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{{Technical data page}}
{{Technical data page}}
The '''quince clan''' [[Tempering out|tempers out]] the [[quince comma]], {{monzo| -15 0 -2 7 }} = 823543/819200. Quince temperaments include:
The '''quince clan''' [[Tempering out|tempers out]] the [[quince comma]] ({{monzo|legend=1| -15 0 -2 7 }}, [[ratio]]: 823543/819200).
* ''[[Casablanca]]'' (+126/125) → [[Starling temperaments #Casablanca|Starling temperaments]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Quincy]]'' (+4375/4374) → [[Ragismic microtemperaments #Quincy|Ragismic microtemperaments]]
* ''[[Birds]]'' (+3136/3125) → [[31st-octave temperaments #Birds|31st-octave temperaments]]
* ''[[Octowerck]]'' (+321489/320000 or 420175/419904) → [[Varunismic temperaments #Octowerck|Varunismic temperaments]]
* [[Cotoneum]] (+10976/10935) → [[Garischismic clan #Cotoneum|Garischismic clan]]


== Mercy ==
== Mercy ==
Mercy is the no-3 version of miracle. It can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]]. Two generators make an [[8/7]]; five generators make a [[7/5]].  
Mercy is the no-3 version of [[miracle]]. Two generators make [[~]][[8/7]], five generators make ~[[7/5]], and seven make ~[[8/5]]. The [[ploidacot]] for this temperament is omega-heptaseph.  


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
Line 17: Line 11:
{{Mapping|legend=2| 1 3 3 | 0 -7 -2 }}
{{Mapping|legend=2| 1 3 3 | 0 -7 -2 }}


{{Mapping|legend=3| 1 0 3 3 | 0 0 -7 -2 }}
: mapping generators: ~2, ~343/320
: mapping generators: ~2, ~343/320


{{Mapping|legend=3| 1 0 3 3 | 0 0 -7 -2 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2169{{c}}, ~343/320 = 116.3116{{c}}
: [[error map]]: {{val| +0.217 +0.156 -0.798 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/320 = 116.2576{{c}}
: error map: {{val| 0.000 -0.117 -1.341 }}
 
{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
 
[[Badness]] (Sintel): 0.428


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~343/320 = 116.291{{c}}
=== Overview to extensions ===
Temperaments discussed elsewhere are:
* ''[[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]]
* ''[[Octowerck]]'' (+321489/320000 or 420175/419904) → [[Varunismic temperaments #Octowerck|Varunismic temperaments]]
* [[Cotoneum]] (+10976/10935) → [[Garischismic clan #Cotoneum|Garischismic clan]]


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
Discussed below is countermiracle.


== Countermiracle ==
== Countermiracle ==
The ''countermiracle'' temperament ({{nowrap| 31 & 145 }}) tempers out the trimyna, 50421/50000 and the [[quince comma]], 823543/819200.
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]]. [[176edo]] makes for a recommendable tuning for both the 7- and 11-limit.


{{Todo|inline=1|expand|comment=Explain relationship to miracle. There's clearly ''some'' relationship, since the extensions are named the same way, but the nature of that relationship isn't obvious from looking at this page.}}
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
[[Subgroup]]: 2.3.5.7
Line 34: Line 47:
[[Comma list]]: 6144/6125, 50421/50000
[[Comma list]]: 6144/6125, 50421/50000


{{Mapping|legend=1| 1 4 3 3 | 0 -25 -7 -2 }}
{{Mapping|legend=1| 1 -21 -4 1 | 0 25 7 2 }}
: mapping generators: ~2, ~343/320
: mapping generators: ~2, ~625/336


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~343/320 = 115.9169{{c}}
[[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, 559cc, 735cc }}
{{Optimal ET sequence|legend=1| 31, 114, 145, 176 }}


[[Badness]] (Smith): 0.102326
[[Badness]] (Sintel): 2.59


=== 11-limit ===
=== 11-limit ===
Line 48: Line 65:
Comma list: 441/440, 3388/3375, 6144/6125
Comma list: 441/440, 3388/3375, 6144/6125


Mapping: {{mapping| 1 4 3 3 8 | 0 -25 -7 -2 -47 }}
Mapping: {{mapping| 1 -21 -4 1 -39 | 0 25 7 2 47 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.9158{{c}}
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 }}
{{Optimal ET sequence|legend=0| 31, 114e, 145, 176 }}


Badness (Smith): 0.039162
Badness (Sintel): 1.29


==== Countermiraculous ====
==== Countermiraculous ====
Line 61: Line 80:
Comma list: 196/195, 352/351, 1001/1000, 6144/6125
Comma list: 196/195, 352/351, 1001/1000, 6144/6125


Mapping: {{mapping| 1 4 3 3 8 1 | 0 -25 -7 -2 -47 28 }}
Mapping: {{mapping| 1 -21 -4 1 -39 29 | 0 25 7 2 47 -28 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.8803{{c}}
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 }}
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145, 321ceff }}


Badness (Smith): 0.039271
Badness (Sintel): 1.62


===== 17-limit =====
===== 17-limit =====
Line 74: Line 95:
Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224
Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224


Mapping: {{mapping| 1 4 3 3 8 1 1 | 0 -25 -7 -2 -47 28 32 }}
Mapping: {{mapping| 1 -21 -4 1 -39 29 33 | 0 25 7 2 47 -28 -32 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.8756{{c}}
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 }}
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145 }}


Badness (Smith): 0.029496
Badness (Sintel): 1.50


==== Counterbenediction ====
==== Counterbenediction ====
Line 87: Line 110:
Comma list: 351/350, 441/440, 3146/3125, 3584/3575
Comma list: 351/350, 441/440, 3146/3125, 3584/3575


Mapping: {{mapping| 1 4 3 3 8 -2 | 0 -25 -7 -2 -47 59 }}
Mapping: {{mapping| 1 -21 -4 1 -39 29 57 | 0 25 7 2 47 -59 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.9335{{c}}
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, 114ef, 145f, 176, 207, 383c, 590cc }}
{{Optimal ET sequence|legend=0| 31, 145f, 176, 207 }}


Badness (Smith): 0.045569
Badness (Sintel): 1.88


===== 17-limit =====
===== 17-limit =====
Line 100: Line 125:
Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125
Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125


Mapping: {{mapping| 1 4 3 3 8 -2 -2 | 0 -25 -7 -2 -47 59 63 }}
Mapping: {{mapping| 1 -21 -4 1 -39 29 57 61 | 0 25 7 2 47 -59 -63 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.9391{{c}}
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, 114efg, 145fg, 176, 207 }}
{{Optimal ET sequence|legend=0| 31, 176, 207 }}


Badness (Smith): 0.036289
Badness (Sintel): 1.85


==== Countermanna ====
==== Countermanna ====
Line 113: Line 140:
Comma list: 364/363, 441/440, 3388/3375, 6144/6125
Comma list: 364/363, 441/440, 3388/3375, 6144/6125


Mapping: {{mapping| 1 4 3 3 8 15 | 0 -25 -7 -2 -47 -117 }}
Mapping: {{mapping| 1 -21 -4 1 -39 -102 | 0 25 7 2 47 117 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.8898{{c}}
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| 145, 176, 321ce }}
{{Optimal ET sequence|legend=0| 31f, 145, 176, 321ce }}


Badness (Smith): 0.053409
Badness (Sintel): 2.21


===== 17-limit =====
===== 17-limit =====
Line 126: Line 155:
Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375
Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375


Mapping: {{mapping| 1 4 3 3 8 15 15 | 0 -25 -7 -2 -47 -117 -113 }}
Mapping: {{mapping| 1 -21 -4 1 -39 -102 -98 | 0 25 7 2 47 117 113 }}
 
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.8832{{c}}
 
{{Optimal ET sequence|legend=0| 145, 321ce }}


Badness (Smith): 0.040898
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}})


=== Counterrevelation ===
{{Optimal ET sequence|legend=0| 31fg, 145, 321ce }}
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 50421/50000
Badness (Sintel): 2.08
 
Mapping: {{mapping| 1 4 3 3 5 | 0 -25 -7 -2 -16 }}
 
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~343/320 = 115.9192{{c}}
 
{{Optimal ET sequence|legend=0| 31, 114, 145e, 176e }}
 
Badness (Smith): 0.064070
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 196/195, 13750/13689
 
Mapping: {{mapping| 1 4 3 3 5 1 | 0 -25 -7 -2 -16 28 }}
 
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~273/256 = 115.8624{{c}}
 
{{Optimal ET sequence|legend=0| 31, 83, 114, 145e }}
 
Badness (Smith): 0.057497
 
==== 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 4 3 3 5 1 1 | 0 -25 -7 -2 -16 28 32 }}
 
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~91/85 = 115.8527{{c}}
 
{{Optimal ET sequence|legend=0| 31, 83, 114, 145e }}
 
Badness (Smith): 0.044043


== Subgroup extensions ==
== Subgroup extensions ==
=== Mercy (2.5.7.13) ===
=== Mercy (2.5.7.13) ===
This extension may be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]].
Subgroup: 2.5.7.13
Subgroup: 2.5.7.13


Line 183: Line 177:
Gencom mapping: {{mapping| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}
Gencom mapping: {{mapping| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/13 = 116.094{{c}}
Optimal tunings:
* WE: ~2 = 1199.2166{{c}}, ~14/13 = 116.0181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 116.2099{{c}}


{{Optimal ET sequence|legend=0| 10, 21, 31}}
{{Optimal ET sequence|legend=0| 10, 21, 31 }}
 
Badness (Sintel): 0.546


==== 2.5.7.13.17 subgroup ====
==== 2.5.7.13.17 subgroup ====
Subgroup: 2.5.7.13.17
Subgroup: 2.5.7.13.17


Comma list: 170/169, 224/221, 640/637
Comma list: 170/169, 224/221, 343/338


Subgroup-val mapping: {{mapping| 1 3 3 4 4 | 0 -7 -2 -3 1 }}
Subgroup-val mapping: {{mapping| 1 3 3 4 4 | 0 -7 -2 -3 1 }}
Line 196: Line 194:
Gencom mapping: {{mapping| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}
Gencom mapping: {{mapping| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/13 = 115.769{{c}}
Optimal tunings:
* WE: ~2 = 1198.5476{{c}}, ~14/13 = 115.6294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.9132{{c}}


{{Optimal ET sequence|legend=0| 10, 21, 31 }}
{{Optimal ET sequence|legend=0| 10, 21, 31, 83fg }}
 
Badness (Sintel): 0.451


==== 2.5.7.13.17.19 subgroup ====
==== 2.5.7.13.17.19 subgroup ====
Subgroup: 2.5.7.13.17.19
Subgroup: 2.5.7.13.17.19


Comma list: 170/169, 343/338, 640/637, 16384/16055
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 }}
Subgroup-val mapping: {{mapping| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}
Line 209: Line 211:
Gencom mapping: {{mapping| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}
Gencom mapping: {{mapping| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/13 = 115.716{{c}}
Optimal tunings:
* WE: ~2 = 1198.4839{{c}}, ~14/13 = 115.5700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.7299{{c}}
 
{{Optimal ET sequence|legend=0| 10h, 21, 31, 52f, 83fg }}


{{Optimal ET sequence|legend=0| 10, 21, 31, 52f }}
Badness (Sintel): 0.759


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Quince clan| ]] <!-- main article -->
[[Category:Quince clan| ]] <!-- main article -->
[[Category:Quince| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Quince_clan"