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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
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{{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:  
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[[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
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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}})
{{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 ==
== 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


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[[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"