Quartonic family

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Temperaments of the quartonic family temper out [3 -18 11⟩ = 390625000/387420489 (saleyo comma).

Quartonic (27 & 53)

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, 292, 345, 398, 451, 504, 955, 1459bc

Badness: 0.117250

Overview to extensions

Apart from the saleyo comma, we also consider other 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.
  • 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]]

Wedgie: ⟨⟨11 18 5 3 -23 -39]]

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

Optimal ET sequence: 26, 27, 53

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, 133d, 186d

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, 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, ~36/35 = 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, 292ddefg

Badness: 0.028112

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]]

Wedgie: ⟨⟨33 54 95 9 58 69]]

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]]

Wedgie: ⟨⟨33 54 -64 9 -194 -300]]

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]]

Wedgie: ⟨⟨44 72 47 12 -49 -93]]

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]]

Wedgie: ⟨⟨33 54 42 9 -26 -54]]

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]]

Wedgie: ⟨⟨44 72 100 12 35 30]]

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, ~77/75 = 45.2243

Optimal ET sequence: 80, 212, 292f, 504eff

Badness: 0.045028

Quintiquart

Subgroup: 2.3.5.7

Comma list: 16875/16807, 390625000/387420489

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

Wedgie: ⟨⟨55 90 105 15 12 -9]]

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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