Quince clan
- 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
- 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:
- Casablanca (+126/125) → Starling temperaments
- Miracle (+225/224) → Gamelismic clan
- Slendi (+4000/3969) → Octagar temperaments
- Quincy (+4375/4374) → Ragismic microtemperaments
- Birds (+3136/3125) → 31st-octave temperaments
- Octowerck (+321489/320000 or 420175/419904) → Varunismic temperaments
- Cotoneum (+10976/10935) → Garischismic clan
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
- 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