Quince clan: Difference between revisions

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@@ Line 9: / Line 9: @@
 [[Comma list]]: 823543/819200
-{{Mapping|legend=2| 1 3 3 | 0 -7 -2 }}
+{{Mapping|legend=3| 1 3 3 | 0 -7 -2 }}
-{{Mapping|legend=3| 1 0 3 3 | 0 0 -7 -2 }}
+{{Mapping|legend=5| 1 0 3 3 | 0 0 -7 -2 }}
 : mapping generators: ~2, ~343/320
@@ Line 37: / Line 37: @@
 == Countermiracle ==
-The countermiracle temperament tempers out the porwell comma ([[6144/6125]]), the trimyna comma ([[50421/50000]]) and the quince comma ([[823543/819200]]). It can be described as the {{nowrap| 31 & 145 }} 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.
+The countermiracle temperament tempers out the porwell comma ([[6144/6125]]), the trimyna comma ([[50421/50000]]) and the quince comma ([[823543/819200]]). 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]].
+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.
-The 31 & 145 extension in the 13- and 17-limit used to be called "countermiraculous" before becoming canonical.
+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
@@ Line 65: / Line 65: @@
 Comma list: 441/440, 3388/3375, 6144/6125
-Mapping: {{mapping| 1 -21 -4 1 -39 | 0 25 7 2 47 }}
+{{Mapping|legend=0| 1 -21 -4 1 -39 | 0 25 7 2 47 }}
 Optimal tunings:
@@ Line 75: / Line 75: @@
 Badness (Sintel): 1.29
-=== 13-limit ===
+==== Countermiraculous ====
 Subgroup: 2.3.5.7.11.13
 Comma list: 196/195, 352/351, 1001/1000, 6144/6125
-Mapping: {{mapping| 1 -21 -4 1 -39 29 | 0 25 7 2 47 -28 }}
+{{Mapping|legend=0| 1 -21 -4 1 -39 29 | 0 25 7 2 47 -28 }}
 Optimal tunings:
@@ Line 90: / Line 90: @@
 Badness (Sintel): 1.62
-=== 17-limit ===
+===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
 Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224
-Mapping: {{mapping| 1 -21 -4 1 -39 29 33 | 0 25 7 2 47 -28 -32 }}
+{{Mapping|legend=0| 1 -21 -4 1 -39 29 33 | 0 25 7 2 47 -28 -32 }}
 Optimal tunings:
@@ Line 104: / Line 104: @@
 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
 == Subgroup extensions ==
@@ Line 113: / Line 173: @@
 Comma list: 343/338, 640/637
-Subgroup-val mapping: {{mapping| 1 3 3 4 | 0 -7 -2 -3 }}
+{{Mapping|legend=2| 1 3 3 4 | 0 -7 -2 -3 }}
-Gencom mapping: {{mapping| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}
+{{Mapping|legend=4| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}
 Optimal tunings:
@@ Line 130: / Line 190: @@
 Comma list: 170/169, 224/221, 343/338
-Subgroup-val mapping: {{mapping| 1 3 3 4 4 | 0 -7 -2 -3 1 }}
+{{Mapping|legend=2| 1 3 3 4 4 | 0 -7 -2 -3 1 }}
-Gencom mapping: {{mapping| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}
+{{Mapping|legend=4| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}
 Optimal tunings:
@@ Line 147: / Line 207: @@
 Comma list: 170/169, 224/221, 343/338, 476/475
-Subgroup-val mapping: {{mapping| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}
+{{Mapping|legend=2| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}
-Gencom mapping: {{mapping| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}
+{{Mapping|legend=4| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}
 Optimal tunings:
@@ Line 159: / Line 219: @@
 Badness (Sintel): 0.759
+[[Category:Quince clan| ]] <!-- main article -->
 [[Category:Temperament clans]]
-[[Category:Quince clan| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]

Quince clan: Difference between revisions

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