Keegic temperaments: Difference between revisions

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Keegic rank-two ~~temperaments~~ temper out the keega, {{monzo|-3 1 -3 3}} = 1029/1000, the difference three [[7/5]]s and perfect eleventh [[8/3]].

'''Keegic''' [[rank two temperament]]s temper out the keega, {{monzo|-3 1 -3 3}} = 1029/1000, the difference three [[7/5]]s and perfect eleventh [[8/3]].

== Aurora ==

The ''aurora'' temperament (13&~~amp;~~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 &~~amp;~~ minor chords.

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

The ''aurora'' temperament ({{nowrap| 13 & 21 }}) is very strongly related to the [[Logarithmic phi|golden oneirotonic tuning]] (generator: {{nowrap| (3 - sqrt(5))/2 }} 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

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[[Mapping]]: [{{val|1 -3 5 7}}, {{val|0 12 -7 -11}}]

~~{{Multival|legend=1|12 -7 -11 -39 -51 -6}}~~

[[POTE generator]]: ~21/16 = 458.348

{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55, 89d, 144cdd }}

[[Badness]]: 0.242134

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POTE generator: ~21/16 = 458.413

Optimal ~~GPV~~ sequence~~: {{Val list~~| 8bd, 13, 21, 34, 55e, 89dee }}

Badness: 0.116654

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POTE generator: ~13/10 = 458.444

Optimal ~~GPV~~ sequence~~: {{Val list~~| 8bd, 13, 21, 34, 55e, 89deef }}

Badness: 0.066541

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POTE generator: ~13/10 = 458.444

Optimal ~~GPV~~ sequence~~: {{Val list~~| 8bd, 13, 21, 34, 55e, 89deef }}

Badness: 0.044148

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POTE generator: ~13/10 = 458.444

Optimal ~~GPV~~ sequence~~: {{Val list~~| 8bd, 13, 21, 34, 55eh, 89deefhh }}

{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55eh, 89deefhh }}

Badness: 0.037621

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POTE generator: ~13/10 = 458.430

Optimal ~~GPV~~ sequence~~: {{Val list~~| 8bd, 13, 21, 34, 55eh, 89deefhh }}

{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55eh, 89deefhh }}

Badness: 0.030236

[[Category:Temperament ~~collection~~]]

=== 29-limit ===

[[Category:~~Mirwomo~~]]

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: [{{val|1 -3 5 7 5 6 6 5 3 6}}, {{val|0 12 -7 -11 -4 -6 -5 -2 4 -3}}]

POTE generator: ~13/10 = 458.426

{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55eh, 89deefhh }}

Badness: 0.025201

[[Category:Temperament collections]]

[[Category:Pages with mostly numerical content]]

[[Category:Keegic]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
-Keegic rank-two temperaments temper out the keega, {{monzo|-3 1 -3 3}} = 1029/1000, the difference three [[7/5]]s and perfect eleventh [[8/3]].
+{{Technical data page}}
+'''Keegic''' [[rank two temperament]]s temper out the keega, {{monzo|-3 1 -3 3}} = 1029/1000, the difference three [[7/5]]s and perfect eleventh [[8/3]].
 Temperaments discussed elsewhere include [[Very low accuracy temperaments|antitonic]], [[Trienstonic clan|kangaroo]], [[Kleismic family|keemun]], [[Meantone family|liese]], [[Archytas clan|progress]], [[Mint temperaments|subklei]], and [[Marvel temperaments|triton]].
 == Aurora ==
-The ''aurora'' temperament (13&amp;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 &amp; minor chords.
+: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''
+The ''aurora'' temperament ({{nowrap| 13 & 21 }}) is very strongly related to the [[Logarithmic phi|golden oneirotonic tuning]] (generator: {{nowrap| (3 - sqrt(5))/2 }} 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
@@ Line 11: / Line 14: @@
 [[Mapping]]: [{{val|1 -3 5 7}}, {{val|0 12 -7 -11}}]
-{{Multival|legend=1|12 -7 -11 -39 -51 -6}}
 [[POTE generator]]: ~21/16 = 458.348
-{{Val list|legend=1| 8bd, 13, 21, 34, 55, 89d, 144cdd }}
+{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55, 89d, 144cdd }}
 [[Badness]]: 0.242134
@@ Line 29: / Line 30: @@
 POTE generator: ~21/16 = 458.413
-Optimal GPV sequence: {{Val list| 8bd, 13, 21, 34, 55e, 89dee }}
+{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55e, 89dee }}
 Badness: 0.116654
@@ Line 42: / Line 43: @@
 POTE generator: ~13/10 = 458.444
-Optimal GPV sequence: {{Val list| 8bd, 13, 21, 34, 55e, 89deef }}
+{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55e, 89deef }}
 Badness: 0.066541
@@ Line 55: / Line 56: @@
 POTE generator: ~13/10 = 458.444
-Optimal GPV sequence: {{Val list| 8bd, 13, 21, 34, 55e, 89deef }}
+{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55e, 89deef }}
 Badness: 0.044148
@@ Line 68: / Line 69: @@
 POTE generator: ~13/10 = 458.444
-Optimal GPV sequence: {{Val list| 8bd, 13, 21, 34, 55eh, 89deefhh }}
+{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55eh, 89deefhh }}
 Badness: 0.037621
@@ Line 81: / Line 82: @@
 POTE generator: ~13/10 = 458.430
-Optimal GPV sequence: {{Val list| 8bd, 13, 21, 34, 55eh, 89deefhh }}
+{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55eh, 89deefhh }}
 Badness: 0.030236
-[[Category:Temperament collection]]
+=== 29-limit ===
-[[Category:Mirwomo]]
+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: [{{val|1 -3 5 7 5 6 6 5 3 6}}, {{val|0 12 -7 -11 -4 -6 -5 -2 4 -3}}]
+POTE generator: ~13/10 = 458.426
+{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55eh, 89deefhh }}
+Badness: 0.025201
+[[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
+[[Category:Keegic]]
 [[Category:Rank 2]]