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Alphaxenic rank three clan: Difference between revisions

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This is a collection of 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.  
{{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.  


== Alpharabian ==
Temperaments discussed elsewhere include:
: ''Not to be confused with [[Alpharabian tuning]].''  
* ''[[Uniwiz]]'' (+385/384) → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]]


Subgroup: 2.3.5.7.11
== Alphaxenic ==
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 131769/131072
[[Comma list]]: 131769/131072


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


[[POTE generator]]s: ~11/8 = 549.3316, ~5/4 = 385.5751, ~7/4 = 967.9328
: mapping generators: ~363/256, ~16/11, ~5, ~7


{{Val list|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee }}
[[Optimal tuning]] ([[POTE]]): ~363/256 = 1\2, ~16/11 = 650.6784, ~5/4 = 385.5751, ~7/4 = 967.9328
 
{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee }}


[[Badness]]: 1.26 × 10<sup>-6</sup>
[[Badness]]: 1.26 × 10<sup>-6</sup>


== Betarabian ==
== Rabic ==
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.
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]].


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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


[[Mapping]]: [{{val| 24 38 0 0 83 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 0 }}]
{{Mapping|legend=1| 24 38 0 0 83 | 0 0 1 0 0 | 0 0 0 1 0 }}
 
Mapping generators: ~33/32, ~5, ~7


{{Multival|legend=1|rank=3| 0 0 0 24 0 0 38 0 0 83 }}
: mapping generators: ~33/32, ~5, ~7


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


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


[[Badness]]: 6.07 × 10<sup>-3</sup>
[[Badness]]: 6.07 × 10<sup>-3</sup>


[[Category:Regular temperament theory]]
[[Category:Temperament clans]]
[[Category:Temperament collection]]
[[Category:Pages with mostly numerical content]]
[[Category:Alpharabian temperaments| ]] <!-- main article -->
[[Category:Alphaxenic rank-3 clan| ]] <!-- main article -->
[[Category:Alphaxenic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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