Cathartic temperaments: Difference between revisions

← Older edit

VisualWikitext

@@ Line 8: / Line 8: @@
 * ''[[Pentaorwell]]'' (+1728/1715) → [[Orwellismic temperaments #Pentaorwell|Orwellismic temperaments]]
 * ''[[Septiquarter]]'' (+5120/5103) → [[Aberschismic temperaments #Septiquarter|Aberschismic temperaments]]
-* ''[[Hemiquindromeda]]'' (+67108864/66976875) → [[Quindromeda family #Hemiquindromeda|Quindromeda family]]
 * ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]]
 * [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]]
-* ''[[Turkey|Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]
+* ''[[Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]
 * ''[[Kaboom]]'' (+65625/65536) → [[Vavoom family #Kaboom|Vavoom family]]
 * ''[[Amicable]]'' (+2401/2400) → [[Amity family #Amicable|Amity family]]
@@ Line 18: / Line 17: @@
 * ''[[Quartiquart]]'' (+390625/388962) → [[Quartonic family #Quartiquart|Quartonic family]]
-Considered below is superlimmal.
+Considered below are hemiquindromeda and superlimmal, in the order of increasing [[badness]].
+== Hemiquindromeda ==
+: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Quindromeda]]''.
+Named by [[Xenllium]] in 2021, hemiquindromeda tempers out the [[decovulture comma]]. It is essentially a 24-form, and may be described as the 193 & 217 temperament. It splits the [[4/3|perfect fourth]] in ten [[~]][[36/35]] quartertone generators, thus having a [[ploidacot]] signature of omega-decacot. Three generator steps reach an accurate ~[[12/11]], from which we can derive an 11-limit extension where [[14700/14641]], [[16384/16335]], and [[19712/19683]] are tempered out, and a 13-limit extension that follows from 67108864/66976875 = ([[676/675]])⋅([[4096/4095]])<sup>2</sup>.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 4802000/4782969, 67108864/66976875
+{{Mapping|legend=1| 1 2 0 6| 0 -10 56 -77 }}
+: mapping generators: ~2, ~36/35
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9255{{c}}, ~36/35 = 49.7534{{c}}
+: [[error map]]: {{val| -0.075 +0.362 -0.126 -0.281 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 49.7566{{c}}
+: error map: {{val| 0.000 +0.479 +0.053 -0.080 }}
+{{Optimal ET sequence|legend=1| 24, 169d, 193, 217, 410 }}
+[[Badness]] (Sintel): 6.19
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 14700/14641, 16384/16335, 19712/19683
+{{Mapping|legend=0| 1 2 0 6 4 |0 -10 56 -77 -13 }}
+Optimal tunings:
+* WE: ~2 = 1199.8344{{c}}, ~36/35 = 49.7571{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 49.7499{{c}}
+{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e }}
+Badness (Sintel): 3.08
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 364/363, 676/675, 4096/4095, 19712/19683
+{{Mapping|legend=0| 1 2 0 6 4 2 | 0 -10 56 -77 -13 41 }}
+Optimal tunings:
+* WE: ~2 = 1199.8852{{c}}, ~36/35 = 49.7543{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 49.7591{{c}}
+{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e }}
+Badness (Sintel): 2.01
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 364/363, 595/594, 676/675, 1156/1155, 4096/4095
+{{Mapping|legend=0| 1 2 0 6 4 2 5 | 0 -10 56 -77 -13 41 -22 }}
+Optimal tunings:
+* WE: ~2 = 1199.8894{{c}}, ~34/33 = 49.7544{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~34/33 = 49.7591{{c}}
+{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e, 627e }}
+Badness (Sintel): 1.43
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 364/363, 595/594, 676/675, 1156/1155, 1216/1215, 1729/1728
+{{Mapping|legend=0| 1 2 0 6 4 2 5 4 | 0 -10 56 -77 -13 41 -22 6 }}
+Optimal tunings:
+* WE: ~2 = 1199.8724{{c}}, ~34/33 = 49.7535{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~34/33 = 49.7589{{c}}
+{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e }}
+Badness (Sintel): 1.18
 == Superlimmal ==

Cathartic temperaments: Difference between revisions

FloraC (talk | contribs)