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. 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.

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:

• Casablanca (+126/125) → Starling temperaments
• Miracle (+225/224) → Gamelismic clan
• 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 (31 & 145) tempers out the trimyna, 50421/50000 and the quince comma, 823543/819200.

Todo: expand 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.

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

Counterrevelation

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 50421/50000

Mapping: [⟨1 -21 -4 1 -11], ⟨0 25 7 2 16]]

Optimal tunings:

• WE: ~2 = 1200.1897 ¢, ~275/147 = 1084.2521 ¢ (~343/320 = 115.9375 ¢)
• CWE: ~2 = 1200.0000 ¢, ~275/147 = 1084.0850 ¢ (~343/320 = 115.9150 ¢)

Optimal ET sequence: 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: [⟨1 -21 -4 1 -11 29], ⟨0 25 7 2 16 -28]]

Optimal tunings:

• WE: ~2 = 1199.7651 ¢, ~220/117 = 1083.9253 ¢ (~117/110 = 115.8398 ¢)
• CWE: ~2 = 1200.0000 ¢, ~220/117 = 1084.1357 ¢ (~117/110 = 115.8643 ¢)

Optimal ET sequence: 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: [⟨1 -21 -4 1 -11 29 33], ⟨0 25 7 2 16 -28 -32]]

Optimal tunings:

• WE: ~2 = 1199.6935 ¢, ~170/91 = 1083.8704 ¢ (~91/85 = 115.8231 ¢)
• CWE: ~2 = 1200.0000 ¢, ~170/91 = 1084.1464 ¢ (~91/85 = 115.8536 ¢)

Optimal ET sequence: 31, 83, 114

Badness (Sintel): 2.24

Subgroup extensions

Mercy (2.5.7.13)

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