Porwell temperaments: Difference between revisions

← Older edit

VisualWikitext

@@ Line 23: / Line 23: @@
 * ''[[Semaja]]'' (+3125/3087) → [[Gariboh clan #Semaja|Gariboh clan]]
 * ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
-* ''[[Countermiracle]]'' (+823543/819200) → [[Quince clan #Countermiracle|Quince clan]]
 * ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]
-Considered below are hendecatonic, nessafof, grendel, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing [[badness]].
+Considered below are hendecatonic, nessafof, grendel, countermiracle, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing [[badness]].
 == Hendecatonic ==
@@ Line 304: / Line 303: @@
 Badness (Sintel): 1.12
+== Countermiracle ==
+Countermiracle tempers out 50421/50000, the [[trimyna comma]], and 823543/819200, the [[quince comma]], in addition to the porwell comma. It can be described as the {{nowrap| 31 & 176 }} temperament, and finds the [[3/2|perfect fifth]] -25 generators away (normalized for 2.5.7) as opposed to miracle's +6 generators. It has the [[ploidacot]] signature of 22-sheared 25-cot (or omega-heptaseph).
+Like miracle, it is naturally an [[11-limit]] temperament with the generator representing [[77/72]], but here the generator does not represent [[15/14]] or [[16/15]]. [[176edo]] makes for a recommendable tuning for both the 7- and 11-limit.
+Countermiracle extends less naturally to the [[13-limit|13-]] and [[17-limit]], with countermiraculous (31 & 145), countermanna (145 & 176), and counterbenediction (176 & 207) being possible candidates, each favoring a tuning range shown by the [[temperament merging|equal-temperament joins]].
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 6144/6125, 50421/50000
+{{Mapping|legend=1| 1 -21 -4 1 | 0 25 7 2 }}
+: mapping generators: ~2, ~625/336
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7944{{c}}, ~625/336 = 1083.8973{{c}} (~343/320 = 115.8970{{c}})
+: [[error map]]: {{val| -0.206 -0.203 +1.790 -1.237 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/336 = 1084.0787{{c}} (~343/320 = 115.9213{{c}})
+: error map: {{val| 0.000 +0.013 +2.237 -0.669 }}
+{{Optimal ET sequence|legend=1| 31, 114, 145, 176 }}
+[[Badness]] (Sintel): 2.59
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 441/440, 3388/3375, 6144/6125
+{{Mapping|legend=0| 1 -21 -4 1 -39 | 0 25 7 2 47 }}
+Optimal tunings:
+* WE: ~2 = 1199.7942{{c}}, ~144/77 = 1083.8983{{c}} (~77/72 = 115.8959{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0798{{c}} (~77/72 = 115.9202{{c}})
+{{Optimal ET sequence|legend=0| 31, 114e, 145, 176 }}
+Badness (Sintel): 1.29
+==== Countermiraculous ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 196/195, 352/351, 1001/1000, 6144/6125
+{{Mapping|legend=0| 1 -21 -4 1 -39 29 | 0 25 7 2 47 -28 }}
+Optimal tunings:
+* WE: ~2 = 1199.4465{{c}}, ~144/77 = 1083.6196{{c}} (~77/72 = 115.8268{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1148{{c}} (~77/72 = 115.8852{{c}})
+{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145, 321ceff }}
+Badness (Sintel): 1.62
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224
+{{Mapping|legend=0| 1 -21 -4 1 -39 29 33 | 0 25 7 2 47 -28 -32 }}
+Optimal tunings:
+* WE: ~2 = 1199.4013{{c}}, ~144/77 = 1083.5835{{c}} (~77/72 = 115.8178{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1219{{c}} (~77/72 = 115.8781{{c}})
+{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145 }}
+Badness (Sintel): 1.50
+==== Counterbenediction ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 351/350, 441/440, 3146/3125, 3584/3575
+{{Mapping|legend=0| 1 -21 -4 1 -39 29 57 | 0 25 7 2 47 -59 }}
+Optimal tunings:
+* WE: ~2 = 1199.9460{{c}}, ~144/77 = 1084.0178{{c}} (~77/72 = 115.9283{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0663{{c}} (~77/72 = 115.9337{{c}})
+{{Optimal ET sequence|legend=0| 31, 145f, 176, 207 }}
+Badness (Sintel): 1.88
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125
+{{Mapping|legend=0| 1 -21 -4 1 -39 29 57 61 | 0 25 7 2 47 -59 -63 }}
+Optimal tunings:
+* WE: ~2 = 1199.9950{{c}}, ~144/77 = 1084.0564{{c}} (~77/72 = 115.9386{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0609{{c}} (~77/72 = 115.9391{{c}})
+{{Optimal ET sequence|legend=0| 31, 176, 207 }}
+Badness (Sintel): 1.85
+==== Countermanna ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 364/363, 441/440, 3388/3375, 6144/6125
+{{Mapping|legend=0| 1 -21 -4 1 -39 -102 | 0 25 7 2 47 117 }}
+Optimal tunings:
+* WE: ~2 = 1199.6687{{c}}, ~144/77 = 1083.8108{{c}} (~77/72 = 115.8578{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1065{{c}} (~77/72 = 115.8935{{c}})
+{{Optimal ET sequence|legend=0| 31f, 145, 176, 321ce }}
+Badness (Sintel): 2.21
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375
+{{Mapping|legend=0| 1 -21 -4 1 -39 -102 -98 | 0 25 7 2 47 117 113 }}
+Optimal tunings:
+* WE: ~2 = 1199.6458{{c}}, ~144/77 = 1083.7968{{c}} (~77/72 = 115.8490{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1132{{c}} (~77/72 = 115.8868{{c}})
+{{Optimal ET sequence|legend=0| 31fg, 145, 321ce }}
+Badness (Sintel): 2.08
 == Aufo ==
-:''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
 Also named by [[Petr Pařízek]] in 2011, ''aufo'' refers to the augmented fourth, which is a generator of this temperament<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>. The functional generator however is the [[64/45]] diminished fifth, and like its [[untriton]] variant, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.

Porwell temperaments: Difference between revisions

FloraC (talk | contribs)