Most Alien Intervals from 13 to 53 tet

Most Alien Intervals: EDT 13 through 53

For each equal division of the octave (EDT), the three intervals whose cent values lie furthest from any interval in 12-TET are identified. Distance (Δ) is measured as the minimum absolute difference in cents from the nearest 12-TET pitch class (0, 100, 200 … 1200¢). These maximally alien intervals represent each tuning system's irreducible sonic identity — the pitches that cannot be heard as mistuned versions of familiar intervals.

Criterion: Peter Thoegersen — intervals maximally distant from 12-TET equivalents as primary cells for polymicrotonal pitch grammar.


EDT

Step (¢)

Rank 1 — Degree (¢) | Δ from 12-TET

Rank 2 — Degree (¢) | Δ from 12-TET

Rank 3 — Degree (¢) | Δ from 12-TET

13-TET

92.308

Deg 6: 553.85¢  |  Δ46.15¢

Deg 7: 646.15¢  |  Δ46.15¢

Deg 5: 461.54¢  |  Δ38.46¢

14-TET

85.714

Deg 3: 257.14¢  |  Δ42.86¢

Deg 4: 342.86¢  |  Δ42.86¢

Deg 10: 857.14¢  |  Δ42.86¢

15-TET

80

Deg 2: 160¢  |  Δ40¢

Deg 3: 240¢  |  Δ40¢

Deg 7: 560¢  |  Δ40¢

16-TET

75

Deg 2: 150¢  |  Δ50¢

Deg 6: 450¢  |  Δ50¢

Deg 10: 750¢  |  Δ50¢

17-TET

70.588

Deg 5: 352.94¢  |  Δ47.06¢

Deg 12: 847.06¢  |  Δ47.06¢

Deg 2: 141.18¢  |  Δ41.18¢

18-TET

66.667

Deg 1: 66.67¢  |  Δ33.33¢

Deg 2: 133.33¢  |  Δ33.33¢

Deg 4: 266.67¢  |  Δ33.33¢

19-TET

63.158

Deg 4: 252.63¢  |  Δ47.37¢

Deg 15: 947.37¢  |  Δ47.37¢

Deg 7: 442.11¢  |  Δ42.11¢

20-TET

60

Deg 1: 60¢  |  Δ40¢

Deg 4: 240¢  |  Δ40¢

Deg 6: 360¢  |  Δ40¢

21-TET

57.143

Deg 1: 57.14¢  |  Δ42.86¢

Deg 6: 342.86¢  |  Δ42.86¢

Deg 8: 457.14¢  |  Δ42.86¢

22-TET

54.545

Deg 1: 54.55¢  |  Δ45.45¢

Deg 10: 545.45¢  |  Δ45.45¢

Deg 12: 654.55¢  |  Δ45.45¢

23-TET

52.174

Deg 1: 52.17¢  |  Δ47.83¢

Deg 22: 1147.83¢  |  Δ47.83¢

Deg 3: 156.52¢  |  Δ43.48¢

24-TET

50

Deg 1: 50¢  |  Δ50¢

Deg 3: 150¢  |  Δ50¢

Deg 5: 250¢  |  Δ50¢

25-TET

48

Deg 1: 48¢  |  Δ48¢

Deg 24: 1152¢  |  Δ48¢

Deg 3: 144¢  |  Δ44¢

26-TET

46.154

Deg 1: 46.15¢  |  Δ46.15¢

Deg 12: 553.85¢  |  Δ46.15¢

Deg 14: 646.15¢  |  Δ46.15¢

27-TET

44.444

Deg 1: 44.44¢  |  Δ44.44¢

Deg 8: 355.56¢  |  Δ44.44¢

Deg 10: 444.44¢  |  Δ44.44¢

28-TET

42.857

Deg 1: 42.86¢  |  Δ42.86¢

Deg 6: 257.14¢  |  Δ42.86¢

Deg 8: 342.86¢  |  Δ42.86¢

29-TET

41.379

Deg 6: 248.28¢  |  Δ48.28¢

Deg 23: 951.72¢  |  Δ48.28¢

Deg 11: 455.17¢  |  Δ44.83¢

30-TET

40

Deg 1: 40¢  |  Δ40¢

Deg 4: 160¢  |  Δ40¢

Deg 6: 240¢  |  Δ40¢

31-TET

38.71

Deg 9: 348.39¢  |  Δ48.39¢

Deg 22: 851.61¢  |  Δ48.39¢

Deg 4: 154.84¢  |  Δ45.16¢

32-TET

37.5

Deg 4: 150¢  |  Δ50¢

Deg 12: 450¢  |  Δ50¢

Deg 20: 750¢  |  Δ50¢

33-TET

36.364

Deg 4: 145.45¢  |  Δ45.45¢

Deg 7: 254.55¢  |  Δ45.45¢

Deg 15: 545.45¢  |  Δ45.45¢

34-TET

35.294

Deg 7: 247.06¢  |  Δ47.06¢

Deg 10: 352.94¢  |  Δ47.06¢

Deg 24: 847.06¢  |  Δ47.06¢

35-TET

34.286

Deg 16: 548.57¢  |  Δ48.57¢

Deg 19: 651.43¢  |  Δ48.57¢

Deg 13: 445.71¢  |  Δ45.71¢

36-TET

33.333

Deg 1: 33.33¢  |  Δ33.33¢

Deg 2: 66.67¢  |  Δ33.33¢

Deg 4: 133.33¢  |  Δ33.33¢

37-TET

32.432

Deg 17: 551.35¢  |  Δ48.65¢

Deg 20: 648.65¢  |  Δ48.65¢

Deg 14: 454.05¢  |  Δ45.95¢

38-TET

31.579

Deg 8: 252.63¢  |  Δ47.37¢

Deg 11: 347.37¢  |  Δ47.37¢

Deg 27: 852.63¢  |  Δ47.37¢

39-TET

30.769

Deg 5: 153.85¢  |  Δ46.15¢

Deg 8: 246.15¢  |  Δ46.15¢

Deg 18: 553.85¢  |  Δ46.15¢

40-TET

30

Deg 5: 150¢  |  Δ50¢

Deg 15: 450¢  |  Δ50¢

Deg 25: 750¢  |  Δ50¢

41-TET

29.268

Deg 12: 351.22¢  |  Δ48.78¢

Deg 29: 848.78¢  |  Δ48.78¢

Deg 5: 146.34¢  |  Δ46.34¢

42-TET

28.571

Deg 2: 57.14¢  |  Δ42.86¢

Deg 5: 142.86¢  |  Δ42.86¢

Deg 9: 257.14¢  |  Δ42.86¢

43-TET

27.907

Deg 9: 251.16¢  |  Δ48.84¢

Deg 34: 948.84¢  |  Δ48.84¢

Deg 16: 446.51¢  |  Δ46.51¢

44-TET

27.273

Deg 2: 54.55¢  |  Δ45.45¢

Deg 9: 245.45¢  |  Δ45.45¢

Deg 13: 354.55¢  |  Δ45.45¢

45-TET

26.667

Deg 2: 53.33¢  |  Δ46.67¢

Deg 13: 346.67¢  |  Δ46.67¢

Deg 17: 453.33¢  |  Δ46.67¢

46-TET

26.087

Deg 2: 52.17¢  |  Δ47.83¢

Deg 21: 547.83¢  |  Δ47.83¢

Deg 25: 652.17¢  |  Δ47.83¢

47-TET

25.532

Deg 2: 51.06¢  |  Δ48.94¢

Deg 45: 1148.94¢  |  Δ48.94¢

Deg 6: 153.19¢  |  Δ46.81¢

48-TET

25

Deg 2: 50¢  |  Δ50¢

Deg 6: 150¢  |  Δ50¢

Deg 10: 250¢  |  Δ50¢

49-TET

24.49

Deg 2: 48.98¢  |  Δ48.98¢

Deg 47: 1151.02¢  |  Δ48.98¢

Deg 6: 146.94¢  |  Δ46.94¢

50-TET

24

Deg 2: 48¢  |  Δ48¢

Deg 23: 552¢  |  Δ48¢

Deg 27: 648¢  |  Δ48¢

51-TET

23.529

Deg 2: 47.06¢  |  Δ47.06¢

Deg 15: 352.94¢  |  Δ47.06¢

Deg 19: 447.06¢  |  Δ47.06¢

52-TET

23.077

Deg 2: 46.15¢  |  Δ46.15¢

Deg 11: 253.85¢  |  Δ46.15¢

Deg 15: 346.15¢  |  Δ46.15¢

53-TET

22.642

Deg 11: 249.06¢  |  Δ49.06¢

Deg 42: 950.94¢  |  Δ49.06¢

Deg 20: 452.83¢  |  Δ47.17¢


Note: Where multiple intervals share the same maximum distance (tied Δ values), all tied intervals are equally alien; the three shown here are ranked by degree order among ties. 12-TET itself has no alien intervals by definition (all Δ = 0). EDTs that are multiples of 12 (24, 36, 48) contain all 12-TET intervals as a subset, so their most alien intervals are the quarter-tone and third-tone intervals introduced by the finer subdivision.

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