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Greenwoodmic temperaments: Difference between revisions

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


=Greenwood=
Temperaments discussed elsewhere are
Commas: 405/392, 1323/1280
* ''[[Schism]]'' (+64/63) → [[Archytas clan #Schism|Archytas clan]]
* ''[[Superpelog]]'' (+49/48) → [[Semaphoresmic clan #Superpelog|Semaphoresmic clan]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[August]]'' (+36/35) → [[Augmented family #August|Augmented family]]
* ''[[Sidi]]'' (+25/24) → [[Dicot family #Sidi|Dicot family]]
* ''[[Ripple]]'' (+126/125) → [[Ripple family #Septimal ripple|Ripple family]]
* ''[[Wesley]]'' (+875/864) → [[Wesley family #Septimal wesley|Wesley family]]
* ''[[Greenwood]]'' (+1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]


POTE generator: ~8/7 = 241.490
Considered below are secund and semishallowtone.  


Map: [<7 11 1 12|, <0 0 2 1|]
== 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]]. Related temperaments include [[progression]] and [[jerome]], which only differ by the mapping of 5.


Wedgie: <<0 14 7 22 11 -23||
The fourth generator step (representing [[45/32]]~[[49/36]]) comes close to [[11/8]]. This allows a simple [[11-limit]] extension in which the generator doubles as [[12/11]], tempering out [[45/44]] and [[99/98]] as is natural for greenwoodmic temperaments. From here, it only makes sense to let the generator take on the additional role of [[13/12]]~[[14/13]], resulting in a full [[13-limit]] temperament.


EDOs: 14c, 21, 35, 84bcd
Possible tunings include [[26edo]], [[35edo]] in the 35f val, or anything in between.


Badness: 0.1218
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 405/392, 525/512
Commas: 45/44, 99/98, 1323/1280


POTE generator: ~8/7 = 242.711
{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}
: mapping generators: ~2, ~16/15


Map: [<7 11 1 12 9|, <0 0 2 1 2|]
[[Optimal tuning]]s:  
* [[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 }}


EDOs: 14c, 21, 35, 49bcde, 84bcde
{{Optimal ET sequence|legend=1| 9, 17, 26, 61bc, 87bcc }}


Badness: 0.0575
[[Badness]] (Sintel): 2.27


==13-limit==
=== 11-limit ===
Commas: 27/26, 45/44, 99/98, 640/637
Subgroup: 2.3.5.7.11


POTE generaor: ~8/7 = 238.607
Comma list: 45/44, 99/98, 385/384


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


EDOs: 14c, 21, 35, 91bf
Optimal tunings:  
* WE: ~2 = 1202.9415{{c}}, ~12/11 = 138.2377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.9970{{c}}


Badness: 0.0540
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}


=Secund=
Badness (Sintel): 1.41
Commas: 405/392, 525/512


POTE generator: ~16/15 = 137.958
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [<1 1 3 2|, <0 5 -6 7|]
Comma list: 45/44, 65/64, 78/77, 99/98


Wedgie: <<5 -6 7 -21 -3 33||
Mapping: {{mapping| 1 1 3 2 3 3 | 0 5 -6 7 4 6 }}


EDOs: 9, 17, 26, 61bc, 87bc
Optimal tunings:  
* WE: ~2 = 1203.0195{{c}}, ~13/12 = 138.2642{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 138.0317{{c}}


Badness: 0.0898
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf }}


==11-limit==
Badness (Sintel): 1.08
Commas: 45/44, 99/98, 385/384


POTE generator: ~12/11 = 137.900
== Semishallowtone ==
{{See also| Shallowtone }}


Map: [<1 1 3 2 3|, <0 5 -6 7 4|]
[[Subgroup]]: 2.3.5.7


EDOs: 9, 17, 26, 61bc, 87bc
[[Comma list]]: 405/392, 35721/32768


Badness: 0.0426
{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}
: mapping generators: ~189/128, ~3


==13-limit==
[[Optimal tuning]]s:
Commas: 27/26, 45/44, 99/98, 385/384
* [[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 }}


POTE generator: ~12/11 = 136.937
{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}


Map: [<1 1 3 2 3 2|, <0 5 -6 7 4 15|]
[[Badness]] (Sintel): 8.42


EDOs: 9, 26f, 35
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0497
Comma list: 45/44, 99/98, 35721/32768
 
Mapping: {{mapping| 2 0 36 15 32 | 0 1 -10 -3 -8 }}
 
Optimal tunings:
* 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}})
 
{{Optimal ET sequence|legend=0| 14c, 44bd, 58bcde, 72bbccddee }}
 
Badness (Sintel): 4.15
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 45/44, 99/98, 35721/32768
 
Mapping: {{mapping| 2 0 36 15 32 -2 | 0 1 -10 -3 -8 3 }}
 
Optimal tunings:
* 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}})
 
{{Optimal ET sequence|legend=0| 14c, 30b }}
 
Badness (Sintel): 4.38
 
[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Catalogs of rank-2 temperaments]]
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