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, {{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]].
-Considered below is secund. Temperaments discussed elsewhere are
+Temperaments discussed elsewhere are
+* ''[[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]]
-* ''[[August]]'' (+36/35) → [[Augmented family #August|Augmented family]]
+* ''[[Ripple]]'' (+126/125) → [[Ripple family #Septimal ripple|Ripple family]]
-* ''[[Superpelog]]'' (+49/48) → [[Slendro clan #Superpelog|Slendro clan]]
+* ''[[Wesley]]'' (+875/864) → [[Wesley family #Septimal wesley|Wesley family]]
-* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
+* ''[[Greenwood]]'' (+1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]
-* ''[[Schism]]'' (+64/63) → [[Archytas clan #Schism|Archytas clan]]
-* ''[[Greenwood]]'' (+1323/1280) → [[Apotome family #Greenwood|Apotome family]]
+Considered below are secund and semishallowtone.
 == 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.
+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.
+Possible tunings include [[26edo]], [[35edo]] in the 35f val, or anything in between.
 [[Subgroup]]: 2.3.5.7
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 {{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:
+* [[WE]]: ~2 = 1203.6786{{c}}, ~16/15 = 138.3805{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/15 = 137.958
+: [[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 }}
-[[Badness]]: 0.089840
+[[Badness]] (Sintel): 2.27
 === 11-limit ===
@@ Line 31: / Line 45: @@
 Mapping: {{mapping| 1 1 3 2 3 | 0 5 -6 7 4 }}
-Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 137.900
+Optimal tunings:
+* WE: ~2 = 1202.9415{{c}}, ~12/11 = 138.2377{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.9970{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 9, 17, 26, 61bc, 87bcc }}
+{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}
-Badness: 0.042562
+Badness (Sintel): 1.41
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 27/26, 45/44, 99/98, 385/384
+Comma list: 45/44, 65/64, 78/77, 99/98
+Mapping: {{mapping| 1 1 3 2 3 3 | 0 5 -6 7 4 6 }}
+Optimal tunings:
+* 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 }}
-Mapping: {{mapping| 1 1 3 2 3 2 | 0 5 -6 7 4 15 }}
+Badness (Sintel): 1.08
-Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 136.937
+== Semishallowtone ==
+{{See also| Shallowtone }}
-Optimal ET sequence: {{Optimal ET sequence| 9, 26f, 35 }}
+[[Subgroup]]: 2.3.5.7
-Badness: 0.049684
+[[Comma list]]: 405/392, 35721/32768
-==== Secundly ====
+{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}
+: mapping generators: ~189/128, ~3
+[[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 }}
+{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}
+[[Badness]] (Sintel): 8.42
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+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: 45/44, 65/64, 78/77, 99/98
+Comma list: 27/26, 45/44, 99/98, 35721/32768
-Mapping: {{mapping| 1 1 3 2 3 3 | 0 5 -6 7 4 6 }}
+Mapping: {{mapping| 2 0 36 15 32 -2 | 0 1 -10 -3 -8 3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}})
-Optimal ET sequence: {{Optimal ET sequence| 9, 17, 26, 61bcf, 87bccf }}
+{{Optimal ET sequence|legend=0| 14c, 30b }}
-Badness: 0.026212
+Badness (Sintel): 4.38
+[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
 [[Category:Temperament collections]]
-[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Greenwoodmic| ]] <!-- key article -->
-[[Category:Rank 2]]