Alphaxenic rank three clan: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Rewriting; categories; temperament info (will add wedgies and badnesses later)
 
(19 intermediate revisions by 7 users not shown)
Line 1: Line 1:
{{todo| expand | 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.
== 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]]
[[Category:Rank 3]]

Latest revision as of 00:31, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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

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

Retrieved from "https://en.xen.wiki/index.php?title=Alphaxenic_rank_three_clan&oldid=200898"