Marvel family

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[[toc|flat]]
=Marvel= 
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.

See also [[Marvel temperaments]].

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

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

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

Lattice basis: secor length 1.256, 3/2 length 1.369
Angle(secor, 3/2) = 106.958 degrees
Map to lattice: [<0 0 -1 -2|, <0 1 -1 0|]

Map: [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]
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
[[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]]

==[[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

==Music== 
[[http://chrisvaisvil.com/?p=314|Semimarvelous Blue Drawf]] by [[Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/marvel10_2.mp3|notJoelTaylorsTreeSpirit-Marvel10-2]] by [[Billy Stiltner]]

==11-limit (Unimarv)== 
[[Comma|Commas]]: 225/224, 385/384
Related linear temperament: [[Kleismic family|catakleismic temperament]]

[[Minimax tuning]]:
[|1 0 0 0 0>, |4/3 8/9 -1/3 0 -1/9>, |8/3 -2/9 1/3 0 -2/9>,
|3 4/3 0 0 -2/3>, |8/3 -2/9 -2/3 0 7/9>]
[[Eigenmonzo subgroup]]: 2.9/5.11/9

Lattice basis: secor length 1.0364 5/4 length 1.0759
Angle(secor, 5/4) = 104.028 degrees
Map to lattice: [<0 -1 0 -2 1|, <0 -1 1 0 -2|]

Map: [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|]
[[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]]

===Hobbit bases=== 
{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

==13-limit== 
Commas: 225/224, 385/384, 351/350

13-limit eigenmonzo subgroup: 2.11/9.13/9
15-limit eigenmonzo subgroup: 2.15/11.15/13

Map: [<1 0 0 -5 12 -4|, <0 1 0 2 -1 -1|, <0 0 1 2 -3 4|]
EDOs: 19, 22, 31, 50, 53, 72, 103, [[175edo|175f]], [[300edo|300cef]], [[403bcef]]
Badness: 0.000690

[[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

==Other eleven limit children== 
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= 
Commas: 225/224, 385/384, 325/324

13-limit eigenmonzo subgroup: 2.7.13/5
15-limit eigenmonzo subgroup: 2.7.15/13

Map: [<1 0 0 -5 12 2|, <0 1 0 2 -1 4|, <0 0 1 2 -3 -2|]
EDOs: 19, 41, 53, 72, 113, 125f, 166, 238cf, 404cef
Badness: 0.000721

Projection pairs: 7 225/32 11 4096/375 13 324/25
[[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== 
Commas: 225/224, 385/384, 325/324, 595/594

Map: [<1 0 0 -5 12 2 18|, <0 1 0 2 -1 4 0|, <0 0 1 2 -3 -2 -6|]
EDOs: 19, 41, 53g, 72, 113, 166g, 238cfg, 351cfg, 404cefg
Badness: 0.000869

==Enodia== 
Commas: 225/224, 385/384, 325/324, 375/374

Map: [<1 0 0 -5 12 2 18|, <0 1 0 2 -1 4 0|, <0 0 1 2 -3 -2 6|]
EDOs: 41g, 53, 72, 125f, 166g, 238cfg, 363cefg, 404cefg
Badness: .000917

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

13-limit eigenmonzo subgroup: 2.9/5.11/9
15-limit eigenmonzo subgroup: 2.7.15/13

Map: [<1 0 0 -5 12 -29|, <0 1 0 2 -1 6|, <0 0 1 2 -3 10|]
EDOs: 9, 31, [[63edo|63]], 72, [[103edo|103]], [[166edo|166]], [[238edo|238cf]], [[269edo|269ce]], 507bcef, 610bcef
Badness: 0.000862

=Isis= 
Commas: 225/224, 385/384, 275/273

Map: [<1 0 0 -5 12 17|, <0 1 0 2 -1 4|, <0 0 1 2 -3 -3|]
EDOs: 10, 22, 31, 41, 53, 94
Badness: 0.000866

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

=Deecee= 
Commas: 225/224, 385/384, 364/363

13-limit eigenmonzo subgroup: 2.9/5.13/9
15-limit eigenmonzo subgroup: 2.3.13/5

Map: [<1 0 0 -5 12 27|, <0 1 0 2 -1 -3|, <0 0 1 2 -3 -8|]
EDOs: 9, 22, 41, 63, 72, [[185edo|185cf]], [[257edo|257cf]]
Badness: 0.000920

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

=Tripod= 
Commas: 105/104, 144/143, 196/195

13-limit eigenmonzo subgroup: 2.9/7.13/11
15-limit eigenmonzo subgroup: 2.5/3.13/11

Map: [<1 0 0 -5 12 -8|, <0 1 0 2 -1 3|, <0 0 1 2 -3 3|]
EDOs: 9, 10, 19, 31, 41, 72f, [[81edo|81]], [[91edo|91]], [[122edo|122f]], [[163edo|163df]]
Badness: 0.000745

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

=Prodigy= 
Prodigy shrinks 1024/1029, 243/242, 384/385 and 2400/2401 down to the same tiny interval. Hence in practice it probably makes the most sense to temper this out as well, leading to [[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
Related linear temperament: [[Gamelismic clan|miracle]]

[[Minimax tuning]]:
[|1 0 0 0 0>, |13/12 1/2 -1/4 0 1/12>,
|13/6 -1 1/2 0 1/6>,
|3/2 -1 1/2 0 1/2>, |0 0 0 0 1>]
[[Eigenmonzo subgroup]]: 2.9/5.11

Lattice basis: secor length 0.9111, 3/2 length 0.9477
Angle(secor, 3/2) = 65.933
Map to lattice: [<0 0 -1 -2 -3|, <0 1 -1 0 3|]

Map: [<1 0 0 -5 -13|, <0 1 0 2 6|, <0 0 1 2 3|]
[[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

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

==Hobbit bases== 
{2, 3, 5} subgroup

31: 81/80, 34171875/33554432
41: 34171875/33554432, 32805/32768

==13-limit== 
Commas: 105/104, 196/195, 352/351

Map: [<1 0 0 -5 -13 -8|, <0 1 0 2 6 3|, <0 0 1 2 3 3|]
EDOs: 10, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233cdef, 274cdef, 305cdef
Badness: 0.000736

==Prodigious== 
Commas: 225/224, 441/440, 364/363

Map: [<1 0 0 -5 -13 -23|, <0 1 0 2 6 11|, <0 0 1 2 3 4|]
EDOs: 29, 41, 72, 113, 185cf, 341cf, 413bcf, 526bcdf
Badness: 0.000900

==Prodigal== 
Commas: 225/224, 441/440, 351/350

Map: [<1 0 0 -5 -13 -4|, <0 1 0 2 6 -1|, <0 0 1 2 3 4|]
EDOs: 31, 72, 103, 175f
Badness: 0.000889

=Mirage= 
Commas: 225/224, 243/242, 385/384

Map: [<1 1 3 3 2 0|, <0 6 -7 -2 15 0|, <0 0 0 0 0 1|]
EDOs: 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef
Badness: 0.000738

=Minerva= 
[[Comma|Commas]]: 99/98, 176/175
Related linear temperament: [[Semicomma family|orwell]]

[[Eigenmonzo subgroup]]: 2.7/5.11/9

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

Map: [<1 0 0 -5 -9|, <0 1 0 2 2|, <0 0 1 2 4|]
[[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

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

==Athene== 
Commas: 99/98, 176/175, 275/273

13-limit eigenmonzo subgroup: 2.11/9.13/7
15-limit eigenmonzo subgroup: 2.11/9.13/7

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

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

=Spectacle= 
[[Comma|Commas]]: 225/224, 243/242
Related linear temperament: [[Marvel temperaments|marvo]]

[[Minimax tuning]]:
[|1 0 0 0 0>, |1/5 0 0 0 2/5>, |2/5 -2 1 0 4/5>, |-19/5 -4 2 0 12/5>, |0 0 0 0 1>]
[[Eigenmonzo subgroup]]: 2.9/5.11

Map: [<1 1 0 -3 2|, <0 2 0 4 5|, <0 0 1 2 0|]
[[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

Scales: [[spectacle31]]

=Apollo= 
[[Comma|Commas]]: 100/99, 225/224
Related linear temperament: [[Magic family|Magic]]

Eigenmonzo subgroup: 2.7/5.11/9

Map: [<1 0 0 -5 2|, <0 1 0 2 -2|, <0 0 1 2 2|]
EDOs: 12, 19, 22, 41, [[104edo]], [[157edo|157ce]], [[198edo|198ce]], [[220edo|220ce]], [[261edo|261ce]]

Projection pairs: 7 225/32 11 100/9

=Potassium= 
Commas: 45/44, 56/55

Eigenmonzo subgroup: 2.9/7.11

Map: [<1 0 0 -5 -2|, <0 1 0 2 2|, <0 0 1 2 1|]
EDOs: 9, 10, 12, 19, 31e, 50e
Badness: 0.000464

Projection pairs: 7 225/32 11 45/4

==13-limit== 
Commas: 45/44, 56/55, 78/77

13-limit eigenmonzo subgroup: 2.9/7.13/9
15-limit eigenmonzo subgroup: 2.9/7.13/9

Map: [<1 0 0 -5 -2 -8|, <0 1 0 2 2 3|, <0 0 1 2 1 3|]
EDOs: 9, 10, 19, 31e, 50e
Badness: 0.000733

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

=Fantastic=
Commas: 225/224, 4375/4356

Map: [<2 0 0 -10 -7|, <0 1 0 2 0|, <0 0 1 2 3|]
EDOs: 12, 22, 50, 72, 166, 238c, 310c
Badness: 0.000743

=Catakleismoid= 
Commas: 225/224, 4375/4374

Map: [<1 0 1 -3 0|, <0 6 5 22 0|, <0 0 0 0 1|]
EDOs: 19, 53, 72, 197e, 269ce
Badness: 0.001275

==13-limit== 
Commas: 169/168, 225/224, 325/324

Map: [<1 0 1 -3 0 0|, <0 6 5 22 0 14|, <0 0 0 0 1 0|]
EDOs: 19, 53, 72, 125f, 197ef, 269cef
Badness: 0.000916

=Hestia= 
Commas: 225/224, 125000/124509

Map: [<1 0 0 -5 9|, <0 2 0 4 -7|, <0 0 1 2 0|]
EDOs: 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce
Badness: 0.00154

==13-limit== 
Commas: 169/168, 225/224, 1001/1000

Map: [<1 0 0 -5 9 -1|, <0 2 0 4 -7 3|, <0 0 1 2 0 1|]
EDOs: 19, 29, 43, 53, 72, 125f, 197ef, 269cef
Badness: 0.001062

=Malcolm= 
Commas: 225/224, 2200/2187

Map: [<1 0 0 -5 -3|, <0 1 0 2 7|, <0 0 1 2 -2|]
EDOs: 41, 53, 94, 229c, 248ce, 289ce, 342ce, 383ce
Badness: 0.001250

==13-limit== 
commas: 225/224, 275/273, 325/324

Map: [<1 0 0 -5 -3 2|, <0 1 0 2 7 4|, <0 0 1 2 -2 -2|]
EDOs: 41, 53, 94, 429cdef, 523cdef
Badness: 0.001075

Original HTML content:

<html><head><title>Marvel family</title></head><body><!-- ws:start:WikiTextTocRule:68:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><a href="#Marvel">Marvel</a><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --> | <a href="#Hecate">Hecate</a><!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --> | <a href="#Marvell">Marvell</a><!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --> | <a href="#Isis">Isis</a><!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --> | <a href="#Deecee">Deecee</a><!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --> | <a href="#Tripod">Tripod</a><!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --> | <a href="#Prodigy">Prodigy</a><!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --> | <a href="#Mirage">Mirage</a><!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --> | <a href="#Minerva">Minerva</a><!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --> | <a href="#Spectacle">Spectacle</a><!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --> | <a href="#Apollo">Apollo</a><!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --> | <a href="#Potassium">Potassium</a><!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --> | <a href="#Fantastic">Fantastic</a><!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --> | <a href="#Catakleismoid">Catakleismoid</a><!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --> | <a href="#Hestia">Hestia</a><!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --> | <a href="#Malcolm">Malcolm</a><!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: -->
<!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Marvel"></a><!-- ws:end:WikiTextHeadingRule:0 -->Marvel</h1>
 The head of the marvel family is marvel, which tempers out <a class="wiki_link" href="/225_224">225/224</a>, the <a class="wiki_link" href="/septimal%20kleisma">septimal kleisma</a> or <a class="wiki_link" href="/marvel%20comma">marvel comma</a>. Marvel has a <a class="wiki_link" href="/Normal%20lists">normal list basis</a> of [2, 3, 5]; hence a <a class="wiki_link" href="/Harmonic%20Limit">5-limit</a> scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by <a class="wiki_link" href="/197edo">197edo</a>.<br />
<br />
Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to <a class="wiki_link" href="/Kleismic%20family">catakleismic temperament</a>. Another temperament which does little damage to tuning accuracy is <a class="wiki_link" href="/Pythagorean%20family">compton temperament</a>, for which <a class="wiki_link" href="/240edo">240edo</a> may be used.<br />
<br />
See also <a class="wiki_link" href="/Marvel%20temperaments">Marvel temperaments</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Marvel-Vital statistics"></a><!-- ws:end:WikiTextHeadingRule:2 -->Vital statistics</h2>
 <a class="wiki_link" href="/Comma">Comma</a> c = 225/224<br />
Related linear temperament: <a class="wiki_link" href="/Kleismic%20family">catakleismic temperament</a><br />
<br />
7-limit minimax: 3 and 5 1/4c flat, 7 just<br />
[|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;]<br />
<a class="wiki_link" href="/Eigenmonzo%20subgroup">Eigenmonzo subgroup</a>: 2.5/3.7<br />
<br />
9-limit minimax: 3 1/6c flat, 5 1/3c flat, 7 just<br />
[|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;]<br />
<a class="wiki_link" href="/Eigenmonzo%20subgroup">Eigenmonzo subgroup</a>: 2.9/5.7<br />
<br />
Lattice basis: secor length 1.256, 3/2 length 1.369<br />
Angle(secor, 3/2) = 106.958 degrees<br />
Map to lattice: [&lt;0 0 -1 -2|, &lt;0 1 -1 0|]<br />
<br />
Map: [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]<br />
Generators: 2, 3, 5<br />
<a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/10edo">10</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/41edo">41</a>,<a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/197edo">197</a>, <a class="wiki_link" href="/269edo">269c</a><br />
Badness: 0.0000365<br />
<br />
<a class="wiki_link" href="/Projection%20pair">Projection pair</a>s: 7 225/32<br />
<a class="wiki_link" href="/Spectrum%20of%20a%20temperament">Spectrum</a>: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9<br />
<br />
Scales: <a class="wiki_link" href="/marvel9">marvel9</a>, <a class="wiki_link" href="/marvel10">marvel10</a>, <a class="wiki_link" href="/marvel11">marvel11</a>, <a class="wiki_link" href="/marvel12">marvel12</a>, <a class="wiki_link" href="/marvel19">marvel19</a>, <a class="wiki_link" href="/marvel22">marvel22</a>, <a class="wiki_link" href="/pump12_1">pump12_1</a>, <a class="wiki_link" href="/pump12_2">pump12_2</a>, <a class="wiki_link" href="/pump13">pump13</a>, <a class="wiki_link" href="/pump14">pump14</a>, <a class="wiki_link" href="/pump15">pump15</a>, <a class="wiki_link" href="/pump16">pump16</a>, <a class="wiki_link" href="/pump17">pump17</a>, <a class="wiki_link" href="/pump18">pump18</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Marvel-Minkowski blocks"></a><!-- ws:end:WikiTextHeadingRule:4 --><a class="wiki_link" href="/Minkowski%20blocks">Minkowski blocks</a></h2>
 {2, 3, 5} subgroup<br />
<br />
8: 16/15, 250/243<br />
9: 135/128, 128/125<br />
10: 25/24, 2048/2025<br />
11: 135/128, 2048/1875<br />
12: 2048/2025, 128/125<br />
15: 128/125, 32768/30375<br />
17: 25/24, 2278125/2097152<br />
19: 16875/16384, 81/80<br />
21: 128/125, 273375/262144<br />
22: 2048/2025, 3125/3072<br />
29: 16875/16384, 32805/32768<br />
31: 81/80, 34171875/33554432<br />
41: 34171875/33554432, 3125/3072<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Marvel-Music"></a><!-- ws:end:WikiTextHeadingRule:6 -->Music</h2>
 <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=314" rel="nofollow">Semimarvelous Blue Drawf</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/marvel10_2.mp3" rel="nofollow">notJoelTaylorsTreeSpirit-Marvel10-2</a> by <a class="wiki_link" href="/Billy%20Stiltner">Billy Stiltner</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Marvel-11-limit (Unimarv)"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit (Unimarv)</h2>
 <a class="wiki_link" href="/Comma">Commas</a>: 225/224, 385/384<br />
Related linear temperament: <a class="wiki_link" href="/Kleismic%20family">catakleismic temperament</a><br />
<br />
<a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br />
[|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;,<br />
|3 4/3 0 0 -2/3&gt;, |8/3 -2/9 -2/3 0 7/9&gt;]<br />
<a class="wiki_link" href="/Eigenmonzo%20subgroup">Eigenmonzo subgroup</a>: 2.9/5.11/9<br />
<br />
Lattice basis: secor length 1.0364 5/4 length 1.0759<br />
Angle(secor, 5/4) = 104.028 degrees<br />
Map to lattice: [&lt;0 -1 0 -2 1|, &lt;0 -1 1 0 -2|]<br />
<br />
Map: [&lt;1 0 0 -5 12|, &lt;0 1 0 2 -1|, &lt;0 0 1 2 -3|]<br />
<a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 5<br />
<a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/10edo">10</a>, <a class="wiki_link" href="/12edo">12e</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/166edo">166</a>, <a class="wiki_link" href="/197edo">197e</a>, <a class="wiki_link" href="/269edo">269ce</a>, <a class="wiki_link" href="/341edo">341ce</a><br />
Badness: 0.000255<br />
<br />
<a class="wiki_link" href="/Projection%20pair">Projection pair</a>s: 7 225/32 11 4096/375<br />
<a class="wiki_link" href="/Spectrum%20of%20a%20temperament">Spectrum</a>: 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<br />
Scales: <a class="wiki_link" href="/marvel22_11">marvel22_11</a>, <a class="wiki_link" href="/unimarv19">unimarv19</a>, <a class="wiki_link" href="/unimarv22">unimarv22</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Marvel-11-limit (Unimarv)-Hobbit bases"></a><!-- ws:end:WikiTextHeadingRule:10 -->Hobbit bases</h3>
 {2, 3, 5} subgroup<br />
<br />
12: 128/125, 2048/2025<br />
15: 128/125, 32768/30375<br />
19: 16875/16384, 81/80<br />
22: 2048/2025, 2109375/2097152<br />
31: 2109375/2097152, 81/80<br />
41: 3125/3072, 34171875/33554432<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Marvel-13-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit</h2>
 Commas: 225/224, 385/384, 351/350<br />
<br />
13-limit eigenmonzo subgroup: 2.11/9.13/9<br />
15-limit eigenmonzo subgroup: 2.15/11.15/13<br />
<br />
Map: [&lt;1 0 0 -5 12 -4|, &lt;0 1 0 2 -1 -1|, &lt;0 0 1 2 -3 4|]<br />
EDOs: 19, 22, 31, 50, 53, 72, 103, <a class="wiki_link" href="/175edo">175f</a>, <a class="wiki_link" href="/300edo">300cef</a>, <a class="wiki_link" href="/403bcef">403bcef</a><br />
Badness: 0.000690<br />
<br />
<a class="wiki_link" href="/Spectrum%20of%20a%20temperament">Spectrum</a>: 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<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Marvel-Other eleven limit children"></a><!-- ws:end:WikiTextHeadingRule:14 -->Other eleven limit children</h2>
 The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Hecate"></a><!-- ws:end:WikiTextHeadingRule:16 -->Hecate</h1>
 Commas: 225/224, 385/384, 325/324<br />
<br />
13-limit eigenmonzo subgroup: 2.7.13/5<br />
15-limit eigenmonzo subgroup: 2.7.15/13<br />
<br />
Map: [&lt;1 0 0 -5 12 2|, &lt;0 1 0 2 -1 4|, &lt;0 0 1 2 -3 -2|]<br />
EDOs: 19, 41, 53, 72, 113, 125f, 166, 238cf, 404cef<br />
Badness: 0.000721<br />
<br />
Projection pairs: 7 225/32 11 4096/375 13 324/25<br />
<a class="wiki_link" href="/Spectrum%20of%20a%20temperament">Spectrum</a>: 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<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Hecate-17-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->17-limit</h2>
 Commas: 225/224, 385/384, 325/324, 595/594<br />
<br />
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|]<br />
EDOs: 19, 41, 53g, 72, 113, 166g, 238cfg, 351cfg, 404cefg<br />
Badness: 0.000869<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Hecate-Enodia"></a><!-- ws:end:WikiTextHeadingRule:20 -->Enodia</h2>
 Commas: 225/224, 385/384, 325/324, 375/374<br />
<br />
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|]<br />
EDOs: 41g, 53, 72, 125f, 166g, 238cfg, 363cefg, 404cefg<br />
Badness: .000917<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Marvell"></a><!-- ws:end:WikiTextHeadingRule:22 -->Marvell</h1>
 Commas: 225/224, 385/384, 1573/1568<br />
<br />
13-limit eigenmonzo subgroup: 2.9/5.11/9<br />
15-limit eigenmonzo subgroup: 2.7.15/13<br />
<br />
Map: [&lt;1 0 0 -5 12 -29|, &lt;0 1 0 2 -1 6|, &lt;0 0 1 2 -3 10|]<br />
EDOs: 9, 31, <a class="wiki_link" href="/63edo">63</a>, 72, <a class="wiki_link" href="/103edo">103</a>, <a class="wiki_link" href="/166edo">166</a>, <a class="wiki_link" href="/238edo">238cf</a>, <a class="wiki_link" href="/269edo">269ce</a>, 507bcef, 610bcef<br />
Badness: 0.000862<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Isis"></a><!-- ws:end:WikiTextHeadingRule:24 -->Isis</h1>
 Commas: 225/224, 385/384, 275/273<br />
<br />
Map: [&lt;1 0 0 -5 12 17|, &lt;0 1 0 2 -1 4|, &lt;0 0 1 2 -3 -3|]<br />
EDOs: 10, 22, 31, 41, 53, 94<br />
Badness: 0.000866<br />
<br />
Projection pairs: 7 225/32 11 4096/375 13 131072/10125<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Deecee"></a><!-- ws:end:WikiTextHeadingRule:26 -->Deecee</h1>
 Commas: 225/224, 385/384, 364/363<br />
<br />
13-limit eigenmonzo subgroup: 2.9/5.13/9<br />
15-limit eigenmonzo subgroup: 2.3.13/5<br />
<br />
Map: [&lt;1 0 0 -5 12 27|, &lt;0 1 0 2 -1 -3|, &lt;0 0 1 2 -3 -8|]<br />
EDOs: 9, 22, 41, 63, 72, <a class="wiki_link" href="/185edo">185cf</a>, <a class="wiki_link" href="/257edo">257cf</a><br />
Badness: 0.000920<br />
<br />
Projection pairs: 7 225/32 11 4096/375 13 134217728/10546875<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Tripod"></a><!-- ws:end:WikiTextHeadingRule:28 -->Tripod</h1>
 Commas: 105/104, 144/143, 196/195<br />
<br />
13-limit eigenmonzo subgroup: 2.9/7.13/11<br />
15-limit eigenmonzo subgroup: 2.5/3.13/11<br />
<br />
Map: [&lt;1 0 0 -5 12 -8|, &lt;0 1 0 2 -1 3|, &lt;0 0 1 2 -3 3|]<br />
EDOs: 9, 10, 19, 31, 41, 72f, <a class="wiki_link" href="/81edo">81</a>, <a class="wiki_link" href="/91edo">91</a>, <a class="wiki_link" href="/122edo">122f</a>, <a class="wiki_link" href="/163edo">163df</a><br />
Badness: 0.000745<br />
<br />
Projection pairs: 7 225/32 11 4096/375 13 3375/256<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="Prodigy"></a><!-- ws:end:WikiTextHeadingRule:30 -->Prodigy</h1>
 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 <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a>. 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.<br />
<br />
<a class="wiki_link" href="/Comma">Commas</a>: 225/224, 441/440<br />
Related linear temperament: <a class="wiki_link" href="/Gamelismic%20clan">miracle</a><br />
<br />
<a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br />
[|1 0 0 0 0&gt;, |13/12 1/2 -1/4 0 1/12&gt;,<br />
|13/6 -1 1/2 0 1/6&gt;,<br />
|3/2 -1 1/2 0 1/2&gt;, |0 0 0 0 1&gt;]<br />
<a class="wiki_link" href="/Eigenmonzo%20subgroup">Eigenmonzo subgroup</a>: 2.9/5.11<br />
<br />
Lattice basis: secor length 0.9111, 3/2 length 0.9477<br />
Angle(secor, 3/2) = 65.933<br />
Map to lattice: [&lt;0 0 -1 -2 -3|, &lt;0 1 -1 0 3|]<br />
<br />
Map: [&lt;1 0 0 -5 -13|, &lt;0 1 0 2 6|, &lt;0 0 1 2 3|]<br />
<a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 5<br />
<a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/10edo">10</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/29edo">29</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/247edo">247c</a>, <a class="wiki_link" href="/319edo">319bcde</a>, <a class="wiki_link" href="/391edo">391bcde</a>, <a class="wiki_link" href="/463edo">463bcde</a><br />
Badness: 0.000344<br />
<br />
<a class="wiki_link" href="/Projection%20pair">Projection pair</a>s: 7 225/32 11 91125/8192<br />
<br />
Scales: <a class="wiki_link" href="/prodigy11">prodigy11</a>, <a class="wiki_link" href="/prodigy12">prodigy12</a>, <a class="wiki_link" href="/prodigy29">prodigy29</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Prodigy-Hobbit bases"></a><!-- ws:end:WikiTextHeadingRule:32 -->Hobbit bases</h2>
 {2, 3, 5} subgroup<br />
<br />
31: 81/80, 34171875/33554432<br />
41: 34171875/33554432, 32805/32768<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Prodigy-13-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->13-limit</h2>
 Commas: 105/104, 196/195, 352/351<br />
<br />
Map: [&lt;1 0 0 -5 -13 -8|, &lt;0 1 0 2 6 3|, &lt;0 0 1 2 3 3|]<br />
EDOs: 10, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233cdef, 274cdef, 305cdef<br />
Badness: 0.000736<br />
<br />
<!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Prodigy-Prodigious"></a><!-- ws:end:WikiTextHeadingRule:36 -->Prodigious</h2>
 Commas: 225/224, 441/440, 364/363<br />
<br />
Map: [&lt;1 0 0 -5 -13 -23|, &lt;0 1 0 2 6 11|, &lt;0 0 1 2 3 4|]<br />
EDOs: 29, 41, 72, 113, 185cf, 341cf, 413bcf, 526bcdf<br />
Badness: 0.000900<br />
<br />
<!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Prodigy-Prodigal"></a><!-- ws:end:WikiTextHeadingRule:38 -->Prodigal</h2>
 Commas: 225/224, 441/440, 351/350<br />
<br />
Map: [&lt;1 0 0 -5 -13 -4|, &lt;0 1 0 2 6 -1|, &lt;0 0 1 2 3 4|]<br />
EDOs: 31, 72, 103, 175f<br />
Badness: 0.000889<br />
<br />
<!-- ws:start:WikiTextHeadingRule:40:&lt;h1&gt; --><h1 id="toc20"><a name="Mirage"></a><!-- ws:end:WikiTextHeadingRule:40 -->Mirage</h1>
 Commas: 225/224, 243/242, 385/384<br />
<br />
Map: [&lt;1 1 3 3 2 0|, &lt;0 6 -7 -2 15 0|, &lt;0 0 0 0 0 1|]<br />
EDOs: 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef<br />
Badness: 0.000738<br />
<br />
<!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc21"><a name="Minerva"></a><!-- ws:end:WikiTextHeadingRule:42 -->Minerva</h1>
 <a class="wiki_link" href="/Comma">Commas</a>: 99/98, 176/175<br />
Related linear temperament: <a class="wiki_link" href="/Semicomma%20family">orwell</a><br />
<br />
<a class="wiki_link" href="/Eigenmonzo%20subgroup">Eigenmonzo subgroup</a>: 2.7/5.11/9<br />
<br />
Lattice basis: 16/15 length 0.8997 5/4 length 1.0457<br />
Angle(16/15, 5/4) = 98.6044 degrees<br />
Map to lattice: [&lt;0 -1 0 -2 -2|, &lt;0 -1 1 0 2|]<br />
<br />
Map: [&lt;1 0 0 -5 -9|, &lt;0 1 0 2 2|, &lt;0 0 1 2 4|]<br />
<a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 5<br />
<a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/9edo">9</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/21edo">21</a>, <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/43edo">43</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/74edo">74</a>, <a class="wiki_link" href="/75edo">75</a>, <a class="wiki_link" href="/96edo">96</a>, <a class="wiki_link" href="/127edo">127</a><br />
Badness: 0.000381<br />
<br />
<a class="wiki_link" href="/Projection%20pair">Projection pair</a>s: 7 225/32 11 5625/512<br />
<br />
Scales: <a class="wiki_link" href="/minerva12">minerva12</a>, <a class="wiki_link" href="/minerva22x">minerva22x</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:44:&lt;h2&gt; --><h2 id="toc22"><a name="Minerva-Athene"></a><!-- ws:end:WikiTextHeadingRule:44 -->Athene</h2>
 Commas: 99/98, 176/175, 275/273<br />
<br />
13-limit eigenmonzo subgroup: 2.11/9.13/7<br />
15-limit eigenmonzo subgroup: 2.11/9.13/7<br />
<br />
Map: [&lt;1 0 0 -5 -9 -4|, &lt;0 1 0 2 2 -1|, &lt;0 0 1 2 4 4|]<br />
EDOs: 22, 31, 53, 84e, 118d, 149df, 171de, 202def<br />
Badness: 0.000818<br />
<br />
<a class="wiki_link" href="/Projection%20pair">Projection pair</a>s: 7 225/32 11 5625/512 13 625/48<br />
<br />
<!-- ws:start:WikiTextHeadingRule:46:&lt;h1&gt; --><h1 id="toc23"><a name="Spectacle"></a><!-- ws:end:WikiTextHeadingRule:46 -->Spectacle</h1>
 <a class="wiki_link" href="/Comma">Commas</a>: 225/224, 243/242<br />
Related linear temperament: <a class="wiki_link" href="/Marvel%20temperaments">marvo</a><br />
<br />
<a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br />
[|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;]<br />
<a class="wiki_link" href="/Eigenmonzo%20subgroup">Eigenmonzo subgroup</a>: 2.9/5.11<br />
<br />
Map: [&lt;1 1 0 -3 2|, &lt;0 2 0 4 5|, &lt;0 0 1 2 0|]<br />
<a class="wiki_link" href="/Generator">Generators</a>: 2, 11/9, 5<br />
<a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/10edo">10</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/240edo">240</a>, <a class="wiki_link" href="/259edo">259b</a>, <a class="wiki_link" href="/269edo">269ce</a>, <a class="wiki_link" href="/310edo">310c</a>, <a class="wiki_link" href="/331edo">331bc</a>, <a class="wiki_link" href="/353edo">353c</a>, <a class="wiki_link" href="/497edo">497bc</a>, <a class="wiki_link" href="/569edo">569bc</a><br />
Badness: 0.000499<br />
<br />
Projection pairs: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9<br />
<br />
Scales: <a class="wiki_link" href="/spectacle31">spectacle31</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:48:&lt;h1&gt; --><h1 id="toc24"><a name="Apollo"></a><!-- ws:end:WikiTextHeadingRule:48 -->Apollo</h1>
 <a class="wiki_link" href="/Comma">Commas</a>: 100/99, 225/224<br />
Related linear temperament: <a class="wiki_link" href="/Magic%20family">Magic</a><br />
<br />
Eigenmonzo subgroup: 2.7/5.11/9<br />
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Map: [&lt;1 0 0 -5 2|, &lt;0 1 0 2 -2|, &lt;0 0 1 2 2|]<br />
EDOs: 12, 19, 22, 41, <a class="wiki_link" href="/104edo">104edo</a>, <a class="wiki_link" href="/157edo">157ce</a>, <a class="wiki_link" href="/198edo">198ce</a>, <a class="wiki_link" href="/220edo">220ce</a>, <a class="wiki_link" href="/261edo">261ce</a><br />
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Projection pairs: 7 225/32 11 100/9<br />
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<!-- ws:start:WikiTextHeadingRule:50:&lt;h1&gt; --><h1 id="toc25"><a name="Potassium"></a><!-- ws:end:WikiTextHeadingRule:50 -->Potassium</h1>
 Commas: 45/44, 56/55<br />
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Eigenmonzo subgroup: 2.9/7.11<br />
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Map: [&lt;1 0 0 -5 -2|, &lt;0 1 0 2 2|, &lt;0 0 1 2 1|]<br />
EDOs: 9, 10, 12, 19, 31e, 50e<br />
Badness: 0.000464<br />
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Projection pairs: 7 225/32 11 45/4<br />
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<!-- ws:start:WikiTextHeadingRule:52:&lt;h2&gt; --><h2 id="toc26"><a name="Potassium-13-limit"></a><!-- ws:end:WikiTextHeadingRule:52 -->13-limit</h2>
 Commas: 45/44, 56/55, 78/77<br />
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13-limit eigenmonzo subgroup: 2.9/7.13/9<br />
15-limit eigenmonzo subgroup: 2.9/7.13/9<br />
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Map: [&lt;1 0 0 -5 -2 -8|, &lt;0 1 0 2 2 3|, &lt;0 0 1 2 1 3|]<br />
EDOs: 9, 10, 19, 31e, 50e<br />
Badness: 0.000733<br />
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Projection pairs: 7 225/32 11 45/4 13 3375/256<br />
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<!-- ws:start:WikiTextHeadingRule:54:&lt;h1&gt; --><h1 id="toc27"><a name="Fantastic"></a><!-- ws:end:WikiTextHeadingRule:54 -->Fantastic</h1>
Commas: 225/224, 4375/4356<br />
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Map: [&lt;2 0 0 -10 -7|, &lt;0 1 0 2 0|, &lt;0 0 1 2 3|]<br />
EDOs: 12, 22, 50, 72, 166, 238c, 310c<br />
Badness: 0.000743<br />
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<!-- ws:start:WikiTextHeadingRule:56:&lt;h1&gt; --><h1 id="toc28"><a name="Catakleismoid"></a><!-- ws:end:WikiTextHeadingRule:56 -->Catakleismoid</h1>
 Commas: 225/224, 4375/4374<br />
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Map: [&lt;1 0 1 -3 0|, &lt;0 6 5 22 0|, &lt;0 0 0 0 1|]<br />
EDOs: 19, 53, 72, 197e, 269ce<br />
Badness: 0.001275<br />
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<!-- ws:start:WikiTextHeadingRule:58:&lt;h2&gt; --><h2 id="toc29"><a name="Catakleismoid-13-limit"></a><!-- ws:end:WikiTextHeadingRule:58 -->13-limit</h2>
 Commas: 169/168, 225/224, 325/324<br />
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Map: [&lt;1 0 1 -3 0 0|, &lt;0 6 5 22 0 14|, &lt;0 0 0 0 1 0|]<br />
EDOs: 19, 53, 72, 125f, 197ef, 269cef<br />
Badness: 0.000916<br />
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<!-- ws:start:WikiTextHeadingRule:60:&lt;h1&gt; --><h1 id="toc30"><a name="Hestia"></a><!-- ws:end:WikiTextHeadingRule:60 -->Hestia</h1>
 Commas: 225/224, 125000/124509<br />
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Map: [&lt;1 0 0 -5 9|, &lt;0 2 0 4 -7|, &lt;0 0 1 2 0|]<br />
EDOs: 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce<br />
Badness: 0.00154<br />
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<!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><h2 id="toc31"><a name="Hestia-13-limit"></a><!-- ws:end:WikiTextHeadingRule:62 -->13-limit</h2>
 Commas: 169/168, 225/224, 1001/1000<br />
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Map: [&lt;1 0 0 -5 9 -1|, &lt;0 2 0 4 -7 3|, &lt;0 0 1 2 0 1|]<br />
EDOs: 19, 29, 43, 53, 72, 125f, 197ef, 269cef<br />
Badness: 0.001062<br />
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<!-- ws:start:WikiTextHeadingRule:64:&lt;h1&gt; --><h1 id="toc32"><a name="Malcolm"></a><!-- ws:end:WikiTextHeadingRule:64 -->Malcolm</h1>
 Commas: 225/224, 2200/2187<br />
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Map: [&lt;1 0 0 -5 -3|, &lt;0 1 0 2 7|, &lt;0 0 1 2 -2|]<br />
EDOs: 41, 53, 94, 229c, 248ce, 289ce, 342ce, 383ce<br />
Badness: 0.001250<br />
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<!-- ws:start:WikiTextHeadingRule:66:&lt;h2&gt; --><h2 id="toc33"><a name="Malcolm-13-limit"></a><!-- ws:end:WikiTextHeadingRule:66 -->13-limit</h2>
 commas: 225/224, 275/273, 325/324<br />
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Map: [&lt;1 0 0 -5 -3 2|, &lt;0 1 0 2 7 4|, &lt;0 0 1 2 -2 -2|]<br />
EDOs: 41, 53, 94, 429cdef, 523cdef<br />
Badness: 0.001075</body></html>