Keenanismic temperaments: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
This is a collection of [[rank-3 temperament|rank-3]] [[temperament]]s [[tempering out]] the keenanisma, [[385/384]]. For the [[rank-4]] keenanismic temperament, see [[Keenanismic family]].  
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:  
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* [[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 (unimarv)|Marvel family]]
* [[Marvel]] (+225/224) → [[Marvel family #Undecimal marvel|Marvel family]]
* ''[[Mirwomo]]'' (+243/242) → [[Rastmic rank-3 clan #Mirwomo|Rastmic rank-3 clan]]
* ''[[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]]
* ''[[Diaschismic rank-3 family#Augustus|Augustus]]'' (+1232/1215) → [[Diaschismic rank-3 family #Augustus|Diaschismic rank-3 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]]
* ''[[Agni]]'' (+1375/1372) → [[Breed family #Agni|Breed family]]
* ''[[Akea]]'' (+2200/2187) → [[Hemifamity family #Akea|Hemifamity family]]
* ''[[Akea]]'' (+2200/2187) → [[Hemifamity family #Akea|Hemifamity family]]
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* ''[[Enlil]]'' (+6250/6237) → [[Kleismic rank-3 family #Enlil|Kleismic rank-3 family]]
* ''[[Enlil]]'' (+6250/6237) → [[Kleismic rank-3 family #Enlil|Kleismic rank-3 family]]


Considered below are ''uniwiz'' and ''tritiwiz''.
Considered below are uniwiz and tritiwiz.


== Uniwiz ==
== Uniwiz ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Uniwiz]].''  
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Uniwiz]].''  


Uniwiz 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. 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]])}.  
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]].  
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{{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]] ([[CTE]]): ~99/70 = 1\2, ~16/11 = 651.1082, ~5/4 = 385.5778
[[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]]: 0.529 × 10<sup>-3</sup>
[[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


Sval mapping: {{mapping| 2 1 0 7 8 6 | 0 2 0 3 -1 2 | 0 0 1 -1 0 0 }}
Subgroup-val mapping: {{mapping| 2 1 0 7 8 6 | 0 2 0 3 -1 2 | 0 0 1 -1 0 0 }}


Optimal tuning (CTE): ~17/12 = 1\2, ~16/11 = 651.2344, ~5/4 = 385.7289
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 ET sequence|legend=1| 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg }}
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: 0.340 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg }}
 
Badness (Sintel): 0.351


== Tritiwiz ==
== Tritiwiz ==
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{{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]] ([[CTE]]): ~2 = 1\1, ~11/10 = 165.8258, ~5/4 = 384.6711
[[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]]: 0.732 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.879


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Keenanismic temperaments| ]] <!-- main article -->
[[Category:Keenanismic temperaments| ]] <!-- main article -->
[[Category:Keenanismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 08:36, 8 June 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-3 temperaments tempering out the keenanisma, 385/384. For the rank-4 keenanismic temperament, see Keenanismic family.

Temperaments discussed elsewhere include:

Considered below are uniwiz and tritiwiz.

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 ([-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 commas, 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 {(S17), S33, S34, S35 = S49 × S50, (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 quartertones. It can be viewed as the temperament described by the three quartertone-size equal temperaments: 22, 24 and 26. In the 11-limit it shares the optimal patent val, 284, with keenanismic.

Subgroup: 2.3.5.7.11

Comma list: 385/384, 9801/9800

Mapping: [⟨2 1 0 7 8], ⟨0 2 0 3 -1], ⟨0 0 1 -1 0]]

mapping generators: ~99/70, ~16/11, ~5

Optimal tunings:

  • WE: ~99/70 = 600.2029 ¢, ~16/11 = 650.9604 ¢, ~5/4 = 385.1616 ¢
error map: ⟨+0.406 +0.169 -0.341 -0.498 -0.655]
  • CWE: ~99/70 = 600.0000 ¢, ~16/11 = 650.8311 ¢, ~5/4 = 385.1658 ¢
error map: ⟨0.000 -0.293 -1.148 -1.499 -2.149]

Optimal ET sequence: 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

Comma list: 289/288, 385/384, 561/560

Subgroup-val mapping: [⟨2 1 0 7 8 6], ⟨0 2 0 3 -1 2], ⟨0 0 1 -1 0 0]]

Gencom 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 ¢, ~16/11 = 651.0126 ¢, ~5/4 = 385.1864 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~16/11 = 650.8664 ¢, ~5/4 = 385.1951 ¢

Optimal ET sequence: 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg

Badness (Sintel): 0.351

Tritiwiz

Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993

Mapping: [⟨1 2 0 8 1], ⟨0 -3 0 -4 1], ⟨0 0 1 -2 1]]

mapping generators: ~2, ~11/10, ~5

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~11/10 = 166.1752 ¢, ~5/4 = 384.3796 ¢
error map: ⟨+0.400 +0.320 -1.133 -0.684 +0.438]
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0419 ¢, ~5/4 = 384.3625 ¢
error map: ⟨0.000 -0.081 -1.951 -1.718 -0.914]

Optimal ET sequence: 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd

Badness (Sintel): 0.879

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