Marvel family: Difference between revisions

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Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[catakleismic temperament]]. Another temperament which does little damage to tuning accuracy is [[compton temperament]], for which [[240edo]] may be used. See [[marvel temperaments]] for some other rank-2 temperaments.  
Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to [[catakleismic temperament]]. Another temperament which does little damage to tuning accuracy is [[compton temperament]], for which [[240edo]] may be used. See [[marvel temperaments]] for some other rank-2 temperaments.  
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 700.4075, ~5/4 = 383.6376
EDO generators: [[19edo|(11, 6)\19]], [[31edo|(18, 10)\31]], [[41edo|(24, 13)\41]]
Scales (Scala files): [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]]
[[Associated temperament]]: [[catakleismic]]
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Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 43: Line 29:
: secor length = 1.256, 3/2 length = 1.369
: secor length = 1.256, 3/2 length = 1.369
: Angle (secor, 3/2) = 106.958 degrees
: Angle (secor, 3/2) = 106.958 degrees
[[POTE generator]]s: ~3/2 = 700.4075, ~5/4 = 383.6376


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7-odd-limit: 3 and 5 1/4c flat, 7 just
* [[7-odd-limit]]: 3 and 5 1/4c flat, 7 just
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/4 1/2 -1/2 1/4 }}, {{monzo| 5/4 -1/2 1/2 1/4 }}, {{monzo| 0 0 0 1 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/4 1/2 -1/2 1/4 }}, {{monzo| 5/4 -1/2 1/2 1/4 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo subgroup]]: 2.5/3.7
: [[Eigenmonzo subgroup]]: 2.5/3.7
* 9-odd-limit: 3 1/6c flat, 5 1/3c flat, 7 just
* [[9-odd-limit]]: 3 1/6c flat, 5 1/3c flat, 7 just
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/6 2/3 -1/3 1/6 }}, {{monzo| 5/3 -2/3 1/3 1/3 }}, {{monzo| 0 0 0 1 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/6 2/3 -1/3 1/6 }}, {{monzo| 5/3 -2/3 1/3 1/3 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo subgroup]]: 2.9/5.7
: [[Eigenmonzo subgroup]]: 2.9/5.7
Line 60: Line 48:
[[Complexity spectrum]]: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9
[[Complexity spectrum]]: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9


</div></div>
[[Associated temperament]]: [[catakleismic]]
 
Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]]


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= 11-limit (unimarv) =
= 11-limit (unimarv) =
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 700.3887, ~5/4 = 383.5403
EDO generators: [[22edo|(13, 7)\22]], [[31edo|(18, 10)\31]], [[41edo|(24, 13)\41]]
Scales (Scala files): [[marvel22_11]], [[unimarv19]], [[unimarv22]]
[[Associated temperament]]: [[catakleismic]]
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Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 117: Line 93:
: secor length = 1.0364, 5/4 length = 1.0759
: secor length = 1.0364, 5/4 length = 1.0759
: Angle (secor, 5/4) = 104.028 degrees
: Angle (secor, 5/4) = 104.028 degrees
[[POTE generator]]s: ~3/2 = 700.3887, ~5/4 = 383.5403


[[Minimax tuning]]:
[[Minimax tuning]]:
Line 131: Line 109:
[[Complexity spectrum]]: 5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11
[[Complexity spectrum]]: 5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11


</div></div>
[[Associated temperament]]: [[catakleismic]]
 
Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]


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== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 699.7367, ~5/4 = 384.0613
EDO generators: [[31edo|(18, 10)\31]], [[53edo|(31, 17)\53]], [[72edo|(42, 23)\72]]
Scales (Scala files):
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Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 0 0 -5 12 -4 }}, {{val| 0 1 0 2 -1 -1 }}, {{val| 0 0 1 2 -3 4 }}]
Mapping: [{{val| 1 0 0 -5 12 -4 }}, {{val| 0 1 0 2 -1 -1 }}, {{val| 0 0 1 2 -3 4 }}]
POTE generators: ~3/2 = 699.7367, ~5/4 = 384.0613


Minimax tuning:  
Minimax tuning:  
Line 175: Line 145:
Badness: 0.690 × 10<sup>-3</sup>
Badness: 0.690 × 10<sup>-3</sup>


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


== Hecate ==
== Hecate ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 700.9779, ~5/4 = 383.1622
EDO generators: [[41edo|(24, 13)\41]], [[53edo|(31, 17)\53]], [[72edo|(42, 23)\72]]
Scales (Scala files):
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Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 0 0 -5 12 2 }}, {{val| 0 1 0 2 -1 4 }}, {{val| 0 0 1 2 -3 -2 }}]
Mapping: [{{val| 1 0 0 -5 12 2 }}, {{val| 0 1 0 2 -1 4 }}, {{val| 0 0 1 2 -3 -2 }}]
POTE generators: ~3/2 = 700.9779, ~5/4 = 383.1622


Minimax tuning:  
Minimax tuning:  
Line 209: Line 167:
Projection pairs: 7 225/32 11 4096/375 13 324/25
Projection pairs: 7 225/32 11 4096/375 13 324/25


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


=== 17-limit ===
=== 17-limit ===
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 700.9619, ~5/4 = 383.0310
EDO generators: [[22edo|(13, 7)\22]], [[41edo|(24, 13)\41]], [[72edo|(42, 23)\72]]
Scales (Scala files):
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Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


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


Mapping: [{{val| 1 0 0 -5 12 2 18 }}, {{val| 0 1 0 2 -1 4 0 }}, {{val| 0 0 1 2 -3 -2 -6 }}]
Mapping: [{{val| 1 0 0 -5 12 2 18 }}, {{val| 0 1 0 2 -1 4 0 }}, {{val| 0 0 1 2 -3 -2 -6 }}]
POTE generators: ~3/2 = 700.9619, ~5/4 = 383.0310


{{Val list|legend=1| 19, 22f, 31fg, 41, 53g, 72, 166g, 238cfg, 404ccefgg }}
{{Val list|legend=1| 19, 22f, 31fg, 41, 53g, 72, 166g, 238cfg, 404ccefgg }}
Line 240: Line 186:


=== Enodia ===
=== Enodia ===
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 700.9658, ~5/4 = 383.3063
EDO generators: [[22edo|(13, 7)\22]], [[53edo|(31, 17)\53]], [[72edo|(42, 23)\72]]
Scales (Scala files):
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Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


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


Mapping: [{{val| 1 0 0 -5 12 2 18 }}, {{val| 0 1 0 2 -1 4 0 }}, {{val| 0 0 1 2 -3 -2 6 }}]
Mapping: [{{val| 1 0 0 -5 12 2 18 }}, {{val| 0 1 0 2 -1 4 0 }}, {{val| 0 0 1 2 -3 -2 6 }}]
POTE generators: ~3/2 = 700.9658, ~5/4 = 383.3063


{{Val list|legend=1| 19g, 22f, 31f, 41g, 53, 72, 166g, 238cfg, 404ccefgg }}
{{Val list|legend=1| 19g, 22f, 31f, 41g, 53, 72, 166g, 238cfg, 404ccefgg }}


Badness: 0.917 × 10<sup>-3</sup>
Badness: 0.917 × 10<sup>-3</sup>
</div></div>


== Marvell ==
== Marvell ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 700.3937, ~5/4 = 383.5725
EDO generators: [[9edo|(5, 3)\9]], [[31edo|(18, 10)\31]], [[72edo|(42, 23)\72]]
Scales (Scala files):
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Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 284: Line 206:


Mapping: [{{val| 1 0 0 -5 12 -29 }}, {{val| 0 1 0 2 -1 6 }}, {{val| 0 0 1 2 -3 10 }}]
Mapping: [{{val| 1 0 0 -5 12 -29 }}, {{val| 0 1 0 2 -1 6 }}, {{val| 0 0 1 2 -3 10 }}]
POTE generators: ~3/2 = 700.3937, ~5/4 = 383.5725


Minimax tuning:  
Minimax tuning:  
Line 296: Line 220:


== Isis ==
== Isis ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 701.9156, ~5/4 = 383.2445
EDO generators: [[22edo|(13, 7)\22]], [[31edo|(18, 10)\31]], [[41edo|(24, 13)\41]]
Scales (Scala files):
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Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 0 0 -5 12 17 }}, {{val| 0 1 0 2 -1 4 }}, {{val| 0 0 1 2 -3 -3 }}]
Mapping: [{{val| 1 0 0 -5 12 17 }}, {{val| 0 1 0 2 -1 4 }}, {{val| 0 0 1 2 -3 -3 }}]
POTE generators: ~3/2 = 701.9156, ~5/4 = 383.2445


{{Val list|legend=1| 10, 19f, 22, 31, 41, 53, 94 }}
{{Val list|legend=1| 10, 19f, 22, 31, 41, 53, 94 }}
Line 320: Line 234:


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


== Deecee ==
== Deecee ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 700.4560, ~5/4 = 382.8177
EDO generators: [[9edo|(5, 3)\9]], [[41edo|(24, 13)\41]], [[72edo|(42, 23)\72]]
Scales (Scala files):
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Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 385/384, 364/363
Comma list: 225/224, 364/363, 385/384


Mapping: [{{val| 1 0 0 -5 12 27 }}, {{val| 0 1 0 2 -1 -3 }}, {{val| 0 0 1 2 -3 -8 }}]
Mapping: [{{val| 1 0 0 -5 12 27 }}, {{val| 0 1 0 2 -1 -3 }}, {{val| 0 0 1 2 -3 -8 }}]
POTE generators: ~3/2 = 700.4560, ~5/4 = 382.8177


Minimax tuning:  
Minimax tuning:  
Line 352: Line 254:


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


== Tripod ==
== Tripod ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 699.2335, ~5/4 = 382.9775
EDO generators: [[9edo|(5, 3)\9]], [[19edo|(11, 6)\19]], [[31edo|(18, 10)\31]]
Scales (Scala files):
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Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 374: Line 262:


Mapping: [{{val| 1 0 0 -5 12 -8 }}, {{val| 0 1 0 2 -1 3 }}, {{val| 0 0 1 2 -3 3 }}]
Mapping: [{{val| 1 0 0 -5 12 -8 }}, {{val| 0 1 0 2 -1 3 }}, {{val| 0 0 1 2 -3 3 }}]
POTE generators: ~3/2 = 699.2335, ~5/4 = 382.9775


Minimax tuning:  
Minimax tuning:  
Line 384: Line 274:


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


== Marvelcat ==
== Marvelcat ==
Period: 1\1
Optimal ([[POTE]]) generators: ~15/13 = 249.7138, ~5/4 = 383.5816
EDO generators: [[9edo|(5, 3)\9]], [[19edo|(11, 6)\19]], [[53edo|(31, 17)\53]]
Scales (Scala files):
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Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 408: Line 284:


Mapping generators: ~2, ~26/15, ~5
Mapping generators: ~2, ~26/15, ~5
POTE generators: ~15/13 = 249.7138, ~5/4 = 383.5816


{{Val list|legend=1| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}
{{Val list|legend=1| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}


Badness: 0.9997 × 10<sup>-3</sup>
Badness: 0.9997 × 10<sup>-3</sup>
</div></div>


= Minerva =
= Minerva =
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 700.2593, ~5/4 = 386.5581
EDO generators: [[12edo|(7, 4)\12]], [[22edo|(13, 7)\22]], [[31edo|(18, 10)\31]]
Scales (Scala files): [[minerva12]], [[minerva22x]]
[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]
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Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 443: Line 305:
: Angle (16/15, 5/4) = 98.6044 degrees
: Angle (16/15, 5/4) = 98.6044 degrees


[[Minimax tuning]]: 11-odd-limit [[eigenmonzo subgroup]]: 2.7/5.11/9
[[POTE generator]]s: ~3/2 = 700.2593, ~5/4 = 386.5581
 
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo subgroup]]: 2.7/5.11/9


{{Val list|legend=1| 9, 12, 19e, 22, 31, 53, 84e, 96, 127 }}
{{Val list|legend=1| 9, 12, 19e, 22, 31, 53, 84e, 96, 127 }}
Line 451: Line 315:
[[Projection pair]]s: 7 225/32 11 5625/512
[[Projection pair]]s: 7 225/32 11 5625/512


</div></div>
Scales (Scala files): [[minerva12]], [[minerva22x]]
 
[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]


== Athene ==
== Athene ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 701.2342, ~5/4 = 385.9594
EDO generators: [[12edo|(7, 4)\12]], [[19edo|(11, 6)\19]], [[22edo|(13, 7)\22]]
Scales (Scala files):
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Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 472: Line 326:


Mapping: [{{val| 1 0 0 -5 -9 -4 }}, {{val| 0 1 0 2 2 -1 }}, {{val| 0 0 1 2 4 4 }}]
Mapping: [{{val| 1 0 0 -5 -9 -4 }}, {{val| 0 1 0 2 2 -1 }}, {{val| 0 0 1 2 4 4 }}]
POTE generators: ~3/2 = 701.2342, ~5/4 = 385.9594


Minimax tuning:  
Minimax tuning:  
Line 486: Line 342:


= Apollo =
= Apollo =
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 703.4846, ~5/4 = 381.6033
EDO generators: [[12edo|(7, 4)\12]], [[19edo|(11, 6)\19]], [[22edo|(13, 7)\22]]
Scales (Scala files):
Associated temperament: [[Magic family #magic|magic]]
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Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 507: Line 349:
[[Mapping]]: [{{val| 1 0 0 -5 2 }}, {{val| 0 1 0 2 -2 }}, {{val| 0 0 1 2 2 }}]
[[Mapping]]: [{{val| 1 0 0 -5 2 }}, {{val| 0 1 0 2 -2 }}, {{val| 0 0 1 2 2 }}]


[[Minimax tuning]]: 11-odd-limit eigenmonzo subgroup: 2.7/5.11/9
[[POTE generator]]s: ~3/2 = 703.4846, ~5/4 = 381.6033
 
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo subgroup]]: 2.7/5.11/9


{{Val list|legend=1| 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce }}
{{Val list|legend=1| 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce }}
Line 513: Line 357:
[[Projection pair]]s: 7 225/32 11 100/9
[[Projection pair]]s: 7 225/32 11 100/9


</div></div>
[[Associated temperament]]: [[Magic family #magic|magic]]


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 703.9984, ~5/4 = 381.5352
EDO generators: [[12edo|(7, 4)\12]], [[22edo|(13, 7)\22]], [[29edo|(17, 9)\29]]
Scales (Scala files):
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<div style="line-height:1.6;">Technical data</div>
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Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 534: Line 366:


Mapping: [{{val| 1 0 0 -5 2 7 }}, {{val| 0 1 0 2 -2 -5 }}, {{val| 0 0 1 2 2 2 }}]
Mapping: [{{val| 1 0 0 -5 2 7 }}, {{val| 0 1 0 2 -2 -5 }}, {{val| 0 0 1 2 2 2 }}]
POTE generators: ~3/2 = 703.9984, ~5/4 = 381.5352


Minimax tuning: 13-odd-limit eigenmonzo subgroup: 2.11/9.13/9
Minimax tuning: 13-odd-limit eigenmonzo subgroup: 2.11/9.13/9
Line 540: Line 374:


Projection pairs: 7 225/32 11 100/9 13 3200/243
Projection pairs: 7 225/32 11 100/9 13 3200/243
</div></div>


= Potassium =
= Potassium =


Period: 1\1
Subgroup: 2.3.5.7.11
 
Optimal ([[POTE]]) generators: ~3/2 = 696.1714, ~5/4 = 385.0500
 
EDO generators: [[7edo|(4, 2)\7]], [[9edo|(5, 3)\9]], [[12edo|(7, 4)\12]]
 
Scales (Scala files):


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 45/44, 56/55
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
[[Mapping]]: [{{val| 1 0 0 -5 -2 }}, {{val| 0 1 0 2 2 }}, {{val| 0 0 1 2 1 }}]


Comma list: 45/44, 56/55
[[POTE generator]]s: ~3/2 = 696.1714, ~5/4 = 385.0500


Mapping: [{{val| 1 0 0 -5 -2 }}, {{val| 0 1 0 2 2 }}, {{val| 0 0 1 2 1 }}]
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo subgroup]]: 2.9/7.11
 
Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.9/7.11


{{Val list|legend=1| 7d, 9, 10, 12, 19, 31e, 50e }}
{{Val list|legend=1| 7d, 9, 10, 12, 19, 31e, 50e }}


Badness: 0.464 × 10<sup>-3</sup>
[[Badness]]: 0.464 × 10<sup>-3</sup>


Projection pairs: 7 225/32 11 45/4
[[Projection pair]]s: 7 225/32 11 45/4
 
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 696.0103, ~5/4 = 384.6785
EDO generators: [[9edo|(5, 3)\9]], [[10edo|(6, 3)\10]], [[12edo|(7, 4)\12]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 592: Line 400:


Mapping: [{{val| 1 0 0 -5 -2 -8 }}, {{val| 0 1 0 2 2 3 }}, {{val| 0 0 1 2 1 3 }}]
Mapping: [{{val| 1 0 0 -5 -2 -8 }}, {{val| 0 1 0 2 2 3 }}, {{val| 0 0 1 2 1 3 }}]
POTE generators: ~3/2 = 696.0103, ~5/4 = 384.6785


Minimax tuning:  
Minimax tuning:  
Line 602: Line 412:


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


= Malcolm =
= Malcolm =


Period: 1\1
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generators: ~3/2 = 701.8913, ~5/4 = 382.4083
[[Comma list]]: 225/224, 2200/2187


EDO generators: [[19edo|(11, 6)\19]], [[41edo|(24, 13)\41]], [[53edo|(31, 17)\53]]
[[Mapping]]: [{{val| 1 0 0 -5 -3 }}, {{val| 0 1 0 2 7 }}, {{val| 0 0 1 2 -2 }}]


Scales (Scala files):  
[[POTE generator]]s: ~3/2 = 701.8913, ~5/4 = 382.4083
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 2200/2187
 
Mapping: [{{val| 1 0 0 -5 -3 }}, {{val| 0 1 0 2 7 }}, {{val| 0 0 1 2 -2 }}]


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


Badness: 1.250 × 10<sup>-3</sup>
[[Badness]]: 1.250 × 10<sup>-3</sup>
 
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 701.8913, ~5/4 = 382.4083
EDO generators: [[19edo|(11, 6)\19]], [[41edo|(24, 13)\41]], [[53edo|(31, 17)\53]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 650: Line 434:


Mapping: [{{val| 1 0 0 -5 -3 2 }}, {{val| 0 1 0 2 7 4 }}, {{val| 0 0 1 2 -2 -2 }}]
Mapping: [{{val| 1 0 0 -5 -3 2 }}, {{val| 0 1 0 2 7 4 }}, {{val| 0 0 1 2 -2 -2 }}]
POTE generators: ~3/2 = 701.8913, ~5/4 = 382.4083


{{Val list|legend=1| 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef }}
{{Val list|legend=1| 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef }}


Badness: 1.075 × 10<sup>-3</sup>
Badness: 1.075 × 10<sup>-3</sup>
</div></div>


= Prodigy =
= 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 [[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.
Prodigy shrinks 1024/1029, 243/242, 384/385 and 2400/2401 down to the same tiny interval. Hence in practice it probably makes the most sense to temper this out as well, leading to [[miracle temperament]]. This, however, does not render it pointless to consider prodigy; for one thing, scales in prodigy such as hobbit scales translate into interesting scales for miracle.
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 699.7981, ~5/4 = 383.5114
EDO generators: [[12edo|(7, 4)\12]], [[31edo|(18, 10)\31]], [[41edo|(24, 13)\41]]
Scales (Scala files): [[prodigy11]], [[prodigy12]], [[prodigy29]]
[[Associated temperament]]: [[Gamelismic clan #Miracle|miracle]]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 685: Line 455:
: secor length = 0.9111, 3/2 length = 0.9477
: secor length = 0.9111, 3/2 length = 0.9477
: Angle (secor, 3/2) = 65.933
: Angle (secor, 3/2) = 65.933
[[POTE generator]]s: ~3/2 = 699.7981, ~5/4 = 383.5114


[[Minimax tuning]]:
[[Minimax tuning]]:
* 11-odd-limit
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 13/12 1/2 -1/4 0 1/12 }}, {{monzo| 13/6 -1 1/2 0 1/6 }}, {{monzo| 3/2 -1 1/2 0 1/2 }}, {{monzo| 0 0 0 0 1 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 13/12 1/2 -1/4 0 1/12 }}, {{monzo| 13/6 -1 1/2 0 1/6 }}, {{monzo| 3/2 -1 1/2 0 1/2 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo subgroup]]: 2.9/5.11
: [[Eigenmonzo subgroup]]: 2.9/5.11
Line 693: Line 465:
{{Val list|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}
{{Val list|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}


Badness: 0.344 × 10<sup>-3</sup>
[[Badness]]: 0.344 × 10<sup>-3</sup>


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


</div></div>
Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]
 
[[Associated temperament]]: [[Gamelismic clan #Miracle|miracle]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Line 710: Line 484:


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 700.4006, ~5/4 = 381.4025
EDO generators: [[10edo|(6, 3)\10]], [[12edo|(7, 4)\12]], [[19edo|(11, 6)\19]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 728: Line 490:


Mapping: [{{val| 1 0 0 -5 -13 -8 }}, {{val| 0 1 0 2 6 3 }}, {{val| 0 0 1 2 3 3 }}]
Mapping: [{{val| 1 0 0 -5 -13 -8 }}, {{val| 0 1 0 2 6 3 }}, {{val| 0 0 1 2 3 3 }}]
POTE generators: ~3/2 = 700.4006, ~5/4 = 381.4025


{{Val list|legend=1| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff }}
{{Val list|legend=1| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff }}


Badness: 0.736 × 10<sup>-3</sup>
Badness: 0.736 × 10<sup>-3</sup>
</div></div>


== Prodigious ==
== Prodigious ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 700.3407, ~5/4 = 383.2592
EDO generators: [[12edo|(7, 4)\12]], [[41edo|(24, 13)\41]], [[72edo|(42, 23)\72]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 754: Line 504:


Mapping: [{{val| 1 0 0 -5 -13 -23 }}, {{val| 0 1 0 2 6 11 }}, {{val| 0 0 1 2 3 4 }}]
Mapping: [{{val| 1 0 0 -5 -13 -23 }}, {{val| 0 1 0 2 6 11 }}, {{val| 0 0 1 2 3 4 }}]
POTE generators: ~3/2 = 700.3407, ~5/4 = 383.2592


{{Val list|legend=1| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}
{{Val list|legend=1| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}


Badness: 0.900 × 10<sup>-3</sup>
Badness: 0.900 × 10<sup>-3</sup>
</div></div>


== Prodigal ==
== Prodigal ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 699.4864, ~5/4 = 384.0998
EDO generators: [[19edo|(11, 6)\19]], [[31edo|(18, 10)\31]], [[72edo|(42, 23)\72]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 780: Line 518:


Mapping: [{{val| 1 0 0 -5 -13 -4 }}, {{val| 0 1 0 2 6 -1 }}, {{val| 0 0 1 2 3 4 }}]
Mapping: [{{val| 1 0 0 -5 -13 -4 }}, {{val| 0 1 0 2 6 -1 }}, {{val| 0 0 1 2 3 4 }}]
POTE generators: ~3/2 = 699.4864, ~5/4 = 384.0998


{{Val list|legend=1| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
{{Val list|legend=1| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}


Badness: 0.889 × 10<sup>-3</sup>
Badness: 0.889 × 10<sup>-3</sup>
</div></div>


= Fantastic =
= Fantastic =
Period: 1\2
Optimal ([[POTE]]) generators: ~3/2 = 700.6242, ~5/4 = 383.2978
EDO generators: [[12edo|(7, 4)\12]], [[22edo|(13, 7)\22]], [[72edo|(42, 23)\72]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 2 0 0 -10 -7 }}, {{val| 0 1 0 2 0 }}, {{val| 0 0 1 2 3 }}]
[[Mapping]]: [{{val| 2 0 0 -10 -7 }}, {{val| 0 1 0 2 0 }}, {{val| 0 0 1 2 3 }}]


Mapping generators: ~99/70, ~3, ~5
Mapping generators: ~99/70, ~3, ~5
[[POTE generator]]s: ~3/2 = 700.6242, ~5/4 = 383.2978


{{Val list|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}
{{Val list|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}


Badness: 0.743 × 10<sup>-3</sup>
[[Badness]]: 0.743 × 10<sup>-3</sup>
 
</div></div>


= Spectacle =
= Spectacle =
Period: 1\1
Optimal ([[POTE]]) generators: ~11/9 = 350.0570, ~5/4 = 383.9323
EDO generators: [[31edo|(9, 10)\31]], [[34edo|(10, 11)\34]], [[41edo|(12, 13)\41]]
Scales (Scala files): [[spectacle31]]
[[Associated temperament]]: [[Marvel temperaments #Marvo|marvo]]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 835: Line 547:
[[Comma list]]: 225/224, 243/242
[[Comma list]]: 225/224, 243/242


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


Mapping generators: ~2, ~11/9, ~5
Mapping generators: ~2, ~11/9, ~5
[[POTE generator]]s: ~11/9 = 350.0570, ~5/4 = 383.9323


[[Minimax tuning]]:
[[Minimax tuning]]:
* 11-odd-limit
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1/5 0 0 0 2/5 }}, {{monzo| 2/5 -2 1 0 4/5 }}, {{monzo| -19/5 -4 2 0 12/5 }}, {{monzo| 0 0 0 0 1 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1/5 0 0 0 2/5 }}, {{monzo| 2/5 -2 1 0 4/5 }}, {{monzo| -19/5 -4 2 0 12/5 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo subgroup]]: 2.9/5.11
: [[Eigenmonzo subgroup]]: 2.9/5.11
Line 846: Line 560:
{{Val list|legend=1| 31, 41, 72, 247c, 281, 353c, 425bc, 497bc }}
{{Val list|legend=1| 31, 41, 72, 247c, 281, 353c, 425bc, 497bc }}


Badness: 0.499 × 10<sup>-3</sup>
[[Badness]]: 0.499 × 10<sup>-3</sup>


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


</div></div>
Scales (Scala files): [[spectacle31]]
 
[[Associated temperament]]: [[Marvel temperaments #Marvo|marvo]]


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generators: ~11/9 = 349.9247, ~5/4 = 384.3505
EDO generators: [[31edo|(9, 10)\31]], [[34edo|(10, 11)\34]], [[72edo|(21, 23)\72]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 873: Line 577:


Mapping generators: ~2, ~11/9, ~5
Mapping generators: ~2, ~11/9, ~5
POTE generators: ~11/9 = 349.9247, ~5/4 = 384.3505


{{Val list|legend=1| 31, 72, 103, 175f, (240) }}
{{Val list|legend=1| 31, 72, 103, 175f, (240) }}


Badness: 1.009 × 10<sup>-3</sup>
Badness: 1.009 × 10<sup>-3</sup>
</div></div>


= Hestia =
= Hestia =
Period: 1\1
Optimal ([[POTE]]) generators: ~231/200 = 249.8526, ~5/4 = 383.6467
EDO generators: [[29edo|(6, 9)\29]], [[53edo|(11, 17)\53]], [[72edo|(15, 23)\72]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 0 0 -5 9 }}, {{val| 0 2 0 4 -7 }}, {{val| 0 0 1 2 0 }}]
[[Mapping]]: [{{val| 1 0 0 -5 9 }}, {{val| 0 2 0 4 -7 }}, {{val| 0 0 1 2 0 }}]


Mapping generators: ~2, ~400/231, ~5
Mapping generators: ~2, ~400/231, ~5
[[POTE generator]]s: ~231/200 = 249.8526, ~5/4 = 383.6467


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


Badness: 1.54 × 10<sup>-3</sup>
[[Badness]]: 1.54 × 10<sup>-3</sup>
 
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generators: ~15/13 = 249.7651, ~5/4 = 383.5558
EDO generators: [[19edo|(4, 6)\19]], [[29edo|(6, 9)\29]], [[53edo|(11, 17)\53]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 929: Line 609:


Mapping generators: ~2, ~26/15, ~5
Mapping generators: ~2, ~26/15, ~5
POTE generators: ~15/13 = 249.7651, ~5/4 = 383.5558


{{Val list|legend=1| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}
{{Val list|legend=1| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}


Badness: 1.062 × 10<sup>-3</sup>
Badness: 1.062 × 10<sup>-3</sup>
</div></div>


= Artemis =
= Artemis =
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 699.8719, ~11/10 = 158.3232
EDO generators: [[7edo|(4, 2)\7]], [[22edo|(13, 7)\22]], [[31edo|(18, 10)\31]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 225/224
[[Comma list]]: 121/120, 225/224


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


Mapping generators: ~2, ~3, ~11/10
Mapping generators: ~2, ~3, ~11/10
[[POTE generator]]s: ~3/2 = 699.8719, ~11/10 = 158.3232


{{Val list|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}
{{Val list|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 698.7090, ~11/10 = 158.7117
EDO generators: [[9edo|(5, 3)\9]], [[29edo|(17, 9)\29]], [[31edo|(18, 10)\31]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 983: Line 639:


Mapping generators: ~2, ~3, ~11/10
Mapping generators: ~2, ~3, ~11/10
POTE generators: ~3/2 = 698.7090, ~11/10 = 158.7117


{{Val list|legend=1| 9, 20, 22f, 29, 31 }}
{{Val list|legend=1| 9, 20, 22f, 29, 31 }}
</div></div>


== Diana ==
== Diana ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 700.9789, ~11/10 = 159.0048
EDO generators: [[22edo|(13, 7)\22]], [[29edo|(17, 9)\29]], [[31edo|(18, 10)\31]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 1,009: Line 653:


Mapping generators: ~2, ~3, ~11/10
Mapping generators: ~2, ~3, ~11/10
POTE generators: ~3/2 = 700.9789, ~11/10 = 159.0048


{{Val list|legend=1| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
{{Val list|legend=1| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
</div></div>


= Catakleismoid =
= Catakleismoid =
Period: 1\1
Optimal ([[POTE]]) generators: ~6/5 = 316.7318, ~11/8 = 549.2528
EDO generators: [[19edo|(5, 8)\19]], [[53edo|(14, 24)\53]], [[72edo|(19, 33)\72]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 4375/4374
[[Comma list]]: 225/224, 4375/4374


Mapping: [{{val| 1 0 1 -3 0 }}, {{val| 0 6 5 22 0 }}, {{val| 0 0 0 0 1 }}]
[[Mapping]]: [{{val| 1 0 1 -3 0 }}, {{val| 0 6 5 22 0 }}, {{val| 0 0 0 0 1 }}]


Mapping generators: ~2, ~6/5, ~11
Mapping generators: ~2, ~6/5, ~11
[[POTE generator]]s: ~6/5 = 316.7318, ~11/8 = 549.2528


{{Val list|legend=1| 19, 34d, 53, 72, 197e, 269ce }}
{{Val list|legend=1| 19, 34d, 53, 72, 197e, 269ce }}


Badness: 1.275 × 10<sup>-3</sup>
[[Badness]]: 1.275 × 10<sup>-3</sup>
 
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generators: ~6/5 = 316.7410, ~11/8 = 548.6028
EDO generators: [[19edo|(5, 8)\19]], [[53edo|(14, 24)\53]], [[72edo|(19, 33)\72]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 1,063: Line 683:


Mapping generators: ~2, ~6/5, ~11
Mapping generators: ~2, ~6/5, ~11
POTE generators: ~6/5 = 316.7410, ~11/8 = 548.6028


{{Val list|legend=1| 19, 34d, 53, 72, 125f, 197ef, 269cef }}
{{Val list|legend=1| 19, 34d, 53, 72, 125f, 197ef, 269cef }}


Badness: 0.916 × 10<sup>-3</sup>
Badness: 0.916 × 10<sup>-3</sup>
</div></div>


= Mirage =
= Mirage =
Period: 1\1
Optimal ([[POTE]]) generators: ~15/14 = 116.6327, ~13/8 = 837.7040
EDO generators: [[31edo|(3, 22)\31]], [[41edo|(4, 29)\41]], [[72edo|(7, 50)\22]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 385/384
[[Comma list]]: 225/224, 243/242, 385/384


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


Mapping generators: ~2, ~15/14, ~13
Mapping generators: ~2, ~15/14, ~13
[[POTE generator]]s: ~15/14 = 116.6327, ~13/8 = 837.7040


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


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


[[Category:Theory]]
[[Category:Theory]]

Revision as of 11:01, 27 March 2021

The marvel family is the set of temperaments that temper out the 7-limit marvel comma (225/224 = [-5 2 2 -1⟩) which is also named septimal kleisma. These temperaments hence equate 16/15 and 15/14, or equivalently they equate two 5/4's and one 14/9. The marvel comma is noteworthy in that it is tempered out by many common edos and rank-2 temperaments.

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

Marvel

Main article: Marvel

The head of the marvel family is marvel, which tempers out 225/224. Marvel has a normal list basis of [2, 3, 5]; hence a 5-limit scale can be converted to marvel simply by tempering it. One way to do that, and an excellent marvel tuning, is given by 197edo.

Little is gained in tuning accuracy by not tempering out 4375/4374 as well as 225/224, leading to catakleismic temperament. Another temperament which does little damage to tuning accuracy is compton temperament, for which 240edo may be used. See marvel temperaments for some other rank-2 temperaments.

Subgroup: 2.3.5.7

Comma list: c = 225/224

Mapping: [⟨1 0 0 -5], ⟨0 1 0 2], ⟨0 0 1 2]]

Mapping generators: ~2, ~3, ~5

Map to lattice: [⟨0 0 -1 -2], ⟨0 1 -1 0]]

Lattice basis:

secor length = 1.256, 3/2 length = 1.369
Angle (secor, 3/2) = 106.958 degrees

POTE generators: ~3/2 = 700.4075, ~5/4 = 383.6376

Minimax tuning:

[[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
[[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

Template:Val list

Badness: 0.0365 × 10-3

Projection pairs: 7 225/32

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

Associated temperament: catakleismic

Scales: marvel9, marvel10, marvel11, marvel12, marvel19, marvel22, pump12_1, pump12_2, pump13, pump14, pump15, pump16, pump17, pump18

{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

Eleven-limit extensions

The second comma of the normal comma list defines which 11-limit family member we are looking at.

  • 4125/4096 gives unidecimal marvel,
  • 91125/90112 gives prodigy,
  • 5632/5625 gives minerva, and
  • 243/242 gives spectacle.

11-limit (unimarv)

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384

Mapping: [⟨1 0 0 -5 12], ⟨0 1 0 2 -1], ⟨0 0 1 2 -3]]

Map to lattice: [⟨0 -1 0 -2 1], ⟨0 -1 1 0 -2]]

Lattice basis:

secor length = 1.0364, 5/4 length = 1.0759
Angle (secor, 5/4) = 104.028 degrees

POTE generators: ~3/2 = 700.3887, ~5/4 = 383.5403

Minimax tuning:

  • 11-odd-limit
[[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

Template:Val list

Badness: 0.255 × 10-3

Projection pairs: 7 225/32 11 4096/375

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

Associated temperament: catakleismic

Scales: marvel22_11, unimarv19, unimarv22

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

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -5 12 -4], ⟨0 1 0 2 -1 -1], ⟨0 0 1 2 -3 4]]

POTE generators: ~3/2 = 699.7367, ~5/4 = 384.0613

Minimax tuning:

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

Template:Val list

Badness: 0.690 × 10-3

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

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -5 12 2], ⟨0 1 0 2 -1 4], ⟨0 0 1 2 -3 -2]]

POTE generators: ~3/2 = 700.9779, ~5/4 = 383.1622

Minimax tuning:

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

Template:Val list

Badness: 0.721 × 10-3

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

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

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 0 -5 12 2 18], ⟨0 1 0 2 -1 4 0], ⟨0 0 1 2 -3 -2 -6]]

POTE generators: ~3/2 = 700.9619, ~5/4 = 383.0310

Template:Val list

Badness: 0.869 × 10-3

Enodia

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 0 -5 12 2 18], ⟨0 1 0 2 -1 4 0], ⟨0 0 1 2 -3 -2 6]]

POTE generators: ~3/2 = 700.9658, ~5/4 = 383.3063

Template:Val list

Badness: 0.917 × 10-3

Marvell

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 385/384, 1573/1568

Mapping: [⟨1 0 0 -5 12 -29], ⟨0 1 0 2 -1 6], ⟨0 0 1 2 -3 10]]

POTE generators: ~3/2 = 700.3937, ~5/4 = 383.5725

Minimax tuning:

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

Template:Val list

Badness: 0.862 × 10-3

Isis

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -5 12 17], ⟨0 1 0 2 -1 4], ⟨0 0 1 2 -3 -3]]

POTE generators: ~3/2 = 701.9156, ~5/4 = 383.2445

Template:Val list

Badness: 0.866 × 10-3

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

Deecee

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 364/363, 385/384

Mapping: [⟨1 0 0 -5 12 27], ⟨0 1 0 2 -1 -3], ⟨0 0 1 2 -3 -8]]

POTE generators: ~3/2 = 700.4560, ~5/4 = 382.8177

Minimax tuning:

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

Template:Val list

Badness: 0.920 × 10-3

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

Tripod

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195

Mapping: [⟨1 0 0 -5 12 -8], ⟨0 1 0 2 -1 3], ⟨0 0 1 2 -3 3]]

POTE generators: ~3/2 = 699.2335, ~5/4 = 382.9775

Minimax tuning:

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

Template:Val list

Badness: 0.745 × 10-3

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

Marvelcat

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 385/384

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

POTE generators: ~15/13 = 249.7138, ~5/4 = 383.5816

Template:Val list

Badness: 0.9997 × 10-3

Minerva

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175

Mapping: [⟨1 0 0 -5 -9], ⟨0 1 0 2 2], ⟨0 0 1 2 4]]

Map to lattice: [⟨0 -1 0 -2 -2], ⟨0 -1 1 0 2]]

Lattice basis:

16/15 length = 0.8997, 5/4 length = 1.0457
Angle (16/15, 5/4) = 98.6044 degrees

POTE generators: ~3/2 = 700.2593, ~5/4 = 386.5581

Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7/5.11/9

Template:Val list

Badness: 0.381 × 10-3

Projection pairs: 7 225/32 11 5625/512

Scales (Scala files): minerva12, minerva22x

Associated temperament: würschmidt

Athene

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 275/273

Mapping: [⟨1 0 0 -5 -9 -4], ⟨0 1 0 2 2 -1], ⟨0 0 1 2 4 4]]

POTE generators: ~3/2 = 701.2342, ~5/4 = 385.9594

Minimax tuning:

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

Template:Val list

Badness: 0.818 × 10-3

Projection pairs: 7 225/32 11 5625/512 13 625/48

Apollo

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224

Mapping: [⟨1 0 0 -5 2], ⟨0 1 0 2 -2], ⟨0 0 1 2 2]]

POTE generators: ~3/2 = 703.4846, ~5/4 = 381.6033

Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.7/5.11/9

Template:Val list

Projection pairs: 7 225/32 11 100/9

Associated temperament: magic

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -5 2 7], ⟨0 1 0 2 -2 -5], ⟨0 0 1 2 2 2]]

POTE generators: ~3/2 = 703.9984, ~5/4 = 381.5352

Minimax tuning: 13-odd-limit eigenmonzo subgroup: 2.11/9.13/9

Template:Val list

Projection pairs: 7 225/32 11 100/9 13 3200/243

Potassium

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55

Mapping: [⟨1 0 0 -5 -2], ⟨0 1 0 2 2], ⟨0 0 1 2 1]]

POTE generators: ~3/2 = 696.1714, ~5/4 = 385.0500

Minimax tuning: 11-odd-limit eigenmonzo subgroup: 2.9/7.11

Template:Val list

Badness: 0.464 × 10-3

Projection pairs: 7 225/32 11 45/4

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 78/77

Mapping: [⟨1 0 0 -5 -2 -8], ⟨0 1 0 2 2 3], ⟨0 0 1 2 1 3]]

POTE generators: ~3/2 = 696.0103, ~5/4 = 384.6785

Minimax tuning:

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

Template:Val list

Badness: 0.733 × 10-3

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

Malcolm

Subgroup: 2.3.5.7.11

Comma list: 225/224, 2200/2187

Mapping: [⟨1 0 0 -5 -3], ⟨0 1 0 2 7], ⟨0 0 1 2 -2]]

POTE generators: ~3/2 = 701.8913, ~5/4 = 382.4083

Template:Val list

Badness: 1.250 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 325/324

Mapping: [⟨1 0 0 -5 -3 2], ⟨0 1 0 2 7 4], ⟨0 0 1 2 -2 -2]]

POTE generators: ~3/2 = 701.8913, ~5/4 = 382.4083

Template:Val list

Badness: 1.075 × 10-3

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 miracle temperament. This, however, does not render it pointless to consider prodigy; for one thing, scales in prodigy such as hobbit scales translate into interesting scales for miracle.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440

Mapping: [⟨1 0 0 -5 -13], ⟨0 1 0 2 6], ⟨0 0 1 2 3]]

Map to lattice: [⟨0 0 -1 -2 -3], ⟨0 1 -1 0 3]]

Lattice basis:

secor length = 0.9111, 3/2 length = 0.9477
Angle (secor, 3/2) = 65.933

POTE generators: ~3/2 = 699.7981, ~5/4 = 383.5114

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

Template:Val list

Badness: 0.344 × 10-3

Projection pairs: 7 225/32 11 91125/8192

Scales: prodigy11, prodigy12, prodigy29

Associated temperament: miracle

Hobbit bases

{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: [⟨1 0 0 -5 -13 -8], ⟨0 1 0 2 6 3], ⟨0 0 1 2 3 3]]

POTE generators: ~3/2 = 700.4006, ~5/4 = 381.4025

Template:Val list

Badness: 0.736 × 10-3

Prodigious

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 364/363, 441/440

Mapping: [⟨1 0 0 -5 -13 -23], ⟨0 1 0 2 6 11], ⟨0 0 1 2 3 4]]

POTE generators: ~3/2 = 700.3407, ~5/4 = 383.2592

Template:Val list

Badness: 0.900 × 10-3

Prodigal

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 441/440

Mapping: [⟨1 0 0 -5 -13 -4], ⟨0 1 0 2 6 -1], ⟨0 0 1 2 3 4]]

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

Template:Val list

Badness: 0.889 × 10-3

Fantastic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 4375/4356

Mapping: [⟨2 0 0 -10 -7], ⟨0 1 0 2 0], ⟨0 0 1 2 3]]

Mapping generators: ~99/70, ~3, ~5

POTE generators: ~3/2 = 700.6242, ~5/4 = 383.2978

Template:Val list

Badness: 0.743 × 10-3

Spectacle

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242

Mapping: [⟨1 1 0 -3 2], ⟨0 2 0 4 5], ⟨0 0 1 2 0]]

Mapping generators: ~2, ~11/9, ~5

POTE generators: ~11/9 = 350.0570, ~5/4 = 383.9323

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

Template:Val list

Badness: 0.499 × 10-3

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

Scales (Scala files): spectacle31

Associated temperament: marvo

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 4 5 -2], ⟨0 0 1 2 0 4]]

Mapping generators: ~2, ~11/9, ~5

POTE generators: ~11/9 = 349.9247, ~5/4 = 384.3505

Template:Val list

Badness: 1.009 × 10-3

Hestia

Subgroup: 2.3.5.7.11

Comma list: 225/224, 125000/124509

Mapping: [⟨1 0 0 -5 9], ⟨0 2 0 4 -7], ⟨0 0 1 2 0]]

Mapping generators: ~2, ~400/231, ~5

POTE generators: ~231/200 = 249.8526, ~5/4 = 383.6467

Template:Val list

Badness: 1.54 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 1001/1000

Mapping: [⟨1 0 0 -5 9 -1], ⟨0 2 0 4 -7 3], ⟨0 0 1 2 0 1]]

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

POTE generators: ~15/13 = 249.7651, ~5/4 = 383.5558

Template:Val list

Badness: 1.062 × 10-3

Artemis

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224

Mapping: [⟨1 0 1 -3 2], ⟨0 1 1 4 1], ⟨0 0 -2 -4 -1]]

Mapping generators: ~2, ~3, ~11/10

POTE generators: ~3/2 = 699.8719, ~11/10 = 158.3232

Template:Val list

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195

Mapping: [⟨1 0 1 -3 2 -5], ⟨0 1 1 4 1 6], ⟨0 0 -2 -4 -1 -6]]

Mapping generators: ~2, ~3, ~11/10

POTE generators: ~3/2 = 698.7090, ~11/10 = 158.7117

Template:Val list

Diana

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273

Mapping: [⟨1 0 1 -3 2 7], ⟨0 1 1 4 1 -2], ⟨0 0 -2 -4 -1 -1]]

Mapping generators: ~2, ~3, ~11/10

POTE generators: ~3/2 = 700.9789, ~11/10 = 159.0048

Template:Val list

Catakleismoid

Subgroup: 2.3.5.7.11

Comma list: 225/224, 4375/4374

Mapping: [⟨1 0 1 -3 0], ⟨0 6 5 22 0], ⟨0 0 0 0 1]]

Mapping generators: ~2, ~6/5, ~11

POTE generators: ~6/5 = 316.7318, ~11/8 = 549.2528

Template:Val list

Badness: 1.275 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324

Mapping: [⟨1 0 1 -3 0 0], ⟨0 6 5 22 0 14], ⟨0 0 0 0 1 0]]

Mapping generators: ~2, ~6/5, ~11

POTE generators: ~6/5 = 316.7410, ~11/8 = 548.6028

Template:Val list

Badness: 0.916 × 10-3

Mirage

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 0], ⟨0 0 0 0 0 1]]

Mapping generators: ~2, ~15/14, ~13

POTE generators: ~15/14 = 116.6327, ~13/8 = 837.7040

Template:Val list

Badness: 0.738 × 10-3

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