Alphaxenic rank three clan: Difference between revisions

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{{~~todo| expand | comment= '~~'~~Wedgie~~'' ~~and~~ ''~~Badness~~'~~' information missing~~ | ~~inline=1~~ }}

The '''alphaxenic rank-3 clan''' of [[temperament]]s [[tempering out|tempers out]] the [[Alpharabian comma]], {{monzo| -17 2 0 0 4 }} = 131769/131072. Both no-5 alphaxenic and no-7 alphaxenic can be the head of this clan, but we can begin with the rank-4 temperament.

~~These are rank-3 temperaments that temper out the~~ [[~~Alpharabian comma~~]]~~, {{monzo~~| ~~-17 2 0 0 4 }} = 131769/131072, but we can begin with the rank-4 temperament.~~

Temperaments discussed elsewhere include:

* ''[[Uniwiz]]'' (+385/384) → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]]

~~Comma~~: ~~131769/131072~~

== Alphaxenic ==

[[Subgroup]]: 2.3.5.7.11

~~POTE generators~~: ~~~11~~/~~8 = 549.3316 or ~33/32 = 50.6784, ~5/4 = 385.5751, ~7/4 = 967.9328~~

[[Comma list]]: 131769/131072

~~Map: [<~~2 1 0 0 8|~~, <~~0 2 0 0 -1|~~, <~~0 0 1 0 0|~~, <~~0 0 0 1 0|]

{{Mapping|legend=1| 2 1 0 0 8 | 0 2 0 0 -1 | 0 0 1 0 0 | 0 0 0 1 0 }}

~~Wedgie~~:

: mapping generators: ~363/256, ~16/11, ~5, ~7

~~EDOs~~: ~~{{EDOs| 22~~, 46, 68, ~~72, 118, 190, 212, 284, 330e, 402de }}~~

[[Optimal tuning]] ([[POTE]]): ~363/256 = 1\2, ~16/11 = 650.6784, ~5/4 = 385.5751, ~7/4 = 967.9328

~~Badness:~~

{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee }}

~~== Betarabian ==~~

[[Badness]]: 1.26 × 10<sup>-6</sup>

Whenever the [[rastma]] is added to the list of commas tempered out among ~~Alpharabian~~ temperaments, you end up with ~~Betarabian~~, which splits the octave into 24 equal parts. This temperament is named as such because tempering out both the Alpharabian comma and the rastma automatically tempers out the [[Betarabian comma]]~~, and although there are technically other Betarabian temperaments, they have different names~~.

== Rabic ==

Whenever the [[rastma]] is added to the list of commas tempered out among alphaxenic temperaments, you end up with rabic, which splits the octave into 24 equal parts. This temperament is named as such because tempering out both the Alpharabian comma and the rastma automatically tempers out the [[Betarabian comma]].

~~Commas~~: ~~243/242, 131769/131072~~

[[Subgroup]]: 2.3.5.7.11

~~POTE generators~~: ~5/~~4 = 385.0647~~, ~7/~~4 = 967.3158~~

[[Comma list]]: 243/242, 131769/131072

~~Map: [<~~24 38 0 0 83|~~, <~~0 0 1 0 0|~~, <~~0 0 0 1 0|]

~~Wedgie~~:

: mapping generators: ~33/32, ~5, ~7

~~EDOs~~: ~~{{EDOs|~~ 24, 48, ~~72, 264 }}~~

[[Optimal tuning]] ([[POTE]]): ~33/32 = 1\24, ~5/4 = 385.0647, ~7/4 = 967.3158

~~Badness:~~

{{Optimal ET sequence|legend=1| 24, 48, 72, 264, 336b, 408b, 480bcde }}

[[Category:~~Theory~~]]

[[Badness]]: 6.07 × 10<sup>-3</sup>

[[Category:~~Temperament~~]]

[[Category:~~Alpharabian~~]]

[[Category:Temperament clans]]

[[Category:Pages with mostly numerical content]]

[[Category:Alphaxenic rank-3 clan| ]]

[[Category:Alphaxenic| ]]

[[Category:Rank 3]]

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-{{todo| expand | comment= ''Wedgie'' and ''Badness'' information missing | inline=1 }}
+{{Technical data page}}
+The '''alphaxenic rank-3 clan''' of [[temperament]]s [[tempering out|tempers out]] the [[Alpharabian comma]], {{monzo| -17 2 0 0 4 }} = 131769/131072. Both no-5 alphaxenic and no-7 alphaxenic can be the head of this clan, but we can begin with the rank-4 temperament.
-These are rank-3 temperaments that temper out the [[Alpharabian comma]], {{monzo| -17 2 0 0 4 }} = 131769/131072, but we can begin with the rank-4 temperament.
+Temperaments discussed elsewhere include:
+* ''[[Uniwiz]]'' (+385/384) → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]]
-Comma: 131769/131072
+== Alphaxenic ==
+[[Subgroup]]: 2.3.5.7.11
-POTE generators: ~11/8 = 549.3316 or ~33/32 = 50.6784, ~5/4 = 385.5751, ~7/4 = 967.9328
+[[Comma list]]: 131769/131072
-Map: [<2 1 0 0 8|, <0 2 0 0 -1|, <0 0 1 0 0|, <0 0 0 1 0|]
+{{Mapping|legend=1| 2 1 0 0 8 | 0 2 0 0 -1 | 0 0 1 0 0 | 0 0 0 1 0 }}
-Wedgie:
+: mapping generators: ~363/256, ~16/11, ~5, ~7
-EDOs: {{EDOs| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de }}
+[[Optimal tuning]] ([[POTE]]): ~363/256 = 1\2, ~16/11 = 650.6784, ~5/4 = 385.5751, ~7/4 = 967.9328
-Badness:
+{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee }}
-== Betarabian ==
+[[Badness]]: 1.26 × 10<sup>-6</sup>
-Whenever the [[rastma]] is added to the list of commas tempered out among Alpharabian temperaments, you end up with Betarabian, which splits the octave into 24 equal parts.  This temperament is named as such because tempering out both the Alpharabian comma and the rastma automatically tempers out the [[Betarabian comma]], and although there are technically other Betarabian temperaments, they have different names.
+== Rabic ==
+Whenever the [[rastma]] is added to the list of commas tempered out among alphaxenic temperaments, you end up with rabic, which splits the octave into 24 equal parts. This temperament is named as such because tempering out both the Alpharabian comma and the rastma automatically tempers out the [[Betarabian comma]].
-Commas: 243/242, 131769/131072
+[[Subgroup]]: 2.3.5.7.11
-POTE generators: ~5/4 = 385.0647, ~7/4 = 967.3158
+[[Comma list]]: 243/242, 131769/131072
-Map: [<24 38 0 0 83|, <0 0 1 0 0|, <0 0 0 1 0|]
+{{Mapping|legend=1| 24 38 0 0 83 | 0 0 1 0 0 | 0 0 0 1 0 }}
-Wedgie:
+: mapping generators: ~33/32, ~5, ~7
-EDOs: {{EDOs| 24, 48, 72, 264 }}
+[[Optimal tuning]] ([[POTE]]): ~33/32 = 1\24, ~5/4 = 385.0647, ~7/4 = 967.3158
-Badness:
+{{Optimal ET sequence|legend=1| 24, 48, 72, 264, 336b, 408b, 480bcde }}
-[[Category:Theory]]
+[[Badness]]: 6.07 × 10<sup>-3</sup>
-[[Category:Temperament]]
-[[Category:Alpharabian]]
+[[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
+[[Category:Alphaxenic rank-3 clan| ]] <!-- main article -->
+[[Category:Alphaxenic| ]] <!-- key article -->
 [[Category:Rank 3]]