Marvel family: Difference between revisions

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Line 9: Line 9:
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.
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 ==
== Marvel ==
{{main| Marvel }}
{{main| Marvel }}


Line 52: Line 52:
Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]]
Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Databox|[[Minkowski blocks]]|
<div style="line-height:1.6;">[[Minkowski blocks]]</div>
<div class="mw-collapsible-content">
 
{2, 3, 5} subgroup
{2, 3, 5} subgroup
* 8: 16/15, 250/243
* 8: 16/15, 250/243
Line 70: Line 67:
* 31: 81/80, 34171875/33554432
* 31: 81/80, 34171875/33554432
* 41: 34171875/33554432, 3125/3072
* 41: 34171875/33554432, 3125/3072
 
}}
</div></div>


=== Eleven-limit extensions  ===
=== Eleven-limit extensions  ===
Line 80: Line 76:
* 243/242 gives spectacle.
* 243/242 gives spectacle.


== 11-limit (unimarv) ==
== Undecimal marvel (unimarv) ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 113: Line 108:
Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]
Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Databox|Hobbit bases|
<div style="line-height:1.6;">Hobbit bases</div>
<div class="mw-collapsible-content">
 
{2, 3, 5} subgroup
{2, 3, 5} subgroup
* 12: 128/125, 2048/2025
* 12: 128/125, 2048/2025
Line 124: Line 116:
* 31: 2109375/2097152, 81/80
* 31: 2109375/2097152, 81/80
* 41: 3125/3072, 34171875/33554432
* 41: 3125/3072, 34171875/33554432
}}


</div></div>
=== 13-limit ===
 
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 141: Line 131:
* 15-odd-limit eigenmonzo subgroup: 2.15/11.15/13
* 15-odd-limit eigenmonzo subgroup: 2.15/11.15/13


{{Val list|legend=1| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}
Vals: {{Val list| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}


Badness: 0.690 × 10<sup>-3</sup>
Badness: 0.690 × 10<sup>-3</sup>
Line 147: Line 137:
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


=== Hecate ===
=== Hecate ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 161: Line 150:
* 15-odd-limit eigenmonzo subgroup: 2.7.15/13
* 15-odd-limit eigenmonzo subgroup: 2.7.15/13


{{Val list|legend=1| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}
Vals: {{Val list| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}


Badness: 0.721 × 10<sup>-3</sup>
Badness: 0.721 × 10<sup>-3</sup>
Line 169: Line 158:
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


==== 17-limit ====
==== 17-limit ====
 
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 179: Line 167:
POTE generators: ~3/2 = 700.9619, ~5/4 = 383.0310
POTE generators: ~3/2 = 700.9619, ~5/4 = 383.0310


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


Badness: 0.869 × 10<sup>-3</sup>
Badness: 0.869 × 10<sup>-3</sup>


</div></div>
==== Enodia ====
 
==== Enodia ====
 
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 195: Line 180:
POTE generators: ~3/2 = 700.9658, ~5/4 = 383.3063
POTE generators: ~3/2 = 700.9658, ~5/4 = 383.3063


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


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


=== Marvell ===
=== Marvell ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 213: Line 197:
* 15-odd-limit eigenmonzo subgroup: 2.7.15/13
* 15-odd-limit eigenmonzo subgroup: 2.7.15/13


{{Val list|legend=1| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff }}
Vals: {{Val list| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff }}


Badness: 0.862 × 10<sup>-3</sup>
Badness: 0.862 × 10<sup>-3</sup>


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


Line 229: Line 210:
POTE generators: ~3/2 = 701.9156, ~5/4 = 383.2445
POTE generators: ~3/2 = 701.9156, ~5/4 = 383.2445


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


Badness: 0.866 × 10<sup>-3</sup>
Badness: 0.866 × 10<sup>-3</sup>
Line 235: Line 216:
Projection pairs: 7 225/32 11 4096/375 13 131072/10125
Projection pairs: 7 225/32 11 4096/375 13 131072/10125


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


Line 249: Line 229:
* 15-odd-limit eigenmonzo subgroup: 2.3.13/5
* 15-odd-limit eigenmonzo subgroup: 2.3.13/5


{{Val list|legend=1| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}
Vals: {{Val list| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}


Badness: 0.920 × 10<sup>-3</sup>
Badness: 0.920 × 10<sup>-3</sup>
Line 255: Line 235:
Projection pairs: 7 225/32 11 4096/375 13 134217728/10546875
Projection pairs: 7 225/32 11 4096/375 13 134217728/10546875


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


Line 269: Line 248:
* 15-odd-limit eigenmonzo subgroup: 2.5/3.13/11
* 15-odd-limit eigenmonzo subgroup: 2.5/3.13/11


{{Val list|legend=1| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}
Vals: {{Val list| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}


Badness: 0.745 × 10<sup>-3</sup>
Badness: 0.745 × 10<sup>-3</sup>
Line 275: Line 254:
Projection pairs: 7 225/32 11 4096/375 13 3375/256
Projection pairs: 7 225/32 11 4096/375 13 3375/256


=== Marvelcat ===
=== Marvelcat ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 287: Line 265:
POTE generators: ~15/13 = 249.7138, ~5/4 = 383.5816
POTE generators: ~15/13 = 249.7138, ~5/4 = 383.5816


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


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


== Minerva ==
== Minerva ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 319: Line 296:
[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]
[[Associated temperament]]: [[Würschmidt family #Würschmidt|würschmidt]]


=== Athene ===
=== Athene ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 333: Line 309:
* 15-odd-limit eigenmonzo subgroup: 2.11/9.13/7
* 15-odd-limit eigenmonzo subgroup: 2.11/9.13/7


{{Val list|legend=1| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}
Vals: {{Val list| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}


Badness: 0.818 × 10<sup>-3</sup>
Badness: 0.818 × 10<sup>-3</sup>
Line 339: Line 315:
Projection pairs: 7 225/32 11 5625/512 13 625/48
Projection pairs: 7 225/32 11 5625/512 13 625/48


</div></div>
== Apollo ==
 
== Apollo ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 359: Line 332:
[[Associated temperament]]: [[Magic family #magic|magic]]
[[Associated temperament]]: [[Magic family #magic|magic]]


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


Line 371: Line 343:
Minimax tuning: 13-odd-limit eigenmonzo subgroup: 2.11/9.13/9
Minimax tuning: 13-odd-limit eigenmonzo subgroup: 2.11/9.13/9


{{Val list|legend=1| 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef }}
Vals: {{Val list| 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef }}


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


== Potassium ==
== Potassium ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 393: Line 364:
[[Projection pair]]s: 7 225/32 11 45/4
[[Projection pair]]s: 7 225/32 11 45/4


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


Line 407: Line 377:
* 15-odd-limit eigenmonzo subgroup: 2.9/7.13/9
* 15-odd-limit eigenmonzo subgroup: 2.9/7.13/9


{{Val list|legend=1| 9, 10, 12f, 19, 31e, 50e }}
Vals: {{Val list| 9, 10, 12f, 19, 31e, 50e }}


Badness: 0.733 × 10<sup>-3</sup>
Badness: 0.733 × 10<sup>-3</sup>
Line 413: Line 383:
Projection pairs: 7 225/32 11 45/4 13 3375/256
Projection pairs: 7 225/32 11 45/4 13 3375/256


== Malcolm ==
== Malcolm ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 427: Line 396:
[[Badness]]: 1.250 × 10<sup>-3</sup>
[[Badness]]: 1.250 × 10<sup>-3</sup>


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


Line 437: Line 405:
POTE generators: ~3/2 = 701.8913, ~5/4 = 382.4083
POTE generators: ~3/2 = 701.8913, ~5/4 = 382.4083


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


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


== 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.


Line 473: Line 441:
[[Associated temperament]]: [[Gamelismic clan #Miracle|miracle]]
[[Associated temperament]]: [[Gamelismic clan #Miracle|miracle]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Databox|Hobbit bases|
<div style="line-height:1.6;">Hobbit bases</div>
<div class="mw-collapsible-content">
 
{2, 3, 5} subgroup
{2, 3, 5} subgroup
* 31: 81/80, 34171875/33554432
* 31: 81/80, 34171875/33554432
* 41: 34171875/33554432, 32805/32768
* 41: 34171875/33554432, 32805/32768
}}


</div></div>
=== 13-limit ===
 
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 493: Line 456:
POTE generators: ~3/2 = 700.4006, ~5/4 = 381.4025
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 }}
Vals: {{Val list| 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>


=== Prodigious ===
=== Prodigious ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 507: Line 469:
POTE generators: ~3/2 = 700.3407, ~5/4 = 383.2592
POTE generators: ~3/2 = 700.3407, ~5/4 = 383.2592


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


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


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


Line 521: Line 482:
POTE generators: ~3/2 = 699.4864, ~5/4 = 384.0998
POTE generators: ~3/2 = 699.4864, ~5/4 = 384.0998


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


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


== Fantastic ==
== Fantastic ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 541: Line 501:
[[Badness]]: 0.743 × 10<sup>-3</sup>
[[Badness]]: 0.743 × 10<sup>-3</sup>


== Spectacle ==
== Spectacle ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 562: Line 521:
[[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 pair]]s: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9


Scales (Scala files): [[spectacle31]]
Scales (Scala files): [[spectacle31]]
Line 568: Line 527:
[[Associated temperament]]: [[Marvel temperaments #Marvo|marvo]]
[[Associated temperament]]: [[Marvel temperaments #Marvo|marvo]]


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


Line 580: Line 538:
POTE generators: ~11/9 = 349.9247, ~5/4 = 384.3505
POTE generators: ~11/9 = 349.9247, ~5/4 = 384.3505


{{Val list|legend=1| 31, 72, 103, 175f, (240) }}
Vals: {{Val list| 31, 72, 103, 175f }} <nowiki>*</nowiki>
 
<nowiki>*</nowiki> optimal patent val: [[240edo|240]]


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


== Hestia ==
== Hestia ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 600: Line 559:
[[Badness]]: 1.54 × 10<sup>-3</sup>
[[Badness]]: 1.54 × 10<sup>-3</sup>


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


Line 612: Line 570:
POTE generators: ~15/13 = 249.7651, ~5/4 = 383.5558
POTE generators: ~15/13 = 249.7651, ~5/4 = 383.5558


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


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


== Artemis ==
== Artemis ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 630: Line 587:
{{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 }}


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


Line 642: Line 598:
POTE generators: ~3/2 = 698.7090, ~11/10 = 158.7117
POTE generators: ~3/2 = 698.7090, ~11/10 = 158.7117


{{Val list|legend=1| 9, 20, 22f, 29, 31 }}
Vals: {{Val list| 9, 20, 22f, 29, 31 }}
 
=== Diana  ===


=== Diana ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 656: Line 611:
POTE generators: ~3/2 = 700.9789, ~11/10 = 159.0048
POTE generators: ~3/2 = 700.9789, ~11/10 = 159.0048


{{Val list|legend=1| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
Vals: {{Val list| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
 
== Catakleismoid  ==


== Catakleismoid ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 674: Line 628:
[[Badness]]: 1.275 × 10<sup>-3</sup>
[[Badness]]: 1.275 × 10<sup>-3</sup>


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


Line 686: Line 639:
POTE generators: ~6/5 = 316.7410, ~11/8 = 548.6028
POTE generators: ~6/5 = 316.7410, ~11/8 = 548.6028


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


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


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


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


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


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]

Revision as of 13:54, 22 June 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

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

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.

Undecimal marvel (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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: Template:Val list *

* optimal patent val: 240

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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: Template:Val list

Badness: 0.738 × 10-3

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