{c}} (~64/63 = 82.5902{{c}})
* CWE: ~91/64 = 600.0000{{c}}, ~3/2 = 682.1676{{c}} (~64/63 = 82.1676{{c}})


Supporting ETs: {{EDOs| 14c, 30b }}
{{Optimal ET sequence|legend=0| 14c, 30b }}


Badness (Dirichlet): 4.377
Badness (Sintel): 4.38


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
[[Category:Greenwoodmic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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