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]] '''greenwoodmic temperaments''', which [[tempering out|temper out]] [[405/392]], the [[greenwoodma]].
-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 ==
+Considered below are secund and semishallowtone.
-{{see also|Apotome family #Greenwood}}
-Subgroup: 2.3.5.7
+== 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]].
-[[Comma list]]: 405/392, 1323/1280
+[[Subgroup]]: 2.3.5.7
-[[Mapping]]: [{{val|7 11 1 12}}, {{val|0 0 2 1}}]
+[[Comma list]]: 405/392, 525/512
-{{Multival|legend=1|0 14 7 22 11 -23}}
+{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}
+: mapping generators: ~2, ~16/15
-[[POTE generator]]: ~8/7 = 241.490
+[[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 }}
-{{Val list|legend=1| 14c, 21, 35 }}
+{{Optimal ET sequence|legend=1| 9, 17, 26, 61bc, 87bcc }}
-[[Badness]]: 0.121752
+[[Badness]] (Sintel): 2.27
 === 11-limit ===
 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:
+* WE: ~2 = 1202.9415{{c}}, ~12/11 = 138.2377{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.9970{{c}}
-Vals: {{Val list| 14c, 21, 35, 49bcde, 84bbccde }}
+{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}
-Badness: 0.057471
+Badness (Sintel): 1.41
-=== 13-limit ===
+==== 13-limit ====
 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:
+* WE: ~2 = 1202.4465{{c}}, ~12/11 = 137.2165{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.0309{{c}}
-Vals: {{Val list| 14c, 21, 35 }}
+{{Optimal ET sequence|legend=0| 9, 35 }}
-Badness: 0.054009
+Badness (Sintel): 2.05
-== 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]]: [{{val|1 1 3 2}}, {{val|0 5 -6 7}}]
+Mapping: {{mapping| 1 1 3 2 3 3 | 0 5 -6 7 4 6 }}
-{{Multival|legend=1|5 -6 7 -21 -3 33}}
+Optimal tunings:
+* WE: ~2 = 1203.0195{{c}}, ~13/12 = 138.2642{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 138.0317{{c}}
-[[POTE generator]]: ~16/15 = 137.958
+{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf }}
-{{Val list|legend=1| 9, 17, 26, 61bc, 87bcc }}
+Badness (Sintel): 1.08
-[[Badness]]: 0.089840
+== Semishallowtone ==
+{{See also| Shallowtone }}
-=== 11-limit ===
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11
-Comma list: 45/44, 99/98, 385/384
+[[Comma list]]: 405/392, 35721/32768
-Mapping: [{{val|1 1 3 2 3}}, {{val|0 5 -6 7 4}}]
+{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}
+: mapping generators: ~189/128, ~3
-POTE generator: ~12/11 = 137.900
+[[Optimal tuning]]s:
+* [[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 }}
-Vals: {{Val list| 9, 17, 26, 61bc, 87bcc }}
+{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}
-Badness: 0.042562
+[[Badness]] (Sintel): 8.42
-=== 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:
+* 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}})
-Vals: {{Val list| 9, 26f, 35 }}
+{{Optimal ET sequence|legend=0| 14c, 44bd, 58bcde, 72bbccddee }}
-Badness: 0.049684
+Badness (Sintel): 4.15
-=== Secundly ===
+=== 13-limit ===
 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:
+* 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}})
-Vals: {{Val list| 9, 17, 26, 61bcf, 87bccf }}
+{{Optimal ET sequence|legend=0| 14c, 30b }}
-Badness: 0.026212
+Badness (Sintel): 4.38
 [[Category:Temperament collections]]
-[[Category:Greenwoodmic]]
+[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
 [[Category:Rank 2]]