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Greenwoodmic temperaments: Difference between revisions

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


Temperaments discussed elsewhere include [[Augmented family #August|august]], [[Meantone family #Injera|injera]], [[Archytas clan #Schism|schism]], [[Dicot family #Sidi|sidi]], and [[Pelogic family #Superpelog|superpelog]].
Temperaments discussed elsewhere are
* ''[[Sidi]]'' (+25/24) → [[Dicot family #Sidi|Dicot family]]
* ''[[August]]'' (+36/35) → [[Augmented family #August|Augmented family]]
* ''[[Superpelog]]'' (+49/48) → [[Semaphoresmic clan #Superpelog|Semaphoresmic clan]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[Schism]]'' (+64/63) → [[Archytas clan #Schism|Archytas clan]]
* ''[[Greenwood]]'' (+1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]


== Greenwood ==
== Secund ==
{{see also|Apotome family #Greenwood}}
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7
[[Comma list]]: 405/392, 525/512


[[Comma list]]: 405/392, 1323/1280
{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}


[[Mapping]]: [{{val|7 11 1 12}}, {{val|0 0 2 1}}]
: mapping generaators: ~2, ~16/15


{{Multival|legend=1|0 14 7 22 11 -23}}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~16/15 = 138.454
: [[error map]]: {{val| 0.000 -9.684 -17.038 +0.353 }}
* [[POTE]]: ~2 = 1200.000, ~16/15 = 137.958
: error map: {{val| 0.000 -12.167 -14.059 -3.123 }}


[[POTE generator]]: ~8/7 = 241.490
{{Optimal ET sequence|legend=1| 9, 17, 26, 61bc, 87bcc }}


{{Val list|legend=1| 14c, 21, 35 }}
[[Badness]] (Smith): 0.089840
 
[[Badness]]: 0.121752


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


Comma list: 45/44, 99/98, 1323/1280
Comma list: 45/44, 99/98, 385/384


Mapping: [{{val|7 11 1 12 9}}, {{val|0 0 2 1 2}}]
Mapping: {{mapping| 1 1 3 2 3 | 0 5 -6 7 4 }}


POTE generator: ~8/7 = 242.711
Optimal tunings:
* CTE: ~2 = 1200.000, ~12/11 = 138.420
* POTE: ~2 = 1200.000, ~12/11 = 137.900


Vals: {{Val list| 14c, 21, 35, 49bcde, 84bbccde }}
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}


Badness: 0.057471
Badness (Smith): 0.042562


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


Comma list: 27/26, 45/44, 99/98, 640/637
Comma list: 27/26, 45/44, 99/98, 385/384


Mapping: [{{val|7 11 1 12 9 26}}, {{val|0 0 2 1 2 0}}]
Mapping: {{mapping| 1 1 3 2 3 2 | 0 5 -6 7 4 15 }}


POTE generator: ~8/7 = 238.607
Optimal tunings:
* CTE: ~2 = 1200.000, ~12/11 = 137.455
* POTE: ~2 = 1200.000, ~12/11 = 136.937


Vals: {{Val list| 14c, 21, 35 }}
{{Optimal ET sequence|legend=0| 9, 35 }}


Badness: 0.054009
Badness (Smith): 0.049684


== Secund ==
==== Secundly ====
{{see also|Avicennmic temperaments #Secund}}
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7
Comma list: 45/44, 65/64, 78/77, 99/98


[[Comma list]]: 405/392, 525/512
Mapping: {{mapping| 1 1 3 2 3 3 | 0 5 -6 7 4 6 }}


[[Mapping]]: [{{val|1 1 3 2}}, {{val|0 5 -6 7}}]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/12 = 138.583
* POTE: ~2 = 1200.000, ~13/12 = 137.917


{{Multival|legend=1|5 -6 7 -21 -3 33}}
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf }}


[[POTE generator]]: ~16/15 = 137.958
Badness (Smith): 0.026212


{{Val list|legend=1| 9, 17, 26, 61bc, 87bcc }}
== Semishallowtone ==
{{See also| Shallowtone }}


[[Badness]]: 0.089840
[[Subgroup]]: 2.3.5.7


=== 11-limit ===
[[Comma list]]: 405/392, 35721/32768
Subgroup: 2.3.5.7.11


Comma list: 45/44, 99/98, 385/384
{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}


Mapping: [{{val|1 1 3 2 3}}, {{val|0 5 -6 7 4}}]
: mapping generators: ~189/128, ~3


POTE generator: ~12/11 = 137.900
[[Optimal tuning]]s:  
* [[CTE]]: ~189/128 = 600.000, ~3/2 = 681.532 (~64/63 = 81.532)
: [[error map]]: {{val| 0.000 -20.423 -1.629 -13.421 }}
* [[CWE]]: ~189/128 = 600.000, ~3/2 = 682.650 (~64/63 = 82.650)
: error map: {{val| 0.000 -19.305 -12.811 -16.775 }}


Vals: {{Val list| 9, 17, 26, 61bc, 87bcc }}
{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}


Badness: 0.042562
[[Badness]]:
* Smith: 0.333
* Dirichlet: 8.420


=== 13-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 27/26, 45/44, 99/98, 385/384
Comma list: 45/44, 99/98, 35721/32768


Mapping: [{{val|1 1 3 2 3 2}}, {{val|0 5 -6 7 4 15}}]
Mapping: {{mapping| 2 0 36 15 32 | 0 1 -10 -3 -8 }}


POTE generator: ~12/11 = 136.937
Optimal tunings:  
* CTE: ~189/128 = 600.000, ~3/2 = 681.438 (~64/63 = 81.438)
* CWE: ~189/128 = 600.000, ~3/2 = 682.724 (~64/63 = 82.724)


Vals: {{Val list| 9, 26f, 35 }}
{{Optimal ET sequence|legend=0| 14c, 44bd, 58bcde, 72bbccddee }}


Badness: 0.049684
Badness:
* Smith: 0.125
* Dirichlet: 4.148


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


Comma list: 45/44, 65/64, 78/77, 99/98
Comma list: 27/26, 45/44, 99/98, 35721/32768


Mapping: [{{val|1 1 3 2 3 3}}, {{val|0 5 -6 7 4 6}}]
Mapping: {{mapping| 2 0 36 15 32 -2 | 0 1 -10 -3 -8 3 }}


POTE generator: ~12/11 = 137.917
Optimal tunings:  
* CTE: ~91/64 = 600.000, ~3/2 = 681.406 (~64/63 = 81.406)
* CWE: ~91/64 = 600.000, ~3/2 = 682.168 (~64/63 = 82.168)


Vals: {{Val list| 9, 17, 26, 61bcf, 87bccf }}
{{Optimal ET sequence|legend=0| 14c, 30b }}


Badness: 0.026212
Badness:
* Smith: 0.106
* Dirichlet: 4.377


[[Category:Theory]]
[[Category:Temperament]]
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Greenwoodmic]]
[[Category:Pages with mostly numerical content]]
[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
[[Category:Greenwoodmic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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