Keegic temperaments: Difference between revisions

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'''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]].

== 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 {{nowrap| 15 & 55 }} temperament. It was named by [[Xenllium]] in 2021.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1000, 6144/6125

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

[[Optimal tuning]]s:

* [[WE]]: ~8/7 = 240.0512{{c}}, ~10/7 = 633.5744{{c}} (~25/24 = 86.5794{{c}})

: [[error map]]: {{val| +0.256 -1.232 +7.722 -8.108 }}

* [[CWE]]: ~8/7 = 240.0000{{c}}, ~10/7 = 633.4834{{c}} (~25/24 = 86.5160{{c}})

: error map: {{val| 0.000 -1.505 +7.170 -8.826 }}

[[Badness]] (Sintel): 4.66

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

Mapping: {{mapping| 5 0 9 14 12 | 0 3 1 0 2 }}

Optimal tunings:

* WE: ~8/7 = 240.0916{{c}}, ~10/7 = 633.6802{{c}} (~22/21 = 86.5947{{c}})

* CWE: ~8/7 = 240.0000{{c}}, ~10/7 = 633.5204{{c}} (~22/21 = 86.4796{{c}})

Badness (Sintel): 2.16

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

Mapping: {{mapping| 5 9 12 14 18 20 | 0 3 1 0 2 4 }}

Optimal tunings:

* WE: ~8/7 = 240.0392{{c}}, ~10/7 = 632.9013{{c}} (~22/21 = 87.2162{{c}})

* CWE: ~8/7 = 240.0000{{c}}, ~10/7 = 632.8351{{c}} (~22/21 = 87.1649{{c}})

Badness (Sintel): 2.00

== Aurora ==

The ''aurora'' temperament (13&~~amp;~~21) is very strongly related to the ~~'''~~[[Logarithmic phi|golden oneirotonic 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

@@ Line 2: / Line 2: @@
 '''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]].
+Temperaments discussed elsewhere include [[Very low accuracy temperaments|antitonic]], [[Trienstonic clan|kangaroo]], [[Kleismic family|keemun]], [[Meantone family|liese]], [[Archytas clan|progress]], [[Mint temperaments|subklei]], [[Marvel temperaments|triton]], [[Augmented family|trisected]], and [[Sensamagic clan|xenia]].
+== 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 {{nowrap| 15 & 55 }} temperament. It was named by [[Xenllium]] in 2021.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 1029/1000, 6144/6125
+{{Mapping|legend=1| 5 0 9 14 | 0 3 1 0 }}
+: mapping generators: ~8/7, ~10/7
+[[Optimal tuning]]s:
+* [[WE]]: ~8/7 = 240.0512{{c}}, ~10/7 = 633.5744{{c}} (~25/24 = 86.5794{{c}})
+: [[error map]]: {{val| +0.256 -1.232 +7.722 -8.108 }}
+* [[CWE]]: ~8/7 = 240.0000{{c}}, ~10/7 = 633.4834{{c}} (~25/24 = 86.5160{{c}})
+: error map: {{val| 0.000 -1.505 +7.170 -8.826 }}
+{{Optimal ET sequence|legend=1| 15, 40, 55 }}
+[[Badness]] (Sintel): 4.66
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 121/120, 176/175, 1029/1000
+Mapping: {{mapping| 5 0 9 14 12 | 0 3 1 0 2 }}
+Optimal tunings:
+* WE: ~8/7 = 240.0916{{c}}, ~10/7 = 633.6802{{c}} (~22/21 = 86.5947{{c}})
+* CWE: ~8/7 = 240.0000{{c}}, ~10/7 = 633.5204{{c}} (~22/21 = 86.4796{{c}})
+{{Optimal ET sequence|legend=0| 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: {{mapping| 5 9 12 14 18 20 | 0 3 1 0 2 4 }}
+Optimal tunings:
+* WE: ~8/7 = 240.0392{{c}}, ~10/7 = 632.9013{{c}} (~22/21 = 87.2162{{c}})
+* CWE: ~8/7 = 240.0000{{c}}, ~10/7 = 632.8351{{c}} (~22/21 = 87.1649{{c}})
+{{Optimal ET sequence|legend=0| 15, 40, 55 }}
+Badness (Sintel): 2.00
 == Aurora ==
-The ''aurora'' temperament (13&amp;21) is very strongly related to the '''[[Logarithmic phi|golden oneirotonic 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 12: / Line 64: @@
 [[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