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Keegic temperaments: Difference between revisions

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{{Technical data page}}
{{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]].
This is a collection of rank-2 '''keegic temperaments''', which [[tempering out|temper out]] the [[keega]] ([[ratio]]: 1029/1000, {{monzo|legend=1| -3 1 -3 3 }}), the difference a stack of three [[10/7]]'s and the [[3/1|3rd harmonic]].


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]], and [[Augmented family|trisected]].
Temperaments discussed elsewhere include:
* ''[[Parakangaroo]]'' (+28/27) → [[Trienstonic clan #Parakangaroo|Trienstonic clan]]
* ''[[Progress]]'' (+64/63) → [[Archytas clan #Progress|Archytas clan]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Triton]]'' (+225/224) → [[Marvel temperaments #Triton|Marvel temperaments]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* ''[[Subklei]]'' (+36/35) → [[Mint temperaments #Subklei|Mint temperaments]]
* ''[[Trisected]]'' (+128/125) → [[Augmented family #Trisected|Augmented family]]
* ''[[Xenia]]'' (+245/243) → [[Sensamagic clan #Xenia|Sensamagic clan]]
* ''[[Antitonic]]'' (+9/8) → [[Very low accuracy temperaments #Antitonic|Very low accuracy temperaments]]
 
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 {{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 ==
== Aurora ==
: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''
: ''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.
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 (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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1000, 28672/28125
[[Comma list]]: 1029/1000, 28672/28125


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


[[POTE generator]]: ~21/16 = 458.348
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6445{{c}}, ~21/16 = 458.5945{{c}}
: [[error map]]: {{val| +0.645 -0.755 +6.748 -8.853 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 458.3493{{c}}
: error map: {{val| 0.000 -1.763 +5.241 -10.668 }}


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


[[Badness]]: 0.242134
[[Badness]] (Sintel): 6.13


=== 11-limit ===
=== 11-limit ===
Line 26: Line 91:
Comma list: 56/55, 1029/1000, 5632/5625
Comma list: 56/55, 1029/1000, 5632/5625


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


POTE generator: ~21/16 = 458.413
Optimal tunings:  
* WE: ~2 = 1199.6148{{c}}, ~21/16 = 458.2655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 458.4131{{c}}


{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55e, 89dee }}
{{Optimal ET sequence|legend=0| 13, 21, 34 }}


Badness: 0.116654
Badness (Sintel): 3.86


=== 13-limit ===
=== 13-limit ===
Line 39: Line 106:
Comma list: 56/55, 105/104, 512/507, 637/625
Comma list: 56/55, 105/104, 512/507, 637/625


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


POTE generator: ~13/10 = 458.444
Optimal tunings:  
* WE: ~2 = 1199.2763{{c}}, ~13/10 = 458.1671{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 458.4462{{c}}


{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55e, 89deef }}
{{Optimal ET sequence|legend=0| 13, 21, 34 }}


Badness: 0.066541
Badness (Sintel): 2.75


=== 17-limit ===
=== 17-limit ===
Line 52: Line 121:
Comma list: 56/55, 105/104, 170/169, 221/220, 375/374
Comma list: 56/55, 105/104, 170/169, 221/220, 375/374


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


POTE generator: ~13/10 = 458.444
Optimal tunings:  
* WE: ~2 = 1199.2758{{c}}, ~13/10 = 458.1670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 458.4472{{c}}


{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55e, 89deef }}
{{Optimal ET sequence|legend=0| 13, 21, 34 }}


Badness: 0.044148
Badness (Sintel): 2.25


=== 19-limit ===
=== 19-limit ===
Line 65: Line 136:
Comma list: 56/55, 76/75, 105/104, 170/169, 190/187, 221/220
Comma list: 56/55, 76/75, 105/104, 170/169, 190/187, 221/220


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


POTE generator: ~13/10 = 458.444
Optimal tunings:  
* WE: ~2 = 1199.7718{{c}}, ~13/10 = 458.3568{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 458.4451{{c}}


{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55eh, 89deefhh }}
{{Optimal ET sequence|legend=0| 13, 21, 34 }}


Badness: 0.037621
Badness (Sintel): 2.29


=== 23-limit ===
=== 23-limit ===
Line 78: Line 151:
Comma list: 56/55, 76/75, 105/104, 161/160, 170/169, 190/187, 221/220
Comma list: 56/55, 76/75, 105/104, 161/160, 170/169, 190/187, 221/220


Mapping: [{{val|1 -3 5 7 5 6 6 5 3}}, {{val|0 12 -7 -11 -4 -6 -5 -2 4}}]
Mapping: {{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 tunings:  
* WE: ~2 = 1199.6512{{c}}, ~13/10 = 458.2963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 458.4307{{c}}


{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55eh, 89deefhh }}
{{Optimal ET sequence|legend=0| 13, 21, 34 }}


Badness: 0.030236
Badness (Sintel): 2.16


=== 29-limit ===
=== 29-limit ===
Line 91: Line 166:
Comma list: 56/55, 76/75, 105/104, 116/115, 145/143, 161/160, 170/169, 221/220
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}}]
Mapping: {{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 tunings:  
* WE: ~2 = 1199.7786{{c}}, ~13/10 = 458.3411{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 458.4265{{c}}


{{Optimal ET sequence|legend=1| 8bd, 13, 21, 34, 55eh, 89deefhh }}
{{Optimal ET sequence|legend=0| 13, 21, 34 }}


Badness: 0.025201
Badness (Sintel): 2.11


[[Category:Keegic temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Keegic]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Keegic_temperaments"