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Keenanismic temperaments: Difference between revisions

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This is a collection of rank-3 temperaments tempering out the keenanisma, [[385/384]]. For the rank-4 keenanismic temperament, see [[Keenanismic family]].  
{{Technical data page}}
This is a collection of [[rank-3 temperament]]s [[tempering out]] the keenanisma, [[385/384]]. For the [[rank-4]] keenanismic temperament, see [[Keenanismic family]].  


== Headquarters ==
Temperaments discussed elsewhere include:
Headquarters tempers out 1500625/1492992 = {{monzo| -11 -6 4 4 }} in the 7-limit, and the keenanisma, 385/384, the kalisma, [[9801/9800]], as well as the [[Alpharabian comma]], 131769/131072 = {{monzo| -17 2 0 0 4 }}, in the 11-limit. It features the split of the whole tone into four equal parts, each representing 36/35~33/32. Therefore, it is most interesting to those who would like to make extensive use of quartertones. It can be viewed as one which is joined by the three quartertone-size equal temperaments: [[22edo|22]], [[24edo|24]], and [[26edo|26]]. It shares the [[optimal patent val]], [[284edo|284]], with [[keenanismic]].
* ''[[Selenium]]'' (+49/48 or 56/55) → [[Semaphoresmic family #Selenium|Semaphoresmic family]]
* ''[[Polyhymnia]]'' (+81/80) → [[Didymus rank-3 family #Polyhymnia|Didymus rank-3 family]]
* ''[[Orwellian]]'' (+99/98) → [[Orwellismic family #Orwellian|Orwellismic family]]
* ''[[Keemic]]'' (+100/99) → [[Keemic family #Undecimal keemic|Keemic family]]
* [[Zeus]] (+121/120 or 176/175) → [[Porwell family #Zeus|Porwell family]] and [[Biyatismic clan #Zeus|Biyatismic clan]]
* [[Starling]] (+126/125) → [[Starling family #Undecimal starling|Starling family]]
* [[Marvel]] (+225/224) → [[Marvel family #Undecimal marvel|Marvel family]]
* ''[[Mirwomo]]'' (+243/242) → [[Rastmic rank-3 clan #Mirwomo|Rastmic rank-3 clan]]
* [[Sensamagic]] (+245/243) → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]
* [[Portent]] (+441/440) → [[Gamelismic family #Portent|Gamelismic family]]
* ''[[Diaschismic rank-3 family #Augustus|Augustus]]'' (+1232/1215) → [[Diaschismic rank-3 family #Augustus|Diaschismic rank-3 family]]
* ''[[Agni]]'' (+1375/1372) → [[Breed family #Agni|Breed family]]
* ''[[Akea]]'' (+2200/2187) → [[Hemifamity family #Akea|Hemifamity family]]
* ''[[Zelda]]'' (+3388/3375) → [[Horwell family #Zelda|Horwell family]]
* ''[[Beyla]]'' (+4375/4374) → [[Ragismic family #Beyla|Ragismic family]]
* ''[[Enlil]]'' (+6250/6237) → [[Kleismic rank-3 family #Enlil|Kleismic rank-3 family]]


=== 7-limit ===
Considered below are uniwiz and tritiwiz.
Subgroup: 2.3.5.7


[[Comma list]]: 1500625/1492992
== Uniwiz ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Uniwiz]].''


[[Mapping]]: [{{val| 2 1 0 7 }}, {{val| 0 2 0 3 }}, {{val| 0 0 1 -1 }}]
Uniwiz tempers out the keenanisma, 385/384, the kalisma, [[9801/9800]], as well as the [[Alpharabian comma]], 131769/131072 ({{monzo| -17 2 0 0 4 }}), in the 11-limit. The [[extension]] to the 2.3.5.7.11.17 [[subgroup]] tempers out two no-13's 17-limit [[PIFE comma]]s, [[561/560]] and [[595/594]], along with their product, [[289/288]], thus also equating the half-octave with [[17/12]][[~]][[24/17]]. The [[S-expression]]-based comma list of this extension is {([[289/288|S17]]), [[1089/1088|S33]], [[1156/1155|S34]], [[1225/1224|S35 = S49 × S50]], ([[9801/9800|S99 = S33/S35]])}.


Mapping generators: ~1225/864, ~35/24, ~5
It features the split of the whole tone into four equal parts, each representing [[36/35]]~[[35/34]]~[[34/33]]~[[33/32]]. Therefore, it is most interesting to those who would like to make extensive use of [[quartertone]]s. It can be viewed as the temperament described by the three quartertone-size equal temperaments: [[22edo|22]], [[24edo|24]] and [[26edo|26]]. In the 11-limit it shares the [[optimal patent val]], [[284edo|284]], with [[keenanismic]].


{{Multival|legend=1|rank=3| 4 -4 -6 11 }}
[[Subgroup]]: 2.3.5.7.11


[[POTE generator]]s: ~36/35 = 50.9529 or ~48/35 = 549.0471, ~5/4 = 385.3326
[[Comma list]]: 385/384, 9801/9800
 
{{Mapping|legend=1| 2 1 0 7 8 | 0 2 0 3 -1 | 0 0 1 -1 0 }}
: mapping generators: ~99/70, ~16/11, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 600.2029{{c}}, ~16/11 = 650.9604{{c}}, ~5/4 = 385.1616{{c}}
: [[error map]]: {{val| +0.406 +0.169 -0.341 -0.498 -0.655 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~16/11 = 650.8311{{c}}, ~5/4 = 385.1658{{c}}
: error map: {{val| 0.000 -0.293 -1.148 -1.499 -2.149 }}
 
{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }}
 
[[Badness]] (Sintel): 0.636
 
=== 2.3.5.7.11.17 subgroup ===
Subgroup: 2.3.5.7.11.17


{{Val list|legend=1| 22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd }}
Comma list: 289/288, 385/384, 561/560


[[Badness]]: 0.7042 × 10<sup>-3</sup>
Subgroup-val mapping: {{mapping| 2 1 0 7 8 6 | 0 2 0 3 -1 2 | 0 0 1 -1 0 0 }}


=== 11-limit ===
Gencom mapping: {{mapping| 2 1 0 7 8 0 6 | 0 2 0 3 -1 0 2 | 0 0 1 -1 0 0 0 }}
Subgroup: 2.3.5.7.11


[[Comma list]]: 385/384, 9801/9800
Optimal tunings:
* WE: ~99/70 = 600.2365{{c}}, ~16/11 = 651.0126{{c}}, ~5/4 = 385.1864{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 650.8664{{c}}, ~5/4 = 385.1951{{c}}
 
{{Optimal ET sequence|legend=0| 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg }}
 
Badness (Sintel): 0.351
 
== Tritiwiz ==
[[Subgroup]]: 2.3.5.7.11


[[Mapping]]: [{{val| 2 1 0 7 8 }}, {{val| 0 2 0 3 -1 }}, {{val| 0 0 1 -1 0 }}]
[[Comma list]]: 385/384, 4000/3993


{{Multival|legend=1|rank=3| 4 -4 0 -6 2 -2 11 17 -17 -31 }}
{{Mapping|legend=1| 1 2 0 8 1 | 0 -3 0 -4 1 | 0 0 1 -2 1 }}
: mapping generators: ~2, ~11/10, ~5


[[POTE generator]]s: ~33/32 = 50.7404 or ~11/8 = 549.2596, ~5/4 = 385.0315
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0000{{c}}, ~11/10 = 166.1752{{c}}, ~5/4 = 384.3796{{c}}
: [[error map]]: {{val| +0.400 +0.320 -1.133 -0.684 +0.438 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/10 = 166.0419{{c}}, ~5/4 = 384.3625{{c}}
: error map: {{val| 0.000 -0.081 -1.951 -1.718 -0.914 }}


{{Val list|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }}
{{Optimal ET sequence|legend=1| 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd }}


[[Badness]]: 0.5292 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.879


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Keenanismic temperaments| ]] <!-- main article -->
[[Category:Keenanismic temperaments| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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