Keenanismic temperaments: Difference between revisions

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: ''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]] ([[CTE]]): ~99/70 = 600.0000{{c}}, ~16/11 = 651.1082{{c}}, ~5/4 = 385.5778{{c}}


{{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]] (Smith): 0.529 × 10<sup>-3</sup>


=== 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 tuning (CTE): ~17/12 = 600.0000{{c}}, ~16/11 = 651.2344{{c}}, ~5/4 = 385.7289{{c}}


Badness: 0.340 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg }}
 
Badness (Smith): 0.340 × 10<sup>-3</sup>


== 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]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~11/10 = 165.8258{{c}}, ~5/4 = 384.6711{{c}}


{{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]] (Smith): 0.732 × 10<sup>-3</sup>


[[Category:Temperament collections]]
[[Category:Temperament collections]]

Revision as of 15:13, 27 November 2025

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 tuning (CTE): ~99/70 = 600.0000 ¢, ~16/11 = 651.1082 ¢, ~5/4 = 385.5778 ¢

Optimal ET sequence: 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de

Badness (Smith): 0.529 × 10-3

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 tuning (CTE): ~17/12 = 600.0000 ¢, ~16/11 = 651.2344 ¢, ~5/4 = 385.7289 ¢

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

Badness (Smith): 0.340 × 10-3

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 tuning (CTE): ~2 = 1200.0000 ¢, ~11/10 = 165.8258 ¢, ~5/4 = 384.6711 ¢

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

Badness (Smith): 0.732 × 10-3

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