Quintaleap family
The quintaleap family tempers out [37 -16 -5⟩, equating a stack of two Pythagorean commas with a stack of five schismas, making it a member of the schismic-Pythagorean equivalence continuum. It is also the temperament where 4/3 is identified by a stack of five 135/128's.
Quintaleap
The name quintaleap comes from "quintans" (Latin for "one fifth") and "leapday", because the generator is 1/5 of the leapday fourth (~4/3, about 496 cents).
Subgroup: 2.3.5
Comma list: [37 -16 -5⟩ = 137438953472/134521003125
Mapping: [⟨1 2 1], ⟨0 -5 16]]
POTE generator: ~135/128 = 99.267
Optimal ET sequence: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc
Badness: 0.444506
Subgroup temperament
The quintaleap temperament works well for the 2.3.5.17.19 subgroup, tempering out 256/255 (equating 16/15 with 17/16), 361/360 (equating 19/18 with 20/19), and 4624/4617. An obvious 17-limit interpretation of the generator is ~18/17, equating three 18/17s with 19/16, five 18/17s with 4/3, and sixteen 18/17s with 5/2.
Subgroup: 2.3.5.17.19
Comma list: 256/255, 361/360, 4624/4617
Sval mapping: [⟨1 2 1 5 4], ⟨0 -5 16 -11 3]]
Gencom: [2 18/17; 256/255 361/360 4624/4617]
POTE generator: ~18/17 = 99.276
Optimal ET sequence: 12, 109, 121, 133
Badness: 0.014623
Quintupole
- Main article: Quintupole
The quintupole temperament tempers out the octagar comma (4000/3969) and the mistisma (458752/455625) in the 7-limit; 896/891 (pentacircle), 1375/1372 (moctdel), and 4375/4356 (fantares, luluzoquadyo) in the 11-limit. The word "quintupole" means five poles, but also a play on the words "quintuple" and "polypyth". It is so named because the generator is 1/5 of the polypyth fourth (~4/3, about 495.8 cents). Xenllium proposes the pronunciation of the word "quintupole" as /'kwɪntʊpəʊl/ or /'kwɪntʊpoʊl/, like as "quin-to-pole". Not to be confused with quintapole temperament (12&85).
Subgroup: 2.3.5.7
Comma list: 4000/3969, 458752/455625
Mapping: [⟨1 2 1 0], ⟨0 -5 16 34]]
Wedgie: ⟨⟨5 -16 -34 -37 -68 -34]]
POTE generator: ~135/128 = 99.175
Optimal ET sequence: 12, 97, 109, 121
Badness: 0.111620
11-limit
Subgroup: 2.3.5.7.11
Comma list: 896/891, 1375/1372, 4375/4356
Mapping: [⟨1 2 1 0 -1], ⟨0 -5 16 34 54]]
POTE generator: ~35/33 = 99.156
Optimal ET sequence: 12, 109, 121, 351bde, 472bdee
Badness: 0.056501
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 364/363, 625/624, 2704/2695
Mapping: [⟨1 2 1 0 -1 -2], ⟨0 -5 16 34 54 69]]
POTE generator: ~35/33 = 99.165
Optimal ET sequence: 12f, 109, 121
Badness: 0.038431
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 256/255, 352/351, 364/363, 375/374, 442/441
Mapping: [⟨1 2 1 0 -1 -2 5], ⟨0 -5 16 34 54 69 -11]]
POTE generator: ~18/17 = 99.172
Optimal ET sequence: 12f, 109, 121
Badness: 0.028721
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374
Mapping: [⟨1 2 1 0 -1 -2 5 4], ⟨0 -5 16 34 54 69 -11 3]]
POTE generator: ~18/17 = 99.164
Optimal ET sequence: 12f, 109, 121
Badness: 0.023818
Quintapole
The quintapole temperament (12&85) tempers out the marvel comma (225/224) and 7812500/7411887 (sepru-atritriyo). In the 11-limit, it tempers out the ptolemisma (100/99) as well as 85184/84035 (trilo-aquinru-agu). It is so named for the following reasons - it has the same commas as the apollo temperament, and its generator is a semitone five of which gives a flat fourth (~4/3, about 495 cents). Xenllium proposes the pronunciation of the word "quintapole" as /'kwɪntəpəʊl/ or /'kwɪntəpoʊl/, like as "quint-a-pole". Not to be confused with quintupole temperament (12&121).
Subgroup: 2.3.5.7
Comma list: 225/224, 7812500/7411887
Mapping: [⟨1 2 1 1], ⟨0 -5 16 22]]
Wedgie: ⟨⟨5 -16 -22 -37 -49 -6]]
POTE generator: ~21/20 = 98.994
Optimal ET sequence: 12, 73c, 85, 97d
Badness: 0.192498
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 85184/84035
Mapping: [⟨1 2 1 1 0], ⟨0 -5 16 22 42]]
POTE generator: ~21/20 = 98.954
Optimal ET sequence: 12, 73ce, 85, 97d
Badness: 0.104353
Galileic
The name galileic comes from "Vincenzo Galilei", because this temperament is strongly related to Galilei tuning.
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 225/224, 275/273, 12168/12005
Mapping: [⟨1 2 1 1 0 -1], ⟨0 -5 16 22 42 57]]
POTE generator: ~21/20 = 98.992
Optimal ET sequence: 12f, 73ceff, 85f, 97d
Badness: 0.078183
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 120/119, 225/224, 275/273, 2431/2401
Mapping: [⟨1 2 1 1 0 -1 5], ⟨0 -5 16 22 42 57 -11]]
POTE generator: ~18/17 = 99.002
Optimal ET sequence: 12f, 73ceffg, 85fg, 97dg
Badness: 0.058826
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 120/119, 225/224, 247/245, 275/273, 361/357
Mapping: [⟨1 2 1 1 0 -1 5 4], ⟨0 -5 16 22 42 57 -11 3]]
POTE generator: ~18/17 = 98.987
Optimal ET sequence: 12f, 73ceffg, 85fg, 97dg
Badness: 0.045459
Catagali
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 225/224, 847/845, 1040/1029
Mapping: [⟨1 2 1 1 0 0], ⟨0 -5 16 22 42 45]]
POTE generator: ~21/20 = 98.882
Optimal ET sequence: 12f, 73ce, 85
Badness: 0.074180
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 120/119, 225/224, 442/441, 847/845
Mapping: [⟨1 2 1 1 0 0 5], ⟨0 -5 16 22 42 45 -11]]
POTE generator: ~18/17 = 98.905
Optimal ET sequence: 12f, 73ceg, 85g
Badness: 0.057924
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 120/119, 209/208, 225/224, 361/357, 442/441
Mapping: [⟨1 2 1 1 0 0 5 4], ⟨0 -5 16 22 42 45 -11 3]]
POTE generator: ~18/17 = 98.888
Optimal ET sequence: 12f, 73ceg, 85g
Badness: 0.044714
Quintain
Subgroup: 2.3.5.7.11
Comma list: 225/224, 245/242, 5000/4851
Mapping: [⟨1 2 1 1 1], ⟨0 -5 16 22 30]]
POTE generator: ~21/20 = 98.816
Optimal ET sequence: 12, 61c, 73c, 85e
Badness: 0.112830
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 245/242, 275/273, 1040/1029
Mapping: [⟨1 2 1 1 1 0], ⟨0 -5 16 22 30 45]]
POTE generator: ~21/20 = 98.796
Optimal ET sequence: 12f, 61cf, 73c, 85e
Badness: 0.079900
Quinticosiennic
The quinticosiennic temperament (12&145) tempers out the hemifamity comma (5120/5103) and 395136/390625 (trizo-aquadbigu) in the 7-limit; 441/440 (werckisma), 896/891 (pentacircle), and 78408/78125 (lolosepgu) in the 11-limit. The word "quinticosiennic" means 5 (quintuple) × 29 (είκοσι εννέα) = 145, and so named because 1/5 of 29edo fourth, i.e. 12\145, is a possible generator.
Subgroup: 2.3.5.7
Comma list: 5120/5103, 395136/390625
Mapping: [⟨1 2 1 -1], ⟨0 -5 16 46]]
Wedgie: ⟨⟨5 -16 -46 -37 -87 -62]]
POTE generator: ~135/128 = 99.345
Optimal ET sequence: 12, 133, 145, 157, 302c, 459bcc
Badness: 0.158041
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 896/891, 78408/78125
Mapping: [⟨1 2 1 -1 -2], ⟨0 -5 16 46 66]]
POTE generator: ~35/33 = 99.318
Optimal ET sequence: 12, 133, 145
Badness: 0.080674
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 364/363, 78408/78125
Mapping: [⟨1 2 1 -1 -2 -3], ⟨0 -5 16 46 66 81]]
POTE generator: ~35/33 = 99.307
Optimal ET sequence: 12f, 133, 145
Badness: 0.052464
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 196/195, 256/255, 352/351, 364/363, 3757/3750
Mapping: [⟨1 2 1 -1 -2 -3 5], ⟨0 -5 16 46 66 81 -11]]
POTE generator: ~18/17 = 99.308
Optimal ET sequence: 12f, 133, 145
Badness: 0.037108
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475
Mapping: [⟨1 2 1 -1 -2 -3 5 4], ⟨0 -5 16 46 66 81 -11 3]]
POTE generator: ~18/17 = 99.303
Optimal ET sequence: 12f, 133, 145
Badness: 0.028440
Decimaleap
The decimaleap temperament (24&121) has a quarter-tone generator and tempers out the porwell comma (6144/6125) and 393379840/387420489 (sasaquadzo-ayo) in the 7-limit; 896/891 and 14700/14641 in the 11-limit. The name decimaleap comes from "decima" (Latin for "one tenth") and "leapday", because the generator is 1/10 of the leapday fourth.
Subgroup: 2.3.5.7
Comma list: 6144/6125, 393379840/387420489
Mapping: [⟨1 2 1 5], ⟨0 -10 32 -53]]
Wedgie: ⟨⟨10 -32 53 -74 56 213]]
POTE generator: ~36/35 = 49.621
Optimal ET sequence: 24, 97d, 121, 145, 266
Badness: 0.314227
11-limit
Subgroup: 2.3.5.7.11
Comma list: 896/891, 6144/6125, 14700/14641
Mapping: [⟨1 2 1 5 4], ⟨0 -10 32 -53 -13]]
POTE generator: ~36/35 = 49.622
Optimal ET sequence: 24, 97d, 121, 145, 266e
Badness: 0.084612
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 364/363, 676/675, 6144/6125
Mapping: [⟨1 2 1 5 4 3], ⟨0 -10 32 -53 -13 17]]
POTE generator: ~36/35 = 49.620
Optimal ET sequence: 24, 97d, 121, 266ef
Badness: 0.041763
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 256/255, 352/351, 364/363, 676/675, 1156/1155
Mapping: [⟨1 2 1 5 4 3 5], ⟨0 -10 32 -53 -13 17 -22]]
POTE generator: ~34/33 = 49.621
Optimal ET sequence: 24, 97dg, 121, 266efg
Badness: 0.025552
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 256/255, 352/351, 361/360, 364/363, 456/455, 665/663
Mapping: [⟨1 2 1 5 4 3 5 4], ⟨0 -10 32 -53 -13 17 -22 6]]
POTE generator: ~34/33 = 49.624
Optimal ET sequence: 24, 97dg, 121, 145, 266efg
Badness: 0.021208