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|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, 56/55} → [[Semiphore family #Selenium]]
* ''[[Selenium]]'' (+49/48 or 56/55) → [[Semiphore family #Selenium|Semiphore family]]
* ''[[Polyhymnia]]'', {81/80, 385/384} → [[Didymus rank three family #Polyhymnia]]
* ''[[Polyhymnia]]'' (+81/80) → [[Didymus rank three family #Polyhymnia|Didymus rank-3 family]]
* ''[[Orwellian]]'', {99/98, 385/384} → [[Orwellismic family #Orwellian]]
* ''[[Orwellian]]'' (+99/98) → [[Orwellismic family #Orwellian|Orwellismic family]]
* ''[[Supermagic]]'', {100/99, 385/384} → [[Keemic family #Undecimal supermagic]]
* ''[[Supermagic]]'' (+100/99) → [[Keemic family #Undecimal supermagic|Keemic family]]
* [[Zeus]], {121/120, 176/175} → [[Porwell family #Zeus]]
* [[Zeus]] (+121/120 or 176/175) → [[Porwell family #Zeus|Porwell family]] and [[Biyatismic clan #Zeus|Biyatismic clan]]
* ''[[Starling]]'', {126/125, 385/384} → [[Starling family #Undecimal starling]]
* [[Starling]] (+126/125) → [[Starling family #Undecimal starling|Starling family]]
* [[Marvel]], {225/224, 385/384} → [[Marvel family #Undecimal marvel (unimarv)]]
* [[Marvel]] (+225/224) → [[Marvel family #Undecimal marvel (unimarv)|Marvel family]]
* ''[[Mirwomo]]'', {243/242, 385/384} → [[Rastmic temperaments #Mirwomo]]
* ''[[Mirwomo]]'' (+243/242) → [[Rastmic rank three clan #Mirwomo|Rastmic rank-3 clan]]
* [[Sensamagic]], {245/243, 385/384} → [[Sensamagic family #Undecimal sensamagic]]
* [[Sensamagic]] (+245/243) → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]
* ''[[Portent]]'', {385/384, 441/440} → [[Gamelismic family #Portent]]
* [[Portent]] (+441/440) → [[Gamelismic family #Portent|Gamelismic family]]
* ''[[Augustus]]'', {385/384, 1232/1215} → [[Diaschismic rank three family #Augustus]]
* ''[[Augustus]]'' (+1232/1215) → [[Diaschismic rank three family #Augustus|Diaschismic rank-3 family]]
* ''[[Agni]]'', {385/384, 1375/1372} → [[Breed family #Agni]]
* ''[[Agni]]'' (+1375/1372) → [[Breed family #Agni|Breed family]]
* ''[[Beyla]]'', {385/384, 4375/4374} → [[Ragismic family #Beyla]]
* ''[[Zelda]]'' (+3388/3375) → [[Horwell family #Zelda|Horwell family]]
* ''[[Enlil]]'', {385/384, 6250/6237} → [[Kleismic rank three family #Enlil]]
* ''[[Beyla]]'' (+4375/4374) → [[Ragismic family #Beyla|Ragismic family]]
* ''[[Enlil]]'' (+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. The [[extension]] to the 2.3.5.7.11.13.17 [[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 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 [[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]].  


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


[[Comma list]]: 1500625/1492992
[[Comma list]]: 1500625/1492992


[[Mapping]]: [{{val| 2 1 0 7 }}, {{val| 0 2 0 3 }}, {{val| 0 0 1 -1 }}]
{{Mapping|legend=1| 2 1 0 7 | 0 2 0 3 | 0 0 1 -1 }}


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


{{Multival|legend=1|rank=3| 4 -4 -6 11 }}
[[Optimal tuning]] ([[CTE]]): ~1225/864 = 1\2, ~35/24 = 651.1974, ~5/4 = 385.6847


[[POTE generator]]s: ~36/35 = 50.9529 or ~48/35 = 549.0471, ~5/4 = 385.3326
{{Optimal ET sequence|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 }}


[[Badness]]: 0.7042 × 10<sup>-3</sup>
[[Badness]]: 0.7042 × 10<sup>-3</sup>


=== 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


[[Mapping]]: [{{val| 2 1 0 7 8 }}, {{val| 0 2 0 3 -1 }}, {{val| 0 0 1 -1 0 }}]
{{Mapping|legend=1| 2 1 0 7 8 | 0 2 0 3 -1 | 0 0 1 -1 0 }}


{{Multival|legend=1|rank=3| 4 -4 0 -6 2 -2 11 17 -17 -31 }}
[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~16/11 = 651.1082, ~5/4 = 385.5778


[[POTE generator]]s: ~33/32 = 50.7404 or ~11/8 = 549.2596, ~5/4 = 385.0315
{{Optimal ET sequence|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 }}


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


[[Mapping]]: [{{val| 2 1 0 7 8 6 }}, {{val| 0 2 0 3 -1 2 }}, {{val| 0 0 1 -1 0 0 }}]
Sval mapping: {{mapping| 2 1 0 7 8 6 | 0 2 0 3 -1 2 | 0 0 1 -1 0 0 }}


[[Wedgie]]: ?
Optimal tuning (CTE): ~17/12 = 1\2, ~16/11 = 651.2344, ~5/4 = 385.7289


[[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
{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg }}


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 }}
Badness: 0.340 × 10<sup>-3</sup>
 
Other EDOs (from previous lists) supporting the temperament: {{ val list|284g, 330eg, 402degg }}
 
[[Badness]]: ?


== Tritiwiz ==
== Tritiwiz ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 385/384, 4000/3993
[[Comma list]]: 385/384, 4000/3993


[[Mapping]]: [{{val| 1 2 0 8 1 }}, {{val| 0 -3 0 -4 1 }}, {{val| 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


{{Multival|legend=1|rank=3| 3 -6 3 -4 1 2 16 5 -26 -12 }}
: mapping generators: ~2, ~11/10, ~5


[[POTE generator]]s: ~11/10 = 166.198, ~5/4 = 384.3513
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~11/10 = 165.8258, ~5/4 = 384.6711


{{Val list|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]]: 0.732 × 10<sup>-3</sup>


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Keenanismic temperaments| ]] <!-- main article -->
[[Category:Keenanismic temperaments| ]] <!-- main article -->
[[Category:Keenanismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 00:35, 24 June 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

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. The extension to the 2.3.5.7.11.13.17 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.

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

Optimal tuning (CTE): ~1225/864 = 1\2, ~35/24 = 651.1974, ~5/4 = 385.6847

Optimal ET sequence: 22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd

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]]

Optimal tuning (CTE): ~99/70 = 1\2, ~16/11 = 651.1082, ~5/4 = 385.5778

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

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): ~17/12 = 1\2, ~16/11 = 651.2344, ~5/4 = 385.7289

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

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

Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 165.8258, ~5/4 = 384.6711

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

Badness: 0.732 × 10-3

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