Quince clan: Difference between revisions

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[[Comma list]]: 823543/819200

: mapping generators: ~2, ~343/320

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* ''[[Casablanca]]'' (+126/125) → [[Starling temperaments #Casablanca|Starling temperaments]]

* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]

* ''[[Slendi]]'' (+4000/3969) → [[Octagar temperaments #Slendi|Octagar temperaments]]

* ''[[Quincy]]'' (+4375/4374) → [[Ragismic microtemperaments #Quincy|Ragismic microtemperaments]]

* ''[[Birds]]'' (+3136/3125) → [[31st-octave temperaments #Birds|31st-octave temperaments]]

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

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

Optimal tunings:

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Comma list: 196/195, 352/351, 1001/1000, 6144/6125

~~Mapping:~~ {{~~mapping~~| 1 -21 -4 1 -39 29 | 0 25 7 2 47 -28 }}

Optimal tunings:

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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:

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Comma list: 351/350, 441/440, 3146/3125, 3584/3575

~~Mapping:~~ {{~~mapping~~| 1 -21 -4 1 -39 29 57 | 0 25 7 2 47 -59 }}

Optimal tunings:

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Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125

~~Mapping:~~ {{~~mapping~~| 1 -21 -4 1 -39 29 57 61 | 0 25 7 2 47 -59 -63 }}

{{Mapping|legend=0| 1 -21 -4 1 -39 29 57 61 | 0 25 7 2 47 -59 -63 }}

Optimal tunings:

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

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

Optimal tunings:

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

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

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

Optimal tunings:

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Badness (Sintel): 2.08

~~=== Counterrevelation ===~~

~~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~~

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

~~Optimal tunings:~~

* WE: ~2 = 1199.7651{{c}}, ~220/117 = 1083.9253{{c}} (~117/110 = 115.8398{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~220/117 = 1084.1357{{c}} (~117/110 = 115.8643{{c}})

~~{{Optimal ET sequence|legend=0| 31, 83, 114 }}~~

~~Badness (Sintel): 2.38~~

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

~~Subgroup: 2.3.5.7.11.13.17~~

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

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

~~Optimal tunings:~~

* WE: ~2 = 1199.6935{{c}}, ~170/91 = 1083.8704{{c}} (~91/85 = 115.8231{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~170/91 = 1084.1464{{c}} (~91/85 = 115.8536{{c}})

~~{{Optimal ET sequence|legend=0| 31, 83, 114 }}~~

~~Badness (Sintel): 2.24~~

== Subgroup extensions ==

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Comma list: 343/338, 640/637

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 3 3 4 | 0 -7 -2 -3 }}

~~Gencom mapping:~~ {{~~mapping~~| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}

Optimal tunings:

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Comma list: 170/169, 224/221, 343/338

~~Subgroup-val mapping:~~ {{~~mapping~~| 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 }}

Optimal tunings:

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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 }}

~~Gencom mapping:~~ {{~~mapping~~| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}

Optimal tunings:

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Badness (Sintel): 0.759

[[Category:Quince clan| ]]

[[Category:Temperament clans]]

[[Category:~~Quince clan| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

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 [[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 28: / Line 28: @@
 * ''[[Casablanca]]'' (+126/125) → [[Starling temperaments #Casablanca|Starling temperaments]]
 * [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
+* ''[[Slendi]]'' (+4000/3969) → [[Octagar temperaments #Slendi|Octagar temperaments]]
 * ''[[Quincy]]'' (+4375/4374) → [[Ragismic microtemperaments #Quincy|Ragismic microtemperaments]]
 * ''[[Birds]]'' (+3136/3125) → [[31st-octave temperaments #Birds|31st-octave temperaments]]
@@ Line 36: / 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
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 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 77: / Line 80: @@
 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 92: / Line 95: @@
 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 107: / Line 110: @@
 Comma list: 351/350, 441/440, 3146/3125, 3584/3575
-Mapping: {{mapping| 1 -21 -4 1 -39 29 57 | 0 25 7 2 47 -59 }}
+{{Mapping|legend=0| 1 -21 -4 1 -39 29 57 | 0 25 7 2 47 -59 }}
 Optimal tunings:
@@ Line 122: / Line 125: @@
 Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125
-Mapping: {{mapping| 1 -21 -4 1 -39 29 57 61 | 0 25 7 2 47 -59 -63 }}
+{{Mapping|legend=0| 1 -21 -4 1 -39 29 57 61 | 0 25 7 2 47 -59 -63 }}
 Optimal tunings:
@@ Line 137: / Line 140: @@
 Comma list: 364/363, 441/440, 3388/3375, 6144/6125
-Mapping: {{mapping| 1 -21 -4 1 -39 -102 | 0 25 7 2 47 117 }}
+{{Mapping|legend=0| 1 -21 -4 1 -39 -102 | 0 25 7 2 47 117 }}
 Optimal tunings:
@@ Line 152: / Line 155: @@
 Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375
-Mapping: {{mapping| 1 -21 -4 1 -39 -102 -98 | 0 25 7 2 47 117 113 }}
+{{Mapping|legend=0| 1 -21 -4 1 -39 -102 -98 | 0 25 7 2 47 117 113 }}
 Optimal tunings:
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 Badness (Sintel): 2.08
-=== Counterrevelation ===
-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
-Mapping: {{mapping| 1 -21 -4 1 -11 29 | 0 25 7 2 16 -28 }}
-Optimal tunings:
-* WE: ~2 = 1199.7651{{c}}, ~220/117 = 1083.9253{{c}} (~117/110 = 115.8398{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~220/117 = 1084.1357{{c}} (~117/110 = 115.8643{{c}})
-{{Optimal ET sequence|legend=0| 31, 83, 114 }}
-Badness (Sintel): 2.38
-==== 17-limit ====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625
-Mapping: {{mapping| 1 -21 -4 1 -11 29 33 | 0 25 7 2 16 -28 -32 }}
-Optimal tunings:
-* WE: ~2 = 1199.6935{{c}}, ~170/91 = 1083.8704{{c}} (~91/85 = 115.8231{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~170/91 = 1084.1464{{c}} (~91/85 = 115.8536{{c}})
-{{Optimal ET sequence|legend=0| 31, 83, 114 }}
-Badness (Sintel): 2.24
 == Subgroup extensions ==
@@ Line 215: / 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 232: / 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 249: / 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 261: / 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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