Marvel family: Difference between revisions

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~~{{interwiki~~

~~| en = Marvel family~~

~~| de = Marvel~~

~~| es =~~

~~| ja =~~

}}

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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The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[normal ~~lists~~|normal generator list]] of {2, 3, 5}; hence a [[5-limit]] scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by [[197edo]].

The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[normal forms|normal generator list]] of {2, 3, 5}; hence a [[5-limit]] scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by [[197edo]].

Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[catakleismic]], with which it shares the [[optimal patent val]]. Another temperament which does little damage to tuning accuracy is [[compton]], for which [[240edo]] may be used. See [[~~marvel~~ temperaments]] for some other rank-2 temperaments.

Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[catakleismic]], with which it shares the [[optimal patent val]]. Another temperament which does little damage to tuning accuracy is [[compton]], for which [[240edo]] may be used. See [[Marvel temperaments]] for some other rank-2 temperaments.

[[Subgroup]]: 2.3.5.7

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=== Overview to extensions ===

The second comma of the [[~~Normal lists~~ #Normal ~~interval list~~|normal comma list]] defines which 11-limit family member we are looking at. 4125/4096 gives unidecimal marvel; 91125/90112 gives prodigy; 5632/5625 gives minerva. These and many others considered below use the same generators as marvel.

The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 11-limit family member we are looking at. 4125/4096 gives unidecimal marvel; 91125/90112 gives prodigy; 5632/5625 gives minerva. These and many others considered below use the same generators as marvel.

Temperaments discussed elsewhere include

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* ''[[Artemis]]'' (+121/120) → [[Biyatismic clan #Artemis|Biyatismic 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]]

* ''[[Catakleismoid]]'' (+4375/4374) → [[Kleismic rank-3 family #Catakleismoid|Kleismic rank-3 family]]

== Undecimal marvel ~~(unimarv)~~ ==

== Undecimal marvel ==

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

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[[Complexity spectrum]]: 5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11

~~[[Associated temperament]]: [[catakleismic]]~~

Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]

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}}

=== ~~13-limit~~ ===

=== Helios ===

Subgroup: 2.3.5.7.11.13

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=== Hecate ===

Hecate tempers out [[325/324]], the marveltwin comma, such that [[16/13]] is found by a stack of two ~[[10/9]]'s, similar to how [[8/7]] is found by a stack of two [[15/14]]~[[16/15]]'s. Hecate has a natural extension to include [[prime interval|prime]] [[19/1|19]], where it further tempers out [[400/399]] and [[513/512]], taking advantage of the factorization 225/224 = (400/399)⋅(513/512). For both of these cases [[166edo]] remains an excellent tuning.

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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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:

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== Prodigy ==

Prodigy shrinks ~~1024/1029,~~ 243/242, 384/385 and 2400/2401 down to the same tiny interval. Hence in practice it probably makes the most sense to temper this out as well, leading to [[miracle]]. This, however, does not render it pointless to consider prodigy; for one thing, scales in prodigy such as hobbit scales translate into interesting scales for miracle.

Prodigy tempers out [[441/440]] and shrinks [[243/242]], [[385/384|384/385]], [[1029/1024|1024/1029]] and [[2401/2400|2400/2401]] down to the same tiny interval. Hence in practice it probably makes the most sense to temper this out as well, leading to [[miracle]]. This, however, does not render it pointless to consider prodigy; for one thing, scales in prodigy such as [[hobbit]] scales translate into interesting scales for miracle.

[[Subgroup]]: 2.3.5.7.11

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Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]

~~[[Associated temperament]]: [[miracle]]~~

{{Databox|Hobbit bases|

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== Minerva ==

Minerva tempers out [[99/98]] as well as [[176/175]]. It may be described as 12 & 22 & 31, and is loosely [[associated temperament|associated]] with [[würschmidt]].

[[Subgroup]]: 2.3.5.7.11

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Scales: [[minerva12]], [[minerva22x]]

~~[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]~~

=== Athene ===

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== Apollo ==

[[File:Lattice Apollo.png|thumb|Lattice for apollo.]]

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 ~[[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

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[[Projection pair]]s: <code>7 225/32 11 100/9</code>

~~[[Associated temperament]]: [[magic]]~~

Scales: [[apollo wholetone]], [[indigo17]]

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Projection pairs: <code>7 225/32 11 100/9 13 3200/243</code>

=== Phoebus ===

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 196/195

Mapping: {{mapping| 1 0 0 -5 2 1 | 0 1 0 2 -2 3 | 0 0 1 2 2 3 }}

Optimal tunings:

* WE: ~2 = 1200.3724{{c}}, ~3/2 = 702.5274{{c}}, ~5/4 = 379.6345{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3357{{c}}, ~5/4 = 379.6830{{c}}

Badness (Sintel): 0.886

=== Musagetes ===

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 225/224

Mapping: {{mapping| 1 0 0 -5 2 2 | 0 1 0 2 -2 4 | 0 0 1 2 2 -2 }}

Optimal tunings:

* WE: ~2 = 1199.2695{{c}}, ~3/2 = 702.5589{{c}}, ~5/4 = 382.7715{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7998{{c}}, ~5/4 = 382.7740{{c}}

{{Optimal ET sequence|legend=0| 19, 22f, 34d, 41, 75e, 94e, 116ef }}

Badness (Sintel): 1.14

== Potassium ==

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[[Badness]] (Sintel): 1.38

~~== Catakleismoid ==~~

~~Catakleismoid is the same as catakleismic in the 2.3.5.7.13 subgroup but with an independent generator for prime 11.~~

~~[[Subgroup]]: 2.3.5.7.11~~

~~[[Comma list]]: 225/224, 4375/4374~~

~~{{Mapping|legend=1| 1 0 1 -3 0 | 0 6 5 22 0 | 0 0 0 0 1 }}~~

~~: mapping generators: ~2, ~6/5, ~11~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}, ~11/8 = 549.5258{{c}}

~~: [[error map]]: {{val| +0.596 -0.619 -1.271 +0.948 -0.003 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}, ~11/8 = 550.0343{{c}}

~~: error map: {{val| 0.000 -1.332 -2.461 +0.126 -1.284 }}~~

~~{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce, 341ce }}~~

~~[[Badness]] (Sintel): 1.53~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 169/168, 225/224, 325/324~~

~~Mapping: {{mapping| 1 0 1 -3 0 0 | 0 6 5 22 0 14 | 0 0 0 0 1 0 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.8238{{c}}, ~6/5 = 316.9478{{c}}, ~11/8 = 548.9611{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}, ~11/8 = 549.5903{{c}}

~~Optimal ET sequence: {{Optimal ET sequence| 19, 34d, 53, 72, 125f, 197ef }}~~

~~Badness (Sintel): 0.853~~

== Subgroup extensions ==

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-{{interwiki
-| en = Marvel family
-| de = Marvel
-| es =
-| ja =
-}}
 {{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.
@@ Line 13: / Line 7: @@
 {{Main| Marvel }}
-The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[normal lists|normal generator list]] of {2, 3, 5}; hence a [[5-limit]] scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by [[197edo]].
+The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[normal forms|normal generator list]] of {2, 3, 5}; hence a [[5-limit]] scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by [[197edo]].
-Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[catakleismic]], with which it shares the [[optimal patent val]]. Another temperament which does little damage to tuning accuracy is [[compton]], for which [[240edo]] may be used. See [[marvel temperaments]] for some other rank-2 temperaments.
+Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[catakleismic]], with which it shares the [[optimal patent val]]. Another temperament which does little damage to tuning accuracy is [[compton]], for which [[240edo]] may be used. See [[Marvel temperaments]] for some other rank-2 temperaments.
 [[Subgroup]]: 2.3.5.7
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 === Overview to extensions ===
-The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 11-limit family member we are looking at. 4125/4096 gives unidecimal marvel;  91125/90112 gives prodigy; 5632/5625 gives minerva. These and many others considered below use the same generators as marvel.
+The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 11-limit family member we are looking at. 4125/4096 gives unidecimal marvel; 91125/90112 gives prodigy; 5632/5625 gives minerva. These and many others considered below use the same generators as marvel.
 Temperaments discussed elsewhere include
@@ Line 78: / Line 72: @@
 * ''[[Artemis]]'' (+121/120) → [[Biyatismic clan #Artemis|Biyatismic 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]]
+* ''[[Catakleismoid]]'' (+4375/4374) → [[Kleismic rank-3 family #Catakleismoid|Kleismic rank-3 family]]
-== Undecimal marvel (unimarv) ==
+== Undecimal marvel ==
 {{Main| Marvel }}
+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
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 [[Complexity spectrum]]: 5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11
-[[Associated temperament]]: [[catakleismic]]
 Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]
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 }}
-=== 13-limit ===
+=== Helios ===
 Subgroup: 2.3.5.7.11.13
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 === Hecate ===
+Hecate tempers out [[325/324]], the marveltwin comma, such that [[16/13]] is found by a stack of two ~[[10/9]]'s, similar to how [[8/7]] is found by a stack of two [[15/14]]~[[16/15]]'s. Hecate has a natural extension to include [[prime interval|prime]] [[19/1|19]], where it further tempers out [[400/399]] and [[513/512]], taking advantage of the factorization 225/224 = (400/399)⋅(513/512). For both of these cases [[166edo]] remains an excellent tuning.
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
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 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:
@@ Line 343: / Line 343: @@
 == Prodigy ==
-Prodigy shrinks 1024/1029, 243/242, 384/385 and 2400/2401 down to the same tiny interval. Hence in practice it probably makes the most sense to temper this out as well, leading to [[miracle]]. This, however, does not render it pointless to consider prodigy; for one thing, scales in prodigy such as hobbit scales translate into interesting scales for miracle.
+Prodigy tempers out [[441/440]] and shrinks [[243/242]], [[385/384|384/385]], [[1029/1024|1024/1029]] and [[2401/2400|2400/2401]] down to the same tiny interval. Hence in practice it probably makes the most sense to temper this out as well, leading to [[miracle]]. This, however, does not render it pointless to consider prodigy; for one thing, scales in prodigy such as [[hobbit]] scales translate into interesting scales for miracle.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 375: / Line 375: @@
 Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]
-[[Associated temperament]]: [[miracle]]
 {{Databox|Hobbit bases|
@@ Line 460: / Line 458: @@
 == Minerva ==
+Minerva tempers out [[99/98]] as well as [[176/175]]. It may be described as 12 & 22 & 31, and is loosely [[associated temperament|associated]] with [[würschmidt]].
 [[Subgroup]]: 2.3.5.7.11
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 Scales: [[minerva12]], [[minerva22x]]
-[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]
 === Athene ===
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 == Apollo ==
 {{See also| Ptolemismic clan #Apollo }}
+[[File:Lattice Apollo.png|thumb|Lattice for apollo.]]
+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 ~[[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
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 [[Projection pair]]s: <code>7 225/32 11 100/9</code>
-[[Associated temperament]]: [[magic]]
 Scales: [[apollo wholetone]], [[indigo17]]
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 Projection pairs: <code>7 225/32 11 100/9 13 3200/243</code>
+=== Phoebus ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 100/99, 105/104, 196/195
+Mapping: {{mapping| 1 0 0 -5 2 1 | 0 1 0 2 -2 3 | 0 0 1 2 2 3 }}
+Optimal tunings:
+* WE: ~2 = 1200.3724{{c}}, ~3/2 = 702.5274{{c}}, ~5/4 = 379.6345{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3357{{c}}, ~5/4 = 379.6830{{c}}
+{{Optimal ET sequence|legend=0| 12f, 19, 22f, 29, 41 }}
+Badness (Sintel): 0.886
+=== Musagetes ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 100/99, 144/143, 225/224
+Mapping: {{mapping| 1 0 0 -5 2 2 | 0 1 0 2 -2 4 | 0 0 1 2 2 -2 }}
+Optimal tunings:
+* WE: ~2 = 1199.2695{{c}}, ~3/2 = 702.5589{{c}}, ~5/4 = 382.7715{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7998{{c}}, ~5/4 = 382.7740{{c}}
+{{Optimal ET sequence|legend=0| 19, 22f, 34d, 41, 75e, 94e, 116ef }}
+Badness (Sintel): 1.14
 == Potassium ==
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 [[Badness]] (Sintel): 1.38
-== Catakleismoid ==
-Catakleismoid is the same as catakleismic in the 2.3.5.7.13 subgroup but with an independent generator for prime 11.
-[[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 225/224, 4375/4374
-{{Mapping|legend=1| 1 0 1 -3 0 | 0 6 5 22 0 | 0 0 0 0 1 }}
-: mapping generators: ~2, ~6/5, ~11
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}, ~11/8 = 549.5258{{c}}
-: [[error map]]: {{val| +0.596 -0.619 -1.271 +0.948 -0.003 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}, ~11/8 = 550.0343{{c}}
-: error map: {{val| 0.000 -1.332 -2.461 +0.126 -1.284 }}
-{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce, 341ce }}
-[[Badness]] (Sintel): 1.53
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 225/224, 325/324
-Mapping: {{mapping| 1 0 1 -3 0 0 | 0 6 5 22 0 14 | 0 0 0 0 1 0 }}
-Optimal tunings:
-* WE: ~2 = 1200.8238{{c}}, ~6/5 = 316.9478{{c}}, ~11/8 = 548.9611{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}, ~11/8 = 549.5903{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 19, 34d, 53, 72, 125f, 197ef }}
-Badness (Sintel): 0.853
 == Subgroup extensions ==