Quartonic 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 quartonic family of temperaments tempers out the quartonic comma (monzo: [3 -18 11⟩, ratio: 390 625 000 / 387 420 489).

Quartonic

The name quartonic refers to the quartertone, since a somewhat flattened quartertone (representing the triptolemaic chromatic semitone, i.e. porcupine comma) is the generator of this temperament. The ploidacot for the temperament is omega-hendecacot, with eleven generator steps giving the perfect fourth. It is a member of the schismic–Mercator equivalence continuum, with equivalence number n = 11/2.

Subgroup: 2.3.5

Comma list: 390625000/387420489

Mapping: [⟨1 2 3], ⟨0 -11 -18]]

mapping generators: ~2, ~250/243

Optimal tunings:

  • WE: ~2 = 1199.9709 ¢, ~250/243 = 45.2318 ¢
error map: ⟨-0.029 +0.437 -0.574]
  • CWE: ~2 = 1200.0000 ¢, ~250/243 = 45.2349 ¢
error map: ⟨0.000 +0.461 -0.541]

Optimal ET sequence: 26, 27, 53, 239, 292, 345, 398, 451

Badness (Sintel): 2.75

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at.

  • 1728/1715 or 4000/3969 gives septimal quartonic, with interpretation of the generator ~36/35. It also tempers out 4375/4374.
  • 10976/10935 gives yarman I (80 & 159) and slices the quartonic generator in three.
  • 5359375/5308416 gives yarman II (79 & 159) and slices the quartonic generator in three.
  • 2401/2400 gives tertiseptisix (27 & 212) with generator ~875/729, three of them give ~12/7, and four give ~250/243 with octave reduction.
  • 250047/250000 gives triquart (27 & 159) with 1/3-octave period.
  • 390625/388962 or 4802000/4782969 gives quartiquart (80 & 212) with 1/4-octave period.
  • 16875/16807 gives quintiquart (80 & 265) with 1/5-octave period.

Septimal quartonic

Septimal quartonic tempers out the orwellisma, the octagar comma, as well as the ragisma, and may be described as the 26 & 27 temperament.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 4000/3969

Mapping: [⟨1 2 3 3], ⟨0 -11 -18 -5]]

Optimal tunings:

  • WE: ~2 = 1199.1424 ¢, ~36/35 = 45.1064 ¢
error map: ⟨-0.858 +0.160 -0.801 +3.069]
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 45.1881 ¢
error map: ⟨0.000 +0.976 +0.301 +5.234]

Optimal ET sequence: 26, 27, 53, 80, 133d, 186d, 319dd

Badness (Sintel): 1.08

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 2200/2187

Mapping: [⟨1 2 3 3 5], ⟨0 -11 -18 -5 -41]]

Optimal tunings:

  • WE: ~2 = 1198.8787 ¢, ~36/35 = 44.9994 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 45.0863 ¢

Optimal ET sequence: 26e, 27e, 53, 80

Badness (Sintel): 1.13

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 540/539

Mapping: [⟨1 2 3 3 5 4], ⟨0 -11 -18 -5 -41 -8]]

Optimal tunings:

  • WE: ~2 = 1199.2991 ¢, ~36/35 = 45.0539 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 45.1049 ¢

Optimal ET sequence: 26e, 27e, 53, 80, 133d

Badness (Sintel): 0.987

Quarto

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 864/847

Mapping: [⟨1 2 3 3 4], ⟨0 -11 -18 -5 -14]]

Optimal tunings:

  • WE: ~2 = 1198.3094 ¢, ~36/35 = 45.0688 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 45.2329 ¢

Optimal ET sequence: 26, 27e, 53e, 80ee

Badness (Sintel): 1.38

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 144/143, 245/242

Mapping: [⟨1 2 3 3 4 4], ⟨0 -11 -18 -5 -14 -8]]

Optimal tunings:

  • WE: ~2 = 1198.9315 ¢, ~36/35 = 45.1650 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 45.2640 ¢

Optimal ET sequence: 26, 27e, 53e

Badness (Sintel): 1.14

Quartz

Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 4000/3969

Mapping: [⟨1 2 3 3 3], ⟨0 -11 -18 -5 12]]

Optimal tunings:

  • WE: ~2 = 1200.5082 ¢, ~33/32 = 45.4047 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 45.3739 ¢

Optimal ET sequence: 26, 53

Badness (Sintel): 1.76

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 275/273, 385/384

Mapping: [⟨1 2 3 3 3 4], ⟨0 -11 -18 -5 12 -8]]

Optimal tunings:

  • WE: ~2 = 1200.5899 ¢, ~33/32 = 45.4096 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 45.3739 ¢

Optimal ET sequence: 26, 53

Badness (Sintel): 1.19

Biquartonic

Subgroup: 2.3.5.7.11

Comma list: 1728/1715, 2420/2401, 2560/2541

Mapping: [⟨2 4 6 6 7], ⟨0 -11 -18 -5 -1]]

mapping generators: ~99/70, ~36/35

Optimal tunings:

  • WE: ~99/70 = 599.5389 ¢, ~36/35 = 45.0947 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~36/35 = 45.1692 ¢

Optimal ET sequence: 26, 54c, 80, 186de, 266dde

Badness (Sintel): 2.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 640/637

Mapping: [⟨2 4 6 6 7 8], ⟨0 -11 -18 -5 -1 -8]]

Optimal tunings:

  • WE: ~55/39 = 599.6870 ¢, ~40/39 = 45.1293 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~40/39 = 45.1789 ¢

Optimal ET sequence: 26, 54c, 80, 106

Badness (Sintel): 1.65

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 544/539

Mapping: [⟨2 4 6 6 7 8 9], ⟨0 -11 -18 -5 -1 -8 -11]]

Optimal tunings:

  • WE: ~17/12 = 599.7706 ¢, ~40/39 = 45.1427 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~40/39 = 45.1790 ¢

Optimal ET sequence: 26, 54c, 80, 106

Badness (Sintel): 1.43

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 289/288, 325/324, 544/539, 400/399

Mapping: [⟨2 4 6 6 7 8 9 10], ⟨0 -11 -18 -5 -1 -8 -11 -20]]

Optimal tunings:

  • WE: ~17/12 = 599.7848 ¢, ~39/38 = 45.1288 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~39/38 = 45.1634 ¢

Optimal ET sequence: 26, 54ch, 80, 106

Badness (Sintel): 1.30

Yarm

Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1728/1715, 4000/3969

Mapping: [⟨1 2 3 3 4], ⟨0 -33 -54 -15 -43]]

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~100/99 = 15.0363 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0617 ¢

Optimal ET sequence: 79, 80, 239dd

Badness (Sintel): 3.30

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 640/637, 1331/1323

Mapping: [⟨1 2 3 3 4 4], ⟨0 -33 -54 -15 -43 -24]]

Optimal tunings:

  • WE: ~2 = 1199.5798 ¢, ~100/99 = 15.0551 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0681 ¢

Optimal ET sequence: 79, 80

Badness (Sintel): 2.55

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 325/324, 561/560, 640/637, 850/847

Mapping: [⟨1 2 3 3 4 4 4], ⟨0 -33 -54 -15 -43 -24 7]]

Optimal tunings:

  • WE: ~2 = 1199.6800 ¢, ~100/99 = 15.0624 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0705 ¢

Optimal ET sequence: 79, 80

Badness (Sintel): 2.38

Yarman I

Yarman I is the first interpretation of Ozan Yarman's proposed tuning for the Turkish makam.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 244140625/243045684

Mapping: [⟨1 2 3 4], ⟨0 -33 -54 -95]]

mapping generators: ~2, ~126/125

Optimal tunings:

  • WE: ~2 = 1199.8726 ¢, ~126/125 = 15.0651 ¢
error map: ⟨-0.127 +0.641 -0.213 -0.523]
  • CWE: ~2 = 1200.0000 ¢, ~126/125 = 15.0689 ¢
error map: ⟨0.000 +0.771 -0.036 -0.374]

Optimal ET sequence: 79d, 80, 159, 239, 637, 876b

Badness (Sintel): 4.89

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 10976/10935

Mapping: [⟨1 2 3 4 4], ⟨0 -33 -54 -95 -43]]

Optimal tunings:

  • WE: ~2 = 1199.8339 ¢, ~100/99 = 15.0637 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0685 ¢

Optimal ET sequence: 79d, 80, 159, 239, 876be

Badness (Sintel): 1.63

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 1001/1000, 10976/10935

Mapping: [⟨1 2 3 4 4 4], ⟨0 -33 -54 -95 -43 -24]]

Optimal tunings:

  • WE: ~2 = 1200.1080 ¢, ~100/99 = 15.0766 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0737 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398f

Badness (Sintel): 1.69

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 595/594, 1001/1000, 10976/10935

Mapping: [⟨1 2 3 4 4 4 4], ⟨0 -33 -54 -95 -43 -24 7]]

Optimal tunings:

  • WE: ~2 = 1200.0222 ¢, ~100/99 = 15.0718 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0713 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398f

Badness (Sintel): 1.58

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 361/360, 364/363, 595/594, 969/968, 1001/1000

Mapping: [⟨1 2 3 4 4 4 4 5], ⟨0 -33 -54 -95 -43 -24 7 -60]]

Optimal tunings:

  • WE: ~2 = 1200.0698 ¢, ~100/99 = 15.0722 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0706 ¢

Optimal ET sequence: 79dh, 80, 159, 239, 398fh

Badness (Sintel): 1.41

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 325/324, 361/360, 364/363, 460/459, 507/506, 529/528, 760/759

Mapping: [⟨1 2 3 4 4 4 4 5 5], ⟨0 -33 -54 -95 -43 -24 7 -60 -38]]

Optimal tunings:

  • WE: ~2 = 1200.0946 ¢, ~100/99 = 15.0732 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0710 ¢

Optimal ET sequence: 79dh, 80, 159, 239, 398fh

Badness (Sintel): 1.27

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 325/324, 361/360, 364/363, 406/405, 460/459, 494/493, 507/506, 529/528

Mapping: [⟨1 2 3 4 4 4 4 5 5 6], ⟨0 -33 -54 -95 -43 -24 7 -60 -38 -91]]

Optimal tunings:

  • WE: ~2 = 1200.0995 ¢, ~100/99 = 15.0726 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0703 ¢

Optimal ET sequence: 79dhj, 80, 159, 239, 398fh

Badness (Sintel): 1.19

Yarman II

Yarman II is the other interpretation of Ozan Yarman's proposed tuning for the Turkish makam.

Subgroup: 2.3.5.7

Comma list: 5359375/5308416, 390625000/387420489

Mapping: [⟨1 2 3 2], ⟨0 -33 -54 64]]

mapping generators: ~2, ~875/864

Optimal tunings:

  • WE: ~2 = 1200.4734 ¢, ~875/864 = 15.1122 ¢
error map: ⟨+0.473 +0.291 -0.950 -0.701]
  • CWE: ~2 = 1200.0000 ¢, ~875/864 = 15.1048 ¢
error map: ⟨0.000 -0.412 -1.971 -2.121]

Optimal ET sequence: 79, 159, 397cd, 556cd, 715ccdd

Badness (Sintel): 16.6

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993, 78121827/77948684

Mapping: [⟨1 2 3 2 4], ⟨0 -33 -54 64 -43]]

Optimal tunings:

  • WE: ~2 = 1200.4357 ¢, ~100/99 = 15.1126 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.1054 ¢

Optimal ET sequence: 79, 159, 397cd, 556cd

Badness (Sintel): 4.74

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1575/1573, 85683/85184

Mapping: [⟨1 2 3 2 4 4], ⟨0 -33 -54 64 -43 -24]]

Optimal tunings:

  • WE: ~2 = 1200.5028 ¢, ~100/99 = 15.1135 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.1052 ¢

Optimal ET sequence: 79, 159, 397cdff, 556cdff

Badness (Sintel): 2.82

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 385/384, 1575/1573, 4928/4913

Mapping: [⟨1 2 3 2 4 4 4], ⟨0 -33 -54 64 -43 -24 7]]

Optimal tunings:

  • WE: ~2 = 1200.3840 ¢, ~100/99 = 15.1086 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.1025 ¢

Optimal ET sequence: 79, 159

Badness (Sintel): 2.60

Tertiseptisix

Subgroup: 2.3.5.7

Comma list: 2401/2400, 390625000/387420489

Mapping: [⟨1 -31 -51 -32], ⟨0 44 72 47]]

mapping generators: ~2, ~1458/875

Optimal tunings:

  • WE: ~2 = 1200.0156 ¢, ~1458/875 = 888.7028 ¢
error map: ⟨+0.016 +0.485 -0.507 -0.293]
  • CWE: ~2 = 1200.0000 ¢, ~1458/875 = 888.6915 ¢
error map: ⟨0.000 +0.472 -0.523 -0.324]

Optimal ET sequence: 27, …, 212, 239, 451

Badness (Sintel): 3.95

Triquart

Subgroup: 2.3.5.7

Comma list: 117649/116640, 250047/250000

Mapping: [⟨3 6 9 10], ⟨0 -11 -18 -14]]

mapping generators: ~63/50, ~250/243

Optimal tunings:

  • WE: ~63/50 = 400.0521 ¢, ~250/243 = 45.2373 ¢
error map: ⟨+0.156 +0.747 -0.117 -1.628]
  • CWE: ~63/50 = 400.0000 ¢, ~250/243 = 45.2204 ¢
error map: ⟨0.000 +0.621 -0.281 -1.911]

Optimal ET sequence: 27, 105cd, 132d, 159, 345d, 506d

Badness (Sintel): 4.30

Quartiquart

Subgroup: 2.3.5.7

Comma list: 390625/388962, 4802000/4782969

Mapping: [⟨4 8 12 15], ⟨0 -11 -18 -25]]

mapping generators: ~25/21, ~250/243

Optimal tunings:

  • WE: ~25/21 = 299.9955 ¢, ~250/243 = 45.2382 ¢
error map: ⟨-0.018 +0.389 -0.655 +0.152]
  • CWE: ~25/21 = 300.0000 ¢, ~250/243 = 45.2400 ¢
error map: ⟨0.000 +0.405 -0.634 +0.174]

Optimal ET sequence: 80, 132d, 212, 292, 504

Badness (Sintel): 5.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 6250/6237, 14641/14580

Mapping: [⟨4 8 12 15 17], ⟨0 -11 -18 -25 -21]]

Optimal tunings:

  • WE: ~25/21 = 300.0140 ¢, ~77/75 = 45.2398 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~77/75 = 45.2342 ¢

Optimal ET sequence: 80, 132de, 212, 292, 504e

Badness (Sintel): 2.06

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 1001/1000, 1375/1372, 10648/10647

Mapping: [⟨4 8 12 15 17 16], ⟨0 -11 -18 -25 -21 -8]]

Optimal tunings:

  • WE: ~25/21 = 300.0805 ¢, ~40/39 = 45.2812 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~40/39 = 45.2517 ¢

Optimal ET sequence: 80, 132de, 212

Badness (Sintel): 1.86

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 561/560, 1001/1000, 10648/10647

Mapping: [⟨4 8 12 15 17 16 18], ⟨0 -11 -18 -25 -21 -8 -11]]

Optimal tunings:

  • WE: ~25/21 = 300.1108 ¢, ~40/39 = 45.2990 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~40/39 = 45.2604 ¢

Optimal ET sequence: 80, 132deg, 212g

Badness (Sintel): 1.59

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 289/288, 325/324, 361/360, 561/560, 1001/1000, 1331/1330

Mapping: [⟨4 8 12 15 17 16 18 20], ⟨0 -11 -18 -25 -21 -8 -11 20]]

Optimal tunings:

  • WE: ~25/21 = 300.1212 ¢, ~39/38 = 45.3015 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~39/38 = 45.2590 ¢

Optimal ET sequence: 80, 132degh, 212gh

Badness (Sintel): 1.36

Quintiquart

Subgroup: 2.3.5.7

Comma list: 16875/16807, 390625000/387420489

Mapping: [⟨5 10 15 18], ⟨0 -11 -18 -21]]

mapping generators: ~35721/31250, ~250/243

Optimal tunings:

  • WE: ~35721/31250 = 239.9950 ¢, ~250/243 = 45.2520 ¢
error map: ⟨-0.025 +0.223 -0.925 +0.792]
  • CWE: ~35721/31250 = 240.0000 ¢, ~250/243 = 45.2546 ¢
error map: ⟨0.000 +0.244 -0.897 +0.826]

Optimal ET sequence: 80, 185c, 265, 610d

Badness (Sintel): 9.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 390625000/387420489

Mapping: [⟨5 10 15 18 19], ⟨0 -11 -18 -21 -9]]

Optimal tunings:

  • WE: ~8019/7000 = 239.9613 ¢, ~250/243 = 45.2270 ¢
  • CWE: ~8019/7000 = 240.0000 ¢, ~250/243 = 45.2460 ¢

Optimal ET sequence: 80, 185c, 265, 345

Badness (Sintel): 3.42

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