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, 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''.

== 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. 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]].

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

: 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 × 10-3

=== 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

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

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

~~[[POTE generator]]s~~: ~~~33/32 = 50.7404 or ~11/~~8 ~~= 549.2596, ~5/4 = 385.0315~~

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

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

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

[[Badness]]: 0.~~529~~ × 10-3

{{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

[[Subgroup]]: 2.3.5.7.11

[[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 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 × 10-3

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

[[Category:Temperament collections]]

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

[[Category:Pages with mostly numerical content]]

[[Category:Keenanismic temperaments| ]]

[[Category:Keenanismic| ]]

[[Category:Rank 3]]

@@ Line 1: / Line 1: @@
-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, 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''.
 == 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. 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]].
+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
+: 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>
 === 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
-{{Multival|legend=1|rank=3| 4 -4 0 -6 2 -2 11 17 -17 -31 }}
+Comma list: 289/288, 385/384, 561/560
-[[POTE generator]]s: ~33/32 = 50.7404 or ~11/8 = 549.2596, ~5/4 = 385.0315
+Sval mapping: {{mapping| 2 1 0 7 8 6 | 0 2 0 3 -1 2 | 0 0 1 -1 0 0 }}
-{{Val list|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }}
+Optimal tuning (CTE): ~17/12 = 1\2, ~16/11 = 651.2344, ~5/4 = 385.7289
-[[Badness]]: 0.529 × 10<sup>-3</sup>
+{{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
+[[Subgroup]]: 2.3.5.7.11
 [[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>
-[[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]]