Keenanismic temperaments: Difference between revisions

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Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Selenium]]'', {49/48, 56/55} → [[Semiphore family #Selenium]]
* ''[[Selenium]]'', {49/48, 56/55} → [[Semiphore family #Selenium|Semiphore family]]
* ''[[Polyhymnia]]'', {81/80, 385/384} → [[Didymus rank three family #Polyhymnia]]
* ''[[Polyhymnia]]'', {81/80, 385/384} → [[Didymus rank three family #Polyhymnia|Didymus rank-3 family]]
* ''[[Orwellian]]'', {99/98, 385/384} → [[Orwellismic family #Orwellian]]
* ''[[Orwellian]]'', {99/98, 385/384} → [[Orwellismic family #Orwellian|Orwellismic family]]
* ''[[Supermagic]]'', {100/99, 385/384} → [[Keemic family #Undecimal supermagic]]
* ''[[Supermagic]]'', {100/99, 385/384} → [[Keemic family #Undecimal supermagic|Keemic family]]
* [[Zeus]], {121/120, 176/175} → [[Porwell family #Zeus]]
* [[Zeus]], {121/120, 176/175} → [[Porwell family #Zeus|Porwell family]]
* ''[[Starling]]'', {126/125, 385/384} → [[Starling family #Undecimal starling]]
* ''[[Starling]]'', {126/125, 385/384} → [[Starling family #Undecimal starling|Starling family]]
* [[Marvel]], {225/224, 385/384} → [[Marvel family #Undecimal marvel (unimarv)]]
* [[Marvel]], {225/224, 385/384} → [[Marvel family #Undecimal marvel (unimarv)|Marvel family]]
* ''[[Mirwomo]]'', {243/242, 385/384} → [[Rastmic temperaments #Mirwomo]]
* ''[[Mirwomo]]'', {243/242, 385/384} → [[Rastmic rank three clan #Mirwomo|Rastmic rank-3 clan]]
* [[Sensamagic]], {245/243, 385/384} → [[Sensamagic family #Undecimal sensamagic]]
* [[Sensamagic]], {245/243, 385/384} → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]
* ''[[Portent]]'', {385/384, 441/440} → [[Gamelismic family #Portent]]
* [[Portent]], {385/384, 441/440} → [[Gamelismic family #Portent|Gamelismic family]]
* ''[[Augustus]]'', {385/384, 1232/1215} → [[Diaschismic rank three family #Augustus]]
* ''[[Augustus]]'', {385/384, 1232/1215} → [[Diaschismic rank three family #Augustus|Diaschismic rank-3 family]]
* ''[[Agni]]'', {385/384, 1375/1372} → [[Breed family #Agni]]
* ''[[Agni]]'', {385/384, 1375/1372} → [[Breed family #Agni|Breed family]]
* ''[[Beyla]]'', {385/384, 4375/4374} → [[Ragismic family #Beyla]]
* ''[[Beyla]]'', {385/384, 4375/4374} → [[Ragismic family #Beyla|Ragismic family]]
* ''[[Enlil]]'', {385/384, 6250/6237} → [[Kleismic rank three family #Enlil]]
* ''[[Enlil]]'', {385/384, 6250/6237} → [[Kleismic rank three family #Enlil|Kleismic rank-3 family]]


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


== Uniwiz ==
== 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. In the 17-limit it tempers 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.
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. In the 17-limit it tempers 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.


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: [[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 quartertones. 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]].  


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1500625/1492992
[[Comma list]]: 1500625/1492992
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{{Multival|legend=1|rank=3| 4 -4 -6 11 }}
{{Multival|legend=1|rank=3| 4 -4 -6 11 }}


[[POTE generator]]s: ~36/35 = 50.9529 or ~48/35 = 549.0471, ~5/4 = 385.3326
Optimal tuning ([[CTE]]): ~35/24 = 651.1974, ~5/4 = 385.6847


{{Val list|legend=1| 22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd }}
{{Val list|legend=1| 22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd }}
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=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 385/384, 9801/9800
[[Comma list]]: 385/384, 9801/9800
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{{Multival|legend=1|rank=3| 4 -4 0 -6 2 -2 11 17 -17 -31 }}
{{Multival|legend=1|rank=3| 4 -4 0 -6 2 -2 11 17 -17 -31 }}


[[POTE generator]]s: ~33/32 = 50.7404 or ~11/8 = 549.2596, ~5/4 = 385.0315
Optimal tuning ([[CTE]]): ~16/11 = 651.1082, ~5/4 = 385.5778


{{Val list|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }}
{{Val list|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }}
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[[Badness]]: 0.529 × 10<sup>-3</sup>
[[Badness]]: 0.529 × 10<sup>-3</sup>


=== No-13's 17-limit ===
=== 2.3.5.7.11.17 subgroup ===
Subgroup 2.3.5.7.11.17
Subgroup: 2.3.5.7.11.17
 
[[Comma list]]: 385/384, 9801/9800, 256/255
 
[[Mapping]]: [{{val| 2 1 0 7 8 6 }}, {{val| 0 2 0 3 -1 2 }}, {{val| 0 0 1 -1 0 0 }}]


[[Wedgie]]: ?
Comma list: 289/288, 385/384, 561/560


[[CTE tuning]]: 17/12~99/70 = 600.000, 33/32~34/33~35/34~36/35 = 51.234 or 16/11~35/24 = 651.234, 5/4 = 385.730
Sval mapping: [{{val| 2 1 0 7 8 6 }}, {{val| 0 2 0 3 -1 2 }}, {{val| 0 0 1 -1 0 0 }}]


All EDOs supporting the temperament by patent val: {{ val list|22, 24, 26, 46, 48, 50, 68, 72, 90, 92, 94, 114, 116, 118, 136, 140, 162, 164, 186 }}
Optimal tuning (CTE): ~16/11 = 651.2344, ~5/4 = 385.7289


Other EDOs (from previous lists) supporting the temperament: {{ val list|284g, 330eg, 402degg }}
Optimal GPV sequence: {{Val list| 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg }}


[[Badness]]: ?
Badness: 0.340 × 10<sup>-3</sup>


== Tritiwiz ==
== Tritiwiz ==
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{{Multival|legend=1|rank=3| 3 -6 3 -4 1 2 16 5 -26 -12 }}
{{Multival|legend=1|rank=3| 3 -6 3 -4 1 2 16 5 -26 -12 }}


[[POTE generator]]s: ~11/10 = 166.198, ~5/4 = 384.3513
Optimal tuning ([[CTE]]): ~11/10 = 165.8258, ~5/4 = 384.6711


{{Val list|legend=1| 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd }}
{{Val list|legend=1| 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd }}

Revision as of 00:25, 26 July 2022

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

Uniwiz tempers out 1500625/1492992 = [-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 = [-17 2 0 0 4⟩, in the 11-limit. In the 17-limit it tempers 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.

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.

7-limit

Subgroup: 2.3.5.7

Comma list: 1500625/1492992

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

Mapping generators: ~1225/864, ~35/24, ~5

Wedgie: ⟨⟨⟨ 4 -4 -6 11 ]]]

Optimal tuning (CTE): ~35/24 = 651.1974, ~5/4 = 385.6847

Template:Val list

Badness: 0.7042 × 10-3

11-limit

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]]

Wedgie: ⟨⟨⟨ 4 -4 0 -6 2 -2 11 17 -17 -31 ]]]

Optimal tuning (CTE): ~16/11 = 651.1082, ~5/4 = 385.5778

Template:Val list

Badness: 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

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

Optimal tuning (CTE): ~16/11 = 651.2344, ~5/4 = 385.7289

Optimal GPV sequence: Template:Val list

Badness: 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

Wedgie: ⟨⟨⟨ 3 -6 3 -4 1 2 16 5 -26 -12 ]]]

Optimal tuning (CTE): ~11/10 = 165.8258, ~5/4 = 384.6711

Template:Val list

Badness: 0.732 × 10-3

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