Quince clan: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,046 edits
added countermiracle, which is the alternative mapping of 3 for quince on the other side of 31edo
→Countermiracle: + alternative ploidacot, more on tuning
 
(9 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{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}}
: gencom: [2 343/320; 823543/819200]
: [[error map]]: {{val| +0.217 +0.156 -0.798 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/320 = 116.2576{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/320 = 116.291
: error map: {{val| 0.000 -0.117 -1.341 }}


{{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 }}


==== 2.5.7.13 subgroup ====
[[Badness]] (Sintel): 0.428
Subgroup: 2.5.7.13


Comma list: 343/338, 640/637
=== 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]]


Sval mapping: {{mapping| 1 3 3 4 | 0 -7 -2 -3 }}
Discussed below is countermiracle.


Gencom mapping: {{mapping| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}
== 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 & 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).  
: gencom: [2 14/13; 343/338 640/637]
 
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 116.094
 
{{Optimal ET sequence|legend=1| 10, 21, 31}}
 
==== 2.5.7.13.17 subgroup ====
Subgroup: 2.5.7.13.17
 
Comma list: 170/169, 224/221, 640/637
 
Sval 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 }}
 
: gencom: [2 14/13; 170/169 224/221 640/637]
 
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 115.769
 
{{Optimal ET sequence|legend=1| 10, 21, 31 }}
 
==== 2.5.7.13.17.19 subgroup ====
Subgroup: 2.5.7.13.17.19
 
Comma list: 170/169, 343/338, 640/637, 16384/16055


Sval mapping: {{mapping| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}
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.


Gencom mapping: {{mapping| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}
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]].  
 
: gencom: [2 14/13; 170/169 343/338 640/637 16384/16055]
 
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 115.716
 
{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}
 
== Countermiracle ==
The ''countermiracle'' temperament (31 & 145) tempers out the trimyna, 50421/50000 and the [[quince comma]], 823543/819200.
{{todo|inline=1|explain relationship to miracle|comment=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.}}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 80: 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, ~625/336


: Mapping generators: ~2, ~343/320
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~343/320 = 115.9169
{{Optimal ET sequence|legend=1| 31, 114, 145, 176 }}


{{Optimal ET sequence|legend=1| 31, 114, 145, 176, 559cc, 735cc }}
[[Badness]] (Sintel): 2.59
 
[[Badness]]: 0.102326


=== 11-limit ===
=== 11-limit ===
Line 95: 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 = 1\1, ~77/72 = 115.9158
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=1| 31, 114e, 145, 176 }}
{{Optimal ET sequence|legend=0| 31, 114e, 145, 176 }}


Badness: 0.039162
Badness (Sintel): 1.29


==== Countermiraculous ====
==== Countermiraculous ====
Line 108: 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 = 1\1, ~77/72 = 115.8803
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=1| 31, 83e, 114e, 145, 321ceff }}
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145, 321ceff }}


Badness: 0.039271
Badness (Sintel): 1.62


===== 17-limit =====
===== 17-limit =====
Line 121: 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 = 1\1, ~77/72 = 115.8756
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=1| 31, 83e, 114e, 145 }}
{{Optimal ET sequence|legend=0| 31, 83e, 114e, 145 }}


Badness: 0.029496
Badness (Sintel): 1.50


==== Counterbenediction ====
==== Counterbenediction ====
Line 134: 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 = 1\1, ~77/72 = 115.9335
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=1| 31, 114ef, 145f, 176, 207, 383c, 590cc }}
{{Optimal ET sequence|legend=0| 31, 145f, 176, 207 }}


Badness: 0.045569
Badness (Sintel): 1.88


===== 17-limit =====
===== 17-limit =====
Line 147: 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 = 1\1, ~77/72 = 115.9391
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=1| 31, 114efg, 145fg, 176, 207 }}
{{Optimal ET sequence|legend=0| 31, 176, 207 }}


Badness: 0.036289
Badness (Sintel): 1.85


==== Countermanna ====
==== Countermanna ====
Line 160: 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 = 1\1, ~77/72 = 115.8898
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=1| 145, 176, 321ce }}
{{Optimal ET sequence|legend=0| 31f, 145, 176, 321ce }}


Badness: 0.053409
Badness (Sintel): 2.21


===== 17-limit =====
===== 17-limit =====
Line 173: 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 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}})
 
{{Optimal ET sequence|legend=0| 31fg, 145, 321ce }}
 
Badness (Sintel): 2.08


Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8832
== 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]].  


{{Optimal ET sequence|legend=1| 145, 321ce }}
Subgroup: 2.5.7.13


Badness: 0.040898
Comma list: 343/338, 640/637


=== Counterrevelation ===
Subgroup-val mapping: {{mapping| 1 3 3 4 | 0 -7 -2 -3 }}
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 50421/50000
Gencom mapping: {{mapping| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}


Mapping: {{mapping| 1 4 3 3 5 | 0 -25 -7 -2 -16 }}
Optimal tunings:  
* WE: ~2 = 1199.2166{{c}}, ~14/13 = 116.0181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 116.2099{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9192
{{Optimal ET sequence|legend=0| 10, 21, 31 }}


{{Optimal ET sequence|legend=1| 31, 114, 145e, 176e }}
Badness (Sintel): 0.546


Badness: 0.064070
==== 2.5.7.13.17 subgroup ====
Subgroup: 2.5.7.13.17


==== 13-limit ====
Comma list: 170/169, 224/221, 343/338
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 176/175, 196/195, 13750/13689
Subgroup-val mapping: {{mapping| 1 3 3 4 4 | 0 -7 -2 -3 1 }}


Mapping: {{mapping| 1 4 3 3 5 1 | 0 -25 -7 -2 -16 28 }}
Gencom mapping: {{mapping| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~273/256 = 115.8624
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=1| 31, 83, 114, 145e }}
{{Optimal ET sequence|legend=0| 10, 21, 31, 83fg }}


Badness: 0.057497
Badness (Sintel): 0.451


==== 17-limit ====
==== 2.5.7.13.17.19 subgroup ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.5.7.13.17.19
 
Comma list: 170/169, 224/221, 343/338, 476/475


Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625
Subgroup-val mapping: {{mapping| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}


Mapping: {{mapping| 1 4 3 3 5 1 1 | 0 -25 -7 -2 -16 28 32 }}
Gencom mapping: {{mapping| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}


Optimal tuning (POTE): ~2 = 1\1, ~91/85 = 115.8527
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=1| 31, 83, 114, 145e }}
{{Optimal ET sequence|legend=0| 10h, 21, 31, 52f, 83fg }}


Badness: 0.044043
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]]

Latest revision as of 10:01, 28 June 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The quince clan tempers out the quince comma (monzo: [-15 0 -2 7⟩, ratio: 823543/819200).

Mercy

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

Comma list: 823543/819200

Subgroup-val mapping: [⟨1 3 3], ⟨0 -7 -2]]

Gencom mapping: [⟨1 0 3 3], ⟨0 0 -7 -2]]

mapping generators: ~2, ~343/320

Optimal tunings:

  • WE: ~2 = 1200.2169 ¢, ~343/320 = 116.3116 ¢
error map: ⟨+0.217 +0.156 -0.798]
  • CWE: ~2 = 1200.0000 ¢, ~343/320 = 116.2576 ¢
error map: ⟨0.000 -0.117 -1.341]

Optimal ET sequence: 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd

Badness (Sintel): 0.428

Overview to extensions

Temperaments discussed elsewhere are:

Discussed below is countermiracle.

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 31 & 176 temperament, and finds the 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.

Countermiracle extends less naturally to the 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 equal-temperament joins.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 50421/50000

Mapping: [⟨1 -21 -4 1], ⟨0 25 7 2]]

mapping generators: ~2, ~625/336

Optimal tunings:

  • WE: ~2 = 1199.7944 ¢, ~625/336 = 1083.8973 ¢ (~343/320 = 115.8970 ¢)
error map: ⟨-0.206 -0.203 +1.790 -1.237]
  • CWE: ~2 = 1200.0000 ¢, ~625/336 = 1084.0787 ¢ (~343/320 = 115.9213 ¢)
error map: ⟨0.000 +0.013 +2.237 -0.669]

Optimal ET sequence: 31, 114, 145, 176

Badness (Sintel): 2.59

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -21 -4 1 -39], ⟨0 25 7 2 47]]

Optimal tunings:

  • WE: ~2 = 1199.7942 ¢, ~144/77 = 1083.8983 ¢ (~77/72 = 115.8959 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.0798 ¢ (~77/72 = 115.9202 ¢)

Optimal ET sequence: 31, 114e, 145, 176

Badness (Sintel): 1.29

Countermiraculous

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -21 -4 1 -39 29], ⟨0 25 7 2 47 -28]]

Optimal tunings:

  • WE: ~2 = 1199.4465 ¢, ~144/77 = 1083.6196 ¢ (~77/72 = 115.8268 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1148 ¢ (~77/72 = 115.8852 ¢)

Optimal ET sequence: 31, 83e, 114e, 145, 321ceff

Badness (Sintel): 1.62

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224

Mapping: [⟨1 -21 -4 1 -39 29 33], ⟨0 25 7 2 47 -28 -32]]

Optimal tunings:

  • WE: ~2 = 1199.4013 ¢, ~144/77 = 1083.5835 ¢ (~77/72 = 115.8178 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1219 ¢ (~77/72 = 115.8781 ¢)

Optimal ET sequence: 31, 83e, 114e, 145

Badness (Sintel): 1.50

Counterbenediction

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -21 -4 1 -39 29 57], ⟨0 25 7 2 47 -59]]

Optimal tunings:

  • WE: ~2 = 1199.9460 ¢, ~144/77 = 1084.0178 ¢ (~77/72 = 115.9283 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.0663 ¢ (~77/72 = 115.9337 ¢)

Optimal ET sequence: 31, 145f, 176, 207

Badness (Sintel): 1.88

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -21 -4 1 -39 29 57 61], ⟨0 25 7 2 47 -59 -63]]

Optimal tunings:

  • WE: ~2 = 1199.9950 ¢, ~144/77 = 1084.0564 ¢ (~77/72 = 115.9386 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.0609 ¢ (~77/72 = 115.9391 ¢)

Optimal ET sequence: 31, 176, 207

Badness (Sintel): 1.85

Countermanna

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -21 -4 1 -39 -102], ⟨0 25 7 2 47 117]]

Optimal tunings:

  • WE: ~2 = 1199.6687 ¢, ~144/77 = 1083.8108 ¢ (~77/72 = 115.8578 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1065 ¢ (~77/72 = 115.8935 ¢)

Optimal ET sequence: 31f, 145, 176, 321ce

Badness (Sintel): 2.21

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -21 -4 1 -39 -102 -98], ⟨0 25 7 2 47 117 113]]

Optimal tunings:

  • WE: ~2 = 1199.6458 ¢, ~144/77 = 1083.7968 ¢ (~77/72 = 115.8490 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.1132 ¢ (~77/72 = 115.8868 ¢)

Optimal ET sequence: 31fg, 145, 321ce

Badness (Sintel): 2.08

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: [⟨1 3 3 4], ⟨0 -7 -2 -3]]

Gencom mapping: [⟨1 0 3 3 0 4], ⟨0 0 -7 -2 0 -3]]

Optimal tunings:

  • WE: ~2 = 1199.2166 ¢, ~14/13 = 116.0181 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 116.2099 ¢

Optimal ET sequence: 10, 21, 31

Badness (Sintel): 0.546

2.5.7.13.17 subgroup

Subgroup: 2.5.7.13.17

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

Subgroup-val mapping: [⟨1 3 3 4 4], ⟨0 -7 -2 -3 1]]

Gencom mapping: [⟨1 0 3 3 0 4 4], ⟨0 0 -7 -2 0 -3 1]]

Optimal tunings:

  • WE: ~2 = 1198.5476 ¢, ~14/13 = 115.6294 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 115.9132 ¢

Optimal ET sequence: 10, 21, 31, 83fg

Badness (Sintel): 0.451

2.5.7.13.17.19 subgroup

Subgroup: 2.5.7.13.17.19

Comma list: 170/169, 224/221, 343/338, 476/475

Subgroup-val mapping: [⟨1 3 3 4 4 3], ⟨0 -7 -2 -3 1 13]]

Gencom mapping: [⟨1 0 3 3 0 4 4 3], ⟨0 0 -7 -2 0 -3 1 13]]

Optimal tunings:

  • WE: ~2 = 1198.4839 ¢, ~14/13 = 115.5700 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 115.7299 ¢

Optimal ET sequence: 10h, 21, 31, 52f, 83fg

Badness (Sintel): 0.759

Retrieved from "https://en.xen.wiki/index.php?title=Quince_clan&oldid=232872"