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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]] which temper out the marvel comma.
 
= 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]].
 
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.
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~3/2 = 700.4075, ~5/4 = 383.6376


EDO generators: [[19edo|(11, 6)\19]], [[31edo|(18, 10)\31]], [[41edo|(24, 13)\41]]
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.


Scales (Scala files): [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]]
== Marvel ==
{{Main| Marvel }}


[[Associated temperament]]: [[catakleismic]]
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]].


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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.  
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: c = 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°


Minimax tuning:  
[[Optimal tuning]]s:
* 7-odd-limit: 3 and 5 1/4c flat, 7 just
* [[WE]]: ~2 = 1200.5971{{c}}, ~3/2 = 700.7560{{c}}, ~5/4 = 383.8285{{c}}
: [{{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 }}]
: [[error map]]: {{val| +0.597 -0.602 -1.291 +0.940 }}
: [[Eigenmonzo subgroup]]: 2.5/3.7
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.6222{{c}}, ~5/4 = 383.8540{{c}}
* 9-odd-limit: 3 1/6c flat, 5 1/3c flat, 7 just
: error map: {{val| 0.000 -1.333 -2.460 +0.127 }}
: [{{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 }}]
 
: [[Eigenmonzo subgroup]]: 2.9/5.7
[[Minimax tuning]]:  
* [[7-odd-limit]]: 3 and 5 1/4-comma flat, 7 just
: {{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 basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
* [[9-odd-limit]]: 3 1/6-comma flat, 5 1/3-comma flat, 7 just
: {{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 basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7


{{Val list|legend=1| 10, 12, 19, 22, 31, 41, 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


</div></div>
Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]], [[marvel wholetone]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">[[Minkowski blocks]]</div>
<div class="mw-collapsible-content">


{2, 3, 5} subgroup
{{Databox|[[Minkowski blocks]]|
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 80: Line 63:
* 31: 81/80, 34171875/33554432
* 31: 81/80, 34171875/33554432
* 41: 34171875/33554432, 3125/3072
* 41: 34171875/33554432, 3125/3072
}}


</div></div>
=== Overview to extensions ===
 
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.  
== Eleven-limit 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, and
* 243/242 gives spectacle.
 
= 11-limit (unimarv) =
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~3/2 = 700.3887, ~5/4 = 383.5403


EDO generators: [[22edo|(13, 7)\22]], [[31edo|(18, 10)\31]], [[41edo|(24, 13)\41]]
Temperaments discussed elsewhere include
* ''[[Supernatural]]'' (+245/243) → [[Keemic family #Supernatural|Keemic family]]
* ''[[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]]


Scales (Scala files): [[marvel22_11]], [[unimarv19]], [[unimarv22]]
== Undecimal marvel ==
{{Main| Marvel }}


[[Associated temperament]]: [[catakleismic]]
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.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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]].
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
[[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 }}


Mapping generators: ~2, ~3, ~5
[[Mapping to lattice]]: [{{val| 0 -1 0 -2 1 }}, {{val| 0 -1 1 0 -2 }}]


Map to lattice: [{{val| 0 -1 0 -2 1 }}, {{val| 0 -1 1 0 -2 }}]
Lattice basis:  
: ~15/14 length = 1.0364, ~5/4 length = 1.0759
: angle (~15/14, ~5/4) = 104.028°


Lattice basis:  
[[Optimal tuning]]s:  
: secor length = 1.0364, 5/4 length = 1.0759
* [[WE]]: ~2 = 1200.6395{{c}}, ~3/2 = 700.7619{{c}}, ~5/4 = 383.7447{{c}}
: Angle (secor, 5/4) = 104.028 degrees
: [[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| 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 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


</div></div>
Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Hobbit bases</div>
<div class="mw-collapsible-content">


{2, 3, 5} subgroup
{{Databox|[[Hobbit]] bases|
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 146: Line 124:
* 31: 2109375/2097152, 81/80
* 31: 2109375/2097152, 81/80
* 41: 3125/3072, 34171875/33554432
* 41: 3125/3072, 34171875/33554432
}}
=== Helios ===
Subgroup: 2.3.5.7.11.13


</div></div>
Comma list: 225/224, 351/350, 385/384


== 13-limit ==
Mapping: {{mapping| 1 0 0 -5 12 -4 | 0 1 0 2 -1 -1 | 0 0 1 2 -3 4 }}


Period: 1\1
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}}


Optimal ([[POTE]]) generators: ~3/2 = 699.7367, ~5/4 = 384.0613
Minimax tuning:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.15/11.15/13


EDO generators: [[31edo|(18, 10)\31]], [[53edo|(31, 17)\53]], [[72edo|(42, 23)\72]]
{{Optimal ET sequence|legend=0| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}


Scales (Scala files):  
Badness (Sintel): 0.645


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== 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, 385/384, 351/350
Comma list: 225/224, 325/324, 385/384
 
Mapping: {{mapping| 1 0 0 -5 12 2 | 0 1 0 2 -1 4 | 0 0 1 2 -3 -2 }}


Mapping: [{{val| 1 0 0 -5 12 -4 }}, {{val| 0 1 0 2 -1 -1 }}, {{val| 0 0 1 2 -3 4 }}]
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: 2.11/9.13/9
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.13/5
* 15-odd-limit eigenmonzo subgroup: 2.15/11.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13


{{Val list|legend=1| 19, 22, 31, 50, 53, 72, 103, 175f, 300cef, 403bcef }}
{{Optimal ET sequence|legend=0| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}


Badness: 0.690 × 10<sup>-3</sup>
Badness (Sintel): 0.674


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


</div></div>
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


== Hecate ==
===== 2.3.5.7.11.13.19 subgroup =====
Subgroup: 2.3.5.7.11.13.19


Period: 1\1
Comma list: 225/224, 325/324, 385/384, 400/399


Optimal ([[POTE]]) generators: ~3/2 = 700.9779, ~5/4 = 383.1622
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 }}


EDO generators: [[41edo|(24, 13)\41]], [[53edo|(31, 17)\53]], [[72edo|(42, 23)\72]]
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}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 41, 53, 72, 94, 113, 166 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.756
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
==== Apotropaia ====
Subgroup: 2.3.5.7.11.13.17


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


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 18 | 0 1 0 2 -1 4 0 | 0 0 1 2 -3 -2 -6 }}


Minimax tuning:  
Optimal tunings:  
* 13-odd-limit eigenmonzo subgroup: 2.7.13/5
* WE: ~2 = 1200.6057{{c}}, ~3/2 = 701.3157{{c}}, ~5/4 = 383.2243{{c}}
* 15-odd-limit eigenmonzo subgroup: 2.7.15/13
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0877{{c}}, ~5/4 = 382.9709{{c}}


{{Val list|legend=1| 19, 22f, 31f, 41, 53, 72, 113, 125f, 166, 238cf, 404cef }}
{{Optimal ET sequence|legend=0| 41, 53g, 72, 166g, 238cfg }}


Badness: 0.721 × 10<sup>-3</sup>
Badness (Sintel): 0.827


Projection pairs: 7 225/32 11 4096/375 13 324/25
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


[[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
Comma list: 225/224, 325/324, 385/384, 400/399, 595/594


</div></div>
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 }}


=== 17-limit ===
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}}


Period: 1\1
{{Optimal ET sequence|legend=0| 41, 53g, 72, 94, 113, 166g }}


Optimal ([[POTE]]) generators: ~3/2 = 700.9619, ~5/4 = 383.0310
Badness (Sintel): 1.01


EDO generators: [[22edo|(13, 7)\22]], [[41edo|(24, 13)\41]], [[72edo|(42, 23)\72]]
==== Enodia ====
 
Subgroup: 2.3.5.7.11.13.17
Scales (Scala files):


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 225/224, 325/324, 375/374, 385/384
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
Mapping: {{mapping| 1 0 0 -5 12 2 -13 | 0 1 0 2 -1 4 2 | 0 0 1 2 -3 -2 6 }}


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


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 }}]
{{Optimal ET sequence|legend=0| 41g, 53, 72, 166g, 238cfg }}


{{Val list|legend=1| 19, 22f, 31fg, 41, 53g, 72, 113, 166g, 238cfg, 351cfg, 404cefg }}
Badness (Sintel): 0.872


Badness: 0.869 × 10<sup>-3</sup>
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


</div></div>
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399


=== Enodia ===
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 }}


Period: 1\1
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 ([[POTE]]) generators: ~3/2 = 700.9658, ~5/4 = 383.3063
{{Optimal ET sequence|legend=0| 41g, 53, 72, 94, 125f, 166g }}


EDO generators: [[22edo|(13, 7)\22]], [[53edo|(31, 17)\53]], [[72edo|(42, 23)\72]]
Badness (Sintel): 1.05


Scales (Scala files):  
=== Marvell ===
Subgroup: 2.3.5.7.11.13


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 225/224, 385/384, 1573/1568
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
Mapping: {{mapping| 1 0 0 -5 12 -29 | 0 1 0 2 -1 6 | 0 0 1 2 -3 10 }}


Comma list: 225/224, 385/384, 325/324, 375/374
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}}


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


{{Val list|legend=1| 19g, 22f, 31f, 41g, 53, 72, 125f, 166g, 238cfg, 363cefg, 404cefg }}
{{Optimal ET sequence|legend=0| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 341cef }}


Badness: 0.917 × 10<sup>-3</sup>
Badness (Sintel): 0.806


</div></div>
=== Isis ===
Subgroup: 2.3.5.7.11.13


== Marvelcat ==
Comma list: 225/224, 275/273, 385/384


Period: 1\1
Mapping: {{mapping| 1 0 0 -5 12 17 | 0 1 0 2 -1 -4 | 0 0 1 2 -3 -3 }}


Optimal ([[POTE]]) generators: ~15/13 = 249.7138, ~5/4 = 383.5816
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}}


EDO generators: [[9edo|(5, 3)\9]], [[19edo|(11, 6)\19]], [[53edo|(31, 17)\53]]
{{Optimal ET sequence|legend=0| 10, 19f, 22, 31, 41, 53, 84e, 94 }}


Scales (Scala files):  
Badness (Sintel): 0.810


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Projection pairs: <code>7 225/32 11 4096/375 13 131072/10125</code>
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Deecee ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 225/224, 385/384
Comma list: 225/224, 364/363, 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 27 | 0 1 0 2 -1 -3 | 0 0 1 2 -3 -8 }}


Mapping generators: ~2, ~26/15, ~5
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}}


{{Val list|legend=1| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269cef }}
Minimax tuning:
* 13-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.9/5.13/9
* 15-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.3.13/5


Badness: 0.9997 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}


</div></div>
Badness (Sintel): 0.861


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


Period: 1\1
=== Tripod ===
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generators: ~3/2 = 700.3937, ~5/4 = 383.5725
Comma list: 105/104, 144/143, 196/195


EDO generators: [[9edo|(5, 3)\9]], [[31edo|(18, 10)\31]], [[72edo|(42, 23)\72]]
Mapping: {{mapping| 1 0 0 -5 12 -8 | 0 1 0 2 -1 3 | 0 0 1 2 -3 3 }}


Scales (Scala files):  
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}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Minimax tuning:
<div style="line-height:1.6;">Technical data</div>
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/11
<div class="mw-collapsible-content">
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/11


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 9, 10, 19, 22f, 31, 41, 72f, 91 }}


Comma list: 225/224, 385/384, 1573/1568
Badness (Sintel): 0.697


Mapping: [{{val| 1 0 0 -5 12 -29 }}, {{val| 0 1 0 2 -1 6 }}, {{val| 0 0 1 2 -3 10 }}]
Projection pairs: <code>7 225/32 11 4096/375 13 3375/256</code>


Minimax tuning:
=== Marvelcat ===
* 13-odd-limit eigenmonzo subgroup: 2.9/5.11/9
Subgroup: 2.3.5.7.11.13
* 15-odd-limit eigenmonzo subgroup: 2.7.15/13


{{Val list|legend=1| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcef, 610bcef }}
Comma list: 169/168, 225/224, 385/384


Badness: 0.862 × 10<sup>-3</sup>
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


</div></div>
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}}


== Isis ==
{{Optimal ET sequence|legend=0| 9, 10, 19, 43e, 44, 53, 72, 125f, 197ef }}


Period: 1\1
Badness (Sintel): 0.935


Optimal ([[POTE]]) generators: ~3/2 = 701.9156, ~5/4 = 383.2445
== 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.


EDO generators: [[22edo|(13, 7)\22]], [[31edo|(18, 10)\31]], [[41edo|(24, 13)\41]]
[[Subgroup]]: 2.3.5.7.11


Scales (Scala files):  
[[Comma list]]: 225/224, 441/440


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 1 0 0 -5 -13 | 0 1 0 2 6 | 0 0 1 2 3 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Map to lattice: [{{val| 0 0 -1 -2 -3 }}, {{val| 0 1 -1 0 3 }}]


Comma list: 225/224, 385/384, 275/273
Lattice basis:  
: ~15/14 length = 0.9111, ~3/2 length = 0.9477
: angle (~15/14, ~3/2) = 65.933°


Mapping: [{{val| 1 0 0 -5 12 17 }}, {{val| 0 1 0 2 -1 4 }}, {{val| 0 0 1 2 -3 -3 }}]
[[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 }}


{{Val list|legend=1| 10, 19f, 22, 31, 41, 53, 94 }}
[[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


Badness: 0.866 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}


Projection pairs: 7 225/32 11 4096/375 13 131072/10125
[[Badness]] (Sintel): 0.402


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


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


Period: 1\1
{{Databox|Hobbit bases|
2.3.5 subgroup
* 31: 81/80, 34171875/33554432
* 41: 34171875/33554432, 32805/32768
}}


Optimal ([[POTE]]) generators: ~3/2 = 700.4560, ~5/4 = 382.8177
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDO generators: [[9edo|(5, 3)\9]], [[41edo|(24, 13)\41]], [[72edo|(42, 23)\72]]
Comma list: 105/104, 196/195, 352/351


Scales (Scala files):  
Mapping: {{mapping| 1 0 0 -5 -13 -8 | 0 1 0 2 6 3 | 0 0 1 2 3 3 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1200.8252{{c}}, ~3/2 = 700.8823{{c}}, ~5/4 = 381.6647{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.4689{{c}}, ~5/4 = 381.6687{{c}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 101cd }}


Comma list: 225/224, 385/384, 364/363
Badness (Sintel): 0.689


Mapping: [{{val| 1 0 0 -5 12 27 }}, {{val| 0 1 0 2 -1 -3 }}, {{val| 0 0 1 2 -3 -8 }}]
=== Prodigious ===
Subgroup: 2.3.5.7.11.13


Minimax tuning:  
Comma list: 225/224, 364/363, 441/440
* 13-odd-limit eigenmonzo subgroup: 2.9/5.13/9
* 15-odd-limit eigenmonzo subgroup: 2.3.13/5


{{Val list|legend=1| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cf }}
Mapping: {{mapping| 1 0 0 -5 -13 -23 | 0 1 0 2 6 11 | 0 0 1 2 3 4 }}


Badness: 0.920 × 10<sup>-3</sup>
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}}


Projection pairs: 7 225/32 11 4096/375 13 134217728/10546875
{{Optimal ET sequence|legend=0| 12f, 29, 31f, 41, 72, 185cf, 257cff }}


</div></div>
Badness (Sintel): 0.841


== Tripod ==
=== Prodigal ===
Subgroup: 2.3.5.7.11.13


Period: 1\1
Comma list: 225/224, 351/350, 441/440


Optimal ([[POTE]]) generators: ~3/2 = 699.2335, ~5/4 = 382.9775
Mapping: {{mapping| 1 0 0 -5 -13 -4 | 0 1 0 2 6 -1 | 0 0 1 2 3 4 }}


EDO generators: [[9edo|(5, 3)\9]], [[19edo|(11, 6)\19]], [[31edo|(18, 10)\31]]
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}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.831
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 105/104, 144/143, 196/195
Comma list: 225/224, 441/440, 1001/1000
 
Mapping: {{mapping| 1 0 0 -5 -13 21 | 0 1 0 2 6 -8 | 0 0 1 2 3 -2 }}


Minimax tuning:  
Optimal tunings:  
* 13-odd-limit eigenmonzo subgroup: 2.9/7.13/11
* WE: ~2 = 1200.9450{{c}}, ~3/2 = 700.1045{{c}}, ~5/4 = 383.8716{{c}}
* 15-odd-limit eigenmonzo subgroup: 2.5/3.13/11
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4224{{c}}, ~5/4 = 383.9828{{c}}


Mapping: [{{val| 1 0 0 -5 12 -8 }}, {{val| 0 1 0 2 -1 3 }}, {{val| 0 0 1 2 -3 3 }}]
{{Optimal ET sequence|legend=0| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}


{{Val list|legend=1| 9, 10, 19, 22f, 31, 41, 72f, 122f, 163df }}
Badness (Sintel): 0.891


Badness: 0.745 × 10<sup>-3</sup>
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Projection pairs: 7 225/32 11 4096/375 13 3375/256
Comma list: 225/224, 273/272, 375/374, 441/440


</div></div>
Mapping: {{mapping| 1 0 0 -5 -13 21 12 | 0 1 0 2 6 -8 -5 | 0 0 1 2 3 -2 0 }}


= Mirage =
Optimal tunings:
Comma list: 225/224, 243/242, 385/384
* 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}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}


Mapping: [{{val| 1 1 3 3 2 0 }}, {{val| 0 6 -7 -2 15 0 }}, {{val| 0 0 0 0 0 1 }}]
Badness (Sintel): 0.734


{{Val list|legend=1| 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
== 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]].


Badness: 0.738 × 10<sup>-3</sup>
[[Subgroup]]: 2.3.5.7.11


= Minerva =
[[Comma list]]: 99/98, 176/175
[[Comma list]]: 99/98, 176/175


Subgroup: 2.3.5.7.11
{{Mapping|legend=1| 1 0 0 -5 -9 | 0 1 0 2 2 | 0 0 1 2 4 }}
 
[[Associated temperament]]: [[Semicomma family #Orwell|orwell]]
 
Mapping: [{{val| 1 0 0 -5 -9 }}, {{val| 0 1 0 2 2 }}, {{val| 0 0 1 2 4 }}]
 
Mapping generators: ~2, ~3, ~5


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


Lattice basis:  
Lattice basis:  
: 16/15 length = 0.8997, 5/4 length = 1.0457
: ~16/15 length = 0.8997, ~5/4 length = 1.0457
: Angle (16/15, 5/4) = 98.6044 degrees
: angle (~16/15, ~5/4) = 98.6044°
 
[[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 }}


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


{{Val list|legend=1| 9, 12, 21, 22, 31, 43, 53, 74, 75, 96, 127 }}
{{Optimal ET sequence|legend=1| 9, 12, 19e, 22, 31, 53, 84e, 96, 127 }}


Badness: 0.381 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.458


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


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


== Athene ==
=== Athene ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 176/175, 275/273
Comma list: 99/98, 176/175, 275/273


Subgroup: 2.3.5.7.11.13
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 -9 -4 }}, {{val| 0 1 0 2 2 -1 }}, {{val| 0 0 1 2 4 4 }}]
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}}


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo subgroup: 2.11/9.13/7
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7
* 15-odd-limit eigenmonzo subgroup: 2.11/9.13/7
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7


{{Val list|legend=1| 22, 31, 53, 84e, 118d, 149df, 171de, 202def }}
{{Optimal ET sequence|legend=0| 12f, 19e, 22, 31, 53, 84e, 118d }}


Badness: 0.818 × 10<sup>-3</sup>
Badness (Sintel): 0.765


[[Projection pair]]s: 7 225/32 11 5625/512 13 625/48
Projection pairs: <code>7 225/32 11 5625/512 13 625/48</code>


= Spectacle =
== Apollo ==
[[Comma list]]: 225/224, 243/242
{{See also| Ptolemismic clan #Apollo }}


Subgroup: 2.3.5.7.11
[[File:Lattice Apollo.png|thumb|Lattice for apollo.]]


[[Associated temperament]]: [[Marvel temperaments #Marvo|marvo]]
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]].


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


Mapping generators: ~2, ~11/9, ~5
[[Subgroup]]: 2.3.5.7.11


[[Minimax tuning]]:
[[Comma list]]: 100/99, 225/224
* 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| 10, 31, 41, 72, 240, 259b, 269ce, 310c, 331bc, 353c, 497bc, 569bc }}


Badness: 0.499 × 10<sup>-3</sup>
{{Mapping|legend=1| 1 0 0 -5 2 | 0 1 0 2 -2 | 0 0 1 2 2 }}


Projection pairs: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9
[[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 }}


Scales: [[spectacle31]]
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9


== 13-limit ==
{{Optimal ET sequence|legend=1| 12, 19, 22, 34d, 41, 104 }}
Comma list: 225/224, 243/242, 351/350


Subgroup: 2.3.5.7.11.13
[[Badness]] (Sintel): 0.637


Mapping: [{{val| 1 1 0 -3 2 -5 }}, {{val| 0 2 0 4 5 -2 }}, {{val| 0 0 1 2 0 4 }}]
[[Projection pair]]s: <code>7 225/32 11 100/9</code>


{{Val list|legend=1| 31, 72, 103, 175f, 209, 240 }}
Scales: [[apollo wholetone]], [[indigo17]]


Badness: 1.009 × 10<sup>-3</sup>
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
= Apollo =
[[Comma list]]: 100/99, 225/224


Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 275/273


Associated temperament: [[Magic family #magic|magic]]
Mapping: {{mapping| 1 0 0 -5 2 7 | 0 1 0 2 -2 -5 | 0 0 1 2 2 2 }}


Mapping: [{{val| 1 0 0 -5 2 }}, {{val| 0 1 0 2 -2 }}, {{val| 0 0 1 2 2 }}]
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}}


Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7/5.11/9
Minimax tuning: 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9


{{Val list|legend=1| 12, 19, 22, 41, 104, 157ce, 198ce, 220ce, 261ce }}
{{Optimal ET sequence|legend=0| 12f, 19f, 22, 29, 34d, 41, 63, 104 }}


Projection pairs: 7 225/32 11 100/9
Badness (Sintel): 0.964


== 13-limit ==
Projection pairs: <code>7 225/32 11 100/9 13 3200/243</code>
[[Comma list]]: 100/99, 225/224, 245/243


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


Mapping: [{{val| 1 0 0 -5 2 7 }}, {{val| 0 1 0 2 -2 -5 }}, {{val| 0 0 1 2 2 2 }}]
Comma list: 100/99, 105/104, 196/195


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


{{Val list|legend=1| 22, 29, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef }}
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}}


Projection pairs: 7 225/32 11 100/9 13 3200/243
{{Optimal ET sequence|legend=0| 12f, 19, 22f, 29, 41 }}


= Artemis =
Badness (Sintel): 0.886
Comma list: 121/120, 225/224


Subgroup: 2.3.5.7.11
=== Musagetes ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 0 1 -3 2 }}, {{val| 0 1 1 4 1 }}, {{val| 0 0 2 4 1 }}]
Comma list: 100/99, 144/143, 225/224


{{Val list|legend=1| 9, 20, 22, 31, 53 }}
Mapping: {{mapping| 1 0 0 -5 2 2 | 0 1 0 2 -2 4 | 0 0 1 2 2 -2 }}


== 13-limit ==
Optimal tunings:
Comma list: 105/104, 121/120, 196/195
* 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}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 19, 22f, 34d, 41, 75e, 94e, 116ef }}


Mapping: [{{val| 1 0 1 -3 2 -5 }}, {{val| 0 1 1 4 1 6 }}, {{val| 0 0 2 4 1 6 }}]
Badness (Sintel): 1.14


{{Val list|legend=1| 9, 20, 29, 31 }}
== Potassium ==
[[Subgroup]]: 2.3.5.7.11


== Diana ==
[[Comma list]]: 45/44, 56/55
Comma list: 121/120, 225/224, 275/273


Subgroup: 2.3.5.7.11.13
{{Mapping|legend=1| 1 0 0 -5 -2 | 0 1 0 2 2 | 0 0 1 2 1 }}


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


{{Val list|legend=1| 22, 29, 31, 53 }}
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11


= Potassium =
{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 31e }}
Comma list: 45/44, 56/55


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


Mapping: [{{val| 1 0 0 -5 -2 }}, {{val| 0 1 0 2 2 }}, {{val| 0 0 1 2 1 }}]
[[Projection pair]]s: <code>7 225/32 11 45/4</code>


Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.9/7.11
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
{{Val list|legend=1| 9, 10, 12, 19, 31e, 50e }}


Badness: 0.464 × 10<sup>-3</sup>
Projection pairs: 7 225/32 11 45/4
== 13-limit ==
Comma list: 45/44, 56/55, 78/77
Comma list: 45/44, 56/55, 78/77


Subgroup: 2.3.5.7.11.13
Mapping: {{mapping| 1 0 0 -5 -2 -8 | 0 1 0 2 2 3 | 0 0 1 2 1 3 }}


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


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzo subgroup: 2.9/7.13/9
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9
* 15-odd-limit eigenmonzo subgroup: 2.9/7.13/9
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9


{{Val list|legend=1| 9, 10, 19, 31e, 50e }}
{{Optimal ET sequence|legend=0| 9, 10, 12f, 19, 31e }}


Badness: 0.733 × 10<sup>-3</sup>
Badness (Sintel): 0.686


Projection pairs: 7 225/32 11 45/4 13 3375/256
Projection pairs: <code>7 225/32 11 45/4 13 3375/256</code>


= Fantastic =
== Malcolm ==
Comma list: 225/224, 4375/4356
{{Redirect|Malcolm|{{w|Alexander Malcolm (writer on music)|Alexander Malcolm}}'s JI scale|Malcolm (scale)}}


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Mapping: [{{val| 2 0 0 -10 -7 }}, {{val| 0 1 0 2 0 }}, {{val| 0 0 1 2 3 }}]
[[Comma list]]: 225/224, 2200/2187


{{Val list|legend=1| 12, 22, 50, 72, 166, 238c, 310c }}
{{Mapping|legend=1| 1 0 0 -5 -3 | 0 1 0 2 7 | 0 0 1 2 -2 }}


Badness: 0.743 × 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 }}


= Catakleismoid =
{{Optimal ET sequence|legend=1| 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce }}
Comma list: 225/224, 4375/4374


Subgroup: 2.3.5.7.11
[[Badness]] (Sintel): 1.50


Mapping: [{{val| 1 0 1 -3 0 }}, {{val| 0 6 5 22 0 }}, {{val| 0 0 0 0 1 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 19, 53, 72, 197e, 269ce }}
Comma list: 225/224, 275/273, 325/324
 
Badness: 1.275 × 10<sup>-3</sup>
 
== 13-limit ==
Comma list: 169/168, 225/224, 325/324
 
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 0 1 -3 0 0 }}, {{val| 0 6 5 22 0 14 }}, {{val| 0 0 0 0 1 0 }}]
Mapping: {{mapping| 1 0 0 -5 -3 2 | 0 1 0 2 7 4 | 0 0 1 2 -2 -2 }}


{{Val list|legend=1| 19, 53, 72, 125f, 197ef, 269cef }}
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}}


Badness: 0.916 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 12e, 19e, 34d, 41, 53, 94 }}


= Hestia =
Badness (Sintel): 1.00
Comma list: 225/224, 125000/124509


Subgroup: 2.3.5.7.11
Scales: [[malco]]


Mapping: [{{val| 1 0 0 -5 9 }}, {{val| 0 2 0 4 -7 }}, {{val| 0 0 1 2 0 }}]
== Fantastic ==
Besides 4375/4356, fantastic also tempers out [[9801/9800]] and splits the octave in two.


{{Val list|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce }}
[[Subgroup]]: 2.3.5.7.11


Badness: 1.54 × 10<sup>-3</sup>
[[Comma list]]: 225/224, 4375/4356


== 13-limit ==
{{Mapping|legend=1| 2 0 0 -10 -7 | 0 1 0 2 0 | 0 0 1 2 3 }}
Comma list: 169/168, 225/224, 1001/1000
: mapping generators: ~99/70, ~3, ~5


Subgroup: 2.3.5.7.11.13
[[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 }}


Mapping: [{{val| 1 0 0 -5 9 -1 }}, {{val| 0 2 0 4 -7 3 }}, {{val| 0 0 1 2 0 1 }}]
{{Optimal ET sequence|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}


{{Val list|legend=1| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}
[[Badness]] (Sintel): 0.893


Badness: 1.062 × 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]].


= Malcolm =
[[Subgroup]]: 2.3.5.7.11
Comma list: 225/224, 2200/2187


Subgroup: 2.3.5.7.11
[[Comma list]]: 225/224, 125000/124509


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


{{Val list|legend=1| 41, 53, 94, 229c, 248ce, 289ce, 342ce, 383ce }}
[[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 }}


Badness: 1.250 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce }}


== 13-limit ==
[[Badness]] (Sintel): 1.845
Comma list: 225/224, 275/273, 325/324


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


Mapping: [{{val| 1 0 0 -5 -3 2 }}, {{val| 0 1 0 2 7 4 }}, {{val| 0 0 1 2 -2 -2 }}]
Comma list: 169/168, 225/224, 1001/1000


{{Val list|legend=1| 41, 53, 94, 429cdef, 523cdef }}
Mapping: {{mapping| 1 0 0 -5 9 -1 | 0 2 0 4 -7 3 | 0 0 1 2 0 1 }}


Badness: 1.075 × 10<sup>-3</sup>
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}}


= Prodigy =
{{Optimal ET sequence|legend=0| 19, 29, 43, 53, 72, 125f, 197ef }}
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.


[[Comma list]]: 225/224, 441/440
Badness (Sintel): 0.993


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


[[Associated temperament]]: [[Gamelismic clan #Miracle|miracle]]
[[Comma list]]: 225/224, 1331/1323


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


Map to lattice: [{{val| 0 0 -1 -2 -3 }}, {{val| 0 1 -1 0 3 }}]
{{Optimal ET sequence|legend=1| 29, 31, 60e, 91e, 94, 125 }}


Lattice basis:
[[Badness]] (Sintel): 1.38
: secor length = 0.9111, 3/2 length = 0.9477
: Angle (secor, 3/2) = 65.933


[[Minimax tuning]]:
== Subgroup extensions ==
* 11-odd-limit
=== Char ===
: [{{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 }}]
Subgroup: 2.3.5.7.17
: [[Eigenmonzo subgroup]]: 2.9/5.11


{{Val list|legend=1| 10, 12, 29, 31, 41, 72, 247c, 319bcde, 391bcde, 463bcde }}
Comma list: 120/119, 225/224


Badness: 0.344 × 10<sup>-3</sup>
Mapping: {{mapping| 1 0 0 -5 8 | 0 1 0 2 -1 | 0 0 1 2 -1 }}


[[Projection pair]]s: 7 225/32 11 91125/8192
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}}


Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]
{{Optimal ET sequence|legend=0| 9, 10, 12, 19, 22, 31, 41, 53 }}


; Hobbit bases
Badness (Sintel): 0.381
{2, 3, 5} subgroup
* 31: 81/80, 34171875/33554432
* 41: 34171875/33554432, 32805/32768
 
== 13-limit ==
Comma list: 105/104, 196/195, 352/351
 
Subgroup: 2.3.5.7.11.13
 
Mapping: [{{val| 1 0 0 -5 -13 -8 }}, {{val| 0 1 0 2 6 3 }}, {{val| 0 0 1 2 3 3 }}]
 
{{Val list|legend=1| 10, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233cdef, 274cdef, 305cdef }}
 
Badness: 0.736 × 10<sup>-3</sup>
 
== Prodigious ==
Comma list: 225/224, 441/440, 364/363


Subgroup: 2.3.5.7.11.13
=== Devimarvel ===
Subgroup: 2.3.5.7.19


Mapping: [{{val| 1 0 0 -5 -13 -23 }}, {{val| 0 1 0 2 6 11 }}, {{val| 0 0 1 2 3 4 }}]
Comma list: 225/224, 400/399


{{Val list|legend=1| 29, 41, 72, 113, 185cf, 341cf, 413bcf, 526bcdf }}
Mapping: {{mapping| 1 0 0 -5 9 | 0 1 0 2 -3 | 0 0 1 2 0 }}


Badness: 0.900 × 10<sup>-3</sup>
Optimal tunings:  
 
* WE: ~2 = 1200.3454{{c}}, ~3/2 = 701.1728{{c}}, ~5/4 = 383.7499{{c}}
== Prodigal ==
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.9546{{c}}, ~5/4 = 383.7912{{c}}
Comma list: 225/224, 441/440, 351/350
 
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 0 0 -5 -13 -4 }}, {{val| 0 1 0 2 6 -1 }}, {{val| 0 0 1 2 3 4 }}]
{{Optimal ET sequence|legend=0| 10, 12, 19, 22, 31, 41, 53, 72, 94, 113, 125, 291c }}


{{Val list|legend=1| 31, 72, 103, 175f }}
Badness (Sintel): 0.310


Badness: 0.889 × 10<sup>-3</sup>
== References ==
<references />


[[Category:Theory]]
[[Category:Marvel family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Temperament families]]
[[Category:Marvel]]
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rank 3]]
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