Keegic 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-2 keegic temperaments, which temper out the keega (ratio: 1029/1000, monzo: [-3 1 -3 3⟩), the difference a stack of three 10/7's and the 3rd harmonic.

Temperaments discussed elsewhere include:

Considered below are quinkee and aurora, in the order of increasing badness.

Quinkee

Quinkee has a period of 1/5 octave and tempers out the keega (1029/1000), from which it derives its name. It may be described as the 15 & 55 temperament. It was named by Xenllium in 2021.

Subgroup: 2.3.5.7

Comma list: 1029/1000, 6144/6125

Mapping: [⟨5 0 9 14], ⟨0 3 1 0]]

mapping generators: ~8/7, ~10/7

Optimal tunings:

  • WE: ~8/7 = 240.0512 ¢, ~10/7 = 633.5744 ¢ (~25/24 = 86.5794 ¢)
error map: ⟨+0.256 -1.232 +7.722 -8.108]
  • CWE: ~8/7 = 240.0000 ¢, ~10/7 = 633.4834 ¢ (~25/24 = 86.5160 ¢)
error map: ⟨0.000 -1.505 +7.170 -8.826]

Optimal ET sequence: 15, 40, 55

Badness (Sintel): 4.66

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 1029/1000

Mapping: [⟨5 0 9 14 12], ⟨0 3 1 0 2]]

Optimal tunings:

  • WE: ~8/7 = 240.0916 ¢, ~10/7 = 633.6802 ¢ (~22/21 = 86.5947 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~10/7 = 633.5204 ¢ (~22/21 = 86.4796 ¢)

Optimal ET sequence: 15, 40, 55

Badness (Sintel): 2.16

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 121/120, 637/625

Mapping: [⟨5 9 12 14 18 20], ⟨0 3 1 0 2 4]]

Optimal tunings:

  • WE: ~8/7 = 240.0392 ¢, ~10/7 = 632.9013 ¢ (~22/21 = 87.2162 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~10/7 = 632.8351 ¢ (~22/21 = 87.1649 ¢)

Optimal ET sequence: 15, 40, 55

Badness (Sintel): 2.00

Aurora

For the 5-limit version, see Diaschismic–gothmic equivalence continuum #Goldis.

The aurora temperament (13 & 21) is very strongly related to the golden oneirotonic tuning (generator: (3 - sqrt(5))/2 octave = 458.3592135001 cents). It has a temperament structure superficially similar to father, with extremely flat fourths (subfourths/ultramajor third) or sharp fifths (superfifths/ultraminor sixths). However, unlike father, 12 of these bad fourths reach a more in-tune fifth, which is useful for creating resolutions when using a large enough gamut, such as the 13L 8s mos which has two good major & minor chords.

Subgroup: 2.3.5.7

Comma list: 1029/1000, 28672/28125

Mapping: [⟨1 -3 5 7], ⟨0 12 -7 -11]]

Optimal tunings:

  • WE: ~2 = 1200.6445 ¢, ~21/16 = 458.5945 ¢
error map: ⟨+0.645 -0.755 +6.748 -8.853]
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 458.3493 ¢
error map: ⟨0.000 -1.763 +5.241 -10.668]

Optimal ET sequence: 13, 21, 34, 55

Badness (Sintel): 6.13

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 1029/1000, 5632/5625

Mapping: [⟨1 -3 5 7 5], ⟨0 12 -7 -11 -4]]

Optimal tunings:

  • WE: ~2 = 1199.6148 ¢, ~21/16 = 458.2655 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 458.4131 ¢

Optimal ET sequence: 13, 21, 34

Badness (Sintel): 3.86

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 105/104, 512/507, 637/625

Mapping: [⟨1 -3 5 7 5 6], ⟨0 12 -7 -11 -4 -6]]

Optimal tunings:

  • WE: ~2 = 1199.2763 ¢, ~13/10 = 458.1671 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/10 = 458.4462 ¢

Optimal ET sequence: 13, 21, 34

Badness (Sintel): 2.75

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 56/55, 105/104, 170/169, 221/220, 375/374

Mapping: [⟨1 -3 5 7 5 6 6], ⟨0 12 -7 -11 -4 -6 -5]]

Optimal tunings:

  • WE: ~2 = 1199.2758 ¢, ~13/10 = 458.1670 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/10 = 458.4472 ¢

Optimal ET sequence: 13, 21, 34

Badness (Sintel): 2.25

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 56/55, 76/75, 105/104, 170/169, 190/187, 221/220

Mapping: [⟨1 -3 5 7 5 6 6 5], ⟨0 12 -7 -11 -4 -6 -5 -2]]

Optimal tunings:

  • WE: ~2 = 1199.7718 ¢, ~13/10 = 458.3568 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/10 = 458.4451 ¢

Optimal ET sequence: 13, 21, 34

Badness (Sintel): 2.29

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 56/55, 76/75, 105/104, 161/160, 170/169, 190/187, 221/220

Mapping: [⟨1 -3 5 7 5 6 6 5 3], ⟨0 12 -7 -11 -4 -6 -5 -2 4]]

Optimal tunings:

  • WE: ~2 = 1199.6512 ¢, ~13/10 = 458.2963 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/10 = 458.4307 ¢

Optimal ET sequence: 13, 21, 34

Badness (Sintel): 2.16

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 56/55, 76/75, 105/104, 116/115, 145/143, 161/160, 170/169, 221/220

Mapping: [⟨1 -3 5 7 5 6 6 5 3 6], ⟨0 12 -7 -11 -4 -6 -5 -2 4 -3]]

Optimal tunings:

  • WE: ~2 = 1199.7786 ¢, ~13/10 = 458.3411 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/10 = 458.4265 ¢

Optimal ET sequence: 13, 21, 34

Badness (Sintel): 2.11

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