Werckismic temperaments

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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.

This is a collection of rank-3 temperaments tempering out the werckisma (ratio: 441/440, monzo: [-3 2 -1 2 -1⟩). For the rank-4 werckismic temperament, see Catalog of rank-4 temperaments #Werckismic (441/440).

Temperaments discussed elsewhere are:

Considered below are history, varuna, and trimyna.

History

For the 7-limit version, see Miscellaneous 7-limit temperaments #History.

History, named by Graham Breed some time before 2011[1], splits the perfect fourth into six, each for 21/20~22/21. A recommendable tuning is given by 289edo.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993

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

mapping generators: ~2, ~21/20, ~5

Optimal tunings:

  • WE: ~2 = 1200.0906 ¢, ~21/20 = 83.0793 ¢, ~5/4 = 386.1759 ¢
error map: ⟨+0.091 -0.250 +0.043 -0.914 +1.288]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0654 ¢, ~5/4 = 386.2710 ¢
error map: ⟨0.000 -0.348 -0.043 -1.097 +1.084]

Minimax tuning:

[[1 0 0 0 0⟩, [4/17 12/17 0 -6/17 6/17⟩, [-27/17 4/17 0 -2/17 19/17⟩, [8/17 -10/17 0 5/17 12/17⟩, [0 0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7.11

Optimal ET sequence: 14c, 15, 29, 43, 57, 58, 72, 130, 202, 217, 289, 491

Badness (Sintel): 0.725

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675

Mapping: [⟨1 2 0 0 1 2], ⟨0 -6 0 7 2 -9], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1200.0242 ¢, ~21/20 = 83.0177 ¢, ~5/4 = 386.6582 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0144 ¢, ~5/4 = 386.6802 ¢

Minimax tuning:

  • 13-odd-limit
[[1 0 0 0 0 0⟩, [5/4 0 0 -3/8 0 3/8⟩, [0 0 1 0 0 0⟩, [7/8 0 1 7/16 0 -7/16⟩, [5/4 0 1 1/8 0 -1/8⟩, [7/8 0 1 -9/16 0 9/16⟩]
unchanged-interval (eigenmonzo) basis: 2.5.13/7

Optimal ET sequence: 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289

Badness (Sintel): 0.505

Complexity spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7

Varuna

For the 7-limit version, see Miscellaneous 7-limit temperaments #Varuna.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000

Mapping: [⟨2 0 0 9 12], ⟨0 1 0 -4 -6], ⟨0 0 1 2 3]]

mapping generators: ~99/70, ~3, ~5

Optimal tunings:

  • WE: ~99/70 = 600.1002 ¢, ~3/2 = 701.3812 ¢, ~5/4 = 386.3126 ¢
error map: ⟨+0.200 -0.373 +0.400 -0.823 +0.536]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.3457 ¢, ~5/4 = 386.5123 ¢
error map: ⟨0.000 -0.609 +0.199 -1.184 +0.144]

Optimal ET sequence: 12, 26, 44ce, 46, 58, 72, 118, 130, 190, 248, 320

Badness (Sintel): 0.503

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440

Mapping: [⟨2 0 0 9 12 11], ⟨0 1 0 -4 -6 -7], ⟨0 0 1 2 3 4]]

Optimal tunings:

  • WE: ~55/39 = 600.0052 ¢, ~3/2 = 701.4439 ¢, ~5/4 = 387.1861 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~3/2 = 701.4418 ¢, ~5/4 = 387.1949 ¢

Optimal ET sequence: 12f, 14cf, 26, 46, 58, 72, 130

Badness (Sintel): 0.595

Trimyna

For the 7-limit version, see Miscellaneous 7-limit temperaments #Trimyna.

Trimyna can be described as the 27e & 29 & 31 temperament. All three equal temperaments have a fairly accurate tritone for ~10/7, and trimyna takes it as one of its generators, five of which get us to the 6th harmonic. The name trimyna is a contraction of tritonic and myna.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375

Mapping: [⟨1 -1 0 1 -3], ⟨0 5 0 -1 8], ⟨0 0 1 1 1]]

mapping generators: ~2, ~10/7, ~5

Optimal tunings:

  • WE: ~2 = 1200.0909 ¢, ~10/7 = 620.4267 ¢, ~5/4 = 387.4364 ¢
error map: ⟨+0.091 +0.088 +1.304 -1.544 -0.559]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 620.3901 ¢, ~5/4 = 387.5065 ¢
error map: ⟨0.000 -0.004 +1.193 -1.710 -0.691]

Optimal ET sequence: 27e, 31, 58, 87, 118, 294, 412d

Badness (Sintel): 0.711

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 1001/1000

Mapping: [⟨1 -1 0 1 -3 5], ⟨0 5 0 -1 8 -7], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.6799 ¢, ~10/7 = 620.4057 ¢, ~5/4 = 387.7593 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 620.5510 ¢, ~5/4 = 387.5132 ¢

Optimal ET sequence: 27e, 29, 31, 56, 58, 87, 118, 145, 232

Badness (Sintel): 0.761

Trimy

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 676/675, 847/845

Mapping: [⟨1 -6 0 2 -11 -10], ⟨0 10 0 -2 16 15], ⟨0 0 1 1 1 1]]

mapping generators: ~2, ~22/13, ~5

Optimal tunings:

  • WE: ~2 = 1200.0924 ¢, ~22/13 = 910.2591 ¢, ~5/4 = 387.4264 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/13 = 910.1939 ¢, ~5/4 = 387.4946 ¢

Optimal ET sequence: 29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294

Badness (Sintel): 1.38

Seminaiad

For the 7-limit version, see Miscellaneous 7-limit temperaments #Seminaiad.

Named by Xenllium in 2026, seminaiad is a weak extension of no-threes naiad. It splits ~13/5 in two to represent ~21/13.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 1127357/1125000

Mapping: [⟨1 5 0 -2 3], ⟨0 -11 0 8 -6], ⟨0 0 1 1 1]]

mapping generators: ~2, ~150/121, ~5

Optimal tunings:

  • WE: ~2 = 1200.203 ¢, ~150/121 = 372.665 ¢, ~5/4 = 386.499 ¢
error map: ⟨+0.203 -0.257 +0.592 -1.004 +0.206]
  • CWE: ~2 = 1200.000 ¢, ~150/121 = 372.591 ¢, ~5/4 = 386.702 ¢
error map: ⟨0.000 -0.453 +0.388 -1.399 -0.161]

Optimal ET sequence: 16, 29, 45e, 58, 87, 103, 145, 161, 190

Badness (Sintel): 2.582

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 847/845, 1001/1000

Mapping: [⟨1 5 0 -2 3 2], ⟨0 -11 0 8 -6 -2], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1200.161 ¢, ~26/21 = 372.652 ¢, ~5/4 = 386.381 ¢
  • CWE: ~2 = 1200.000 ¢, ~26/21 = 372.593 ¢, ~5/4 = 386.546 ¢

Optimal ET sequence: 16, 29, 45ef, 58, 87, 103, 145, 161, 190

Badness (Sintel): 0.944

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 221/220, 441/440, 847/845

Mapping: [⟨1 5 0 -2 3 2 3], ⟨0 -11 0 8 -6 -2 -4], ⟨0 0 1 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1200.375 ¢, ~21/17 = 372.711 ¢, ~5/4 = 386.747 ¢
  • CWE: ~2 = 1200.000 ¢, ~21/17 = 372.573 ¢, ~5/4 = 387.138 ¢

Optimal ET sequence: 16, 29g, 45efg, 58, 87g

Badness (Sintel): 1.748

Seminaiadia

Subgroup: 2.3.5.7.11.13.17

Comma list: 441/440, 833/832, 847/845, 1001/1000

Mapping: [⟨1 5 0 -2 3 2 12], ⟨0 -11 0 8 -6 -2 -18], ⟨0 0 1 1 1 1 -1]]

Optimal tunings:

  • WE: ~2 = 1199.925 ¢, ~26/21 = 372.566 ¢, ~5/4 = 387.117 ¢
  • CWE: ~2 = 1200.000 ¢, ~26/21 = 372.594 ¢, ~5/4 = 387.080 ¢

Optimal ET sequence: 16, 29, 45ef, 58, 87, 103, 145, 161, 190

Badness (Sintel): 1.284

References

  1. ↑ Yahoo! Tuning Group | Madagascar[19]
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