Xenharmonic Wiki

Quartonic family

Revision as of 05:32, 3 August 2026 by FloraC (talk | contribs) (Switch to Sintel's badness, WE & CWE tunings (1/))
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

Subgroup: 2.3.5.7

Comma list: 10976/10935, 244140625/243045684

Mapping: [⟨1 2 3 4], ⟨0 -33 -54 -95]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~126/125 = 15.0714 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398, 637

Badness (Smith): 0.193315

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 tuning (CTE): ~2 = 1200.0000 ¢, ~100/99 = 15.0724 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398, 637, 1035bd

Badness (Smith): 0.049170

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 tuning (CTE): ~2 = 1200.0000 ¢, ~91/90 = 15.0707 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398f, 637ff

Badness (Smith): 0.040929

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 tuning (CTE): ~2 = 1200.0000 ¢, ~91/90 = 15.0706 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398f, 637ff

Badness (Smith): 0.031015

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 tuning (CTE): ~2 = 1200.0000 ¢, ~91/90 = 15.0683 ¢

Optimal ET sequence: 79dh, 80, 159, 239, 637ffh

Badness (Smith): 0.023193

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 tuning (CTE): ~2 = 1200.0000 ¢, ~91/90 = 15.0676 ¢

Optimal ET sequence: 79dh, 80, 159, 239, 637ffhi

Badness (Smith): 0.017682

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 tuning (CTE): ~2 = 1200.0000 ¢, ~91/90 = 15.0667 ¢

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

Badness (Smith): 0.014289

Yarman II

Subgroup: 2.3.5.7

Comma list: 5359375/5308416, 390625000/387420489

Mapping: [⟨1 2 3 2], ⟨0 -33 -54 64]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~875/864 = 15.0995 ¢

Optimal ET sequence: 79, 159

Badness (Smith): 0.655487

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 tuning (CTE): ~2 = 1200.0000 ¢, ~100/99 = 15.0982 ¢

Optimal ET sequence: 79, 159

Badness (Smith): 0.143477

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 tuning (CTE): ~2 = 1200.0000 ¢, ~100/99 = 15.0952 ¢

Optimal ET sequence: 79, 159

Badness (Smith): 0.068150

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 tuning (CTE): ~2 = 1200.0000 ¢, ~100/99 = 15.0950 ¢

Optimal ET sequence: 79, 159

Badness (Smith): 0.051019

Tertiseptisix

Subgroup: 2.3.5.7

Comma list: 2401/2400, 390625000/387420489

Mapping: [⟨1 13 21 15], ⟨0 -44 -72 -47]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~875/729 = 311.308 ¢

Optimal ET sequence: 27, 131bccd, 158cd, 185c, 212, 239, 451

Badness (Smith): 0.155952

Triquart

Subgroup: 2.3.5.7

Comma list: 117649/116640, 250047/250000

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

Optimal tuning (CTE): ~63/50 = 400.0000 ¢, ~250/243 = 45.2083 ¢

Optimal ET sequence: 27, 105cd, 132d, 159, 186, 345d

Badness (Smith): 0.170062

Quartiquart

Subgroup: 2.3.5.7

Comma list: 390625/388962, 4802000/4782969

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

Optimal tuning (CTE): ~25/21 = 300.0000 ¢, ~250/243 = 45.2411 ¢

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

Badness (Smith): 0.199116

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 tuning (CTE): ~25/21 = 300.0000 ¢, ~77/75 = 45.2303 ¢

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

Badness (Smith): 0.062450

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 tuning (CTE): ~25/21 = 300.0000 ¢, ~40/39 = 45.2243 ¢

Optimal ET sequence: 80, 132de, 212

Badness (Smith): 0.045028

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 tuning (CTE): ~25/21 = 300.000 ¢, ~40/39 = 45.218 ¢

Optimal ET sequence: 52cdeg, 80, 132deg, 212g

Badness (Smith): 0.0312

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 tuning (CTE): ~25/21 = 300.000 ¢, ~39/38 = 45.210 ¢

Optimal ET sequence: 52cdegh, 80, 132degh, 212gh

Badness (Smith): 0.0224

Quintiquart

Subgroup: 2.3.5.7

Comma list: 16875/16807, 390625000/387420489

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

Optimal tuning (CTE): ~35721/31250 = 240.0000 ¢, ~250/243 = 45.2563 ¢

Optimal ET sequence: 80, 185c, 265, 610d, 875cd

Badness (Smith): 0.357387

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 tuning (CTE): ~8019/7000 = 240.0000 ¢, ~250/243 = 45.2624 ¢

Optimal ET sequence: 80, 185c, 265, 610de, 875cde

Badness (Smith): 0.103496

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