Keegic temperaments: Difference between revisions
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== Aurora == | == Aurora == | ||
The ''aurora'' temperament (13&21) is very strongly related to the '''golden father tuning''' (generator: <math>\frac{3-\sqrt{5}}{2}</math> octave = 458.3592135001 cents). It has a temperament structure superficially similar to [[Father family|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. | The ''aurora'' temperament (13&21) is very strongly related to the '''[[Logarithmic phi|golden father tuning]]''' (generator: <math>\frac{3-\sqrt{5}}{2}</math> octave = 458.3592135001 cents). It has a temperament structure superficially similar to [[Father family|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 | Subgroup: 2.3.5.7 | ||
Revision as of 03:41, 24 January 2022
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
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 GPV sequence: Template:Val list
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 GPV sequence: Template:Val list
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 GPV sequence: Template:Val list
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 GPV sequence: Template:Val list
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 GPV sequence: Template:Val list
Badness: 0.030236