Alphaxenic rank three clan: Difference between revisions
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The '''alphaxenic rank-3 clan''' of | 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. | ||
Temperaments discussed elsewhere include: | Temperaments discussed elsewhere include: | ||
* [[Uniwiz]] (+385/384) → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]] | * ''[[Uniwiz]]'' (+385/384) → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]] | ||
== Alphaxenic == | == Alphaxenic == | ||
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[[Comma list]]: 131769/131072 | [[Comma list]]: 131769/131072 | ||
{{Mapping|legend=1| 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 | : mapping generators: ~363/256, ~16/11, ~5, ~7 | ||
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[[Comma list]]: 243/242, 131769/131072 | [[Comma list]]: 243/242, 131769/131072 | ||
{{Mapping|legend=1| 24 38 0 0 83 | 0 0 1 0 0 | 0 0 0 1 0 }} | |||
: mapping generators: ~33/32, ~5, ~7 | : mapping generators: ~33/32, ~5, ~7 | ||
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[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||
[[Category:Alphaxenic rank-3 clan| ]] <!-- main article --> | [[Category:Alphaxenic rank-3 clan| ]] <!-- main article --> | ||
[[Category:Alphaxenic| ]] <!-- key article --> | |||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||
Revision as of 08:16, 14 September 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:
- Uniwiz (+385/384) → Keenanismic temperaments
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