Rastmic rank-3 clan

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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 rastmic rank-3 clan of temperaments tempers out the rastma, 243/242. Both no-5 rastmic and no-7 rastmic can be the head of this clan. These temperaments divide the fifth in half and use it as an 11/9 neutral third.

Temperaments discussed elsewhere include:

Spectacle

Spectacle, named by Gene Ward Smith in 2010[1], can be described as the 31 & 34 & 41 temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds. It is therefore generated by octaves, major thirds, and neutral thirds. 3/2 is equated with a stack of two 11/9's as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242

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

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 350.377 ¢, ~5/4 = 384.370 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/9 = 350.176 ¢, ~5/4 = 384.095 ¢

Minimax tuning:

  • 11-odd-limit: ~2 = [1 0 0 0 0⟩, ~11/9 = [-2/5 0 0 0 1/5⟩, ~5 = [2/5 -2 1 0 4/5⟩
unchanged-interval (eigenmonzo) basis: 2.9/5.11

Optimal ET sequence: 31, 41, 72, 247c, 281, 353c, 425bc, 497bc

Badness (Smith): 0.499 × 10-3

Projection pairs: 3 242/81, 7 366025/52488, 11 644204/59049 to 2.5.11/9

Scales: spectacle31

Associated temperament: marvo

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 350.239 ¢, ~5/4 = 384.998 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/9 = 350.039 ¢, ~5/4 = 384.587 ¢

Optimal ET sequence: 31, 72, 103, 175f *

* optimal patent val: 240

Badness (Smith): 1.009 × 10-3

Cuckoo

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242

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

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 350.355 ¢, ~5/4 = 389.555 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/9 = 350.421 ¢, ~5/4 = 389.731 ¢

Optimal ET sequence: 24d, 27e, 31, 58, 89, 154, 185

Badness (Smith): 0.933 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 -4 5 -10], ⟨0 0 1 3 0 5]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 350.585 ¢, ~5/4 = 389.302 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/9 = 350.568 ¢, ~5/4 = 389.610 ¢

Optimal ET sequence: 27e, 31, 58, 96d, 154

Mirwomo

The mirwomo temperament tunes ~36/35 to exactly half of an apotome ~2187/2048, and tunes ~128/105 to exactly half of a perfect fifth ~3/2.

7-limit

Subgroup: 2.3.5.7

Comma list: 33075/32768

Mapping: [⟨1 1 0 6], ⟨0 2 0 -3], ⟨0 0 1 -1]]

mapping generators: ~2, ~128/105, ~5

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~128/105 = 350.174 ¢, ~5/4 = 384.014 ¢

Optimal ET sequence: 17, 21, 24, 31, 41, 72, 281d

Badness (Smith): 0.770 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 -3 5], ⟨0 0 1 -1 0]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 350.217 ¢, ~5/4 = 383.962 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/9 = 350.003 ¢, ~5/4 = 384.078 ¢

Optimal ET sequence: 17, 24, 31, 41, 72, 312bd, 384bcd, 456bcde, 528bcde, 631bcde

Badness (Smith): 0.641 × 10-3

Mirage

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384

Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 0], ⟨0 0 0 0 0 1]]

mapping generators: ~2, ~15/14, ~13

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~15/14 = 116.711 ¢, ~13/8 = 840.528 ¢
  • CWE: ~2 = 1200.000 ¢, ~15/14 = 116.647 ¢, ~13/8 = 838.212 ¢

Optimal ET sequence: 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef

Badness (Smith): 0.738 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 385/384

Mapping: [⟨1 1 3 3 2 0 0], ⟨0 6 -7 -2 15 0 4], ⟨0 0 0 0 0 1 1]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~15/14 = 116.704 ¢, ~13/8 = 839.452 ¢
  • CWE: ~2 = 1200.000 ¢, ~15/14 = 116.640 ¢, ~13/8 = 837.142 ¢

Optimal ET sequence: 10, 31, 41, 62, 72, 103, 175f

Mandos

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 5 5], ⟨0 0 1 -2 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 350.145 ¢, ~5/4 = 389.708 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/9 = 350.555 ¢, ~5/4 = 390.269 ¢

Optimal ET sequence: 7, 24, 31, 58, 89

Badness (Smith): 0.751 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 243/242

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 350.293 ¢, ~5/4 = 389.979 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/9 = 351.155 ¢, ~5/4 = 391.123 ¢

Optimal ET sequence: 7, 24, 31, 58

Badness (Smith): 0.923 × 10-3

Parahemif

7-limit

Subgroup: 2.3.5.7

Comma list: 1605632/1594323

Mapping: [⟨1 1 0 -1], ⟨0 2 0 13], ⟨0 0 1 0]]

mapping generators: ~2, ~896/729, ~5

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~896/729 = 351.416 ¢, ~5/4 = 386.314 ¢

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99, 239, 338

Badness (Smith): 0.607 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 896/891

Mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5], ⟨0 0 1 0 0]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 351.320 ¢, ~5/4 = 386.314 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/9 = 351.459 ¢, ~5/4 = 387.423 ¢

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99e *

* optimal patent val: 123

Badness (Smith): 1.34547 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 243/242, 364/363

Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1], ⟨0 0 1 0 0 0]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 351.343 ¢, ~5/4 = 386.314 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/9 = 351.604 ¢, ~5/4 = 388.472 ¢

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99ef, 157eff, 290cdeeefff

Badness (Smith): 1.19366 × 10-3

Urania

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120

Mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5], ⟨0 0 0 1 0]]

Mapping to lattice: [⟨0 2 8 0 5], ⟨0 0 0 -1 0]]

Lattice basis:

11/9 length = 0.2536, 8/7 length = 2.807
Angle (11/9, 8/7) = 90 degrees

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 348.830 ¢, ~7/4 = 968.826 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/9 = 348.379 ¢, ~7/4 = 965.630 ¢

Optimal ET sequence: 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde

Badness (Smith): 0.842 × 10-3

Complexity spectrum: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5

Scales: urania24

References

  1. ↑ Yahoo! Tuning Group | Spectacle temperament?
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