Keegic temperaments

Keegic rank-two temperaments temper out the keega, [-3 1 -3 3⟩ = 1029/1000, the difference three 7/5s and perfect eleventh 8/3.

Temperaments discussed elsewhere include antitonic, kangaroo, keemun, liese, progress, subklei, and triton.

Aurora

The aurora temperament (13&21) is very strongly related to the golden father tuning (generator: [math]\displaystyle{ \frac{3-\sqrt{5}}{2} }[/math] octave = 458.3592135001 cents). It has a temperament structure superficially similar to father, with extremely flat fourths (sub-fourths/ultra-major third) or sharp fifths (super-fifths/ultra-minor 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]]

Wedgie: ⟨⟨ 12 -7 -11 -39 -51 -6 ]]

POTE generator: ~21/16 = 458.348

Optimal ET sequence: 8bd, 13, 21, 34, 55, 89d, 144cdd

Badness: 0.242134

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]]

POTE generator: ~21/16 = 458.413

Optimal ET sequence: 8bd, 13, 21, 34, 55e, 89dee

Badness: 0.116654

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]]

POTE generator: ~13/10 = 458.444

Optimal ET sequence: 8bd, 13, 21, 34, 55e, 89deef

Badness: 0.066541

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]]

POTE generator: ~13/10 = 458.444

Optimal ET sequence: 8bd, 13, 21, 34, 55e, 89deef

Badness: 0.044148

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]]

POTE generator: ~13/10 = 458.444

Optimal ET sequence: 8bd, 13, 21, 34, 55eh, 89deefhh

Badness: 0.037621

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]]

POTE generator: ~13/10 = 458.430

Optimal ET sequence: 8bd, 13, 21, 34, 55eh, 89deefhh

Badness: 0.030236

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]]

POTE generator: ~13/10 = 458.426

Optimal ET sequence: 8bd, 13, 21, 34, 55eh, 89deefhh

Badness: 0.025201