Marvel family: Difference between revisions
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Decanonicalize tridecimal marvel |
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{{Technical data page}} | {{Technical data page}} | ||
The '''marvel family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[7-limit]] [[marvel comma]] ([[ratio]]: [[225/224]], {{monzo|legend=1| -5 2 2 -1 }}), also known as ''septimal kleisma''. These temperaments hence equate [[16/15]] and [[15/14]], or equivalently they equate two [[5/4]]'s and one [[14/9]]. The marvel comma is noteworthy in that it is tempered out by many common [[equal temperament|equal]] and [[rank-2 temperament]]s. | The '''marvel family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[7-limit]] [[marvel comma]] ([[ratio]]: [[225/224]], {{monzo|legend=1| -5 2 2 -1 }}), also known as ''septimal kleisma''. These temperaments hence equate [[16/15]] and [[15/14]], or equivalently they equate two [[5/4]]'s and one [[14/9]]. The marvel comma is noteworthy in that it is tempered out by many common [[equal temperament|equal]] and [[rank-2 temperament]]s. | ||
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* ''[[Artemis]]'' (+121/120) → [[Biyatismic clan #Artemis|Biyatismic clan]] | * ''[[Artemis]]'' (+121/120) → [[Biyatismic clan #Artemis|Biyatismic clan]] | ||
* ''[[Spectacle]]'' (+243/242) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]] | * ''[[Spectacle]]'' (+243/242) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]] | ||
* ''[[Marvelpine]]'' (+4000/3993) → [[Wizardharry clan #Marvelpine|Wizardharry clan]] | |||
* ''[[Mirage]]'' (+243/242, +385/384) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]] | * ''[[Mirage]]'' (+243/242, +385/384) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]] | ||
* ''[[Catakleismoid]]'' (+4375/4374) → [[Kleismic rank-3 family #Catakleismoid|Kleismic rank-3 family]] | |||
== Undecimal marvel == | == Undecimal marvel == | ||
{{Main| Marvel }} | {{Main| Marvel }} | ||
Undecimal marvel tempers out [[385/384]] as well as [[540/539]], and is loosely [[associated temperament|associated]] with [[wizard]]. [[197edo]] remains useful as a tuning, with the 197e val, but [[166edo]], which among other things has a virtually pure 7, works as well. | Undecimal marvel tempers out [[385/384]] as well as [[540/539]], and is loosely [[associated temperament|associated]] with [[wizard]]. This extension is natural because of the factorization 225/224 = (385/384)⋅(540/539). [[197edo]] remains useful as a tuning, with the 197e val, but [[166edo]], which among other things has a virtually pure 7, works as well. | ||
In the 13-limit, 225/224 factors as ([[351/350]])⋅([[625/624]]) or ([[325/324]])⋅([[729/728]]). Tempering out 351/350 and 625/624 leads to helios, tempering out 325/324 and 729/728 leads to hecate. Tempering out all of them leads to the 13-limit version of [[catakleismic]]. | |||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
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}} | }} | ||
=== | === Helios === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Comma list: 225/224, 275/273, 385/384 | Comma list: 225/224, 275/273, 385/384 | ||
Mapping: {{mapping| 1 0 0 -5 12 17 | 0 1 0 2 -1 4 | 0 0 1 2 -3 -3 }} | Mapping: {{mapping| 1 0 0 -5 12 17 | 0 1 0 2 -1 -4 | 0 0 1 2 -3 -3 }} | ||
Optimal tunings: | Optimal tunings: | ||
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Apollo tempers out not only [[100/99]] but [[896/891]]. Note that marvel tempers together [[25/24]] and [[28/27]], and apollo further equates it with [[33/32]] via the vanishing of 100/99. This makes it a weak extension of [[parapyth]], and [[associated temperament|associates]] it with [[magic]]. The lattice structure is very compact, comparable to that of [[ares]], from which apollo only differs in the mapping of [[prime interval|prime]] [[7/1|7]]. | Apollo tempers out not only [[100/99]] but [[896/891]]. Note that marvel tempers together [[25/24]] and [[28/27]], and apollo further equates it with [[33/32]] via the vanishing of 100/99. This makes it a weak extension of [[parapyth]], and [[associated temperament|associates]] it with [[magic]]. The lattice structure is very compact, comparable to that of [[ares]], from which apollo only differs in the mapping of [[prime interval|prime]] [[7/1|7]]. | ||
The canonical [[13-limit]] extension is implied by parapyth, tempering out [[352/351]] and [[364/363]], but there are a number of other extenions to consider, these being called phoebus and musagetes, after epithets of Apollo. Phoebus tempers out [[105/104]] and finds ~[[13 | The canonical [[13-limit]] extension is implied by parapyth, tempering out [[352/351]] and [[364/363]], but there are a number of other extenions to consider, these being called phoebus and musagetes, after epithets of Apollo. Phoebus tempers out [[105/104]] and finds ~[[16/13]] as a stack of three [[secor]]s. Musagetes tempers out [[144/143]] and conflates 16/13 and [[11/9]], which in this case is simply a stack of two ~[[10/9]]'s. These extensions are [[support]]ed by 13-limit [[magic]], unlike the canonical one. | ||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
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[[Badness]] (Sintel): 1.38 | [[Badness]] (Sintel): 1.38 | ||
== Subgroup extensions == | == Subgroup extensions == | ||