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Quince clan: Difference between revisions

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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 16: Line 10:


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


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 25: Line 19:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/320 = 116.2576{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/320 = 116.2576{{c}}
: error map: {{val| 0.000 -0.117 -1.341 }}
: error map: {{val| 0.000 -0.117 -1.341 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~343/320 = 116.291{{c}} -->


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


[[Badness]] (Sintel): 0.428
[[Badness]] (Sintel): 0.428
=== 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]]
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 48: Line 55:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/336 = 1084.0787{{c}} (~343/320 = 115.9213{{c}})
* [[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 }}
: error map: {{val| 0.000 +0.013 +2.237 -0.669 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~343/320 = 115.9169{{c}} -->


{{Optimal ET sequence|legend=1| 31, 114, 145, 176 }}
{{Optimal ET sequence|legend=1| 31, 114, 145, 176 }}
Line 64: Line 70:
* WE: ~2 = 1199.7942{{c}}, ~144/77 = 1083.8983{{c}} (~77/72 = 115.8959{{c}})
* 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}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0798{{c}} (~77/72 = 115.9202{{c}})
<!-- * POTE: ~2 = 1200.0000{{c}}, ~77/72 = 115.9158{{c}} -->


{{Optimal ET sequence|legend=0| 31, 114e, 145, 176 }}
{{Optimal ET sequence|legend=0| 31, 114e, 145, 176 }}
Line 80: Line 85:
* WE: ~2 = 1199.4465{{c}}, ~144/77 = 1083.6196{{c}} (~77/72 = 115.8268{{c}})
* 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}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1148{{c}} (~77/72 = 115.8852{{c}})
<!-- * POTE: ~2 = 1200.0000{{c}}, ~77/72 = 115.8803{{c}} -->


{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145, 321ceff }}
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145, 321ceff }}
Line 96: Line 100:
* WE: ~2 = 1199.4013{{c}}, ~144/77 = 1083.5835{{c}} (~77/72 = 115.8178{{c}})
* 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}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1219{{c}} (~77/72 = 115.8781{{c}})
<!-- * POTE: ~2 = 1200.0000{{c}}, ~77/72 = 115.8756{{c}} -->


{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145 }}
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145 }}
Line 112: Line 115:
* WE: ~2 = 1199.9460{{c}}, ~144/77 = 1084.0178{{c}} (~77/72 = 115.9283{{c}})
* 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}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0663{{c}} (~77/72 = 115.9337{{c}})
<!-- * POTE: ~2 = 1200.0000{{c}}, ~77/72 = 115.9335{{c}} -->


{{Optimal ET sequence|legend=0| 31, 145f, 176, 207 }}
{{Optimal ET sequence|legend=0| 31, 145f, 176, 207 }}
Line 128: Line 130:
* WE: ~2 = 1199.9950{{c}}, ~144/77 = 1084.0564{{c}} (~77/72 = 115.9386{{c}})
* 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}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.0609{{c}} (~77/72 = 115.9391{{c}})
<!-- * POTE: ~2 = 1200.0000{{c}}, ~77/72 = 115.9391{{c}} -->


{{Optimal ET sequence|legend=0| 31, 176, 207 }}
{{Optimal ET sequence|legend=0| 31, 176, 207 }}
Line 144: Line 145:
* WE: ~2 = 1199.6687{{c}}, ~144/77 = 1083.8108{{c}} (~77/72 = 115.8578{{c}})
* 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}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1065{{c}} (~77/72 = 115.8935{{c}})
<!-- * POTE: ~2 = 1200.0000{{c}}, ~77/72 = 115.8898{{c}} -->


{{Optimal ET sequence|legend=0| 31f, 145, 176, 321ce }}
{{Optimal ET sequence|legend=0| 31f, 145, 176, 321ce }}
Line 160: Line 160:
* WE: ~2 = 1199.6458{{c}}, ~144/77 = 1083.7968{{c}} (~77/72 = 115.8490{{c}})
* 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}})
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.1132{{c}} (~77/72 = 115.8868{{c}})
<!-- * POTE: ~2 = 1200.0000{{c}}, ~77/72 = 115.8832{{c}} -->


{{Optimal ET sequence|legend=0| 31fg, 145, 321ce }}
{{Optimal ET sequence|legend=0| 31fg, 145, 321ce }}


Badness (Sintel): 2.08
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}})
<!-- * POTE: ~2 = 1200.0000{{c}}, ~343/320 = 115.9192{{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}})
<!-- * POTE: ~2 = 1200.0000{{c}}, ~273/256 = 115.8624{{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}})
<!-- * POTE: ~2 = 1200.0000{{c}}, ~91/85 = 115.8527{{c}} -->
{{Optimal ET sequence|legend=0| 31, 83, 114 }}
Badness (Sintel): 2.24


== 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 227: Line 180:
* WE: ~2 = 1199.2166{{c}}, ~14/13 = 116.0181{{c}}
* WE: ~2 = 1199.2166{{c}}, ~14/13 = 116.0181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 116.2099{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 116.2099{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~14/13 = 116.094{{c}} -->


{{Optimal ET sequence|legend=0| 10, 21, 31 }}
{{Optimal ET sequence|legend=0| 10, 21, 31 }}
Line 245: Line 197:
* WE: ~2 = 1198.5476{{c}}, ~14/13 = 115.6294{{c}}
* WE: ~2 = 1198.5476{{c}}, ~14/13 = 115.6294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.9132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.9132{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~14/13 = 115.769{{c}} -->


{{Optimal ET sequence|legend=0| 10, 21, 31, 83fg }}
{{Optimal ET sequence|legend=0| 10, 21, 31, 83fg }}
Line 263: Line 214:
* WE: ~2 = 1198.4839{{c}}, ~14/13 = 115.5700{{c}}
* WE: ~2 = 1198.4839{{c}}, ~14/13 = 115.5700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.7299{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.7299{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~14/13 = 115.716{{c}} -->


{{Optimal ET sequence|legend=0| 10h, 21, 31, 52f, 83fg }}
{{Optimal ET sequence|legend=0| 10h, 21, 31, 52f, 83fg }}
Line 271: Line 221:
[[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"