Quince clan: Difference between revisions

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== 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.

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

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Badness (Sintel): 1.85

=== ~~Counterrevelation~~ ===

==== Countermanna ====

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 121/120, 176/175, 50421/50000~~

~~Mapping: {{mapping| 1 -21 -4 1 -11 | 0 25 7 2 16 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.1897{{c}}, ~275/147 = 1084.2521{{c}} (~343/320 = 115.9375{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~275/147 = 1084.0850{{c}} (~343/320 = 115.9150{{c}})

~~{{Optimal ET sequence|legend=0| 31, 114, 145e }}~~

~~Badness (Sintel): 2.12~~

~~==== 13-limit~~ ====

Subgroup: 2.3.5.7.11.13

Comma list: ~~121~~/~~120~~, ~~176~~/~~175~~, ~~196~~/~~195~~, ~~13750~~/~~13689~~

Comma list: 364/363, 441/440, 3388/3375, 6144/6125

Mapping: {{mapping| 1 -21 -4 1 -~~11 29~~ | 0 25 7 2 ~~16 -28~~ }}

Mapping: {{mapping| 1 -21 -4 1 -39 -102 | 0 25 7 2 47 117 }}

Optimal tunings:

* WE: ~2 = 1199.~~7651~~{{c}}, ~~~220~~/~~117~~ = 1083.~~9253~~{{c}} (~~~117~~/~~110~~ = 115.~~8398~~{{c}})

* WE: ~2 = 1199.6687{{c}}, ~144/77 = 1083.8108{{c}} (~77/72 = 115.8578{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~~~220~~/~~117~~ = 1084.~~1357~~{{c}} (~~~117~~/~~110~~ = 115.~~8643~~{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1065{{c}} (~77/72 = 115.8935{{c}})

Badness (Sintel): 2.38

Badness (Sintel): 2.21

==== 17-limit ====

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: ~~121~~/~~120~~, ~~154~~/~~153~~, ~~176~~/~~175~~, ~~196~~/~~195~~, ~~10647~~/~~10625~~

Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375

Mapping: {{mapping| 1 -21 -4 1 -~~11 29 33~~ | 0 25 7 2 ~~16 -28 -32~~ }}

Mapping: {{mapping| 1 -21 -4 1 -39 -102 -98 | 0 25 7 2 47 117 113 }}

Optimal tunings:

* WE: ~2 = 1199.~~6935~~{{c}}, ~~~170~~/91 = 1083.~~8704~~{{c}} (~91/85 = 115.~~8231~~{{c}})

* WE: ~2 = 1199.6458{{c}}, ~144/77 = 1083.7968{{c}} (~77/72 = 115.8490{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~~~170~~/91 = 1084.~~1464~~{{c}} (~91/85 = 115.~~8536~~{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1132{{c}} (~77/72 = 115.8868{{c}})

Badness (Sintel): 2.24

Badness (Sintel): 2.08

== Subgroup extensions ==

@@ 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.
+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 133: / Line 135: @@
 Badness (Sintel): 1.85
-=== Counterrevelation ===
+==== Countermanna ====
-Subgroup: 2.3.5.7.11
-Comma list: 121/120, 176/175, 50421/50000
-Mapping: {{mapping| 1 -21 -4 1 -11 | 0 25 7 2 16 }}
-Optimal tunings:
-* WE: ~2 = 1200.1897{{c}}, ~275/147 = 1084.2521{{c}} (~343/320 = 115.9375{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~275/147 = 1084.0850{{c}} (~343/320 = 115.9150{{c}})
-{{Optimal ET sequence|legend=0| 31, 114, 145e }}
-Badness (Sintel): 2.12
-==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 176/175, 196/195, 13750/13689
+Comma list: 364/363, 441/440, 3388/3375, 6144/6125
-Mapping: {{mapping| 1 -21 -4 1 -11 29 | 0 25 7 2 16 -28 }}
+Mapping: {{mapping| 1 -21 -4 1 -39 -102 | 0 25 7 2 47 117 }}
 Optimal tunings:
-* WE: ~2 = 1199.7651{{c}}, ~220/117 = 1083.9253{{c}} (~117/110 = 115.8398{{c}})
+* WE: ~2 = 1199.6687{{c}}, ~144/77 = 1083.8108{{c}} (~77/72 = 115.8578{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~220/117 = 1084.1357{{c}} (~117/110 = 115.8643{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1065{{c}} (~77/72 = 115.8935{{c}})
-{{Optimal ET sequence|legend=0| 31, 83, 114 }}
+{{Optimal ET sequence|legend=0| 31f, 145, 176, 321ce }}
-Badness (Sintel): 2.38
+Badness (Sintel): 2.21
-==== 17-limit ====
+===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625
+Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375
-Mapping: {{mapping| 1 -21 -4 1 -11 29 33 | 0 25 7 2 16 -28 -32 }}
+Mapping: {{mapping| 1 -21 -4 1 -39 -102 -98 | 0 25 7 2 47 117 113 }}
 Optimal tunings:
-* WE: ~2 = 1199.6935{{c}}, ~170/91 = 1083.8704{{c}} (~91/85 = 115.8231{{c}})
+* WE: ~2 = 1199.6458{{c}}, ~144/77 = 1083.7968{{c}} (~77/72 = 115.8490{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~170/91 = 1084.1464{{c}} (~91/85 = 115.8536{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1132{{c}} (~77/72 = 115.8868{{c}})
-{{Optimal ET sequence|legend=0| 31, 83, 114 }}
+{{Optimal ET sequence|legend=0| 31fg, 145, 321ce }}
-Badness (Sintel): 2.24
+Badness (Sintel): 2.08
 == Subgroup extensions ==