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{{interwiki
| en = Marvel family
| de = Marvel
| es =
| ja =
}}
{{Technical data page}}
{{Technical data page}}
The '''marvel family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[7-limit]] [[marvel comma]] ([[ratio]]: [[225/224]], {{monzo|legend=1| -5 2 2 -1 }}), also known as ''septimal kleisma''. These temperaments hence equate [[16/15]] and [[15/14]], or equivalently they equate two [[5/4]]'s and one [[14/9]]. The marvel comma is noteworthy in that it is tempered out by many common [[equal temperament|equal]] and [[rank-2 temperament]]s.
The '''marvel family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[7-limit]] [[marvel comma]] ([[ratio]]: [[225/224]], {{monzo|legend=1| -5 2 2 -1 }}), also known as ''septimal kleisma''. These temperaments hence equate [[16/15]] and [[15/14]], or equivalently they equate two [[5/4]]'s and one [[14/9]]. The marvel comma is noteworthy in that it is tempered out by many common [[equal temperament|equal]] and [[rank-2 temperament]]s.
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[[Projection pair]]s: <code>7 225/32</code>
[[Projection pair]]s: <code>7 225/32</code>
[[Complexity spectrum]]: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9


Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]], [[marvel wholetone]]
Scales: [[marvel9]], [[marvel10]], [[marvel11]], [[marvel12]], [[marvel19]], [[marvel22]], [[pump12_1]], [[pump12_2]], [[pump13]], [[pump14]], [[pump15]], [[pump16]], [[pump17]], [[pump18]], [[marvel wholetone]]
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* ''[[Marvelpine]]'' (+4000/3993) → [[Wizardharry clan #Marvelpine|Wizardharry clan]]
* ''[[Marvelpine]]'' (+4000/3993) → [[Wizardharry clan #Marvelpine|Wizardharry clan]]
* ''[[Mirage]]'' (+243/242, +385/384) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]
* ''[[Mirage]]'' (+243/242, +385/384) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]
* ''[[Catakleismoid]]'' (+4375/4374) → [[Kleismic rank-3 family #Catakleismoid|Kleismic rank-3 family]]


== Undecimal marvel ==
== Undecimal marvel ==
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Undecimal marvel tempers out [[385/384]] as well as [[540/539]], and is loosely [[associated temperament|associated]] with [[wizard]]. This extension is natural because of the factorization 225/224 = (385/384)⋅(540/539). [[197edo]] remains useful as a tuning, with the 197e val, but [[166edo]], which among other things has a virtually pure 7, works as well.
Undecimal marvel tempers out [[385/384]] as well as [[540/539]], and is loosely [[associated temperament|associated]] with [[wizard]]. This extension is natural because of the factorization 225/224 = (385/384)⋅(540/539). [[197edo]] remains useful as a tuning, with the 197e val, but [[166edo]], which among other things has a virtually pure 7, works as well.


In the 13-limit, 225/224 factors as ([[351/350]])⋅([[625/624]]) or ([[325/324]])⋅([[729/728]]). Tempering out 351/350 and 625/624 leads to tridecimal marvel, tempering out 325/324 and 729/728 leads to hecate. Tempering out all of them leads to the 13-limit version of [[catakleismic]].  
In the 13-limit, 225/224 factors as ([[351/350]])⋅([[625/624]]) or ([[325/324]])⋅([[729/728]]). Tempering out 351/350 and 625/624 leads to helios, tempering out 325/324 and 729/728 leads to hecate. Tempering out all of them leads to the 13-limit version of [[catakleismic]].  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
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[[Projection pair]]s: <code>7 225/32 11 4096/375</code>
[[Projection pair]]s: <code>7 225/32 11 4096/375</code>
[[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


Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]
Scales: [[marvel22_11]], [[unimarv19]], [[unimarv22]]
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}}
}}


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


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


Mapping: {{mapping| 1 0 0 -5 12 -4 | 0 1 0 2 -1 -1 | 0 0 1 2 -3 4 }}
{{Mapping|legend=0| 1 0 0 -5 12 -4 | 0 1 0 2 -1 -1 | 0 0 1 2 -3 4 }}


Optimal tunings:  
Optimal tunings:  
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Badness (Sintel): 0.645
Badness (Sintel): 0.645
Complexity spectrum: 5/4, 4/3, 16/15, 15/14, 9/7, 6/5, 7/6, 11/8, 7/5, 9/8, 8/7, 10/9, 12/11, 13/10, 11/10, 15/11, 16/13, 11/9, 15/13, 14/13, 13/12, 14/11, 18/13, 13/11


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


Mapping: {{mapping| 1 0 0 -5 12 2 | 0 1 0 2 -1 4 | 0 0 1 2 -3 -2 }}
{{Mapping|legend=0| 1 0 0 -5 12 2 | 0 1 0 2 -1 4 | 0 0 1 2 -3 -2 }}


Optimal tunings:  
Optimal tunings:  
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Projection pairs: <code>7 225/32 11 4096/375 13 324/25</code>
Projection pairs: <code>7 225/32 11 4096/375 13 324/25</code>
Complexity spectrum: 4/3, 5/4, 16/15, 15/14, 6/5, 9/8, 7/5, 9/7, 7/6, 10/9, 8/7, 18/13, 11/8, 12/11, 13/12, 11/9, 11/10, 15/13, 15/11, 16/13, 13/11, 14/13, 13/10, 14/11


===== 2.3.5.7.11.13.19 subgroup =====
===== 2.3.5.7.11.13.19 subgroup =====
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Comma list: 225/224, 325/324, 385/384, 400/399
Comma list: 225/224, 325/324, 385/384, 400/399


Subgroup-val mapping: {{mapping| 1 0 0 -5 12 2 9 | 0 1 0 2 -1 4 -3 | 0 0 1 2 -3 -2 0 }}
{{Mapping|legend=2| 1 0 0 -5 12 2 9 | 0 1 0 2 -1 4 -3 | 0 0 1 2 -3 -2 0 }}


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


Mapping: {{mapping| 1 0 0 -5 12 2 18 | 0 1 0 2 -1 4 0 | 0 0 1 2 -3 -2 -6 }}
{{Mapping|legend=0| 1 0 0 -5 12 2 18 | 0 1 0 2 -1 4 0 | 0 0 1 2 -3 -2 -6 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 225/224, 325/324, 385/384, 400/399, 595/594
Comma list: 225/224, 325/324, 385/384, 400/399, 595/594


Mapping: {{mapping| 1 0 0 -5 12 2 18 9 | 0 1 0 2 -1 4 0 -3 | 0 0 1 2 -3 -2 -6 0 }}
{{Mapping|legend=0| 1 0 0 -5 12 2 18 9 | 0 1 0 2 -1 4 0 -3 | 0 0 1 2 -3 -2 -6 0 }}


Optimal tunings:  
Optimal tunings:  
Line 228: Line 215:
Comma list: 225/224, 325/324, 375/374, 385/384
Comma list: 225/224, 325/324, 375/374, 385/384


Mapping: {{mapping| 1 0 0 -5 12 2 -13 | 0 1 0 2 -1 4 2 | 0 0 1 2 -3 -2 6 }}
{{Mapping|legend=0| 1 0 0 -5 12 2 -13 | 0 1 0 2 -1 4 2 | 0 0 1 2 -3 -2 6 }}


Optimal tunings:  
Optimal tunings:  
Line 243: Line 230:
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399


Mapping: {{mapping| 1 0 0 -5 12 2 -13 9 | 0 1 0 2 -1 4 2 -3 | 0 0 1 2 -3 -2 6 0 }}
{{Mapping|legend=0| 1 0 0 -5 12 2 -13 9 | 0 1 0 2 -1 4 2 -3 | 0 0 1 2 -3 -2 6 0 }}


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


Mapping: {{mapping| 1 0 0 -5 12 -29 | 0 1 0 2 -1 6 | 0 0 1 2 -3 10 }}
{{Mapping|legend=0| 1 0 0 -5 12 -29 | 0 1 0 2 -1 6 | 0 0 1 2 -3 10 }}


Optimal tunings:  
Optimal tunings:  
Line 277: Line 264:
Comma list: 225/224, 275/273, 385/384
Comma list: 225/224, 275/273, 385/384


Mapping: {{mapping| 1 0 0 -5 12 17 | 0 1 0 2 -1 4 | 0 0 1 2 -3 -3 }}
{{Mapping|legend=0| 1 0 0 -5 12 17 | 0 1 0 2 -1 -4 | 0 0 1 2 -3 -3 }}


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


Mapping: {{mapping| 1 0 0 -5 12 27 | 0 1 0 2 -1 -3 | 0 0 1 2 -3 -8 }}
{{Mapping|legend=0| 1 0 0 -5 12 27 | 0 1 0 2 -1 -3 | 0 0 1 2 -3 -8 }}


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


Mapping: {{mapping| 1 0 0 -5 12 -8 | 0 1 0 2 -1 3 | 0 0 1 2 -3 3 }}
{{Mapping|legend=0| 1 0 0 -5 12 -8 | 0 1 0 2 -1 3 | 0 0 1 2 -3 3 }}


Optimal tunings:  
Optimal tunings:  
Line 336: Line 323:
Comma list: 169/168, 225/224, 385/384
Comma list: 169/168, 225/224, 385/384


Mapping: {{mapping| 1 0 0 -5 12 -1 | 0 2 0 4 -2 3 | 0 0 1 2 -3 1 }}
{{Mapping|legend=0| 1 0 0 -5 12 -1 | 0 2 0 4 -2 3 | 0 0 1 2 -3 1 }}
: mapping generators: ~2, ~26/15, ~5
: mapping generators: ~2, ~26/15, ~5


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Comma list: 105/104, 196/195, 352/351
Comma list: 105/104, 196/195, 352/351


Mapping: {{mapping| 1 0 0 -5 -13 -8 | 0 1 0 2 6 3 | 0 0 1 2 3 3 }}
{{Mapping|legend=0| 1 0 0 -5 -13 -8 | 0 1 0 2 6 3 | 0 0 1 2 3 3 }}


Optimal tunings:  
Optimal tunings:  
Line 407: Line 394:
Comma list: 225/224, 364/363, 441/440
Comma list: 225/224, 364/363, 441/440


Mapping: {{mapping| 1 0 0 -5 -13 -23 | 0 1 0 2 6 11 | 0 0 1 2 3 4 }}
{{Mapping|legend=0| 1 0 0 -5 -13 -23 | 0 1 0 2 6 11 | 0 0 1 2 3 4 }}


Optimal tunings:  
Optimal tunings:  
Line 422: Line 409:
Comma list: 225/224, 351/350, 441/440
Comma list: 225/224, 351/350, 441/440


Mapping: {{mapping| 1 0 0 -5 -13 -4 | 0 1 0 2 6 -1 | 0 0 1 2 3 4 }}
{{Mapping|legend=0| 1 0 0 -5 -13 -4 | 0 1 0 2 6 -1 | 0 0 1 2 3 4 }}


Optimal tunings:  
Optimal tunings:  
Line 437: Line 424:
Comma list: 225/224, 441/440, 1001/1000
Comma list: 225/224, 441/440, 1001/1000


Mapping: {{mapping| 1 0 0 -5 -13 21 | 0 1 0 2 6 -8 | 0 0 1 2 3 -2 }}
{{Mapping|legend=0| 1 0 0 -5 -13 21 | 0 1 0 2 6 -8 | 0 0 1 2 3 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 452: Line 439:
Comma list: 225/224, 273/272, 375/374, 441/440
Comma list: 225/224, 273/272, 375/374, 441/440


Mapping: {{mapping| 1 0 0 -5 -13 21 12 | 0 1 0 2 6 -8 -5 | 0 0 1 2 3 -2 0 }}
{{Mapping|legend=0| 1 0 0 -5 -13 21 12 | 0 1 0 2 6 -8 -5 | 0 0 1 2 3 -2 0 }}


Optimal tunings:  
Optimal tunings:  
Line 498: Line 485:
Comma list: 99/98, 176/175, 275/273
Comma list: 99/98, 176/175, 275/273


Mapping: {{mapping| 1 0 0 -5 -9 -4 | 0 1 0 2 2 -1 | 0 0 1 2 4 4 }}
{{Mapping|legend=0| 1 0 0 -5 -9 -4 | 0 1 0 2 2 -1 | 0 0 1 2 4 4 }}


Optimal tunings:  
Optimal tunings:  
Line 550: Line 537:
Comma list: 100/99, 225/224, 275/273
Comma list: 100/99, 225/224, 275/273


Mapping: {{mapping| 1 0 0 -5 2 7 | 0 1 0 2 -2 -5 | 0 0 1 2 2 2 }}
{{Mapping|legend=0| 1 0 0 -5 2 7 | 0 1 0 2 -2 -5 | 0 0 1 2 2 2 }}


Optimal tunings:  
Optimal tunings:  
Line 569: Line 556:
Comma list: 100/99, 105/104, 196/195
Comma list: 100/99, 105/104, 196/195


Mapping: {{mapping| 1 0 0 -5 2 1 | 0 1 0 2 -2 3 | 0 0 1 2 2 3 }}
{{Mapping|legend=0| 1 0 0 -5 2 1 | 0 1 0 2 -2 3 | 0 0 1 2 2 3 }}


Optimal tunings:  
Optimal tunings:  
Line 584: Line 571:
Comma list: 100/99, 144/143, 225/224
Comma list: 100/99, 144/143, 225/224


Mapping: {{mapping| 1 0 0 -5 2 2 | 0 1 0 2 -2 4 | 0 0 1 2 2 -2 }}
{{Mapping|legend=0| 1 0 0 -5 2 2 | 0 1 0 2 -2 4 | 0 0 1 2 2 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 620: Line 607:
Comma list: 45/44, 56/55, 78/77
Comma list: 45/44, 56/55, 78/77


Mapping: {{mapping| 1 0 0 -5 -2 -8 | 0 1 0 2 2 3 | 0 0 1 2 1 3 }}
{{Mapping|legend=0| 1 0 0 -5 -2 -8 | 0 1 0 2 2 3 | 0 0 1 2 1 3 }}


Optimal tunings:  
Optimal tunings:  
Line 660: Line 647:
Comma list: 225/224, 275/273, 325/324
Comma list: 225/224, 275/273, 325/324


Mapping: {{mapping| 1 0 0 -5 -3 2 | 0 1 0 2 7 4 | 0 0 1 2 -2 -2 }}
{{Mapping|legend=0| 1 0 0 -5 -3 2 | 0 1 0 2 7 4 | 0 0 1 2 -2 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 717: Line 704:
Comma list: 169/168, 225/224, 1001/1000
Comma list: 169/168, 225/224, 1001/1000


Mapping: {{mapping| 1 0 0 -5 9 -1 | 0 2 0 4 -7 3 | 0 0 1 2 0 1 }}
{{Mapping|legend=0| 1 0 0 -5 9 -1 | 0 2 0 4 -7 3 | 0 0 1 2 0 1 }}


Optimal tunings:  
Optimal tunings:  
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[[Badness]] (Sintel): 1.38
[[Badness]] (Sintel): 1.38
== Catakleismoid ==
Catakleismoid is the same as catakleismic in the 2.3.5.7.13 subgroup but with an independent generator for prime 11.
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 225/224, 4375/4374
{{Mapping|legend=1| 1 0 1 -3 0 | 0 6 5 22 0 | 0 0 0 0 1 }}
: mapping generators: ~2, ~6/5, ~11
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}, ~11/8 = 549.5258{{c}}
: [[error map]]: {{val| +0.596 -0.619 -1.271 +0.948 -0.003 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}, ~11/8 = 550.0343{{c}}
: error map: {{val| 0.000 -1.332 -2.461 +0.126 -1.284 }}
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce, 341ce }}
[[Badness]] (Sintel): 1.53
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 225/224, 325/324
Mapping: {{mapping| 1 0 1 -3 0 0 | 0 6 5 22 0 14 | 0 0 0 0 1 0 }}
Optimal tunings:
* WE: ~2 = 1200.8238{{c}}, ~6/5 = 316.9478{{c}}, ~11/8 = 548.9611{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}, ~11/8 = 549.5903{{c}}
Optimal ET sequence: {{Optimal ET sequence| 19, 34d, 53, 72, 125f, 197ef }}
Badness (Sintel): 0.853


== Subgroup extensions ==
== Subgroup extensions ==
Line 786: Line 738:
Comma list: 120/119, 225/224
Comma list: 120/119, 225/224


Mapping: {{mapping| 1 0 0 -5 8 | 0 1 0 2 -1 | 0 0 1 2 -1 }}
{{Mapping|legend=0| 1 0 0 -5 8 | 0 1 0 2 -1 | 0 0 1 2 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 801: Line 753:
Comma list: 225/224, 400/399
Comma list: 225/224, 400/399


Mapping: {{mapping| 1 0 0 -5 9 | 0 1 0 2 -3 | 0 0 1 2 0 }}
{{Mapping|legend=0| 1 0 0 -5 9 | 0 1 0 2 -3 | 0 0 1 2 0 }}


Optimal tunings:  
Optimal tunings:  
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<references />
<references />


[[Category:Marvel family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Marvel family| ]] <!-- main article -->
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Marvel_family"