Marvel family
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The head of the marvel family is marvel, which tempers out 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.
===Vital statistics===
[[Comma]] c = 225/224
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|Eigenmonzos]]: 2, 5/4, 6/5
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|Eigenmonzos]]: 2, 8/7, 10/9
Lattice basis: secor length 1.256, 3/2 length 1.369
Angle(secor, 3/2) = 106.958 cents
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]]: 9, 10, 11, 12, 19, 22, 29, 31, 41, 50, 53, 72, 84, 94, 125, [[197edo|197]], 228
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]]
==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.
===Unidecimal marvel===
[[Comma|Commas]]: 225/224, 385/384
[[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|Eigenmonzos]]: 2, 10/9, 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]]: [[19edo|19]], [[22edo|22]], [[31edo|31]], [[41edo|41]], [[50edo|50]], [[53edo|53]], [[72edo|72]], [[166edo|166]], [[197edo|197]]
Scales: [[marvel22_11]], [[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
====Marvell====
Commas: 225/224, 385/384, 1573/1568
Map: [<1 0 0 -5 12 -29|, <0 1 0 2 -1 6|, <0 0 1 2 -3 10|]
EDOs: 9, 31, 63, 72, 103, 166
====Deecee====
Commas: 225/224, 385/384, 364/363
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
===Prodigy===
[[Comma|Commas]]: 225/224, 441/440
[[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|Eigenmonzos]]: 2, 10/9, 11/8
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]]
Scales: [[prodigy29]]
===Hobbit bases===
{2, 3, 5} subgroup
31: 81/80, 34171875/33554432
41: 34171875/33554432, 32805/32768
===Minerva===
[[Comma|Commas]]: 99/98, 225/224
[[Minimax tuning]]:
[|1 0 0 0 0>, |0 1 0 0 0>, |9/4 -1/2 0 0 1/4>, |-1/2 1 0 0 1/2>, |0 0 0 0 1>]
[[Eigenmonzo|Eigenmonzos]]: 2, 3/2, 11/8
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]]
Scales: [[minerva22x]]
===Spectacle===
[[Comma|Commas]]: 225/224, 243/242
[[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|Eigenmonzos]]: 2, 10/9, 11/8
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]], [[250edo|250]]
Scales: [[spectacle31]]Original HTML content:
<html><head><title>Marvel family</title></head><body>The head of the marvel family is marvel, which tempers out 225/224, the septimal kleisma or marvel comma. 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 />
<!-- ws:start:WikiTextHeadingRule:0:<h3> --><h3 id="toc0"><a name="x--Vital statistics"></a><!-- ws:end:WikiTextHeadingRule:0 -->Vital statistics</h3>
<a class="wiki_link" href="/Comma">Comma</a> c = 225/224<br />
<br />
7-limit minimax: 3 and 5 1/4c flat, 7 just<br />
[|1 0 0 0>, |5/4 1/2 -1/2 1/4>, |5/4 -1/2 1/2 1/4>, |0 0 0 1>]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 5/4, 6/5<br />
<br />
9-limit minimax: 3 1/6c flat, 5 1/3c flat, 7 just<br />
[|1 0 0 0>, |5/6 2/3 -1/3 1/6>, |5/3 -2/3 1/3 1/3>, |0 0 0 1>]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 8/7, 10/9<br />
<br />
Lattice basis: secor length 1.256, 3/2 length 1.369<br />
Angle(secor, 3/2) = 106.958 cents <br />
Map to lattice: [<0 0 -1 -2|, <0 1 -1 0|]<br />
<br />
Map: [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]<br />
Generators: 2, 3, 5<br />
<a class="wiki_link" href="/edo">EDOs</a>: 9, 10, 11, 12, 19, 22, 29, 31, 41, 50, 53, 72, 84, 94, 125, <a class="wiki_link" href="/197edo">197</a>, 228<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:2:<h3> --><h3 id="toc1"><a name="x--Minkowski blocks"></a><!-- ws:end:WikiTextHeadingRule:2 --><a class="wiki_link" href="/Minkowski%20blocks">Minkowski blocks</a></h3>
{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:4:<h3> --><h3 id="toc2"><a name="x--Music"></a><!-- ws:end:WikiTextHeadingRule:4 -->Music</h3>
<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 />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x-Eleven limit children"></a><!-- ws:end:WikiTextHeadingRule:6 -->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:8:<h3> --><h3 id="toc4"><a name="x-Eleven limit children-Unidecimal marvel"></a><!-- ws:end:WikiTextHeadingRule:8 -->Unidecimal marvel</h3>
<a class="wiki_link" href="/Comma">Commas</a>: 225/224, 385/384<br />
<br />
<a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>: <br />
[|1 0 0 0 0>, |4/3 8/9 -1/3 0 -1/9>, |8/3 -2/9 1/3 0 -2/9>,<br />
|3 4/3 0 0 -2/3>, |8/3 -2/9 -2/3 0 7/9>]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 10/9, 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: [<0 -1 0 -2 1|, <0 -1 1 0 -2|]<br />
<br />
Map: [<1 0 0 -5 12|, <0 1 0 2 -1|, <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="/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="/50edo">50</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">197</a><br />
Scales: <a class="wiki_link" href="/marvel22_11">marvel22_11</a>, <a class="wiki_link" href="/unimarv22">unimarv22</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="x-Eleven limit children-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:<h4> --><h4 id="toc6"><a name="x-Eleven limit children-Hobbit bases-Marvell"></a><!-- ws:end:WikiTextHeadingRule:12 -->Marvell</h4>
Commas: 225/224, 385/384, 1573/1568<br />
<br />
Map: [<1 0 0 -5 12 -29|, <0 1 0 2 -1 6|, <0 0 1 2 -3 10|]<br />
EDOs: 9, 31, 63, 72, 103, 166<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:<h4> --><h4 id="toc7"><a name="x-Eleven limit children-Hobbit bases-Deecee"></a><!-- ws:end:WikiTextHeadingRule:14 -->Deecee</h4>
<br />
Commas: 225/224, 385/384, 364/363<br />
<br />
Map: [<1 0 0 -5 12 27|, <0 1 0 2 -1 -3|, <0 0 1 2 -3 -8|]<br />
EDOs: 9, 22, 41, 63, 72<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="x-Eleven limit children-Prodigy"></a><!-- ws:end:WikiTextHeadingRule:16 -->Prodigy</h3>
<a class="wiki_link" href="/Comma">Commas</a>: 225/224, 441/440<br />
<br />
<a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br />
[|1 0 0 0 0>, |13/12 1/2 -1/4 0 1/12>, <br />
|13/6 -1 1/2 0 1/6>,<br />
|3/2 -1 1/2 0 1/2>, |0 0 0 0 1>]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 10/9, 11/8<br />
<br />
Lattice basis: secor length 0.9111, 3/2 length 0.9477<br />
Angle(secor, 3/2) = 65.933<br />
Map to lattice: [<0 0 -1 -2 -3|, <0 1 -1 0 3|]<br />
<br />
Map: [<1 0 0 -5 -13|, <0 1 0 2 6|, <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><br />
Scales: <a class="wiki_link" href="/prodigy29">prodigy29</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:<h3> --><h3 id="toc9"><a name="x-Eleven limit children-Hobbit bases"></a><!-- ws:end:WikiTextHeadingRule:18 -->Hobbit bases</h3>
{2, 3, 5} subgroup<br />
<br />
31: 81/80, 34171875/33554432<br />
41: 34171875/33554432, 32805/32768<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:<h3> --><h3 id="toc10"><a name="x-Eleven limit children-Minerva"></a><!-- ws:end:WikiTextHeadingRule:20 -->Minerva</h3>
<a class="wiki_link" href="/Comma">Commas</a>: 99/98, 225/224<br />
<br />
<a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br />
[|1 0 0 0 0>, |0 1 0 0 0>, |9/4 -1/2 0 0 1/4>, |-1/2 1 0 0 1/2>, |0 0 0 0 1>]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 3/2, 11/8<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: [<0 -1 0 -2 -2|, <0 -1 1 0 2|]<br />
<br />
Map: [<1 0 0 -5 -9|, <0 1 0 2 2|, <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 />
Scales: <a class="wiki_link" href="/minerva22x">minerva22x</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:<h3> --><h3 id="toc11"><a name="x-Eleven limit children-Spectacle"></a><!-- ws:end:WikiTextHeadingRule:22 -->Spectacle</h3>
<a class="wiki_link" href="/Comma">Commas</a>: 225/224, 243/242<br />
<br />
<a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br />
[|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>]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 10/9, 11/8<br />
<br />
Map: [<1 1 0 -3 2|, <0 2 0 4 5|, <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="/250edo">250</a><br />
Scales: <a class="wiki_link" href="/spectacle31">spectacle31</a></body></html>