Quince clan: Difference between revisions

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 {{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 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
@@ Line 15: / Line 9: @@
 [[Comma list]]: 823543/819200
-{{Mapping|legend=2| 1 3 3 | 0 -7 -2 }}
+{{Mapping|legend=3| 1 3 3 | 0 -7 -2 }}
+{{Mapping|legend=5| 1 0 3 3 | 0 0 -7 -2 }}
 : mapping generators: ~2, ~343/320
-{{Mapping|legend=3| 1 0 3 3 | 0 0 -7 -2 }}
 [[Optimal tuning]]s:
@@ Line 29: / Line 23: @@
 [[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 ==
-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).
-{{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.}}
+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 57: / Line 65: @@
 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 72: / 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 87: / 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 102: / 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 117: / 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 132: / 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 147: / 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:
@@ Line 156: / Line 164: @@
 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 ==
 === 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
 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 225: / 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 242: / 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 254: / 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:Quince| ]] <!-- key article -->
-[[Category:Rank 2]]

Quince clan: Difference between revisions

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