Alphaxenic rank three clan: Difference between revisions

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m FloraC moved page Alpharabian rank three clan to Alphaxenic rank three clan: We agreed the corresponding temperament of the Alpharabian comma shouldn't be "alpharabian"
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The '''alpharabian rank-3 clan''' of temperaments tempers out the [[Alpharabian comma]], {{monzo| -17 2 0 0 4 }} = 131769/131072. Both no-5 alpharabian and no-7 alpharabian can be the head of this clan, but we can begin with the rank-4 temperament.  
The '''alphaxenic rank-3 clan''' of temperaments 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.  


Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* [[Uniwiz]], {385/384, 9801/9800} → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]]
* [[Uniwiz]] (+385/384) → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]]


== Alpharabian ==
== Alphaxenic ==
: ''Not to be confused with [[Alpharabian tuning]].''
[[Subgroup]]: 2.3.5.7.11
 
Subgroup: 2.3.5.7.11


[[Comma list]]: 131769/131072
[[Comma list]]: 131769/131072
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[[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]]: [{{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 generators: ~363/256, ~16/11, ~5, ~7
: mapping generators: ~363/256, ~16/11, ~5, ~7


[[POTE generator]]s: ~16/11 = 650.6784, ~5/4 = 385.5751, ~7/4 = 967.9328
[[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 }}
{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee }}
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== Rabic ==
== Rabic ==
Whenever the [[rastma]] is added to the list of commas tempered out among Alpharabian 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]].
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
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[[Mapping]]: [{{val| 24 38 0 0 83 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 0 }}]
[[Mapping]]: [{{val| 24 38 0 0 83 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 0 }}]


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


{{Multival|legend=1|rank=3| 0 0 0 24 0 0 38 0 0 83 }}
{{Multival|legend=1|rank=3| 0 0 0 24 0 0 38 0 0 83 }}


[[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


{{Optimal ET sequence|legend=1| 24, 48, 72, 264, 336b, 408b, 480bcde }}
{{Optimal ET sequence|legend=1| 24, 48, 72, 264, 336b, 408b, 480bcde }}
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[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Alpharabian rank-3 clan| ]] <!-- main article -->
[[Category:Alphaxenic rank-3 clan| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 06:43, 17 June 2023

The alphaxenic rank-3 clan of temperaments tempers out the Alpharabian comma, [-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.

Temperaments discussed elsewhere include:

Alphaxenic

Subgroup: 2.3.5.7.11

Comma list: 131769/131072

Mapping: [⟨2 1 0 0 8], ⟨0 2 0 0 -1], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]

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

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

Optimal ET sequence: 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee

Badness: 1.26 × 10-6

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.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 131769/131072

Mapping: [⟨24 38 0 0 83], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]

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

Wedgie: ⟨⟨⟨ 0 0 0 24 0 0 38 0 0 83 ]]]

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

Optimal ET sequence: 24, 48, 72, 264, 336b, 408b, 480bcde

Badness: 6.07 × 10-3

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