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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-20 14:11:05 UTC</tt>.<br>
: The original revision id was <tt>277408660</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]


=Marvel=
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.
The head of the marvel family is marvel, which tempers out [[225_224|225/224]], the [[septimal kleisma]] or [[marvel comma]]. Marvel has a [[Normal lists|normal list basis]] of [2, 3, 5]; hence a [[Harmonic Limit|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 [[Kleismic family|catakleismic temperament]]. Another temperament which does little damage to tuning accuracy is [[Pythagorean family|compton temperament]], for which [[240edo]] may be used.
== Marvel ==
{{Main| Marvel }}


==Vital statistics==
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]].
[[Comma]] c = 225/224
Related linear temperament: [[Kleismic family|catakleismic temperament]]


7-limit minimax: 3 and 5 1/4c flat, 7 just
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.  
[|1 0 0 0&gt;, |5/4 1/2 -1/2 1/4&gt;, |5/4 -1/2 1/2 1/4&gt;, |0 0 0 1&gt;]
[[Eigenmonzo subgroup]]: 2.5/3.7


9-limit minimax: 3 1/6c flat, 5 1/3c flat, 7 just
[[Subgroup]]: 2.3.5.7
[|1 0 0 0&gt;, |5/6 2/3 -1/3 1/6&gt;, |5/3 -2/3 1/3 1/3&gt;, |0 0 0 1&gt;]
[[Eigenmonzo subgroup]]: 2.9/5.7


Lattice basis: secor length 1.256, 3/2 length 1.369
[[Comma list]]: 225/224
Angle(secor, 3/2) = 106.958 degrees
Map to lattice: [&lt;0 0 -1 -2|, &lt;0 1 -1 0|]


Map: [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]
{{Mapping|legend=1| 1 0 0 -5 | 0 1 0 2 | 0 0 1 2 }}
Generators: 2, 3, 5
: mapping generators: ~2, ~3, ~5
[[edo|EDOs]]: [[10edo|10]], [[12edo|12]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[41edo|41]],[[72edo|72]], [[197edo|197]], [[269edo|269c]]
Badness: 0.0000365


[[Projection pair]]s: 7 225/32
Map to lattice: [{{val| 0 0 -1 -2 }}, {{val| 0 1 -1 0 }}]
[[Spectrum of a temperament|Spectrum]]: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9


Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]]
Lattice basis:  
: ~15/14 length = 1.256, ~3/2 length = 1.369
: angle (~15/14, ~3/2) = 106.958°


==[[Minkowski blocks]]==  
[[Optimal tuning]]s:
{2, 3, 5} subgroup
* [[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 }}


8: 16/15, 250/243
[[Minimax tuning]]:
9: 135/128, 128/125
* [[7-odd-limit]]: 3 and 5 1/4-comma flat, 7 just
10: 25/24, 2048/2025
: {{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 }}
11: 135/128, 2048/1875
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
12: 2048/2025, 128/125
* [[9-odd-limit]]: 3 1/6-comma flat, 5 1/3-comma flat, 7 just
15: 128/125, 32768/30375
: {{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 }}
17: 25/24, 2278125/2097152
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7
19: 16875/16384, 81/80
 
21: 128/125, 273375/262144
{{Optimal ET sequence|legend=1| 9, 10, 12, 19, 31, 41, 53, 72, 197, 269c }}
22: 2048/2025, 3125/3072
 
29: 16875/16384, 32805/32768
[[Badness]] (Sintel): 0.161
31: 81/80, 34171875/33554432
 
41: 34171875/33554432, 3125/3072
[[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
 
Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]], [[marvel wholetone]]
 
{{Databox|[[Minkowski blocks]]|
2.3.5 subgroup
* 8: 16/15, 250/243
* 9: 135/128, 128/125
* 10: 25/24, 2048/2025
* 11: 135/128, 2048/1875
* 12: 2048/2025, 128/125
* 15: 128/125, 32768/30375
* 17: 25/24, 2278125/2097152
* 19: 16875/16384, 81/80
* 21: 128/125, 273375/262144
* 22: 2048/2025, 3125/3072
* 29: 16875/16384, 32805/32768
* 31: 81/80, 34171875/33554432
* 41: 34171875/33554432, 3125/3072
}}
 
=== 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.
 
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]]
 
== Undecimal marvel ==
{{Main| Marvel }}
 
Undecimal marvel tempers out [[385/384]] as well as [[540/539]], and is loosely [[associated temperament|associated]] with [[wizard]]. This extension is natural because of the factorization 225/224 = (385/384)⋅(540/539). [[197edo]] remains useful as a tuning, with the 197e val, but [[166edo]], which among other things has a virtually pure 7, works as well.
 
In the 13-limit, 225/224 factors as ([[351/350]])⋅([[625/624]]) or ([[325/324]])⋅([[729/728]]). Tempering out 351/350 and 625/624 leads to helios, tempering out 325/324 and 729/728 leads to hecate. Tempering out all of them leads to the 13-limit version of [[catakleismic]].
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 225/224, 385/384
 
{{Mapping|legend=1| 1 0 0 -5 12 | 0 1 0 2 -1 | 0 0 1 2 -3 }}
 
[[Mapping to lattice]]: [{{val| 0 -1 0 -2 1 }}, {{val| 0 -1 1 0 -2 }}]


==Music==
Lattice basis:
[[http://chrisvaisvil.com/?p=314|Semimarvelous Blue Drawf]] by [[Chris Vaisvil]]
: ~15/14 length = 1.0364, ~5/4 length = 1.0759
: angle (~15/14, ~5/4) = 104.028°


[[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 }}


==11-limit (Unimarv)==
[[Minimax tuning]]:
[[Comma|Commas]]: 225/224, 385/384
* [[11-odd-limit]]
Related linear temperament: [[Kleismic family|catakleismic temperament]]
: [{{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 basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9


[[Minimax tuning]]:
{{Optimal ET sequence|legend=1| 9, 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 238c, 269ce, 341ce }}
[|1 0 0 0 0&gt;, |4/3 8/9 -1/3 0 -1/9&gt;, |8/3 -2/9 1/3 0 -2/9&gt;,
 
|3 4/3 0 0 -2/3&gt;, |8/3 -2/9 -2/3 0 7/9&gt;]
[[Badness]] (Sintel): 0.306
[[Eigenmonzo subgroup]]: 2.9/5.11/9


Lattice basis: secor length 1.0364 5/4 length 1.0759
[[Projection pair]]s: <code>7 225/32 11 4096/375</code>
Angle(secor, 5/4) = 104.028 degrees
Map to lattice: [&lt;0 -1 0 -2 1|, &lt;0 -1 1 0 -2|]


Map: [&lt;1 0 0 -5 12|, &lt;0 1 0 2 -1|, &lt;0 0 1 2 -3|]
[[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
[[Generator|Generators]]: 2, 3, 5
[[edo|Edos]]: [[10edo|10]], [[12edo|12e]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[166edo|166]], [[197edo|197e]], [[269edo|269ce]], [[341edo|341ce]]
Badness: 0.000255


[[Projection pair]]s: 7 225/32 11 4096/375
[[Spectrum of a temperament|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
Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]
Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]


===Hobbit bases===
{{Databox|[[Hobbit]] bases|
{2, 3, 5} subgroup
2.3.5 subgroup
* 12: 128/125, 2048/2025
* 15: 128/125, 32768/30375
* 19: 16875/16384, 81/80
* 22: 2048/2025, 2109375/2097152
* 31: 2109375/2097152, 81/80
* 41: 3125/3072, 34171875/33554432
}}
 
=== Helios ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 351/350, 385/384
 
Mapping: {{mapping| 1 0 0 -5 12 -4 | 0 1 0 2 -1 -1 | 0 0 1 2 -3 4 }}
 
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:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.15/11.15/13
 
{{Optimal ET sequence|legend=0| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}
 
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
 
=== 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
 
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 }}
 
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:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.13/5
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13
 
{{Optimal ET sequence|legend=0| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}
 
Badness (Sintel): 0.674
 
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
 
===== 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
 
Comma list: 225/224, 325/324, 385/384, 595/594
 
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 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 41, 53g, 72, 94, 113, 166g }}
 
Badness (Sintel): 1.01
 
==== Enodia ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 325/324, 375/374, 385/384
 
Mapping: {{mapping| 1 0 0 -5 12 2 -13 | 0 1 0 2 -1 4 2 | 0 0 1 2 -3 -2 6 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 41g, 53, 72, 166g, 238cfg }}
 
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


12: 128/125, 2048/2025
=== Marvell ===
15: 128/125, 32768/30375
Subgroup: 2.3.5.7.11.13
19: 16875/16384, 81/80
22: 2048/2025, 2109375/2097152
31: 2109375/2097152, 81/80
41: 3125/3072, 34171875/33554432


==13-limit==
Comma list: 225/224, 385/384, 1573/1568
Commas: 225/224, 385/384, 351/350


13-limit eigenmonzo subgroup: 2.11/9.13/9
Mapping: {{mapping| 1 0 0 -5 12 -29 | 0 1 0 2 -1 6 | 0 0 1 2 -3 10 }}
15-limit eigenmonzo subgroup: 2.15/11.15/13


Map: [&lt;1 0 0 -5 12 -4|, &lt;0 1 0 2 -1 -1|, &lt;0 0 1 2 -3 4|]
Optimal tunings:  
EDOs: 19, 22, 31, 50, 53, 72, 103, [[175edo|175f]], [[300edo|300cef]], [[403bcef]]
* WE: ~2 = 1200.6404{{c}}, ~3/2 = 700.7675{{c}}, ~5/4 = 383.7772{{c}}
Badness: 0.000690
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6113{{c}}, ~5/4 = 383.4925{{c}}


[[Spectrum of a temperament|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
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


==Other eleven limit children==
{{Optimal ET sequence|legend=0| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 341cef }}
The second comma of the [[Normal lists|normal comma list]] defines which 11-limit family member we are looking at. Adding 4125/4096 gives unidecimal marvel, 91125/90112 gives prodigy, 5632/5625 minerva and 243/242 spectacle.


=Hecate=
Badness (Sintel): 0.806
Commas: 225/224, 385/384, 325/324


13-limit eigenmonzo subgroup: 2.7.13/5
=== Isis ===
15-limit eigenmonzo subgroup: 2.7.15/13
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 -5 12 2|, &lt;0 1 0 2 -1 4|, &lt;0 0 1 2 -3 -2|]
Comma list: 225/224, 275/273, 385/384
EDOs: 19, 41, 53, 72, 113, 125f, 166, 238cf, 404cef
Badness: 0.000721


Projection pairs: 7 225/32 11 4096/375 13 324/25
Mapping: {{mapping| 1 0 0 -5 12 17 | 0 1 0 2 -1 -4 | 0 0 1 2 -3 -3 }}
[[Spectrum of a temperament|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==
Optimal tunings:
Commas: 225/224, 385/384, 325/324, 595/594
* 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}}


Map: [&lt;1 0 0 -5 12 2 18|, &lt;0 1 0 2 -1 4 0|, &lt;0 0 1 2 -3 -2 -6|]
{{Optimal ET sequence|legend=0| 10, 19f, 22, 31, 41, 53, 84e, 94 }}
EDOs: 19, 41, 53g, 72, 113, 166g, 238cfg, 351cfg, 404cefg
Badness: 0.000869


==Enodia==
Badness (Sintel): 0.810
Commas: 225/224, 385/384, 325/324, 375/374


Map: [&lt;1 0 0 -5 12 2 18|, &lt;0 1 0 2 -1 4 0|, &lt;0 0 1 2 -3 -2 6|]
Projection pairs: <code>7 225/32 11 4096/375 13 131072/10125</code>
EDOs: 41g, 53, 72, 125f, 166g, 238cfg, 363cefg, 404cefg
Badness: .000917


=Marvell=
=== Deecee ===
Commas: 225/224, 385/384, 1573/1568
Subgroup: 2.3.5.7.11.13


13-limit eigenmonzo subgroup: 2.9/5.11/9
Comma list: 225/224, 364/363, 385/384
15-limit eigenmonzo subgroup: 2.7.15/13


Map: [&lt;1 0 0 -5 12 -29|, &lt;0 1 0 2 -1 6|, &lt;0 0 1 2 -3 10|]
Mapping: {{mapping| 1 0 0 -5 12 27 | 0 1 0 2 -1 -3 | 0 0 1 2 -3 -8 }}
EDOs: 9, 31, [[63edo|63]], 72, [[103edo|103]], [[166edo|166]], [[238edo|238cf]], [[269edo|269ce]], 507bcef, 610bcef
Badness: 0.000862


=Isis=
Optimal tunings:
Commas: 225/224, 385/384, 275/273
* 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}}


Map: [&lt;1 0 0 -5 12 17|, &lt;0 1 0 2 -1 4|, &lt;0 0 1 2 -3 -3|]
Minimax tuning:  
EDOs: 10, 22, 31, 41, 53, 94
* 13-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.9/5.13/9
Badness: 0.000866
* 15-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.3.13/5


Projection pairs: 7 225/32 11 4096/375 13 131072/10125
{{Optimal ET sequence|legend=0| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}


=Deecee=
Badness (Sintel): 0.861
Commas: 225/224, 385/384, 364/363


13-limit eigenmonzo subgroup: 2.9/5.13/9
Projection pairs: <code>7 225/32 11 4096/375 13 134217728/10546875</code>
15-limit eigenmonzo subgroup: 2.3.13/5


Map: [&lt;1 0 0 -5 12 27|, &lt;0 1 0 2 -1 -3|, &lt;0 0 1 2 -3 -8|]
=== Tripod ===
EDOs: 9, 22, 41, 63, 72, [[185edo|185cf]], [[257edo|257cf]]
Subgroup: 2.3.5.7.11.13
Badness: 0.000920


Projection pairs: 7 225/32 11 4096/375 13 134217728/10546875
Comma list: 105/104, 144/143, 196/195


=Tripod=
Mapping: {{mapping| 1 0 0 -5 12 -8 | 0 1 0 2 -1 3 | 0 0 1 2 -3 3 }}
Commas: 105/104, 144/143, 196/195


13-limit eigenmonzo subgroup: 2.9/7.13/11
Optimal tunings:  
15-limit eigenmonzo subgroup: 2.5/3.13/11
* 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}}


Map: [&lt;1 0 0 -5 12 -8|, &lt;0 1 0 2 -1 3|, &lt;0 0 1 2 -3 3|]
Minimax tuning:  
EDOs: 9, 10, 19, 31, 41, 72f, [[81edo|81]], [[91edo|91]], [[122edo|122f]], [[163edo|163df]]
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/11
Badness: 0.000745
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/11


Projection pairs: 7 225/32 11 4096/375 13 3375/256
{{Optimal ET sequence|legend=0| 9, 10, 19, 22f, 31, 41, 72f, 91 }}


=Prodigy=
Badness (Sintel): 0.697
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 [[Gamelismic clan|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|Commas]]: 225/224, 441/440
Projection pairs: <code>7 225/32 11 4096/375 13 3375/256</code>
Related linear temperament: [[Gamelismic clan|miracle]]
 
=== Marvelcat ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 385/384
 
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
 
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}}
 
{{Optimal ET sequence|legend=0| 9, 10, 19, 43e, 44, 53, 72, 125f, 197ef }}
 
Badness (Sintel): 0.935
 
== 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.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 225/224, 441/440
 
{{Mapping|legend=1| 1 0 0 -5 -13 | 0 1 0 2 6 | 0 0 1 2 3 }}
 
Map to lattice: [{{val| 0 0 -1 -2 -3 }}, {{val| 0 1 -1 0 3 }}]
 
Lattice basis:
: ~15/14 length = 0.9111, ~3/2 length = 0.9477
: angle (~15/14, ~3/2) = 65.933°
 
[[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]]:
[[Minimax tuning]]:
[|1 0 0 0 0&gt;, |13/12 1/2 -1/4 0 1/12&gt;,  
* [[11-odd-limit]]
|13/6 -1 1/2 0 1/6&gt;,
: [{{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 }}]
|3/2 -1 1/2 0 1/2&gt;, |0 0 0 0 1&gt;]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11
[[Eigenmonzo subgroup]]: 2.9/5.11


Lattice basis: secor length 0.9111, 3/2 length 0.9477
{{Optimal ET sequence|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}
Angle(secor, 3/2) = 65.933
Map to lattice: [&lt;0 0 -1 -2 -3|, &lt;0 1 -1 0 3|]


Map: [&lt;1 0 0 -5 -13|, &lt;0 1 0 2 6|, &lt;0 0 1 2 3|]
[[Badness]] (Sintel): 0.402
[[Generator|Generators]]: 2, 3, 5
[[edo|EDOs]]: [[10edo|10]], [[12edo|12]], [[29edo|29]], [[31edo|31]], [[41edo|41]], [[72edo|72]], [[247edo|247c]], [[319edo|319bcde]], [[391edo|391bcde]], [[463edo|463bcde]]
Badness: 0.000344


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


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


==Hobbit bases==
{{Databox|Hobbit bases|
{2, 3, 5} subgroup
2.3.5 subgroup
* 31: 81/80, 34171875/33554432
* 41: 34171875/33554432, 32805/32768
}}
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 196/195, 352/351
 
Mapping: {{mapping| 1 0 0 -5 -13 -8 | 0 1 0 2 6 3 | 0 0 1 2 3 3 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 101cd }}
 
Badness (Sintel): 0.689
 
=== Prodigious ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 0 0 -5 -13 -23 | 0 1 0 2 6 11 | 0 0 1 2 3 4 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 12f, 29, 31f, 41, 72, 185cf, 257cff }}
 
Badness (Sintel): 0.841
 
=== Prodigal ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 351/350, 441/440
 
Mapping: {{mapping| 1 0 0 -5 -13 -4 | 0 1 0 2 6 -1 | 0 0 1 2 3 4 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
 
Badness (Sintel): 0.831
 
=== Protannic ===
Subgroup: 2.3.5.7.11.13
 
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 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}
 
Badness (Sintel): 0.891
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


31: 81/80, 34171875/33554432
Comma list: 225/224, 273/272, 375/374, 441/440
41: 34171875/33554432, 32805/32768


==13-limit==
Mapping: {{mapping| 1 0 0 -5 -13 21 12 | 0 1 0 2 6 -8 -5 | 0 0 1 2 3 -2 0 }}
Commas: 105/104, 196/195, 352/351


Map: [&lt;1 0 0 -5 -13 -8|, &lt;0 1 0 2 6 3|, &lt;0 0 1 2 3 3|]
Optimal tunings:  
EDOs: 10, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233cdef, 274cdef, 305cdef
* WE: ~2 = 1200.9342{{c}}, ~3/2 = 700.1708{{c}}, ~5/4 = 383.7444{{c}}
Badness: 0.000736
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4849{{c}}, ~5/4 = 383.8744{{c}}


=Minerva=
{{Optimal ET sequence|legend=0| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}
[[Comma|Commas]]: 99/98, 176/175
Related linear temperament: [[Semicomma family|orwell]]


[[Eigenmonzo subgroup]]: 2.7/5.11/9
Badness (Sintel): 0.734


Lattice basis: 16/15 length 0.8997 5/4 length 1.0457
== Minerva ==
Angle(16/15, 5/4) = 98.6044 degrees
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]].
Map to lattice: [&lt;0 -1 0 -2 -2|, &lt;0 -1 1 0 2|]


Map: [&lt;1 0 0 -5 -9|, &lt;0 1 0 2 2|, &lt;0 0 1 2 4|]
[[Subgroup]]: 2.3.5.7.11
[[Generator|Generators]]: 2, 3, 5
[[edo|EDOs]]: [[9edo|9]], [[12edo|12]], [[21edo|21]], [[22edo|22]], [[31edo|31]], [[43edo|43]], [[53edo|53]], [[74edo|74]], [[75edo|75]], [[96edo|96]], [[127edo|127]]
Badness: 0.000381


[[Projection pair]]s: 7 225/32 11 5625/512  
[[Comma list]]: 99/98, 176/175
 
{{Mapping|legend=1| 1 0 0 -5 -9 | 0 1 0 2 2 | 0 0 1 2 4 }}
 
Map to lattice: [{{val| 0 -1 0 -2 -2 }}, {{val| 0 -1 1 0 2 }}]
 
Lattice basis:
: ~16/15 length = 0.8997, ~5/4 length = 1.0457
: 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 basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9
 
{{Optimal ET sequence|legend=1| 9, 12, 19e, 22, 31, 53, 84e, 96, 127 }}
 
[[Badness]] (Sintel): 0.458
 
[[Projection pair]]s: <code>7 225/32 11 5625/512</code>


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


==Athene==
=== Athene ===
Commas: 99/98, 176/175, 275/273
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 176/175, 275/273
 
Mapping: {{mapping| 1 0 0 -5 -9 -4 | 0 1 0 2 2 -1 | 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:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7
 
{{Optimal ET sequence|legend=0| 12f, 19e, 22, 31, 53, 84e, 118d }}
 
Badness (Sintel): 0.765
 
Projection pairs: <code>7 225/32 11 5625/512 13 625/48</code>
 
== Apollo ==
{{See also| Ptolemismic clan #Apollo }}
 
[[File:Lattice Apollo.png|thumb|Lattice for apollo.]]
 
Apollo tempers out not only [[100/99]] but [[896/891]]. Note that marvel tempers together [[25/24]] and [[28/27]], and apollo further equates it with [[33/32]] via the vanishing of 100/99. This makes it a weak extension of [[parapyth]], and [[associated temperament|associates]] it with [[magic]]. The lattice structure is very compact, comparable to that of [[ares]], from which apollo only differs in the mapping of [[prime interval|prime]] [[7/1|7]].
 
The canonical [[13-limit]] extension is implied by parapyth, tempering out [[352/351]] and [[364/363]], but there are a number of other extenions to consider, these being called phoebus and musagetes, after epithets of Apollo. Phoebus tempers out [[105/104]] and finds ~[[16/13]] as a stack of three [[secor]]s. Musagetes tempers out [[144/143]] and conflates 16/13 and [[11/9]], which in this case is simply a stack of two ~[[10/9]]'s. These extensions are [[support]]ed by 13-limit [[magic]], unlike the canonical one.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 100/99, 225/224
 
{{Mapping|legend=1| 1 0 0 -5 2 | 0 1 0 2 -2 | 0 0 1 2 2 }}
 
[[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 }}
 
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9
 
{{Optimal ET sequence|legend=1| 12, 19, 22, 34d, 41, 104 }}
 
[[Badness]] (Sintel): 0.637
 
[[Projection pair]]s: <code>7 225/32 11 100/9</code>
 
Scales: [[apollo wholetone]], [[indigo17]]
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 225/224, 275/273
 
Mapping: {{mapping| 1 0 0 -5 2 7 | 0 1 0 2 -2 -5 | 0 0 1 2 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: 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9
 
{{Optimal ET sequence|legend=0| 12f, 19f, 22, 29, 34d, 41, 63, 104 }}
 
Badness (Sintel): 0.964
 
Projection pairs: <code>7 225/32 11 100/9 13 3200/243</code>
 
=== Phoebus ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 105/104, 196/195
 
Mapping: {{mapping| 1 0 0 -5 2 1 | 0 1 0 2 -2 3 | 0 0 1 2 2 3 }}
 
Optimal tunings:
* WE: ~2 = 1200.3724{{c}}, ~3/2 = 702.5274{{c}}, ~5/4 = 379.6345{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3357{{c}}, ~5/4 = 379.6830{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 19, 22f, 29, 41 }}
 
Badness (Sintel): 0.886
 
=== Musagetes ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 144/143, 225/224
 
Mapping: {{mapping| 1 0 0 -5 2 2 | 0 1 0 2 -2 4 | 0 0 1 2 2 -2 }}
 
Optimal tunings:
* WE: ~2 = 1199.2695{{c}}, ~3/2 = 702.5589{{c}}, ~5/4 = 382.7715{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7998{{c}}, ~5/4 = 382.7740{{c}}
 
{{Optimal ET sequence|legend=0| 19, 22f, 34d, 41, 75e, 94e, 116ef }}
 
Badness (Sintel): 1.14
 
== Potassium ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 45/44, 56/55
 
{{Mapping|legend=1| 1 0 0 -5 -2 | 0 1 0 2 2 | 0 0 1 2 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 }}
 
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11
 
{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 31e }}
 
[[Badness]] (Sintel): 0.557
 
[[Projection pair]]s: <code>7 225/32 11 45/4</code>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 56/55, 78/77
 
Mapping: {{mapping| 1 0 0 -5 -2 -8 | 0 1 0 2 2 3 | 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:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9
 
{{Optimal ET sequence|legend=0| 9, 10, 12f, 19, 31e }}
 
Badness (Sintel): 0.686
 
Projection pairs: <code>7 225/32 11 45/4 13 3375/256</code>


13-limit eigenmonzo subgroup: 2.11/9.13/7
== Malcolm ==
15-limit eigenmonzo subgroup: 2.11/9.13/7
{{Redirect|Malcolm|{{w|Alexander Malcolm (writer on music)|Alexander Malcolm}}'s JI scale|Malcolm (scale)}}


Map: [&lt;1 0 0 -5 -9 -4|, &lt;0 1 0 2 2 -1|, &lt;0 0 1 2 4 4|]
[[Subgroup]]: 2.3.5.7.11
EDOs: 22, 31, 53, 84e, 118d, 149df, 171de, 202def
Badness: 0.000818


[[Projection pair]]s: 7 225/32 11 5625/512 13 625/48
[[Comma list]]: 225/224, 2200/2187


=Spectacle=
{{Mapping|legend=1| 1 0 0 -5 -3 | 0 1 0 2 7 | 0 0 1 2 -2 }}
[[Comma|Commas]]: 225/224, 243/242
Related linear temperament: [[Marvel temperaments|marvo]]


[[Minimax tuning]]:
[[Optimal tuning]]s:  
[|1 0 0 0 0&gt;, |1/5 0 0 0 2/5&gt;, |2/5 -2 1 0 4/5&gt;, |-19/5 -4 2 0 12/5&gt;, |0 0 0 0 1&gt;]
* [[WE]]: ~2 = 1199.5261{{c}}, ~3/2 = 702.1990{{c}}, ~5/4 = 382.5759{{c}}
[[Eigenmonzo subgroup]]: 2.9/5.11
: [[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 }}
 
{{Optimal ET sequence|legend=1| 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce }}
 
[[Badness]] (Sintel): 1.50
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 275/273, 325/324
 
Mapping: {{mapping| 1 0 0 -5 -3 2 | 0 1 0 2 7 4 | 0 0 1 2 -2 -2 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 12e, 19e, 34d, 41, 53, 94 }}
 
Badness (Sintel): 1.00
 
Scales: [[malco]]
 
== Fantastic ==
Besides 4375/4356, fantastic also tempers out [[9801/9800]] and splits the octave in two.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 225/224, 4375/4356
 
{{Mapping|legend=1| 2 0 0 -10 -7 | 0 1 0 2 0 | 0 0 1 2 3 }}
: mapping generators: ~99/70, ~3, ~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 }}
 
{{Optimal ET sequence|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}
 
[[Badness]] (Sintel): 0.893
 
== 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]].
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 225/224, 125000/124509
 
{{Mapping|legend=1| 1 0 0 -5 9 | 0 2 0 4 -7 | 0 0 1 2 0 }}
: mapping generators: ~2, ~400/231, ~5
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce }}
 
[[Badness]] (Sintel): 1.845
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 1001/1000
 
Mapping: {{mapping| 1 0 0 -5 9 -1 | 0 2 0 4 -7 3 | 0 0 1 2 0 1 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 19, 29, 43, 53, 72, 125f, 197ef }}
 
Badness (Sintel): 0.993
 
== Morfil ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 225/224, 1331/1323
 
{{Mapping|legend=1| 1 0 -2 -9 -6 | 0 1 2 6 5 | 0 0 3 6 4 }}
: mapping generators: ~2, ~3, ~55/42
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 29, 31, 60e, 91e, 94, 125 }}


Map: [&lt;1 1 0 -3 2|, &lt;0 2 0 4 5|, &lt;0 0 1 2 0|]
[[Badness]] (Sintel): 1.38
[[Generator|Generators]]: 2, 11/9, 5
[[edo|EDOs]]: [[10edo|10]], [[31edo|31]], [[41edo|41]], [[72edo|72]], [[240edo|240]], [[259edo|259b]], [[269edo|269ce]], [[310edo|310c]], [[331edo|331bc]], [[353edo|353c]], [[497edo|497bc]], [[569edo|569bc]]
Badness: 0.000499


Projection pairs: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9
== Subgroup extensions ==
=== Char ===
Subgroup: 2.3.5.7.17


Scales: [[spectacle31]]
Comma list: 120/119, 225/224


=Apollo=
Mapping: {{mapping| 1 0 0 -5 8 | 0 1 0 2 -1 | 0 0 1 2 -1 }}
[[Comma|Commas]]: 100/99, 225/224
Related linear temperament: [[Magic family|Magic]]


Eigenmonzo subgroup: 2.7/5.11/9
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}}


Map: [&lt;1 0 0 -5 2|, &lt;0 1 0 2 -2|, &lt;0 0 1 2 2|]
{{Optimal ET sequence|legend=0| 9, 10, 12, 19, 22, 31, 41, 53 }}
EDOs: 12, 19, 22, 41, [[104edo]], [[157edo|157ce]], [[198edo|198ce]], [[220edo|220ce]], [[261edo|261ce]]


Projection pairs: 7 225/32 11 100/9
Badness (Sintel): 0.381


=Potassium=
=== Devimarvel ===
Commas: 45/44, 56/55
Subgroup: 2.3.5.7.19


Eigenmonzo subgroup: 2.9/7.11
Comma list: 225/224, 400/399


Map: [&lt;1 0 0 -5 -2|, &lt;0 1 0 2 2|, &lt;0 0 1 2 1|]
Mapping: {{mapping| 1 0 0 -5 9 | 0 1 0 2 -3 | 0 0 1 2 0 }}
EDOs: 9, 10, 12, 19, 31e, 50e
Badness: 0.000464


Projection pairs: 7 225/32 11 45/4
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}}


==13-limit==
{{Optimal ET sequence|legend=0| 10, 12, 19, 22, 31, 41, 53, 72, 94, 113, 125, 291c }}
Commas: 45/44, 56/55, 78/77


13-limit eigenmonzo subgroup: 2.9/7.13/9
Badness (Sintel): 0.310
15-limit eigenmonzo subgroup: 2.9/7.13/9


Map: [&lt;1 0 0 -5 -2 -8|, &lt;0 1 0 2 2 3|, &lt;0 0 1 2 1 3|]
== References ==
EDOs: 9, 10, 19, 31e, 50e
<references />
Badness: 0.000733


Projection pairs: 7 225/32 11 45/4 13 3375/256</pre></div>
[[Category:Marvel family| ]] <!-- main article -->
<h4>Original HTML content:</h4>
[[Category:Temperament families]]
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Marvel family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:48:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;a href="#Marvel"&gt;Marvel&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt; | &lt;a href="#Hecate"&gt;Hecate&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt; | &lt;a href="#Marvell"&gt;Marvell&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt; | &lt;a href="#Isis"&gt;Isis&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt; | &lt;a href="#Deecee"&gt;Deecee&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt; | &lt;a href="#Tripod"&gt;Tripod&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt; | &lt;a href="#Prodigy"&gt;Prodigy&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;!-- ws:start:WikiTextTocRule:65: --&gt;&lt;!-- ws:end:WikiTextTocRule:65 --&gt;&lt;!-- ws:start:WikiTextTocRule:66: --&gt;&lt;!-- ws:end:WikiTextTocRule:66 --&gt;&lt;!-- ws:start:WikiTextTocRule:67: --&gt; | &lt;a href="#Minerva"&gt;Minerva&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:67 --&gt;&lt;!-- ws:start:WikiTextTocRule:68: --&gt;&lt;!-- ws:end:WikiTextTocRule:68 --&gt;&lt;!-- ws:start:WikiTextTocRule:69: --&gt; | &lt;a href="#Spectacle"&gt;Spectacle&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt; | &lt;a href="#Apollo"&gt;Apollo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:70 --&gt;&lt;!-- ws:start:WikiTextTocRule:71: --&gt; | &lt;a href="#Potassium"&gt;Potassium&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:71 --&gt;&lt;!-- ws:start:WikiTextTocRule:72: --&gt;&lt;!-- ws:end:WikiTextTocRule:72 --&gt;&lt;!-- ws:start:WikiTextTocRule:73: --&gt;
[[Category:Catalogs of rank-3 temperaments]]
&lt;!-- ws:end:WikiTextTocRule:73 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Marvel"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Marvel&lt;/h1&gt;
The head of the marvel family is marvel, which tempers out &lt;a class="wiki_link" href="/225_224"&gt;225/224&lt;/a&gt;, the &lt;a class="wiki_link" href="/septimal%20kleisma"&gt;septimal kleisma&lt;/a&gt; or &lt;a class="wiki_link" href="/marvel%20comma"&gt;marvel comma&lt;/a&gt;. Marvel has a &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal list basis&lt;/a&gt; of [2, 3, 5]; hence a &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;5-limit&lt;/a&gt; scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by &lt;a class="wiki_link" href="/197edo"&gt;197edo&lt;/a&gt;. &lt;br /&gt;
&lt;br /&gt;
Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to &lt;a class="wiki_link" href="/Kleismic%20family"&gt;catakleismic temperament&lt;/a&gt;. Another temperament which does little damage to tuning accuracy is &lt;a class="wiki_link" href="/Pythagorean%20family"&gt;compton temperament&lt;/a&gt;, for which &lt;a class="wiki_link" href="/240edo"&gt;240edo&lt;/a&gt; may be used.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Marvel-Vital statistics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Vital statistics&lt;/h2&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt; c = 225/224&lt;br /&gt;
Related linear temperament: &lt;a class="wiki_link" href="/Kleismic%20family"&gt;catakleismic temperament&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
7-limit minimax: 3 and 5 1/4c flat, 7 just&lt;br /&gt;
[|1 0 0 0&amp;gt;, |5/4 1/2 -1/2 1/4&amp;gt;, |5/4 -1/2 1/2 1/4&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo%20subgroup"&gt;Eigenmonzo subgroup&lt;/a&gt;: 2.5/3.7&lt;br /&gt;
&lt;br /&gt;
9-limit minimax: 3 1/6c flat, 5 1/3c flat, 7 just&lt;br /&gt;
[|1 0 0 0&amp;gt;, |5/6 2/3 -1/3 1/6&amp;gt;, |5/3 -2/3 1/3 1/3&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo%20subgroup"&gt;Eigenmonzo subgroup&lt;/a&gt;: 2.9/5.7&lt;br /&gt;
&lt;br /&gt;
Lattice basis: secor length 1.256, 3/2 length 1.369&lt;br /&gt;
Angle(secor, 3/2) = 106.958 degrees&lt;br /&gt;
Map to lattice: [&amp;lt;0 0 -1 -2|, &amp;lt;0 1 -1 0|]&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5|, &amp;lt;0 1 0 2|, &amp;lt;0 0 1 2|]&lt;br /&gt;
Generators: 2, 3, 5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/10edo"&gt;10&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;,&lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/197edo"&gt;197&lt;/a&gt;, &lt;a class="wiki_link" href="/269edo"&gt;269c&lt;/a&gt;&lt;br /&gt;
Badness: 0.0000365&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;s: 7 225/32&lt;br /&gt;
&lt;a class="wiki_link" href="/Spectrum%20of%20a%20temperament"&gt;Spectrum&lt;/a&gt;: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9&lt;br /&gt;
&lt;br /&gt;
Scales: &lt;a class="wiki_link" href="/marvel9"&gt;marvel9&lt;/a&gt;, &lt;a class="wiki_link" href="/marvel10"&gt;marvel10&lt;/a&gt;, &lt;a class="wiki_link" href="/marvel11"&gt;marvel11&lt;/a&gt;, &lt;a class="wiki_link" href="/marvel12"&gt;marvel12&lt;/a&gt;, &lt;a class="wiki_link" href="/marvel19"&gt;marvel19&lt;/a&gt;, &lt;a class="wiki_link" href="/marvel22"&gt;marvel22&lt;/a&gt;, &lt;a class="wiki_link" href="/pump12_1"&gt;pump12_1&lt;/a&gt;, &lt;a class="wiki_link" href="/pump12_2"&gt;pump12_2&lt;/a&gt;, &lt;a class="wiki_link" href="/pump13"&gt;pump13&lt;/a&gt;, &lt;a class="wiki_link" href="/pump14"&gt;pump14&lt;/a&gt;, &lt;a class="wiki_link" href="/pump15"&gt;pump15&lt;/a&gt;, &lt;a class="wiki_link" href="/pump16"&gt;pump16&lt;/a&gt;, &lt;a class="wiki_link" href="/pump17"&gt;pump17&lt;/a&gt;, &lt;a class="wiki_link" href="/pump18"&gt;pump18&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Marvel-Minkowski blocks"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;a class="wiki_link" href="/Minkowski%20blocks"&gt;Minkowski blocks&lt;/a&gt;&lt;/h2&gt;
{2, 3, 5} subgroup&lt;br /&gt;
&lt;br /&gt;
8: 16/15, 250/243&lt;br /&gt;
9: 135/128, 128/125&lt;br /&gt;
10: 25/24, 2048/2025&lt;br /&gt;
11: 135/128, 2048/1875&lt;br /&gt;
12: 2048/2025, 128/125&lt;br /&gt;
15: 128/125, 32768/30375&lt;br /&gt;
17: 25/24, 2278125/2097152&lt;br /&gt;
19: 16875/16384, 81/80&lt;br /&gt;
21: 128/125, 273375/262144&lt;br /&gt;
22: 2048/2025, 3125/3072&lt;br /&gt;
29: 16875/16384, 32805/32768&lt;br /&gt;
31: 81/80, 34171875/33554432&lt;br /&gt;
41: 34171875/33554432, 3125/3072&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Marvel-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Music&lt;/h2&gt;
&lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=314" rel="nofollow"&gt;Semimarvelous Blue Drawf&lt;/a&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Marvel-11-limit (Unimarv)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit (Unimarv)&lt;/h2&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 225/224, 385/384&lt;br /&gt;
Related linear temperament: &lt;a class="wiki_link" href="/Kleismic%20family"&gt;catakleismic temperament&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;: &lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |4/3 8/9 -1/3 0 -1/9&amp;gt;, |8/3 -2/9 1/3 0 -2/9&amp;gt;,&lt;br /&gt;
|3 4/3 0 0 -2/3&amp;gt;, |8/3 -2/9 -2/3 0 7/9&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo%20subgroup"&gt;Eigenmonzo subgroup&lt;/a&gt;: 2.9/5.11/9&lt;br /&gt;
&lt;br /&gt;
Lattice basis: secor length 1.0364 5/4 length 1.0759&lt;br /&gt;
Angle(secor, 5/4) = 104.028 degrees&lt;br /&gt;
Map to lattice: [&amp;lt;0 -1 0 -2 1|, &amp;lt;0 -1 1 0 -2|]&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 12|, &amp;lt;0 1 0 2 -1|, &amp;lt;0 0 1 2 -3|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/10edo"&gt;10&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12e&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/166edo"&gt;166&lt;/a&gt;, &lt;a class="wiki_link" href="/197edo"&gt;197e&lt;/a&gt;, &lt;a class="wiki_link" href="/269edo"&gt;269ce&lt;/a&gt;, &lt;a class="wiki_link" href="/341edo"&gt;341ce&lt;/a&gt;&lt;br /&gt;
Badness: 0.000255&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;s: 7 225/32 11 4096/375&lt;br /&gt;
&lt;a class="wiki_link" href="/Spectrum%20of%20a%20temperament"&gt;Spectrum&lt;/a&gt;: 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&lt;br /&gt;
Scales: &lt;a class="wiki_link" href="/marvel22_11"&gt;marvel22_11&lt;/a&gt;, &lt;a class="wiki_link" href="/unimarv19"&gt;unimarv19&lt;/a&gt;, &lt;a class="wiki_link" href="/unimarv22"&gt;unimarv22&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Marvel-11-limit (Unimarv)-Hobbit bases"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Hobbit bases&lt;/h3&gt;
{2, 3, 5} subgroup&lt;br /&gt;
&lt;br /&gt;
12: 128/125, 2048/2025&lt;br /&gt;
15: 128/125, 32768/30375&lt;br /&gt;
19: 16875/16384, 81/80&lt;br /&gt;
22: 2048/2025, 2109375/2097152&lt;br /&gt;
31: 2109375/2097152, 81/80&lt;br /&gt;
41: 3125/3072, 34171875/33554432&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Marvel-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit&lt;/h2&gt;
Commas: 225/224, 385/384, 351/350&lt;br /&gt;
&lt;br /&gt;
13-limit eigenmonzo subgroup: 2.11/9.13/9&lt;br /&gt;
15-limit eigenmonzo subgroup: 2.15/11.15/13&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 12 -4|, &amp;lt;0 1 0 2 -1 -1|, &amp;lt;0 0 1 2 -3 4|]&lt;br /&gt;
EDOs: 19, 22, 31, 50, 53, 72, 103, &lt;a class="wiki_link" href="/175edo"&gt;175f&lt;/a&gt;, &lt;a class="wiki_link" href="/300edo"&gt;300cef&lt;/a&gt;, &lt;a class="wiki_link" href="/403bcef"&gt;403bcef&lt;/a&gt;&lt;br /&gt;
Badness: 0.000690&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Spectrum%20of%20a%20temperament"&gt;Spectrum&lt;/a&gt;: 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&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Marvel-Other eleven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Other eleven limit children&lt;/h2&gt;
The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 11-limit family member we are looking at. Adding 4125/4096 gives unidecimal marvel, 91125/90112 gives prodigy, 5632/5625 minerva and 243/242 spectacle.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Hecate"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Hecate&lt;/h1&gt;
Commas: 225/224, 385/384, 325/324&lt;br /&gt;
&lt;br /&gt;
13-limit eigenmonzo subgroup: 2.7.13/5&lt;br /&gt;
15-limit eigenmonzo subgroup: 2.7.15/13&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 12 2|, &amp;lt;0 1 0 2 -1 4|, &amp;lt;0 0 1 2 -3 -2|]&lt;br /&gt;
EDOs: 19, 41, 53, 72, 113, 125f, 166, 238cf, 404cef&lt;br /&gt;
Badness: 0.000721&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 7 225/32 11 4096/375 13 324/25&lt;br /&gt;
&lt;a class="wiki_link" href="/Spectrum%20of%20a%20temperament"&gt;Spectrum&lt;/a&gt;: 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&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Hecate-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;17-limit&lt;/h2&gt;
Commas: 225/224, 385/384, 325/324, 595/594&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 12 2 18|, &amp;lt;0 1 0 2 -1 4 0|, &amp;lt;0 0 1 2 -3 -2 -6|]&lt;br /&gt;
EDOs: 19, 41, 53g, 72, 113, 166g, 238cfg, 351cfg, 404cefg&lt;br /&gt;
Badness: 0.000869&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Hecate-Enodia"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Enodia&lt;/h2&gt;
Commas: 225/224, 385/384, 325/324, 375/374&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 12 2 18|, &amp;lt;0 1 0 2 -1 4 0|, &amp;lt;0 0 1 2 -3 -2 6|]&lt;br /&gt;
EDOs: 41g, 53, 72, 125f, 166g, 238cfg, 363cefg, 404cefg&lt;br /&gt;
Badness: .000917&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Marvell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Marvell&lt;/h1&gt;
Commas: 225/224, 385/384, 1573/1568&lt;br /&gt;
&lt;br /&gt;
13-limit eigenmonzo subgroup: 2.9/5.11/9&lt;br /&gt;
15-limit eigenmonzo subgroup: 2.7.15/13&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 12 -29|, &amp;lt;0 1 0 2 -1 6|, &amp;lt;0 0 1 2 -3 10|]&lt;br /&gt;
EDOs: 9, 31, &lt;a class="wiki_link" href="/63edo"&gt;63&lt;/a&gt;, 72, &lt;a class="wiki_link" href="/103edo"&gt;103&lt;/a&gt;, &lt;a class="wiki_link" href="/166edo"&gt;166&lt;/a&gt;, &lt;a class="wiki_link" href="/238edo"&gt;238cf&lt;/a&gt;, &lt;a class="wiki_link" href="/269edo"&gt;269ce&lt;/a&gt;, 507bcef, 610bcef&lt;br /&gt;
Badness: 0.000862&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Isis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Isis&lt;/h1&gt;
Commas: 225/224, 385/384, 275/273&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 12 17|, &amp;lt;0 1 0 2 -1 4|, &amp;lt;0 0 1 2 -3 -3|]&lt;br /&gt;
EDOs: 10, 22, 31, 41, 53, 94&lt;br /&gt;
Badness: 0.000866&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 7 225/32 11 4096/375 13 131072/10125&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="Deecee"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Deecee&lt;/h1&gt;
Commas: 225/224, 385/384, 364/363&lt;br /&gt;
&lt;br /&gt;
13-limit eigenmonzo subgroup: 2.9/5.13/9&lt;br /&gt;
15-limit eigenmonzo subgroup: 2.3.13/5&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 12 27|, &amp;lt;0 1 0 2 -1 -3|, &amp;lt;0 0 1 2 -3 -8|]&lt;br /&gt;
EDOs: 9, 22, 41, 63, 72, &lt;a class="wiki_link" href="/185edo"&gt;185cf&lt;/a&gt;, &lt;a class="wiki_link" href="/257edo"&gt;257cf&lt;/a&gt;&lt;br /&gt;
Badness: 0.000920&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 7 225/32 11 4096/375 13 134217728/10546875&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Tripod"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Tripod&lt;/h1&gt;
Commas: 105/104, 144/143, 196/195&lt;br /&gt;
&lt;br /&gt;
13-limit eigenmonzo subgroup: 2.9/7.13/11&lt;br /&gt;
15-limit eigenmonzo subgroup: 2.5/3.13/11&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 12 -8|, &amp;lt;0 1 0 2 -1 3|, &amp;lt;0 0 1 2 -3 3|]&lt;br /&gt;
EDOs: 9, 10, 19, 31, 41, 72f, &lt;a class="wiki_link" href="/81edo"&gt;81&lt;/a&gt;, &lt;a class="wiki_link" href="/91edo"&gt;91&lt;/a&gt;, &lt;a class="wiki_link" href="/122edo"&gt;122f&lt;/a&gt;, &lt;a class="wiki_link" href="/163edo"&gt;163df&lt;/a&gt;&lt;br /&gt;
Badness: 0.000745&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 7 225/32 11 4096/375 13 3375/256&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="Prodigy"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Prodigy&lt;/h1&gt;
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 &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;miracle temperament&lt;/a&gt;. 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.&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 225/224, 441/440&lt;br /&gt;
Related linear temperament: &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;miracle&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;:&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |13/12 1/2 -1/4 0 1/12&amp;gt;, &lt;br /&gt;
|13/6 -1 1/2 0 1/6&amp;gt;,&lt;br /&gt;
|3/2 -1 1/2 0 1/2&amp;gt;, |0 0 0 0 1&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo%20subgroup"&gt;Eigenmonzo subgroup&lt;/a&gt;: 2.9/5.11&lt;br /&gt;
&lt;br /&gt;
Lattice basis: secor length 0.9111, 3/2 length 0.9477&lt;br /&gt;
Angle(secor, 3/2) = 65.933&lt;br /&gt;
Map to lattice: [&amp;lt;0 0 -1 -2 -3|, &amp;lt;0 1 -1 0 3|]&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 -13|, &amp;lt;0 1 0 2 6|, &amp;lt;0 0 1 2 3|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/10edo"&gt;10&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/29edo"&gt;29&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/247edo"&gt;247c&lt;/a&gt;, &lt;a class="wiki_link" href="/319edo"&gt;319bcde&lt;/a&gt;, &lt;a class="wiki_link" href="/391edo"&gt;391bcde&lt;/a&gt;, &lt;a class="wiki_link" href="/463edo"&gt;463bcde&lt;/a&gt;&lt;br /&gt;
Badness: 0.000344&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;s: 7 225/32 11 91125/8192 &lt;br /&gt;
&lt;br /&gt;
Scales: &lt;a class="wiki_link" href="/prodigy11"&gt;prodigy11&lt;/a&gt;, &lt;a class="wiki_link" href="/prodigy12"&gt;prodigy12&lt;/a&gt;, &lt;a class="wiki_link" href="/prodigy29"&gt;prodigy29&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Prodigy-Hobbit bases"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Hobbit bases&lt;/h2&gt;
{2, 3, 5} subgroup&lt;br /&gt;
&lt;br /&gt;
31: 81/80, 34171875/33554432&lt;br /&gt;
41: 34171875/33554432, 32805/32768&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Prodigy-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;13-limit&lt;/h2&gt;
Commas: 105/104, 196/195, 352/351&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 -13 -8|, &amp;lt;0 1 0 2 6 3|, &amp;lt;0 0 1 2 3 3|]&lt;br /&gt;
EDOs: 10, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233cdef, 274cdef, 305cdef&lt;br /&gt;
Badness: 0.000736&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc18"&gt;&lt;a name="Minerva"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Minerva&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 99/98, 176/175&lt;br /&gt;
Related linear temperament: &lt;a class="wiki_link" href="/Semicomma%20family"&gt;orwell&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo%20subgroup"&gt;Eigenmonzo subgroup&lt;/a&gt;: 2.7/5.11/9&lt;br /&gt;
&lt;br /&gt;
Lattice basis: 16/15 length 0.8997 5/4 length 1.0457 &lt;br /&gt;
Angle(16/15, 5/4) = 98.6044 degrees&lt;br /&gt;
Map to lattice: [&amp;lt;0 -1 0 -2 -2|, &amp;lt;0 -1 1 0 2|]&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 -9|, &amp;lt;0 1 0 2 2|, &amp;lt;0 0 1 2 4|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/9edo"&gt;9&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/21edo"&gt;21&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/43edo"&gt;43&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/74edo"&gt;74&lt;/a&gt;, &lt;a class="wiki_link" href="/75edo"&gt;75&lt;/a&gt;, &lt;a class="wiki_link" href="/96edo"&gt;96&lt;/a&gt;, &lt;a class="wiki_link" href="/127edo"&gt;127&lt;/a&gt;&lt;br /&gt;
Badness: 0.000381&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;s: 7 225/32 11 5625/512 &lt;br /&gt;
&lt;br /&gt;
Scales: &lt;a class="wiki_link" href="/minerva12"&gt;minerva12&lt;/a&gt;, &lt;a class="wiki_link" href="/minerva22x"&gt;minerva22x&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Minerva-Athene"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Athene&lt;/h2&gt;
Commas: 99/98, 176/175, 275/273&lt;br /&gt;
&lt;br /&gt;
13-limit eigenmonzo subgroup: 2.11/9.13/7&lt;br /&gt;
15-limit eigenmonzo subgroup: 2.11/9.13/7&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 -9 -4|, &amp;lt;0 1 0 2 2 -1|, &amp;lt;0 0 1 2 4 4|]&lt;br /&gt;
EDOs: 22, 31, 53, 84e, 118d, 149df, 171de, 202def&lt;br /&gt;
Badness: 0.000818&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;s: 7 225/32 11 5625/512 13 625/48&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc20"&gt;&lt;a name="Spectacle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;Spectacle&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 225/224, 243/242&lt;br /&gt;
Related linear temperament: &lt;a class="wiki_link" href="/Marvel%20temperaments"&gt;marvo&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;:&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |1/5 0 0 0 2/5&amp;gt;, |2/5 -2 1 0 4/5&amp;gt;, |-19/5 -4 2 0 12/5&amp;gt;, |0 0 0 0 1&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo%20subgroup"&gt;Eigenmonzo subgroup&lt;/a&gt;: 2.9/5.11&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 1 0 -3 2|, &amp;lt;0 2 0 4 5|, &amp;lt;0 0 1 2 0|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 11/9, 5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/10edo"&gt;10&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/240edo"&gt;240&lt;/a&gt;, &lt;a class="wiki_link" href="/259edo"&gt;259b&lt;/a&gt;, &lt;a class="wiki_link" href="/269edo"&gt;269ce&lt;/a&gt;, &lt;a class="wiki_link" href="/310edo"&gt;310c&lt;/a&gt;, &lt;a class="wiki_link" href="/331edo"&gt;331bc&lt;/a&gt;, &lt;a class="wiki_link" href="/353edo"&gt;353c&lt;/a&gt;, &lt;a class="wiki_link" href="/497edo"&gt;497bc&lt;/a&gt;, &lt;a class="wiki_link" href="/569edo"&gt;569bc&lt;/a&gt;&lt;br /&gt;
Badness: 0.000499&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9&lt;br /&gt;
&lt;br /&gt;
Scales: &lt;a class="wiki_link" href="/spectacle31"&gt;spectacle31&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc21"&gt;&lt;a name="Apollo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;Apollo&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 100/99, 225/224&lt;br /&gt;
Related linear temperament: &lt;a class="wiki_link" href="/Magic%20family"&gt;Magic&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
Eigenmonzo subgroup: 2.7/5.11/9&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 2|, &amp;lt;0 1 0 2 -2|, &amp;lt;0 0 1 2 2|]&lt;br /&gt;
EDOs: 12, 19, 22, 41, &lt;a class="wiki_link" href="/104edo"&gt;104edo&lt;/a&gt;, &lt;a class="wiki_link" href="/157edo"&gt;157ce&lt;/a&gt;, &lt;a class="wiki_link" href="/198edo"&gt;198ce&lt;/a&gt;, &lt;a class="wiki_link" href="/220edo"&gt;220ce&lt;/a&gt;, &lt;a class="wiki_link" href="/261edo"&gt;261ce&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 7 225/32 11 100/9&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc22"&gt;&lt;a name="Potassium"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;Potassium&lt;/h1&gt;
Commas: 45/44, 56/55&lt;br /&gt;
&lt;br /&gt;
Eigenmonzo subgroup: 2.9/7.11&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 -2|, &amp;lt;0 1 0 2 2|, &amp;lt;0 0 1 2 1|]&lt;br /&gt;
EDOs: 9, 10, 12, 19, 31e, 50e&lt;br /&gt;
Badness: 0.000464&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 7 225/32 11 45/4&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Potassium-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;13-limit&lt;/h2&gt;
Commas: 45/44, 56/55, 78/77&lt;br /&gt;
&lt;br /&gt;
13-limit eigenmonzo subgroup: 2.9/7.13/9&lt;br /&gt;
15-limit eigenmonzo subgroup: 2.9/7.13/9&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5 -2 -8|, &amp;lt;0 1 0 2 2 3|, &amp;lt;0 0 1 2 1 3|]&lt;br /&gt;
EDOs: 9, 10, 19, 31e, 50e&lt;br /&gt;
Badness: 0.000733&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 7 225/32 11 45/4 13 3375/256&lt;/body&gt;&lt;/html&gt;</pre></div>
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