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, [3 -18 11⟩ = 390625000/387420489.

Quartonic

The name "quartonic" means quarter-tone, which is the generator of this temperament.

Subgroup: 2.3.5

Comma list: 390625000/387420489

Mapping: [⟨1 2 3], ⟨0 -11 -18]]

Optimal tuning (CTE): 2 = 1\1, ~250/243 = 45.2368

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

Badness: 0.117250

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

Subgroup: 2.3.5.7

Comma list: 1728/1715, 4000/3969

Mapping: [⟨1 2 3 3], ⟨0 -11 -18 -5]]

Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 45.2652

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

Badness: 0.042632

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 tuning (CTE): ~2 = 1\1, ~36/35 = 45.1674

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

Badness: 0.034031

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 tuning (CTE): ~2 = 1\1, ~36/35 = 45.1632

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

Badness: 0.023875

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 tuning (CTE): ~2 = 1\1, ~36/35 = 45.4022

Optimal ET sequence: 26, 53e, 132ee

Badness: 0.041786

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 tuning (CTE): ~2 = 1\1, ~36/35 = 45.3857

Optimal ET sequence: 26, 53e, 132ee

Badness: 0.027692

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 tuning (CTE): ~2 = 1\1, ~33/32 = 45.3313

Optimal ET sequence: 26, 53

Badness: 0.053285

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 tuning (CTE): ~2 = 1\1, ~33/32 = 45.3168

Optimal ET sequence: 26, 53

Badness: 0.028818

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]]

Optimal tuning (CTE): ~99/70 = 1\2, ~36/35 = 45.2678

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

Badness: 0.060737

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 tuning (CTE): ~55/39 = 1\2, ~40/39 = 45.2544

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

Badness: 0.039891

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 tuning (CTE): ~17/12 = 1\2, ~34/33 = 45.2397

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

Badness: 0.028112

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 tuning (CTE): ~17/12 = 1\2, ~39/38 = 45.222

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

Badness: 0.0213

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 tuning (CTE): ~2 = 1\1, ~100/99 = 15.0880

Optimal ET sequence: 79, 80, 159d

Badness: 0.099950

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 tuning (CTE): ~2 = 1\1, ~100/99 = 15.0842

Optimal ET sequence: 79, 80, 159d

Badness: 0.061645

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 tuning (CTE): ~2 = 1\1, ~100/99 = 15.0840

Optimal ET sequence: 79, 80, 159d

Badness: 0.046718

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 = 1\1, ~126/125 = 15.0714

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

Badness: 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 = 1\1, ~100/99 = 15.0724

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

Badness: 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 = 1\1, ~91/90 = 15.0707

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

Badness: 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 = 1\1, ~91/90 = 15.0706

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

Badness: 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 = 1\1, ~91/90 = 15.0683

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

Badness: 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 = 1\1, ~91/90 = 15.0676

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

Badness: 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 = 1\1, ~91/90 = 15.0667

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

Badness: 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 = 1\1, ~875/864 = 15.0995

Optimal ET sequence: 79, 159

Badness: 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 = 1\1, ~100/99 = 15.0982

Optimal ET sequence: 79, 159

Badness: 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 = 1\1, ~100/99 = 15.0952

Optimal ET sequence: 79, 159

Badness: 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 = 1\1, ~100/99 = 15.0950

Optimal ET sequence: 79, 159

Badness: 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 = 1\1, ~875/729 = 311.308

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

Badness: 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 = 1\3, ~250/243 = 45.2083

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

Badness: 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 = 1\4, ~250/243 = 45.2411

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

Badness: 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 = 1\4, ~77/75 = 45.2303

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

Badness: 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 = 1\4, ~40/39 = 45.2243

Optimal ET sequence: 80, 132de, 212

Badness: 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 = 1\4, ~40/39 = 45.218

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

Badness: 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 = 1\4, ~39/38 = 45.210

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

Badness: 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 = 1\5, ~250/243 = 45.2563

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

Badness: 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 = 1\5, ~250/243 = 45.2624

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

Badness: 0.103496

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