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]].

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]].

== ~~Headquarters~~ ==

Temperaments discussed elsewhere include:

~~Headquarters~~ 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. It features the split of the whole tone into four equal parts, each representing 36/35~33/32. Therefore, it is most interesting to those who would like to make extensive use of ~~quartertones~~. It can be viewed as ~~one which is joined~~ by the three quartertone-size equal temperaments: [[22edo|22]], [[24edo|24]], and [[26edo|26]]. It shares the [[optimal patent val]], [[284edo|284]], with [[keenanismic]].

* ''[[Selenium]]'' (+49/48 or 56/55) → [[Semiphore family #Selenium|Semiphore family]]

* ''[[Polyhymnia]]'' (+81/80) → [[Didymus rank three family #Polyhymnia|Didymus rank-3 family]]

* ''[[Orwellian]]'' (+99/98) → [[Orwellismic family #Orwellian|Orwellismic family]]

* ''[[Supermagic]]'' (+100/99) → [[Keemic family #Undecimal supermagic|Keemic family]]

* [[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]]

* [[Marvel]] (+225/224) → [[Marvel family #Undecimal marvel (unimarv)|Marvel family]]

* ''[[Mirwomo]]'' (+243/242) → [[Rastmic rank three clan #Mirwomo|Rastmic rank-3 clan]]

* [[Sensamagic]] (+245/243) → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]

* [[Portent]] (+441/440) → [[Gamelismic family #Portent|Gamelismic family]]

* ''[[Augustus]]'' (+1232/1215) → [[Diaschismic rank three family #Augustus|Diaschismic rank-3 family]]

* ''[[Agni]]'' (+1375/1372) → [[Breed family #Agni|Breed family]]

* ''[[Zelda]]'' (+3388/3375) → [[Horwell family #Zelda|Horwell family]]

* ''[[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''.

== 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 2.3.5.7.11.13.17 [[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 [[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 ===

Subgroup: 2.3.5.7

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1500625/1492992

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

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

~~{{Multival|legend=1|rank=3| 4 -4 -6 11 }}~~

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

[[~~POTE generator~~]]s: ~36/35 = ~~50.9529 or~~ ~48/35 = ~~549~~.~~0471~~, ~5/4 = 385.~~3326~~

[[Optimal tuning]] ([[CTE]]): ~1225/864 = 1\2, ~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 }}

{{Optimal ET sequence|legend=1| 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

[[Subgroup]]: 2.3.5.7.11

[[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 }}]

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

{{Optimal ET sequence|legend=1| 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: {{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|legend=1| 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

~~{{Multival|legend=1|rank=3| 4 -4 0 -6 2 -~~2 11 ~~17 -17 -31 }}~~

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

[[~~POTE generator~~]]s: ~~~33/32~~ = ~~50.7404 or~~ ~11/8 = ~~549~~.~~2596~~, ~5/4 = ~~385~~.~~0315~~

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

{{~~Val list~~|legend=1| 22, 46, 68, 72, ~~118~~, ~~190~~, ~~212~~, ~~284~~, ~~330e~~, ~~402de~~ }}

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

[[Badness]]: 0.~~5292~~ × 10-3

[[Badness]]: 0.732 × 10-3

[[Category:~~Regular temperament theory~~]]

[[Category:Temperament collections]]

[[Category:~~Temperament collection~~]]

[[Category:Pages with mostly numerical content]]

[[Category:Keenanismic temperaments| ]]

[[Category:Keenanismic| ]]

[[Category:Rank 3]]

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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]].
-== Headquarters ==
+Temperaments discussed elsewhere include:
-Headquarters 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. It features the split of the whole tone into four equal parts, each representing 36/35~33/32. Therefore, it is most interesting to those who would like to make extensive use of quartertones. It can be viewed as one which is joined by the three quartertone-size equal temperaments: [[22edo|22]], [[24edo|24]], and [[26edo|26]]. It shares the [[optimal patent val]], [[284edo|284]], with [[keenanismic]].
+* ''[[Selenium]]'' (+49/48 or 56/55) → [[Semiphore family #Selenium|Semiphore family]]
+* ''[[Polyhymnia]]'' (+81/80) → [[Didymus rank three family #Polyhymnia|Didymus rank-3 family]]
+* ''[[Orwellian]]'' (+99/98) → [[Orwellismic family #Orwellian|Orwellismic family]]
+* ''[[Supermagic]]'' (+100/99) → [[Keemic family #Undecimal supermagic|Keemic family]]
+* [[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]]
+* [[Marvel]] (+225/224) → [[Marvel family #Undecimal marvel (unimarv)|Marvel family]]
+* ''[[Mirwomo]]'' (+243/242) → [[Rastmic rank three clan #Mirwomo|Rastmic rank-3 clan]]
+* [[Sensamagic]] (+245/243) → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]
+* [[Portent]] (+441/440) → [[Gamelismic family #Portent|Gamelismic family]]
+* ''[[Augustus]]'' (+1232/1215) → [[Diaschismic rank three family #Augustus|Diaschismic rank-3 family]]
+* ''[[Agni]]'' (+1375/1372) → [[Breed family #Agni|Breed family]]
+* ''[[Zelda]]'' (+3388/3375) → [[Horwell family #Zelda|Horwell family]]
+* ''[[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''.
+== 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 2.3.5.7.11.13.17 [[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 [[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 ===
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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
-{{Multival|legend=1|rank=3| 4 -4 -6 11 }}
+: mapping generators: ~1225/864, ~35/24, ~5
-[[POTE generator]]s: ~36/35 = 50.9529 or ~48/35 = 549.0471, ~5/4 = 385.3326
+[[Optimal tuning]] ([[CTE]]): ~1225/864 = 1\2, ~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 }}
+{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd }}
 [[Badness]]: 0.7042 × 10<sup>-3</sup>
 === 11-limit ===
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[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 }}
+[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~16/11 = 651.1082, ~5/4 = 385.5778
+{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }}
+[[Badness]]: 0.529 × 10<sup>-3</sup>
+=== 2.3.5.7.11.17 subgroup ===
+Subgroup: 2.3.5.7.11.17
+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 }}
+Optimal tuning (CTE): ~17/12 = 1\2, ~16/11 = 651.2344, ~5/4 = 385.7289
+{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190g, 212g, 284g, 330eg, 402degg }}
+Badness: 0.340 × 10<sup>-3</sup>
+== Tritiwiz ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 385/384, 4000/3993
+{{Mapping|legend=1| 1 2 0 8 1 | 0 -3 0 -4 1 | 0 0 1 -2 1 }}
-{{Multival|legend=1|rank=3| 4 -4 0 -6 2 -2 11 17 -17 -31 }}
+: mapping generators: ~2, ~11/10, ~5
-[[POTE generator]]s: ~33/32 = 50.7404 or ~11/8 = 549.2596, ~5/4 = 385.0315
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~11/10 = 165.8258, ~5/4 = 384.6711
-{{Val list|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }}
+{{Optimal ET sequence|legend=1| 15, 22, 37, 50, 65, 72, 159, 231, 253, 325c, 412cd, 484cd }}
-[[Badness]]: 0.5292 × 10<sup>-3</sup>
+[[Badness]]: 0.732 × 10<sup>-3</sup>
-[[Category:Regular temperament theory]]
+[[Category:Temperament collections]]
-[[Category:Temperament collection]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Keenanismic temperaments| ]] <!-- main article -->
+[[Category:Keenanismic| ]] <!-- key article -->
 [[Category:Rank 3]]