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{{interwiki
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| de = Marvel
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.
| en = Marvel family
| es =
| ja =
}}
The '''marvel family''' is the set of temperaments that temper out the [[7-limit]] ''marvel comma'' ([[225/224]] = {{monzo|-5 2 2 -1}}) which is also named ''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 edos and rank-2 temperaments.


The marvel comma can also be viewed as a comma of the 2.9.25.7 subgroup. Hence it is tempered out by any subset edos of marvel-supporting edos that have this subgroup, such as [[11edo]] and [[17edo]] which are subsets of [[22edo]] and [[34edo]] (when using the 34b val) which temper out the marvel comma.
The marvel comma can also be viewed as a comma of the 2.9.25.7 [[subgroup]]. Hence it is tempered out by any subset edos of marvel-supporting edos that have this subgroup, such as [[11edo]] and [[17edo]] which are subsets of [[22edo]] and [[34edo]] (when using the 34d val) which temper out the marvel comma.


== Marvel ==
== Marvel ==
{{main| Marvel }}
{{Main| Marvel }}


The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[Normal lists|normal list basis]] 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 temperament]]. Another temperament which does little damage to tuning accuracy is [[compton temperament]], 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224
[[Comma list]]: 225/224


[[Mapping]]: [{{val| 1 0 0 -5 }}, {{val| 0 1 0 2 }}, {{val| 0 0 1 2 }}]
{{Mapping|legend=1| 1 0 0 -5 | 0 1 0 2 | 0 0 1 2 }}
 
: mapping generators: ~2, ~3, ~5
Mapping generators: ~2, ~3, ~5


Map to lattice: [{{val| 0 0 -1 -2 }}, {{val| 0 1 -1 0 }}]
Map to lattice: [{{val| 0 0 -1 -2 }}, {{val| 0 1 -1 0 }}]


Lattice basis:  
Lattice basis:  
: secor length = 1.256, 3/2 length = 1.369
: ~15/14 length = 1.256, ~3/2 length = 1.369
: Angle (secor, 3/2) = 106.958 degrees
: angle (~15/14, ~3/2) = 106.958°


[[POTE generator]]s: ~3/2 = 700.4075, ~5/4 = 383.6376
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5971{{c}}, ~3/2 = 700.7560{{c}}, ~5/4 = 383.8285{{c}}
: [[error map]]: {{val| +0.597 -0.602 -1.291 +0.940 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.6222{{c}}, ~5/4 = 383.8540{{c}}
: error map: {{val| 0.000 -1.333 -2.460 +0.127 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: 3 and 5 1/4c flat, 7 just
* [[7-odd-limit]]: 3 and 5 1/4-comma flat, 7 just
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/4 1/2 -1/2 1/4 }}, {{monzo| 5/4 -1/2 1/2 1/4 }}, {{monzo| 0 0 0 1 }}]
: {{monzo list| 1 0 0 0 | 5/4 1/2 -1/2 1/4 | 5/4 -1/2 1/2 1/4 | 0 0 0 1 }}
: [[Eigenmonzo subgroup]]: 2.5/3.7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
* [[9-odd-limit]]: 3 1/6c flat, 5 1/3c flat, 7 just
* [[9-odd-limit]]: 3 1/6-comma flat, 5 1/3-comma flat, 7 just
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/6 2/3 -1/3 1/6 }}, {{monzo| 5/3 -2/3 1/3 1/3 }}, {{monzo| 0 0 0 1 }}]
: {{monzo list| 1 0 0 0 | 5/6 2/3 -1/3 1/6 | 5/3 -2/3 1/3 1/3 | 0 0 0 1 }}
: [[Eigenmonzo subgroup]]: 2.9/5.7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7


{{Val list|legend=1| 9, 10, 12, 19, 31, 41, 53, 72, 197, 269c }}
{{Optimal ET sequence|legend=1| 9, 10, 12, 19, 31, 41, 53, 72, 197, 269c }}


[[Badness]]: 0.0365 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.161


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


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


[[Associated temperament]]: [[catakleismic]]
Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]], [[marvel wholetone]]
 
Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]]


{{Databox|[[Minkowski blocks]]|
{{Databox|[[Minkowski blocks]]|
{2, 3, 5} subgroup
2.3.5 subgroup
* 8: 16/15, 250/243
* 8: 16/15, 250/243
* 9: 135/128, 128/125
* 9: 135/128, 128/125
Line 69: Line 65:
}}
}}


=== Eleven-limit extensions ===
=== 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.  
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.
* 4125/4096 gives unidecimal marvel,
 
* 91125/90112 gives prodigy,
Temperaments discussed elsewhere include
* 5632/5625 gives minerva, and
* ''[[Supernatural]]'' (+245/243) → [[Keemic family #Supernatural|Keemic family]]
* 243/242 gives spectacle.
* ''[[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 ==
Subgroup: 2.3.5.7.11
{{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


[[Comma list]]: 225/224, 385/384
[[Comma list]]: 225/224, 385/384


[[Mapping]]: [{{val| 1 0 0 -5 12 }}, {{val| 0 1 0 2 -1 }}, {{val| 0 0 1 2 -3 }}]
{{Mapping|legend=1| 1 0 0 -5 12 | 0 1 0 2 -1 | 0 0 1 2 -3 }}


Map to lattice: [{{val| 0 -1 0 -2 1 }}, {{val| 0 -1 1 0 -2 }}]
[[Mapping to lattice]]: [{{val| 0 -1 0 -2 1 }}, {{val| 0 -1 1 0 -2 }}]


Lattice basis:  
Lattice basis:  
: secor length = 1.0364, 5/4 length = 1.0759
: ~15/14 length = 1.0364, ~5/4 length = 1.0759
: Angle (secor, 5/4) = 104.028 degrees
: angle (~15/14, ~5/4) = 104.028°


[[POTE generator]]s: ~3/2 = 700.3887, ~5/4 = 383.5403
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6395{{c}}, ~3/2 = 700.7619{{c}}, ~5/4 = 383.7447{{c}}
: [[error map]]: {{val| +0.639 -0.554 -1.290 +0.827 -0.117 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.6048{{c}}, ~5/4 = 383.4538{{c}}
: error map: {{val| 0.000 -1.350 -2.860 -0.709 -2.284 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* 11-odd-limit
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 8/9 -1/3 0 -1/9 }}, {{monzo| 8/3 -2/9 1/3 0 -2/9 }}, {{monzo| 3 4/3 0 0 -2/3 }}, {{monzo| 8/3 -2/9 -2/3 0 7/9 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 8/9 -1/3 0 -1/9 }}, {{monzo| 8/3 -2/9 1/3 0 -2/9 }}, {{monzo| 3 4/3 0 0 -2/3 }}, {{monzo| 8/3 -2/9 -2/3 0 7/9 }}]
: [[Eigenmonzo subgroup]]: 2.9/5.11/9
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9


{{Val list|legend=1| 9, 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 238c, 269ce, 341ce }}
{{Optimal ET sequence|legend=1| 9, 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 238c, 269ce, 341ce }}


[[Badness]]: 0.255 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.306


[[Projection pair]]s: 7 225/32 11 4096/375
[[Projection pair]]s: <code>7 225/32 11 4096/375</code>


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


{{Databox|Hobbit bases|
{{Databox|[[Hobbit]] bases|
{2, 3, 5} subgroup
2.3.5 subgroup
* 12: 128/125, 2048/2025
* 12: 128/125, 2048/2025
* 15: 128/125, 32768/30375
* 15: 128/125, 32768/30375
Line 118: Line 126:
}}
}}


=== 13-limit ===
=== Helios ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 351/350, 385/384
Comma list: 225/224, 351/350, 385/384


Mapping: [{{val| 1 0 0 -5 12 -4 }}, {{val| 0 1 0 2 -1 -1 }}, {{val| 0 0 1 2 -3 4 }}]
Mapping: {{mapping| 1 0 0 -5 12 -4 | 0 1 0 2 -1 -1 | 0 0 1 2 -3 4 }}


POTE generators: ~3/2 = 699.7367, ~5/4 = 384.0613
Optimal tunings:  
* WE: ~2 = 1200.8043{{c}}, ~3/2 = 700.2057{{c}}, ~5/4 = 384.3188{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.8109{{c}}, ~5/4 = 384.1177{{c}}


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo subgroup: 2.11/9.13/9
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9
* 15-odd-limit eigenmonzo subgroup: 2.15/11.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.15/11.15/13


Vals: {{Val list| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}
{{Optimal ET sequence|legend=0| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}


Badness: 0.690 × 10<sup>-3</sup>
Badness (Sintel): 0.645


Complexity spectrum: 5/4, 4/3, 16/15, 15/14, 9/7, 6/5, 7/6, 11/8, 7/5, 9/8, 8/7, 10/9, 12/11, 13/10, 11/10, 15/11, 16/13, 11/9, 15/13, 14/13, 13/12, 14/11, 18/13, 13/11
Complexity spectrum: 5/4, 4/3, 16/15, 15/14, 9/7, 6/5, 7/6, 11/8, 7/5, 9/8, 8/7, 10/9, 12/11, 13/10, 11/10, 15/11, 16/13, 11/9, 15/13, 14/13, 13/12, 14/11, 18/13, 13/11


=== Hecate ===
=== 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
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 325/324, 385/384
Comma list: 225/224, 325/324, 385/384


Mapping: [{{val| 1 0 0 -5 12 2 }}, {{val| 0 1 0 2 -1 4 }}, {{val| 0 0 1 2 -3 -2 }}]
Mapping: {{mapping| 1 0 0 -5 12 2 | 0 1 0 2 -1 4 | 0 0 1 2 -3 -2 }}


POTE generators: ~3/2 = 700.9779, ~5/4 = 383.1622
Optimal tunings:  
* WE: ~2 = 1200.5788{{c}}, ~3/2 = 701.3161{{c}}, ~5/4 = 383.3471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1003{{c}}, ~5/4 = 383.1315{{c}}


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo subgroup: 2.7.13/5
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.13/5
* 15-odd-limit eigenmonzo subgroup: 2.7.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13


Vals: {{Val list| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}
{{Optimal ET sequence|legend=0| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}


Badness: 0.721 × 10<sup>-3</sup>
Badness (Sintel): 0.674


Projection pairs: 7 225/32 11 4096/375 13 324/25
Projection pairs: <code>7 225/32 11 4096/375 13 324/25</code>


Complexity spectrum: 4/3, 5/4, 16/15, 15/14, 6/5, 9/8, 7/5, 9/7, 7/6, 10/9, 8/7, 18/13, 11/8, 12/11, 13/12, 11/9, 11/10, 15/13, 15/11, 16/13, 13/11, 14/13, 13/10, 14/11
Complexity spectrum: 4/3, 5/4, 16/15, 15/14, 6/5, 9/8, 7/5, 9/7, 7/6, 10/9, 8/7, 18/13, 11/8, 12/11, 13/12, 11/9, 11/10, 15/13, 15/11, 16/13, 13/11, 14/13, 13/10, 14/11


==== 17-limit ====
===== 2.3.5.7.11.13.19 subgroup =====
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 225/224, 325/324, 385/384, 400/399
 
Subgroup-val mapping: {{mapping| 1 0 0 -5 12 2 9 | 0 1 0 2 -1 4 -3 | 0 0 1 2 -3 -2 0 }}
 
Optimal tunings:
* WE: ~2 = 1200.4716{{c}}, ~3/2 = 701.4395{{c}}, ~5/4 = 383.2779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2002{{c}}, ~5/4 = 383.1136{{c}}
 
{{Optimal ET sequence|legend=0| 41, 53, 72, 94, 113, 166 }}
 
Badness (Sintel): 0.756
 
==== Apotropaia ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 325/324, 385/384, 595/594
Comma list: 225/224, 325/324, 385/384, 595/594


Mapping: [{{val| 1 0 0 -5 12 2 18 }}, {{val| 0 1 0 2 -1 4 0 }}, {{val| 0 0 1 2 -3 -2 -6 }}]
Mapping: {{mapping| 1 0 0 -5 12 2 18 | 0 1 0 2 -1 4 0 | 0 0 1 2 -3 -2 -6 }}
 
Optimal tunings:
* WE: ~2 = 1200.6057{{c}}, ~3/2 = 701.3157{{c}}, ~5/4 = 383.2243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0877{{c}}, ~5/4 = 382.9709{{c}}
 
{{Optimal ET sequence|legend=0| 41, 53g, 72, 166g, 238cfg }}
 
Badness (Sintel): 0.827
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 385/384, 400/399, 595/594
 
Mapping: {{mapping| 1 0 0 -5 12 2 18 9 | 0 1 0 2 -1 4 0 -3 | 0 0 1 2 -3 -2 -6 0 }}


POTE generators: ~3/2 = 700.9619, ~5/4 = 383.0310
Optimal tunings:
* WE: ~2 = 1200.4927{{c}}, ~3/2 = 701.4506{{c}}, ~5/4 = 383.1291{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2007{{c}}, ~5/4 = 382.9404{{c}}


Vals: {{Val list| 19, 22f, 31fg, 41, 53g, 72, 166g, 238cfg, 404ccefgg }}
{{Optimal ET sequence|legend=0| 41, 53g, 72, 94, 113, 166g }}


Badness: 0.869 × 10<sup>-3</sup>
Badness (Sintel): 1.01


==== Enodia ====
==== Enodia ====
Line 176: Line 223:
Comma list: 225/224, 325/324, 375/374, 385/384
Comma list: 225/224, 325/324, 375/374, 385/384


Mapping: [{{val| 1 0 0 -5 12 2 18 }}, {{val| 0 1 0 2 -1 4 0 }}, {{val| 0 0 1 2 -3 -2 6 }}]
Mapping: {{mapping| 1 0 0 -5 12 2 -13 | 0 1 0 2 -1 4 2 | 0 0 1 2 -3 -2 6 }}


POTE generators: ~3/2 = 700.9658, ~5/4 = 383.3063
Optimal tunings:  
* WE: ~2 = 1200.6165{{c}}, ~3/2 = 701.3259{{c}}, ~5/4 = 383.5032{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0949{{c}}, ~5/4 = 383.3140{{c}}


Vals: {{Val list| 19g, 22f, 31f, 41g, 53, 72, 166g, 238cfg, 404ccefgg }}
{{Optimal ET sequence|legend=0| 41g, 53, 72, 166g, 238cfg }}


Badness: 0.917 × 10<sup>-3</sup>
Badness (Sintel): 0.872
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399
 
Mapping: {{mapping| 1 0 0 -5 12 2 -13 9 | 0 1 0 2 -1 4 2 -3 | 0 0 1 2 -3 -2 6 0 }}
 
Optimal tunings:
* WE: ~2 = 1200.5038{{c}}, ~3/2 = 701.4654{{c}}, ~5/4 = 383.4549{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2113{{c}}, ~5/4 = 383.3052{{c}}
 
{{Optimal ET sequence|legend=0| 41g, 53, 72, 94, 125f, 166g }}
 
Badness (Sintel): 1.05


=== Marvell ===
=== Marvell ===
Line 189: Line 253:
Comma list: 225/224, 385/384, 1573/1568
Comma list: 225/224, 385/384, 1573/1568


Mapping: [{{val| 1 0 0 -5 12 -29 }}, {{val| 0 1 0 2 -1 6 }}, {{val| 0 0 1 2 -3 10 }}]
Mapping: {{mapping| 1 0 0 -5 12 -29 | 0 1 0 2 -1 6 | 0 0 1 2 -3 10 }}


POTE generators: ~3/2 = 700.3937, ~5/4 = 383.5725
Optimal tunings:  
* WE: ~2 = 1200.6404{{c}}, ~3/2 = 700.7675{{c}}, ~5/4 = 383.7772{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6113{{c}}, ~5/4 = 383.4925{{c}}


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo subgroup: 2.9/5.11/9
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.11/9
* 15-odd-limit eigenmonzo subgroup: 2.7.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13


Vals: {{Val list| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff }}
{{Optimal ET sequence|legend=0| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 341cef }}


Badness: 0.862 × 10<sup>-3</sup>
Badness (Sintel): 0.806


=== Isis ===
=== Isis ===
Line 206: Line 272:
Comma list: 225/224, 275/273, 385/384
Comma list: 225/224, 275/273, 385/384


Mapping: [{{val| 1 0 0 -5 12 17 }}, {{val| 0 1 0 2 -1 4 }}, {{val| 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 }}


POTE generators: ~3/2 = 701.9156, ~5/4 = 383.2445
Optimal tunings:  
* WE: ~2 = 1200.1824{{c}}, ~3/2 = 702.0223{{c}}, ~5/4 = 383.3028{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9276{{c}}, ~5/4 = 383.2283{{c}}


Vals: {{Val list| 10, 19f, 22, 31, 41, 53, 94 }}
{{Optimal ET sequence|legend=0| 10, 19f, 22, 31, 41, 53, 84e, 94 }}


Badness: 0.866 × 10<sup>-3</sup>
Badness (Sintel): 0.810


Projection pairs: 7 225/32 11 4096/375 13 131072/10125
Projection pairs: <code>7 225/32 11 4096/375 13 131072/10125</code>


=== Deecee ===
=== Deecee ===
Line 221: Line 289:
Comma list: 225/224, 364/363, 385/384
Comma list: 225/224, 364/363, 385/384


Mapping: [{{val| 1 0 0 -5 12 27 }}, {{val| 0 1 0 2 -1 -3 }}, {{val| 0 0 1 2 -3 -8 }}]
Mapping: {{mapping| 1 0 0 -5 12 27 | 0 1 0 2 -1 -3 | 0 0 1 2 -3 -8 }}


POTE generators: ~3/2 = 700.4560, ~5/4 = 382.8177
Optimal tunings:  
* WE: ~2 = 1200.7533{{c}}, ~3/2 = 700.8957{{c}}, ~5/4 = 383.0580{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.7184{{c}}, ~5/4 = 382.6611{{c}}


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo subgroup: 2.9/5.13/9
* 13-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.9/5.13/9
* 15-odd-limit eigenmonzo subgroup: 2.3.13/5
* 15-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.3.13/5


Vals: {{Val list| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}
{{Optimal ET sequence|legend=0| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}


Badness: 0.920 × 10<sup>-3</sup>
Badness (Sintel): 0.861


Projection pairs: 7 225/32 11 4096/375 13 134217728/10546875
Projection pairs: <code>7 225/32 11 4096/375 13 134217728/10546875</code>


=== Tripod ===
=== Tripod ===
Line 240: Line 310:
Comma list: 105/104, 144/143, 196/195
Comma list: 105/104, 144/143, 196/195


Mapping: [{{val| 1 0 0 -5 12 -8 }}, {{val| 0 1 0 2 -1 3 }}, {{val| 0 0 1 2 -3 3 }}]
Mapping: {{mapping| 1 0 0 -5 12 -8 | 0 1 0 2 -1 3 | 0 0 1 2 -3 3 }}


POTE generators: ~3/2 = 699.2335, ~5/4 = 382.9775
Optimal tunings:  
* WE: ~2 = 1200.4789{{c}}, ~3/2 = 699.5125{{c}}, ~5/4 = 383.1304{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4041{{c}}, ~5/4 = 382.9166{{c}}


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo subgroup: 2.9/7.13/11
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/11
* 15-odd-limit eigenmonzo subgroup: 2.5/3.13/11
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/11


Vals: {{Val list| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}
{{Optimal ET sequence|legend=0| 9, 10, 19, 22f, 31, 41, 72f, 91 }}


Badness: 0.745 × 10<sup>-3</sup>
Badness (Sintel): 0.697


Projection pairs: 7 225/32 11 4096/375 13 3375/256
Projection pairs: <code>7 225/32 11 4096/375 13 3375/256</code>


=== Marvelcat ===
=== Marvelcat ===
Line 259: Line 331:
Comma list: 169/168, 225/224, 385/384
Comma list: 169/168, 225/224, 385/384


Mapping: [{{val| 1 0 0 -5 12 -1 }}, {{val| 0 2 0 4 -2 3 }}, {{val| 0 0 1 2 -3 1 }}]
Mapping: {{mapping| 1 0 0 -5 12 -1 | 0 2 0 4 -2 3 | 0 0 1 2 -3 1 }}
: mapping generators: ~2, ~26/15, ~5


Mapping generators: ~2, ~26/15, ~5
Optimal tunings:
* WE: ~2 = 1200.7720{{c}}, ~3/2 = 950.8976{{c}}, ~5/4 = 383.8283{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.4265{{c}}, ~5/4 = 383.4790{{c}}


POTE generators: ~15/13 = 249.7138, ~5/4 = 383.5816
{{Optimal ET sequence|legend=0| 9, 10, 19, 43e, 44, 53, 72, 125f, 197ef }}


Vals: {{Val list| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}
Badness (Sintel): 0.935


Badness: 0.9997 × 10<sup>-3</sup>
== Prodigy ==
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.


== Minerva ==
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 99/98, 176/175
[[Comma list]]: 225/224, 441/440


[[Mapping]]: [{{val| 1 0 0 -5 -9 }}, {{val| 0 1 0 2 2 }}, {{val| 0 0 1 2 4 }}]
{{Mapping|legend=1| 1 0 0 -5 -13 | 0 1 0 2 6 | 0 0 1 2 3 }}


Map to lattice: [{{val| 0 -1 0 -2 -2 }}, {{val| 0 -1 1 0 2 }}]
Map to lattice: [{{val| 0 0 -1 -2 -3 }}, {{val| 0 1 -1 0 3 }}]


Lattice basis:  
Lattice basis:  
: 16/15 length = 0.8997, 5/4 length = 1.0457
: ~15/14 length = 0.9111, ~3/2 length = 0.9477
: Angle (16/15, 5/4) = 98.6044 degrees
: angle (~15/14, ~3/2) = 65.933°


[[POTE generator]]s: ~3/2 = 700.2593, ~5/4 = 386.5581
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7854{{c}}, ~3/2 = 700.2562{{c}}, ~5/4 = 383.7624{{c}}
: [[error map]]: {{val| +0.785 -0.913 -0.980 -0.003 +0.721 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.8610{{c}}, ~5/4 = 383.7724{{c}}
: error map: {{val| 0.000 -2.094 -2.541 -1.559 -0.835 }}


[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo subgroup]]: 2.7/5.11/9
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 13/12 1/2 -1/4 0 1/12 }}, {{monzo| 13/6 -1 1/2 0 1/6 }}, {{monzo| 3/2 -1 1/2 0 1/2 }}, {{monzo| 0 0 0 0 1 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11


{{Val list|legend=1| 9, 12, 19e, 22, 31, 53, 84e, 96, 127 }}
{{Optimal ET sequence|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}


[[Badness]]: 0.381 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.402


[[Projection pair]]s: 7 225/32 11 5625/512
[[Projection pair]]s: <code>7 225/32 11 91125/8192</code>


Scales (Scala files): [[minerva12]], [[minerva22x]]
Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]


[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]
{{Databox|Hobbit bases|
2.3.5 subgroup
* 31: 81/80, 34171875/33554432
* 41: 34171875/33554432, 32805/32768
}}


=== Athene ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 176/175, 275/273
Comma list: 105/104, 196/195, 352/351


Mapping: [{{val| 1 0 0 -5 -9 -4 }}, {{val| 0 1 0 2 2 -1 }}, {{val| 0 0 1 2 4 4 }}]
Mapping: {{mapping| 1 0 0 -5 -13 -8 | 0 1 0 2 6 3 | 0 0 1 2 3 3 }}


POTE generators: ~3/2 = 701.2342, ~5/4 = 385.9594
Optimal tunings:  
* WE: ~2 = 1200.8252{{c}}, ~3/2 = 700.8823{{c}}, ~5/4 = 381.6647{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.4689{{c}}, ~5/4 = 381.6687{{c}}


Minimax tuning:
{{Optimal ET sequence|legend=0| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 101cd }}
* 13-odd-limit eigenmonzo subgroup: 2.11/9.13/7
* 15-odd-limit eigenmonzo subgroup: 2.11/9.13/7


Vals: {{Val list| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}
Badness (Sintel): 0.689


Badness: 0.818 × 10<sup>-3</sup>
=== Prodigious ===
Subgroup: 2.3.5.7.11.13


Projection pairs: 7 225/32 11 5625/512 13 625/48
Comma list: 225/224, 364/363, 441/440


== Apollo ==
Mapping: {{mapping| 1 0 0 -5 -13 -23 | 0 1 0 2 6 11 | 0 0 1 2 3 4 }}
Subgroup: 2.3.5.7.11


[[Comma list]]: 100/99, 225/224
Optimal tunings:  
* WE: ~2 = 1200.6284{{c}}, ~3/2 = 700.7075{{c}}, ~5/4 = 383.4599{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.3302{{c}}, ~5/4 = 383.5030{{c}}


[[Mapping]]: [{{val| 1 0 0 -5 2 }}, {{val| 0 1 0 2 -2 }}, {{val| 0 0 1 2 2 }}]
{{Optimal ET sequence|legend=0| 12f, 29, 31f, 41, 72, 185cf, 257cff }}


[[POTE generator]]s: ~3/2 = 703.4846, ~5/4 = 381.6033
Badness (Sintel): 0.841


[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo subgroup]]: 2.7/5.11/9
=== Prodigal ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce }}
Comma list: 225/224, 351/350, 441/440


[[Projection pair]]s: 7 225/32 11 100/9
Mapping: {{mapping| 1 0 0 -5 -13 -4 | 0 1 0 2 6 -1 | 0 0 1 2 3 4 }}


[[Associated temperament]]: [[Magic family #magic|magic]]
Optimal tunings:  
* WE: ~2 = 1200.7798{{c}}, ~3/2 = 699.9410{{c}}, ~5/4 = 384.3494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.5538{{c}}, ~5/4 = 384.3496{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
Subgroup: 2.3.5.7.11.13


Comma list: 100/99, 225/224, 275/273
Badness (Sintel): 0.831


Mapping: [{{val| 1 0 0 -5 2 7 }}, {{val| 0 1 0 2 -2 -5 }}, {{val| 0 0 1 2 2 2 }}]
=== Protannic ===
Subgroup: 2.3.5.7.11.13


POTE generators: ~3/2 = 703.9984, ~5/4 = 381.5352
Comma list: 225/224, 441/440, 1001/1000


Minimax tuning: 13-odd-limit eigenmonzo subgroup: 2.11/9.13/9
Mapping: {{mapping| 1 0 0 -5 -13 21 | 0 1 0 2 6 -8 | 0 0 1 2 3 -2 }}


Vals: {{Val list| 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef }}
Optimal tunings:  
* WE: ~2 = 1200.9450{{c}}, ~3/2 = 700.1045{{c}}, ~5/4 = 383.8716{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4224{{c}}, ~5/4 = 383.9828{{c}}


Projection pairs: 7 225/32 11 100/9 13 3200/243
{{Optimal ET sequence|legend=0| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}


== Potassium ==
Badness (Sintel): 0.891
Subgroup: 2.3.5.7.11


[[Comma list]]: 45/44, 56/55
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


[[Mapping]]: [{{val| 1 0 0 -5 -2 }}, {{val| 0 1 0 2 2 }}, {{val| 0 0 1 2 1 }}]
Comma list: 225/224, 273/272, 375/374, 441/440


[[POTE generator]]s: ~3/2 = 696.1714, ~5/4 = 385.0500
Mapping: {{mapping| 1 0 0 -5 -13 21 12 | 0 1 0 2 6 -8 -5 | 0 0 1 2 3 -2 0 }}


[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo subgroup]]: 2.9/7.11
Optimal tunings:  
* WE: ~2 = 1200.9342{{c}}, ~3/2 = 700.1708{{c}}, ~5/4 = 383.7444{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4849{{c}}, ~5/4 = 383.8744{{c}}


{{Val list|legend=1| 7d, 9, 10, 12, 19, 31e, 50e }}
{{Optimal ET sequence|legend=0| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}


[[Badness]]: 0.464 × 10<sup>-3</sup>
Badness (Sintel): 0.734


[[Projection pair]]s: 7 225/32 11 45/4
== 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]].


=== 13-limit ===
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 56/55, 78/77
[[Comma list]]: 99/98, 176/175


Mapping: [{{val| 1 0 0 -5 -2 -8 }}, {{val| 0 1 0 2 2 3 }}, {{val| 0 0 1 2 1 3 }}]
{{Mapping|legend=1| 1 0 0 -5 -9 | 0 1 0 2 2 | 0 0 1 2 4 }}


POTE generators: ~3/2 = 696.0103, ~5/4 = 384.6785
Map to lattice: [{{val| 0 -1 0 -2 -2 }}, {{val| 0 -1 1 0 2 }}]


Minimax tuning:  
Lattice basis:  
* 13-odd-limit eigenmonzo subgroup: 2.9/7.13/9
: ~16/15 length = 0.8997, ~5/4 length = 1.0457
* 15-odd-limit eigenmonzo subgroup: 2.9/7.13/9
: angle (~16/15, ~5/4) = 98.6044°
 
Vals: {{Val list| 9, 10, 12f, 19, 31e, 50e }}
 
Badness: 0.733 × 10<sup>-3</sup>


Projection pairs: 7 225/32 11 45/4 13 3375/256
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1086{{c}}, ~3/2 = 700.3226{{c}}, ~5/4 = 386.5931{{c}}
: [[error map]]: {{val| +0.109 -1.524 +0.497 +5.114 -4.192 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.3006{{c}}, ~5/4 = 386.5785{{c}}
: error map: {{val| 0.000 -1.654 +0.265 +4.932 -4.403 }}


== Malcolm ==
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 2200/2187
{{Optimal ET sequence|legend=1| 9, 12, 19e, 22, 31, 53, 84e, 96, 127 }}


[[Mapping]]: [{{val| 1 0 0 -5 -3 }}, {{val| 0 1 0 2 7 }}, {{val| 0 0 1 2 -2 }}]
[[Badness]] (Sintel): 0.458


[[POTE generator]]s: ~3/2 = 701.8913, ~5/4 = 382.4083
[[Projection pair]]s: <code>7 225/32 11 5625/512</code>


{{Val list|legend=1| 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce, 342ccee, 383cce }}
Scales: [[minerva12]], [[minerva22x]]


[[Badness]]: 1.250 × 10<sup>-3</sup>
=== Athene ===
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
Comma list: 99/98, 176/175, 275/273
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 275/273, 325/324
Mapping: {{mapping| 1 0 0 -5 -9 -4 | 0 1 0 2 2 -1 | 0 0 1 2 4 4 }}


Mapping: [{{val| 1 0 0 -5 -3 2 }}, {{val| 0 1 0 2 7 4 }}, {{val| 0 0 1 2 -2 -2 }}]
Optimal tunings:  
* WE: ~2 = 1199.9127{{c}}, ~3/2 = 701.1832{{c}}, ~5/4 = 385.9313{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2143{{c}}, ~5/4 = 385.9336{{c}}


POTE generators: ~3/2 = 701.8913, ~5/4 = 382.4083
Minimax tuning:  
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7


Vals: {{Val list| 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef }}
{{Optimal ET sequence|legend=0| 12f, 19e, 22, 31, 53, 84e, 118d }}


Badness: 1.075 × 10<sup>-3</sup>
Badness (Sintel): 0.765


== Prodigy ==
Projection pairs: <code>7 225/32 11 5625/512 13 625/48</code>
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 temperament]]. 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
== Apollo ==
{{See also| Ptolemismic clan #Apollo }}


[[Comma list]]: 225/224, 441/440
[[File:Lattice Apollo.png|thumb|Lattice for apollo.]]


[[Mapping]]: [{{val| 1 0 0 -5 -13 }}, {{val| 0 1 0 2 6 }}, {{val| 0 0 1 2 3 }}]
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]].


Map to lattice: [{{val| 0 0 -1 -2 -3 }}, {{val| 0 1 -1 0 3 }}]
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.


Lattice basis:  
[[Subgroup]]: 2.3.5.7.11
: secor length = 0.9111, 3/2 length = 0.9477
: Angle (secor, 3/2) = 65.933


[[POTE generator]]s: ~3/2 = 699.7981, ~5/4 = 383.5114
[[Comma list]]: 100/99, 225/224


[[Minimax tuning]]:
{{Mapping|legend=1| 1 0 0 -5 2 | 0 1 0 2 -2 | 0 0 1 2 2 }}
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 13/12 1/2 -1/4 0 1/12 }}, {{monzo| 13/6 -1 1/2 0 1/6 }}, {{monzo| 3/2 -1 1/2 0 1/2 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo subgroup]]: 2.9/5.11


{{Val list|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8250{{c}}, ~3/2 = 703.3820{{c}}, ~5/4 = 381.5476{{c}}
: [[error map]]: {{val| -0.175 +1.252 -5.116 +0.858 +4.313 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4612{{c}}, ~5/4 = 381.5071{{c}}
: error map: {{val| 0.000 +1.506 -4.807 +1.111 +4.774 }}


[[Badness]]: 0.344 × 10<sup>-3</sup>
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9


[[Projection pair]]s: 7 225/32 11 91125/8192
{{Optimal ET sequence|legend=1| 12, 19, 22, 34d, 41, 104 }}


Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]
[[Badness]] (Sintel): 0.637


[[Associated temperament]]: [[Gamelismic clan #Miracle|miracle]]
[[Projection pair]]s: <code>7 225/32 11 100/9</code>


{{Databox|Hobbit bases|
Scales: [[apollo wholetone]], [[indigo17]]
{2, 3, 5} subgroup
* 31: 81/80, 34171875/33554432
* 41: 34171875/33554432, 32805/32768
}}


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 105/104, 196/195, 352/351
Comma list: 100/99, 225/224, 275/273
 
Mapping: [{{val| 1 0 0 -5 -13 -8 }}, {{val| 0 1 0 2 6 3 }}, {{val| 0 0 1 2 3 3 }}]
 
POTE generators: ~3/2 = 700.4006, ~5/4 = 381.4025
 
Vals: {{Val list| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff }}
 
Badness: 0.736 × 10<sup>-3</sup>


=== Prodigious ===
Mapping: {{mapping| 1 0 0 -5 2 7 | 0 1 0 2 -2 -5 | 0 0 1 2 2 2 }}
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 364/363, 441/440
Optimal tunings:  
* WE: ~2 = 1199.6919{{c}}, ~3/2 = 703.8176{{c}}, ~5/4 = 381.4372{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.9853{{c}}, ~5/4 = 381.3579{{c}}


Mapping: [{{val| 1 0 0 -5 -13 -23 }}, {{val| 0 1 0 2 6 11 }}, {{val| 0 0 1 2 3 4 }}]
Minimax tuning: 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9


POTE generators: ~3/2 = 700.3407, ~5/4 = 383.2592
{{Optimal ET sequence|legend=0| 12f, 19f, 22, 29, 34d, 41, 63, 104 }}


Vals: {{Val list| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}
Badness (Sintel): 0.964


Badness: 0.900 × 10<sup>-3</sup>
Projection pairs: <code>7 225/32 11 100/9 13 3200/243</code>


=== Prodigal ===
=== Phoebus ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 351/350, 441/440
Comma list: 100/99, 105/104, 196/195


Mapping: [{{val| 1 0 0 -5 -13 -4 }}, {{val| 0 1 0 2 6 -1 }}, {{val| 0 0 1 2 3 4 }}]
Mapping: {{mapping| 1 0 0 -5 2 1 | 0 1 0 2 -2 3 | 0 0 1 2 2 3 }}


POTE generators: ~3/2 = 699.4864, ~5/4 = 384.0998
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}}


Vals: {{Val list| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
{{Optimal ET sequence|legend=0| 12f, 19, 22f, 29, 41 }}


Badness: 0.889 × 10<sup>-3</sup>
Badness (Sintel): 0.886


=== Protannic ===
=== Musagetes ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 441/440, 1001/1000
Comma list: 100/99, 144/143, 225/224


Mapping: [{{val| 1 0 0 -5 -13 21 }}, {{val| 0 1 0 2 6 -8 }}, {{val| 0 0 1 2 3 -2 }}]
Mapping: {{mapping| 1 0 0 -5 2 2 | 0 1 0 2 -2 4 | 0 0 1 2 2 -2 }}


POTE generators: ~3/2 = 699.5536, ~5/4 = 383.5696
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}}


Vals: {{val list| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}
{{Optimal ET sequence|legend=0| 19, 22f, 34d, 41, 75e, 94e, 116ef }}


Badness: 0.953 × 10<sup>-3</sup>
Badness (Sintel): 1.14


==== 17-limit ====
== Potassium ==
Subgroup: 2.3.5.7.11.13.17
[[Subgroup]]: 2.3.5.7.11


Comma list: 225/224, 273/272, 375/374, 441/440
[[Comma list]]: 45/44, 56/55


Mapping: [{{val| 1 0 0 -5 -13 21 12 }}, {{val| 0 1 0 2 6 -8 -5 }}, {{val| 0 0 1 2 3 -2 0 }}]
{{Mapping|legend=1| 1 0 0 -5 -2 | 0 1 0 2 2 | 0 0 1 2 1 }}


POTE generators: ~3/2 = 699.6262, ~5/4 = 383.4458
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7106{{c}}, ~3/2 = 696.0036{{c}}, ~5/4 = 384.9572{{c}}
: [[error map]]: {{val| -0.289 -6.241 -1.935 -7.194 +25.068 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.0586{{c}}, ~5/4 = 384.9472{{c}}
: error map: {{val| 0.000 -5.896 -1.367 -6.814 +25.746 }}


Vals: {{val list| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11


Badness: 0.772 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 31e }}


== Fantastic ==
[[Badness]] (Sintel): 0.557
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 4375/4356
[[Projection pair]]s: <code>7 225/32 11 45/4</code>


[[Mapping]]: [{{val| 2 0 0 -10 -7 }}, {{val| 0 1 0 2 0 }}, {{val| 0 0 1 2 3 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping generators: ~99/70, ~3, ~5
Comma list: 45/44, 56/55, 78/77


[[POTE generator]]s: ~3/2 = 700.6242, ~5/4 = 383.2978
Mapping: {{mapping| 1 0 0 -5 -2 -8 | 0 1 0 2 2 3 | 0 0 1 2 1 3 }}


{{Val list|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}
Optimal tunings:
* WE: ~2 = 1199.8192{{c}}, ~3/2 = 695.9054{{c}}, ~5/4 = 384.6205{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.9480{{c}}, ~5/4 = 384.6372{{c}}


[[Badness]]: 0.743 × 10<sup>-3</sup>
Minimax tuning:  
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9


== Spectacle ==
{{Optimal ET sequence|legend=0| 9, 10, 12f, 19, 31e }}
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 243/242
Badness (Sintel): 0.686


[[Mapping]]: [{{val| 1 1 0 -3 2 }}, {{val| 0 2 0 4 5 }}, {{val| 0 0 1 2 0 }}]
Projection pairs: <code>7 225/32 11 45/4 13 3375/256</code>


Mapping generators: ~2, ~11/9, ~5
== Malcolm ==
{{Redirect|Malcolm|{{w|Alexander Malcolm (writer on music)|Alexander Malcolm}}'s JI scale|Malcolm (scale)}}


[[POTE generator]]s: ~11/9 = 350.0570, ~5/4 = 383.9323
[[Subgroup]]: 2.3.5.7.11


[[Minimax tuning]]:
[[Comma list]]: 225/224, 2200/2187
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1/5 0 0 0 2/5 }}, {{monzo| 2/5 -2 1 0 4/5 }}, {{monzo| -19/5 -4 2 0 12/5 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo subgroup]]: 2.9/5.11


{{Val list|legend=1| 31, 41, 72, 247c, 281, 353c, 425bc, 497bc }}
{{Mapping|legend=1| 1 0 0 -5 -3 | 0 1 0 2 7 | 0 0 1 2 -2 }}


[[Badness]]: 0.499 × 10<sup>-3</sup>
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.5261{{c}}, ~3/2 = 702.1990{{c}}, ~5/4 = 382.5759{{c}}
: [[error map]]: {{val| +0.526 +0.770 -2.686 +1.250 -1.077 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0535{{c}}, ~5/4 = 382.6222{{c}}
: error map: {{val| 0.000 +0.098 -3.692 +0.525 -2.188 }}


[[Projection pair]]s: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9
{{Optimal ET sequence|legend=1| 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce }}


Scales (Scala files): [[spectacle31]]
[[Badness]] (Sintel): 1.50
 
[[Associated temperament]]: [[Marvel temperaments #Marvo|marvo]]


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 351/350
Comma list: 225/224, 275/273, 325/324


Mapping: [{{val| 1 1 0 -3 2 -5 }}, {{val| 0 2 0 4 5 -2 }}, {{val| 0 0 1 2 0 4 }}]
Mapping: {{mapping| 1 0 0 -5 -3 2 | 0 1 0 2 7 4 | 0 0 1 2 -2 -2 }}


Mapping generators: ~2, ~11/9, ~5
Optimal tunings:
* WE: ~2 = 1200.2427{{c}}, ~3/2 = 702.1575{{c}}, ~5/4 = 383.0704{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0882{{c}}, ~5/4 = 383.0630{{c}}


POTE generators: ~11/9 = 349.9247, ~5/4 = 384.3505
{{Optimal ET sequence|legend=0| 12e, 19e, 34d, 41, 53, 94 }}


Vals: {{Val list| 31, 72, 103, 175f }} <nowiki>*</nowiki>
Badness (Sintel): 1.00


<nowiki>*</nowiki> optimal patent val: [[240edo|240]]
Scales: [[malco]]


Badness: 1.009 × 10<sup>-3</sup>
== Fantastic ==
Besides 4375/4356, fantastic also tempers out [[9801/9800]] and splits the octave in two.  


== Hestia ==
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 125000/124509
[[Comma list]]: 225/224, 4375/4356


[[Mapping]]: [{{val| 1 0 0 -5 9 }}, {{val| 0 2 0 4 -7 }}, {{val| 0 0 1 2 0 }}]
{{Mapping|legend=1| 2 0 0 -10 -7 | 0 1 0 2 0 | 0 0 1 2 3 }}
: mapping generators: ~99/70, ~3, ~5


Mapping generators: ~2, ~400/231, ~5
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 600.2896{{c}}, ~3/2 = 700.9624{{c}}, ~5/4 = 383.4829{{c}}
: [[error map]]: {{val| +0.579 -0.413 -1.672 +0.644 +0.579 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 700.8160{{c}}, ~5/4 = 383.5350{{c}}
: error map: {{val| 0.000 -1.139 -2.779 -0.124 -0.713 }}


[[POTE generator]]s: ~231/200 = 249.8526, ~5/4 = 383.6467
{{Optimal ET sequence|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}


{{Val list|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce }}
[[Badness]] (Sintel): 0.893


[[Badness]]: 1.54 × 10<sup>-3</sup>
== Hestia ==
Named by [[Graham Breed]] in 2011, hestia was found to be locally efficient in the higher limits among all rank-3 extensions of marvel<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19673.html Yahoo! Tuning Group | ''Artemis and friends'']</ref>, although it is a [[weak extension]].


=== 13-limit ===
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 225/224, 1001/1000
[[Comma list]]: 225/224, 125000/124509
 
Mapping: [{{val| 1 0 0 -5 9 -1 }}, {{val| 0 2 0 4 -7 3 }}, {{val| 0 0 1 2 0 1 }}]
 
Mapping generators: ~2, ~26/15, ~5
 
POTE generators: ~15/13 = 249.7651, ~5/4 = 383.5558
 
Vals: {{Val list| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}
 
Badness: 1.062 × 10<sup>-3</sup>
 
== Artemis ==
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 121/120, 225/224


[[Mapping]]: [{{val| 1 0 1 -3 2 }}, {{val| 0 1 1 4 1 }}, {{val| 0 0 -2 -4 -1 }}]
{{Mapping|legend=1| 1 0 0 -5 9 | 0 2 0 4 -7 | 0 0 1 2 0 }}
: mapping generators: ~2, ~400/231, ~5


Mapping generators: ~2, ~3, ~11/10
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6417{{c}}, ~400/231 = 950.6555{{c}}, ~5/4 = 383.8518{{c}}
: [[error map]]: {{val| +0.642 -0.644 -1.179 +0.858 -0.131 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 950.1211{{c}}, ~5/4 = 383.9530{{c}}
: error map: {{val| 0.000 -1.712 -2.361 -0.436 -2.165 }}


[[POTE generator]]s: ~3/2 = 699.8719, ~11/10 = 158.3232
{{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce }}


{{Val list|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}
[[Badness]] (Sintel): 1.845


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 105/104, 121/120, 196/195
Comma list: 169/168, 225/224, 1001/1000
 
Mapping: [{{val| 1 0 1 -3 2 -5 }}, {{val| 0 1 1 4 1 6 }}, {{val| 0 0 -2 -4 -1 -6 }}]
 
Mapping generators: ~2, ~3, ~11/10


POTE generators: ~3/2 = 698.7090, ~11/10 = 158.7117
Mapping: {{mapping| 1 0 0 -5 9 -1 | 0 2 0 4 -7 3 | 0 0 1 2 0 1 }}


Vals: {{Val list| 9, 20, 22f, 29, 31 }}
Optimal tunings:  
* WE: ~2 = 1200.8238{{c}}, ~26/15 = 950.8873{{c}}, ~5/4 = 383.8191{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.2108{{c}}, ~5/4 = 383.9486{{c}}


=== Diana ===
{{Optimal ET sequence|legend=0| 19, 29, 43, 53, 72, 125f, 197ef }}
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 225/224, 275/273
Badness (Sintel): 0.993
 
Mapping: [{{val| 1 0 1 -3 2 7 }}, {{val| 0 1 1 4 1 -2 }}, {{val| 0 0 -2 -4 -1 -1 }}]
 
Mapping generators: ~2, ~3, ~11/10
 
POTE generators: ~3/2 = 700.9789, ~11/10 = 159.0048
 
Vals: {{Val list| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}


== Morfil ==
== Morfil ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 225/224, 1331/1323
[[Comma list]]: 225/224, 1331/1323


[[Mapping]]: [{{val| 1 0 1 -3 -2 }}, {{val| 0 1 2 6 5 }}, {{val| 0 0 -3 -6 -4 }}]
{{Mapping|legend=1| 1 0 -2 -9 -6 | 0 1 2 6 5 | 0 0 3 6 4 }}
: mapping generators: ~2, ~3, ~55/42


Mapping generators: ~2, ~3, ~84/55
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5931{{c}}, ~3/2 = 701.2447{{c}}, ~55/42 = 460.8465{{c}}
: [[error map]]: {{val| +0.593 -0.117 -1.285 +1.942 -2.302 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.1108{{c}}, ~55/42 = 460.5488{{c}}
: error map: {{val| 0.000 -0.844 -2.446 +1.131 -3.569 }}


[[POTE generator]]s: ~3/2 = 700.8983, ~84/55 = 739.3812
{{Optimal ET sequence|legend=1| 29, 31, 60e, 91e, 94, 125 }}


{{Val list|legend=1| 29, 31, 60e, 91e, 94, 125 }}
[[Badness]] (Sintel): 1.38


[[Badness]]: 1.152 × 10<sup>-3</sup>
== Subgroup extensions ==
=== Char ===
Subgroup: 2.3.5.7.17


== Catakleismoid ==
Comma list: 120/119, 225/224
Subgroup: 2.3.5.7.11


[[Comma list]]: 225/224, 4375/4374
Mapping: {{mapping| 1 0 0 -5 8 | 0 1 0 2 -1 | 0 0 1 2 -1 }}


[[Mapping]]: [{{val| 1 0 1 -3 0 }}, {{val| 0 6 5 22 0 }}, {{val| 0 0 0 0 1 }}]
Optimal tunings:  
* WE: ~2 = 1199.3681{{c}}, ~3/2 = 701.3669{{c}}, ~5/4 = 384.9641{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6286{{c}}, ~5/4 = 385.0841{{c}}


Mapping generators: ~2, ~6/5, ~11
{{Optimal ET sequence|legend=0| 9, 10, 12, 19, 22, 31, 41, 53 }}


[[POTE generator]]s: ~6/5 = 316.7318, ~11/8 = 549.2528
Badness (Sintel): 0.381


{{Val list|legend=1| 19, 34d, 53, 72, 197e, 269ce }}
=== Devimarvel ===
 
Subgroup: 2.3.5.7.19
[[Badness]]: 1.275 × 10<sup>-3</sup>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 325/324
 
Mapping: [{{val| 1 0 1 -3 0 0 }}, {{val| 0 6 5 22 0 14 }}, {{val| 0 0 0 0 1 0 }}]
 
Mapping generators: ~2, ~6/5, ~11
 
POTE generators: ~6/5 = 316.7410, ~11/8 = 548.6028
 
Vals: {{Val list| 19, 34d, 53, 72, 125f, 197ef, 269cef }}
 
Badness: 0.916 × 10<sup>-3</sup>
 
== Mirage ==
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 385/384
Comma list: 225/224, 400/399


Mapping: [{{val| 1 1 3 3 2 0 }}, {{val| 0 6 -7 -2 15 0 }}, {{val| 0 0 0 0 0 1 }}]
Mapping: {{mapping| 1 0 0 -5 9 | 0 1 0 2 -3 | 0 0 1 2 0 }}


Mapping generators: ~2, ~15/14, ~13
Optimal tunings:
* WE: ~2 = 1200.3454{{c}}, ~3/2 = 701.1728{{c}}, ~5/4 = 383.7499{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.9546{{c}}, ~5/4 = 383.7912{{c}}


POTE generators: ~15/14 = 116.6327, ~13/8 = 837.7040
{{Optimal ET sequence|legend=0| 10, 12, 19, 22, 31, 41, 53, 72, 94, 113, 125, 291c }}


Vals: {{Val list| 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
Badness (Sintel): 0.310


Badness: 0.738 × 10<sup>-3</sup>
== References ==
<references />


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Marvel family| ]] <!-- main article -->
[[Category:Marvel family| ]] <!-- main article -->
[[Category:Marvel]]
[[Category:Rank 3]]
[[Category:Rank 3]]
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