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]].

Temperaments discussed elsewhere include:

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

* ''[[Polyhymnia]]'' (+81/80 ~~or 385/384~~) → [[Didymus rank three family #Polyhymnia|Didymus rank-3 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]]

* [[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]]

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* ''[[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. 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.

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 ===

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: 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

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~~{{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

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

~~S-expression-based comma list: {([[289/288|S17]],) [[1089/1088|S33 = S35*S99]], [[1156/1155|S34]], [[1225/1224|S35 = S49*S50]](, [[9801/9800|S99]])}~~

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

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

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

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

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[[Category:Temperament collections]]

[[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]].
 Temperaments discussed elsewhere include:
 * ''[[Selenium]]'' (+49/48 or 56/55) → [[Semiphore family #Selenium|Semiphore family]]
-* ''[[Polyhymnia]]'' (+81/80 or 385/384) → [[Didymus rank three family #Polyhymnia|Didymus rank-3 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]]
+* [[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]]
@@ Line 14: / Line 15: @@
 * ''[[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. 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.
+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 ===
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 : 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
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 {{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
@@ Line 60: / Line 58: @@
 Comma list: 289/288, 385/384, 561/560
-S-expression-based comma list: {([[289/288|S17]],) [[1089/1088|S33 = S35*S99]], [[1156/1155|S34]], [[1225/1224|S35 = S49*S50]](, [[9801/9800|S99]])}
 Sval mapping: {{mapping| 2 1 0 7 8 6 | 0 2 0 3 -1 2 | 0 0 1 -1 0 0 }}
@@ Line 79: / Line 75: @@
 : mapping generators: ~2, ~11/10, ~5
-{{Multival|legend=1|rank=3| 3 -6 3 -4 1 2 16 5 -26 -12 }}
 [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~11/10 = 165.8258, ~5/4 = 384.6711
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 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Keenanismic temperaments| ]] <!-- main article -->
+[[Category:Keenanismic| ]] <!-- key article -->
 [[Category:Rank 3]]