Quintaleap family

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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 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, making it a member of omega-pentacot temperaments.

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]]

Optimal tunings:

  • WE: ~2 = 1199.359¢, ~135/128 = 99.214¢
  • CWE: ~2 = 1200.000¢, ~135/128 = 99.256¢

Optimal ET sequence: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc

Badness (Sintel): 10.427

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.

2.3.5.17

Subgroup: 2.3.5.17

Comma list: 256/255, 1419857/1417176

Gencom: [2 18/17; 256/255 1419857/1417176]

Gencom mapping: [⟨1 2 1 0 0 0 5], ⟨0 -5 16 0 0 0 -11]]

Sval mapping: [⟨1 2 1 5], ⟨0 -5 16 -11]]

Optimal tunings:

  • WE: ~2 = 1199.328¢, ~18/17 = 99.217¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.270¢

Optimal ET sequence: 12, 109, 121, 133

Badness (Sintel): 1.305

2.3.5.17.19

Subgroup: 2.3.5.17.19

Comma list: 256/255, 361/360, 4624/4617

Gencom: [2 18/17; 256/255 361/360 4624/4617]

Gencom mapping: [⟨1 2 1 0 0 0 5 4], ⟨0 -5 16 0 0 0 -11 3]]

Sval mapping: [⟨1 2 1 5 4], ⟨0 -5 16 -11 3]]

Optimal tunings:

  • WE: ~2 = 1199.446¢, ~18/17 = 99.230¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.273¢

Optimal ET sequence: 12, 109, 121, 133

Badness (Sintel): 0.712

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]]

Optimal tunings:

  • WE: ~2 = 1199.324¢, ~135/128 = 99.120¢
  • CWE: ~2 = 1200.000¢, ~135/128 = 99.162¢

Optimal ET sequence: 12, 97, 109, 121

Badness (Sintel): 2.825

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]]

Optimal tunings:

  • WE: ~2 = 1199.342¢, ~35/33 = 99.101¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.144¢

Optimal ET sequence: 12, 109, 121, 351bde, 472bdee

Badness (Sintel): 1.868

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]]

Optimal tunings:

  • WE: ~2 = 1199.322¢, ~35/33 = 99.109¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.154¢

Optimal ET sequence: 12f, 109, 121

Badness (Sintel): 1.588

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]]

Optimal tunings:

  • WE: ~2 = 1199.223¢, ~18/17 = 99.108¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.163¢

Optimal ET sequence: 12f, 109, 121

Badness (Sintel): 1.463

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]]

Optimal tunings:

  • WE: ~2 = 1199.368¢, ~18/17 = 99.112¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.158¢

Optimal ET sequence: 12f, 109, 121

Badness (Sintel): 1.449

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]]

Optimal tunings:

  • WE: ~2 = 1198.959¢, ~21/20 = 98.908¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.968¢

Optimal ET sequence: 12, 73c, 85, 97d

Badness (Sintel): 4.872

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]]

Optimal tunings:

  • WE: ~2 = 1198.982¢, ~21/20 = 98.870¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.931¢

Optimal ET sequence: 12, 73ce, 85, 97d

Badness (Sintel): 3.450

Galileic

The name galileic comes from "Vincenzo Galilei", because this temperament is strongly related to Galilei's 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]]

Optimal tunings:

  • WE: ~2 = 1198.912¢, ~21/20 = 98.902¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.970¢

Optimal ET sequence: 12f, 73ceff, 85f, 97d

Badness (Sintel): 3.231

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]]

Optimal tunings:

  • WE: ~2 = 1198.798¢, ~18/17 = 98.903¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 98.986¢

Optimal ET sequence: 12f, 73ceffg, 85fg, 97dg

Badness (Sintel): 2.997

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]]

Optimal tunings:

  • WE: ~2 = 1199.031¢, ~18/17 = 98.907¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 98.976¢

Optimal ET sequence: 12f, 73ceffg, 85fg, 97dg

Badness (Sintel): 2.765

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]]

Optimal tunings:

  • WE: ~2 = 1199.021¢, ~21/20 = 98.801¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.860¢

Optimal ET sequence: 12f, 73ce, 85

Badness (Sintel): 3.065

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]]

Optimal tunings:

  • WE: ~2 = 1198.792¢, ~18/17 = 98.805¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 98.887¢

Optimal ET sequence: 12f, 73ceg, 85g

Badness (Sintel): 2.951

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]]

Optimal tunings:

  • WE: ~2 = 1199.037¢, ~18/17 = 98.808¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 98.875¢

Optimal ET sequence: 12f, 73ceg, 85g

Badness (Sintel): 2.720

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]]

Optimal tunings:

  • WE: ~2 = 1198.804¢, ~21/20 = 98.718¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.784¢

Optimal ET sequence: 12, 61c, 73c, 85e

Badness (Sintel): 3.730

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]]

Optimal tunings:

  • WE: ~2 = 1198.830¢, ~21/20 = 98.700¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.768¢

Optimal ET sequence: 12f, 61cf, 73c, 85e

Badness (Sintel): 3.302

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]]

Optimal tunings:

  • WE: ~2 = 1199.352¢, ~135/128 = 99.291¢
  • CWE: ~2 = 1200.000¢, ~135/128 = 99.334¢

Optimal ET sequence: 12, 133, 145, 157, 302c, 459bcc

Badness (Sintel): 4.000

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]]

Optimal tunings:

  • WE: ~2 = 1199.380¢, ~35/33 = 99.266¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.308¢

Optimal ET sequence: 12, 133, 145

Badness (Sintel): 2.667

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]]

Optimal tunings:

  • WE: ~2 = 1199.406¢, ~35/33 = 99.258¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.299¢

Optimal ET sequence: 12f, 133, 145

Badness (Sintel): 2.168

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]]

Optimal tunings:

  • WE: ~2 = 1199.389¢, ~18/17 = 99.257¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.302¢

Optimal ET sequence: 12f, 133, 145

Badness (Sintel): 1.890

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]]

Optimal tunings:

  • WE: ~2 = 1199.488¢, ~18/17 = 99.261¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.299¢

Optimal ET sequence: 12f, 133, 145

Badness (Sintel): 1.730

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]]

Optimal tunings:

  • WE: ~2 = 1199.430¢, ~36/35 = 49.598¢
  • CWE: ~2 = 1200.000¢, ~36/35 = 49.623¢

Optimal ET sequence: 24, 97d, 121, 145, 266

Badness (Sintel): 7.952

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]]

Optimal tunings:

  • WE: ~2 = 1199.337¢, ~36/35 = 49.594¢
  • CWE: ~2 = 1200.000¢, ~36/35 = 49.625¢

Optimal ET sequence: 24, 97d, 121, 145, 266e

Badness (Sintel): 2.797

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]]

Optimal tunings:

  • WE: ~2 = 1199.308¢, ~36/35 = 49.591¢
  • CWE: ~2 = 1200.000¢, ~36/35 = 49.622¢

Optimal ET sequence: 24, 97d, 121, 266ef

Badness (Sintel): 1.726

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]]

Optimal tunings:

  • WE: ~2 = 1199.289¢, ~34/33 = 49.591¢
  • CWE: ~2 = 1200.000¢, ~34/33 = 49.623¢

Optimal ET sequence: 24, 97dg, 121, 266efg

Badness (Sintel): 1.302

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]]

Optimal tunings:

  • WE: ~2 = 1199.378¢, ~34/33 = 49.598¢
  • CWE: ~2 = 1200.000¢, ~34/33 = 49.626¢

Optimal ET sequence: 24, 97dg, 121, 145, 266efg

Badness (Sintel): 1.290

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