Keenanismic temperaments: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
This is a collection of [[rank-3 | This is a collection of [[rank-3 temperament]]s [[tempering out]] the keenanisma, [[385/384]]. For the [[rank-4]] keenanismic temperament, see [[Keenanismic family]]. | ||
Temperaments discussed elsewhere include: | Temperaments discussed elsewhere include: | ||
* ''[[Selenium]]'' (+49/48 or 56/55) → [[ | * ''[[Selenium]]'' (+49/48 or 56/55) → [[Semaphoresmic family #Selenium|Semaphoresmic family]] | ||
* ''[[Polyhymnia]]'' (+81/80) → [[Didymus rank | * ''[[Polyhymnia]]'' (+81/80) → [[Didymus rank-3 family #Polyhymnia|Didymus rank-3 family]] | ||
* ''[[Orwellian]]'' (+99/98) → [[Orwellismic family #Orwellian|Orwellismic 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]] | * [[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]] | * [[Starling]] (+126/125) → [[Starling family #Undecimal starling|Starling family]] | ||
* [[Marvel]] (+225/224) → [[Marvel family #Undecimal marvel | * [[Marvel]] (+225/224) → [[Marvel family #Undecimal marvel|Marvel family]] | ||
* ''[[Mirwomo]]'' (+243/242) → [[Rastmic rank | * ''[[Mirwomo]]'' (+243/242) → [[Rastmic rank-3 clan #Mirwomo|Rastmic rank-3 clan]] | ||
* [[Sensamagic]] (+245/243) → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]] | * [[Sensamagic]] (+245/243) → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]] | ||
* [[Portent]] (+441/440) → [[Gamelismic family #Portent|Gamelismic family]] | * [[Portent]] (+441/440) → [[Gamelismic family #Portent|Gamelismic family]] | ||
* ''[[Augustus]]'' (+1232/1215) → [[Diaschismic rank | * ''[[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]] | * ''[[Agni]]'' (+1375/1372) → [[Breed family #Agni|Breed family]] | ||
* ''[[Akea]]'' (+2200/2187) → [[Hemifamity family #Akea|Hemifamity family]] | |||
* ''[[Zelda]]'' (+3388/3375) → [[Horwell family #Zelda|Horwell family]] | * ''[[Zelda]]'' (+3388/3375) → [[Horwell family #Zelda|Horwell family]] | ||
* ''[[Beyla]]'' (+4375/4374) → [[Ragismic family #Beyla|Ragismic family]] | * ''[[Beyla]]'' (+4375/4374) → [[Ragismic family #Beyla|Ragismic family]] | ||
* ''[[Enlil]]'' (+6250/6237) → [[Kleismic rank | * ''[[Enlil]]'' (+6250/6237) → [[Kleismic rank-3 family #Enlil|Kleismic rank-3 family]] | ||
Considered below are | Considered below are uniwiz and tritiwiz. | ||
== Uniwiz == | == Uniwiz == | ||
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Uniwiz]].'' | |||
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]])}. | |||
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]]. | 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]]. | ||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
| Line 47: | Line 34: | ||
{{Mapping|legend=1| 2 1 0 7 8 | 0 2 0 3 -1 | 0 0 1 -1 0 }} | {{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]] | [[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 }} | {{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.636 | ||
=== 2.3.5.7.11.17 subgroup === | === 2.3.5.7.11.17 subgroup === | ||
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Comma list: 289/288, 385/384, 561/560 | Comma list: 289/288, 385/384, 561/560 | ||
Subgroup-val mapping: {{mapping| 2 1 0 7 8 6 | 0 2 0 3 -1 2 | 0 0 1 -1 0 0 }} | |||
Gencom mapping: {{mapping| 2 1 0 7 8 0 6 | 0 2 0 3 -1 0 2 | 0 0 1 -1 0 0 0 }} | |||
{{ | 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}} | |||
Badness: 0. | {{Optimal ET sequence|legend=0| 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg }} | ||
Badness (Sintel): 0.351 | |||
== Tritiwiz == | == Tritiwiz == | ||
| Line 73: | Line 69: | ||
{{Mapping|legend=1| 1 2 0 8 1 | 0 -3 0 -4 1 | 0 0 1 -2 1 }} | {{Mapping|legend=1| 1 2 0 8 1 | 0 -3 0 -4 1 | 0 0 1 -2 1 }} | ||
: mapping generators: ~2, ~11/10, ~5 | : mapping generators: ~2, ~11/10, ~5 | ||
[[Optimal tuning]] | [[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 }} | |||
{{Optimal ET sequence|legend=1| 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd }} | {{Optimal ET sequence|legend=1| 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.879 | ||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category:Keenanismic temperaments| ]] <!-- main article --> | [[Category:Keenanismic temperaments| ]] <!-- main article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||