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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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~343/320 = 116.291{{c}}
[[Badness]] (Sintel): 0.428


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


=== Counterrevelation ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* 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}})


Comma list: 121/120, 176/175, 50421/50000
{{Optimal ET sequence|legend=0| 31fg, 145, 321ce }}


Mapping: {{mapping| 1 4 3 3 5 | 0 -25 -7 -2 -16 }}
Badness (Sintel): 2.08
 
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:Quince clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Quince clan| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Quince| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 07:39, 15 August 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

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