Marvel family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Tags: Mobile edit Mobile web edit
m Units & misc. cleanup
Line 1: Line 1:
{{interwiki
{{interwiki
| en = Marvel family
| de = Marvel
| de = Marvel
| en = Marvel family
| es =  
| es =  
| ja =  
| ja =  
}}
}}
{{Technical data page}}
{{Technical data page}}
The '''marvel family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] tempers out the [[7-limit]] marvel comma, [[225/224]] = {{monzo| -5 2 2 -1 }}, which is 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.


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]] (when using the 34d val) 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]] (when using the 34d val) which temper out the marvel comma.
Line 13: Line 13:
{{Main| Marvel }}
{{Main| Marvel }}


The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[normal lists|normal generator list]] 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]].
The head of the marvel family is marvel, which tempers out [[225/224]]. Marvel has a [[normal lists|normal generator list]] 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]], with which it shares the [[optimal patent val]]. Another temperament which does little damage to tuning accuracy is [[compton]], 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]], with which it shares the [[optimal patent val]]. Another temperament which does little damage to tuning accuracy is [[compton]], for which [[240edo]] may be used. See [[marvel temperaments]] for some other rank-2 temperaments.  
Line 22: Line 22:


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


Line 28: Line 27:


Lattice basis:  
Lattice basis:  
: secor length = 1.256, 3/2 length = 1.369
: ~15/14 length = 1.256, ~3/2 length = 1.369
: Angle (secor, 3/2) = 106.958 degrees
: Angle (~15/14, ~3/2) = 106.958 degrees


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 700.974, ~5/4 = 384.208
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.974{{c}}, ~5/4 = 384.208{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 700.622, ~5/4 = 383.854
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.622{{c}}, ~5/4 = 383.854{{c}}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 45: Line 44:
{{Optimal ET sequence|legend=1| 9, 10, 12, 19, 31, 41, 53, 72, 197, 269c }}
{{Optimal ET sequence|legend=1| 9, 10, 12, 19, 31, 41, 53, 72, 197, 269c }}


[[Badness]]: 0.0365 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.0365 × 10<sup>-3</sup>


[[Projection pair]]s: 7 225/32
[[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
[[Complexity spectrum]]: 4/3, 5/4, 7/5, 7/6, 8/7, 6/5, 9/8, 9/7, 10/9
Line 91: Line 90:


Lattice basis:  
Lattice basis:  
: secor length = 1.0364, 5/4 length = 1.0759
: ~15/14 length = 1.0364, ~5/4 length = 1.0759
: Angle (secor, 5/4) = 104.028 degrees
: angle (~15/14, ~5/4) = 104.028 degrees


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.373, ~5/4 = 383.146
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.373{{c}}, ~5/4 = 383.146{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 700.605, ~5/4 = 383.454
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.605{{c}}, ~5/4 = 383.454{{c}}


[[Minimax tuning]]:
[[Minimax tuning]]:
Line 105: Line 104:
{{Optimal ET sequence|legend=1| 9, 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 238c, 269ce, 341ce }}
{{Optimal ET sequence|legend=1| 9, 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 238c, 269ce, 341ce }}


[[Badness]]: 0.255 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.255 × 10<sup>-3</sup>


[[Projection pair]]s: 7 225/32 11 4096/375
[[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
[[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
Line 133: Line 132:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 700.182, ~5/4 = 384.400
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.182{{c}}, ~5/4 = 384.400{{c}}
* CWE: ~2 = 1\1, ~3/2 = 699.811, ~5/4 = 384.118
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 699.811{{c}}, ~5/4 = 384.118{{c}}


Minimax tuning:  
Minimax tuning:  
Line 140: Line 139:
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.15/11.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.15/11.15/13


Optimal ET sequence: {{Optimal ET sequence| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}
{{Optimal ET sequence|legend=0| 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff }}


Badness: 0.690 × 10<sup>-3</sup>
Badness (Smith): 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
Line 154: Line 153:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.563, ~5/4 = 383.015
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.563{{c}}, ~5/4 = 383.015{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.100, ~5/4 = 383.132
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.100{{c}}, ~5/4 = 383.132{{c}}


Minimax tuning:  
Minimax tuning:  
Line 161: Line 160:
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13


Optimal ET sequence: {{Optimal ET sequence| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}
{{Optimal ET sequence|legend=0| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}


Badness: 0.721 × 10<sup>-3</sup>
Badness (Smith): 0.721 × 10<sup>-3</sup>


Projection pairs: 7 225/32 11 4096/375 13 324/25
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
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
Line 174: Line 173:
Comma list: 225/224, 325/324, 385/384, 400/399
Comma list: 225/224, 325/324, 385/384, 400/399


Sval mapping: {{mapping| 1 0 0 -5 12 2 9 | 0 1 0 2 -1 4 -3 | 0 0 1 2 -3 -2 0 }}
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 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.424, ~5/4 = 383.029
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.424{{c}}, ~5/4 = 383.029{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.200, ~5/4 = 383.114
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.200{{c}}, ~5/4 = 383.114{{c}}


Optimal ET sequence: {{Optimal ET sequence| 41, 53, 72, 94, 113, 166 }}
Optimal ET sequence: {{Optimal ET sequence| 41, 53, 72, 94, 113, 166 }}


Badness: 0.773 × 10<sup>-3</sup>
Badness (Smith): 0.773 × 10<sup>-3</sup>


==== Apotropaia ====
==== Apotropaia ====
Line 192: Line 191:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.586, ~5/4 = 382.733
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.586{{c}}, ~5/4 = 382.733{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.088, ~5/4 = 382.971
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.088{{c}}, ~5/4 = 382.971{{c}}


Optimal ET sequence: {{Optimal ET sequence| 41, 53g, 72, 166g, 238cfg }}
{{Optimal ET sequence|legend=0| 41, 53g, 72, 166g, 238cfg }}


Badness: 0.869 × 10<sup>-3</sup>
Badness (Smith): 0.869 × 10<sup>-3</sup>


===== 19-limit =====
===== 19-limit =====
Line 207: Line 206:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.443, ~5/4 = 382.739
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.443{{c}}, ~5/4 = 382.739{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.201, ~5/4 = 382.940
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.201{{c}}, ~5/4 = 382.940{{c}}


Optimal ET sequence: {{Optimal ET sequence| 41, 53g, 72, 94, 113, 166g }}
{{Optimal ET sequence|legend=0| 41, 53g, 72, 94, 113, 166g }}


Badness: 0.978 × 10<sup>-3</sup>
Badness (Smith): 0.978 × 10<sup>-3</sup>


==== Enodia ====
==== Enodia ====
Line 222: Line 221:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.627, ~5/4 = 383.346
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.627{{c}}, ~5/4 = 383.346{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.095, ~5/4 = 383.314
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.095{{c}}, ~5/4 = 383.314{{c}}


Optimal ET sequence: {{Optimal ET sequence| 41g, 53, 72, 166g, 238cfg }}
{{Optimal ET sequence|legend=0| 41g, 53, 72, 166g, 238cfg }}


Badness: 0.917 × 10<sup>-3</sup>
Badness (Smith): 0.917 × 10<sup>-3</sup>


===== 19-limit =====
===== 19-limit =====
Line 237: Line 236:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.479, ~5/4 = 383.379
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.479{{c}}, ~5/4 = 383.379{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.211, ~5/4 = 383.305
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.211{{c}}, ~5/4 = 383.305{{c}}


Optimal ET sequence: {{Optimal ET sequence| 41g, 53, 72, 94, 125f, 166g }}
{{Optimal ET sequence|legend=0| 41g, 53, 72, 94, 125f, 166g }}


Badness: 1.03 × 10<sup>-3</sup>
Badness (Smith): 1.03 × 10<sup>-3</sup>


=== Marvell ===
=== Marvell ===
Line 252: Line 251:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.385, ~5/4 = 383.208
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.385{{c}}, ~5/4 = 383.208{{c}}
* CWE: ~2 = 1\1, ~3/2 = 700.611, ~5/4 = 383.493
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 700.611{{c}}, ~5/4 = 383.493{{c}}


Minimax tuning:  
Minimax tuning:  
Line 259: Line 258:
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13


Optimal ET sequence: {{Optimal ET sequence| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff }}
{{Optimal ET sequence|legend=0| 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff }}


Badness: 0.862 × 10<sup>-3</sup>
Badness (Smith): 0.862 × 10<sup>-3</sup>


=== Isis ===
=== Isis ===
Line 271: Line 270:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.997, ~5/4 = 383.134
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.997{{c}}, ~5/4 = 383.134{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.928, ~5/4 = 383.228
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.928{{c}}, ~5/4 = 383.228{{c}}


Optimal ET sequence: {{Optimal ET sequence| 10, 19f, 22, 31, 41, 53, 94 }}
{{Optimal ET sequence|legend=0| 10, 19f, 22, 31, 41, 53, 94 }}


Badness: 0.866 × 10<sup>-3</sup>
Badness (Smith): 0.866 × 10<sup>-3</sup>


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


=== Deecee ===
=== Deecee ===
Line 288: Line 287:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.702, ~5/4 = 382.074
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.702{{c}}, ~5/4 = 382.074{{c}}
* CWE: ~2 = 1\1, ~3/2 = 700.718, ~5/4 = 382.661
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 700.718{{c}}, ~5/4 = 382.661{{c}}


Minimax tuning:  
Minimax tuning:  
Line 295: Line 294:
* 15-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.3.13/5
* 15-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.3.13/5


Optimal ET sequence: {{Optimal ET sequence| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}
{{Optimal ET sequence|legend=0| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}


Badness: 0.920 × 10<sup>-3</sup>
Badness (Smith): 0.920 × 10<sup>-3</sup>


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


=== Tripod ===
=== Tripod ===
Line 309: Line 308:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 700.028, ~5/4 = 382.694
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.028{{c}}, ~5/4 = 382.694{{c}}
* CWE: ~2 = 1\1, ~3/2 = 699.404, ~5/4 = 382.917
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 699.404{{c}}, ~5/4 = 382.917{{c}}


Minimax tuning:  
Minimax tuning:  
Line 316: Line 315:
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/11
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/11


Optimal ET sequence: {{Optimal ET sequence| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}
{{Optimal ET sequence|legend=0| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}


Badness: 0.745 × 10<sup>-3</sup>
Badness (Smith): 0.745 × 10<sup>-3</sup>


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


=== Marvelcat ===
=== Marvelcat ===
Line 328: Line 327:


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~26/15 = 950.961, ~5/4 = 383.088
* CTE: ~2 = 1200.000{{c}}, ~26/15 = 950.961{{c}}, ~5/4 = 383.088{{c}}
* CWE: ~2 = 1\1, ~26/15 = 950.426, ~5/4 = 383.479
* CWE: ~2 = 1200.000{{c}}, ~26/15 = 950.426{{c}}, ~5/4 = 383.479{{c}}


Optimal ET sequence: {{Optimal ET sequence| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}
{{Optimal ET sequence|legend=0| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}


Badness: 0.9997 × 10<sup>-3</sup>
Badness (Smith): 0.9997 × 10<sup>-3</sup>


== Minerva ==
== Minerva ==
Line 349: Line 347:


Lattice basis:  
Lattice basis:  
: 16/15 length = 0.8997, 5/4 length = 1.0457
: ~16/15 length = 0.8997, ~5/4 length = 1.0457
: Angle (16/15, 5/4) = 98.6044 degrees
: angle (~16/15, ~5/4) = 98.6044 degrees


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 700.379, ~5/4 = 386.617
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.379{{c}}, ~5/4 = 386.617{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 700.301, ~5/4 = 386.579
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.301{{c}}, ~5/4 = 386.579{{c}}


[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9
Line 360: Line 358:
{{Optimal ET sequence|legend=1| 9, 12, 19e, 22, 31, 53, 84e, 96, 127 }}
{{Optimal ET sequence|legend=1| 9, 12, 19e, 22, 31, 53, 84e, 96, 127 }}


[[Badness]]: 0.381 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.381 × 10<sup>-3</sup>


[[Projection pair]]s: 7 225/32 11 5625/512
[[Projection pair]]s: <code>7 225/32 11 5625/512</code>


Scales: [[minerva12]], [[minerva22x]]
Scales: [[minerva12]], [[minerva22x]]
Line 376: Line 374:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.179, ~5/4 = 385.888
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.179{{c}}, ~5/4 = 385.888{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.214, ~5/4 = 385.934
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.214{{c}}, ~5/4 = 385.934{{c}}


Minimax tuning:  
Minimax tuning:  
Line 383: Line 381:
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7


Optimal ET sequence: {{Optimal ET sequence| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}
{{Optimal ET sequence|legend=0| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}


Badness: 0.818 × 10<sup>-3</sup>
Badness (Smith): 0.818 × 10<sup>-3</sup>


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


== Apollo ==
== Apollo ==
Line 399: Line 397:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 703.418, ~5/4 = 381.331
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.418{{c}}, ~5/4 = 381.331{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 703.461, ~5/4 = 381.507
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.461{{c}}, ~5/4 = 381.507{{c}}


[[Minimax tuning]]: [[11-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9


{{Optimal ET sequence|legend=1| 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce }}
{{Optimal ET sequence|legend=1| 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce }}


[[Projection pair]]s: 7 225/32 11 100/9
[[Projection pair]]s: <code>7 225/32 11 100/9</code>


[[Associated temperament]]: [[magic]]
[[Associated temperament]]: [[magic]]
Line 420: Line 418:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 703.959, ~5/4 = 381.001
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.959{{c}}, ~5/4 = 381.001{{c}}
* CWE: ~2 = 1\1, ~3/2 = 703.985, ~5/4 = 381.358
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 703.985{{c}}, ~5/4 = 381.358{{c}}


Minimax tuning: 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9
Minimax tuning: 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9


Optimal ET sequence: {{Optimal ET sequence| 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef }}
{{Optimal ET sequence|legend=0| 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: <code>7 225/32 11 100/9 13 3200/243</code>


== Potassium ==
== Potassium ==
Line 437: Line 435:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 695.937, ~5/4 = 384.836
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 695.937{{c}}, ~5/4 = 384.836{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 696.059, ~5/4 = 384.947
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 696.059{{c}}, ~5/4 = 384.947{{c}}


[[Minimax tuning]]: [[11-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11
[[Minimax tuning]]: [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11


{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 31e, 50e }}
{{Optimal ET sequence|legend=1| 7d, 9, 10, 12, 19, 31e, 50e }}


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


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


=== 13-limit ===
=== 13-limit ===
Line 456: Line 454:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 695.894, ~5/4 = 384.602
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.894{{c}}, ~5/4 = 384.602{{c}}
* CWE: ~2 = 1\1, ~3/2 = 695.948, ~5/4 = 384.637
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 695.948{{c}}, ~5/4 = 384.637{{c}}


Minimax tuning:  
Minimax tuning:  
Line 463: Line 461:
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9


Optimal ET sequence: {{Optimal ET sequence| 9, 10, 12f, 19, 31e, 50e }}
{{Optimal ET sequence|legend=0| 9, 10, 12f, 19, 31e, 50e }}


Badness: 0.733 × 10<sup>-3</sup>
Badness (Smith): 0.733 × 10<sup>-3</sup>


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


== Malcolm ==
== Malcolm ==
Line 479: Line 477:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.322, ~5/4 = 382.976
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.322{{c}}, ~5/4 = 382.976{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 702.054, ~5/4 = 382.622
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.054{{c}}, ~5/4 = 382.622{{c}}


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


[[Badness]]: 1.250 × 10<sup>-3</sup>
[[Badness]] (Smith): 1.250 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 494: Line 492:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 702.235, ~5/4 = 383.205
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.235{{c}}, ~5/4 = 383.205{{c}}
* CWE: ~2 = 1\1, ~3/2 = 702.088, ~5/4 = 383.063
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.088{{c}}, ~5/4 = 383.063{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef }}
{{Optimal ET sequence|legend=0| 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef }}


Badness: 1.075 × 10<sup>-3</sup>
Badness (Smith): 1.075 × 10<sup>-3</sup>


Scales: [[malco]]
Scales: [[malco]]
Line 515: Line 513:


Lattice basis:  
Lattice basis:  
: secor length = 0.9111, 3/2 length = 0.9477
: ~15/14 length = 0.9111, ~3/2 length = 0.9477
: Angle (secor, 3/2) = 65.933
: angle (~15/14, ~3/2) = 65.933


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 699.995, ~5/4 = 384.330
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 699.995{{c}}, ~5/4 = 384.330{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 699.861, ~5/4 = 383.772
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 699.861{{c}}, ~5/4 = 383.772{{c}}


[[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 basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11


{{Optimal ET sequence|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}
{{Optimal ET sequence|legend=1| 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde }}


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


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


Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]
Scales: [[prodigy11]], [[prodigy12]], [[prodigy29]]
Line 551: Line 549:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 700.617, ~5/4 = 382.244
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.617{{c}}, ~5/4 = 382.244{{c}}
* CWE: ~2 = 1\1, ~3/2 = 700.469, ~5/4 = 381.669
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 700.469{{c}}, ~5/4 = 381.669{{c}}


Optimal ET sequence: {{Optimal ET sequence| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff }}
{{Optimal ET sequence|legend=0| 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff }}


Badness: 0.736 × 10<sup>-3</sup>
Badness (Smith): 0.736 × 10<sup>-3</sup>


=== Prodigious ===
=== Prodigious ===
Line 566: Line 564:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 700.305, ~5/4 = 384.075
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.305{{c}}, ~5/4 = 384.075{{c}}
* CWE: ~2 = 1\1, ~3/2 = 700.330, ~5/4 = 383.503
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 700.330{{c}}, ~5/4 = 383.503{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}
{{Optimal ET sequence|legend=0| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}


Badness: 0.900 × 10<sup>-3</sup>
Badness (Smith): 0.900 × 10<sup>-3</sup>


=== Prodigal ===
=== Prodigal ===
Line 581: Line 579:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 699.698, ~5/4 = 384.884
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 699.698{{c}}, ~5/4 = 384.884{{c}}
* CWE: ~2 = 1\1, ~3/2 = 699.554, ~5/4 = 384.350
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 699.554{{c}}, ~5/4 = 384.350{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}
{{Optimal ET sequence|legend=0| 12f, 19e, 31, 53e, 60eff, 72, 103, 175f }}


Badness: 0.889 × 10<sup>-3</sup>
Badness (Smith): 0.889 × 10<sup>-3</sup>


=== Protannic ===
=== Protannic ===
Line 596: Line 594:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 698.947, ~5/4 = 385.480
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 698.947{{c}}, ~5/4 = 385.480{{c}}
* CWE: ~2 = 1\1, ~3/2 = 699.422, ~5/4 = 383.983
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 699.422{{c}}, ~5/4 = 383.983{{c}}


Optimal ET sequence: {{Optimal ET sequence| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}
{{Optimal ET sequence|legend=0| 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff }}


Badness: 0.953 × 10<sup>-3</sup>
Badness (Smith): 0.953 × 10<sup>-3</sup>


==== 17-limit ====
==== 17-limit ====
Line 611: Line 609:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 698.969, ~5/4 = 385.440
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 698.969{{c}}, ~5/4 = 385.440{{c}}
* CWE: ~2 = 1\1, ~3/2 = 699.485, ~5/4 = 383.874
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 699.485{{c}}, ~5/4 = 383.874{{c}}


Optimal ET sequence: {{Optimal ET sequence| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}
{{Optimal ET sequence|legend=0| 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg }}


Badness: 0.772 × 10<sup>-3</sup>
Badness (Smith): 0.772 × 10<sup>-3</sup>


== Fantastic ==
== Fantastic ==
Line 626: Line 624:


{{Mapping|legend=1| 2 0 0 -10 -7 | 0 1 0 2 0 | 0 0 1 2 3 }}
{{Mapping|legend=1| 2 0 0 -10 -7 | 0 1 0 2 0 | 0 0 1 2 3 }}
: mapping generators: ~99/70, ~3, ~5
: mapping generators: ~99/70, ~3, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~99/70 = 1\2, ~3/2 = 701.132, ~5/4 = 383.926
* [[CTE]]: ~99/70 = 600.000{{c}}, ~3/2 = 701.132{{c}}, ~5/4 = 383.926{{c}}
* [[CWE]]: ~99/70 = 1\2, ~3/2 = 700.816, ~5/4 = 383.535
* [[CWE]]: ~99/70 = 600.000{{c}}, ~3/2 = 700.816{{c}}, ~5/4 = 383.535{{c}}


{{Optimal ET sequence|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}
{{Optimal ET sequence|legend=1| 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c }}


[[Badness]]: 0.743 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.743 × 10<sup>-3</sup>


== Hestia ==
== Hestia ==
Line 645: Line 642:


{{Mapping|legend=1| 1 0 0 -5 9 | 0 2 0 4 -7 | 0 0 1 2 0 }}
{{Mapping|legend=1| 1 0 0 -5 9 | 0 2 0 4 -7 | 0 0 1 2 0 }}
: mapping generators: ~2, ~400/231, ~5
: mapping generators: ~2, ~400/231, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~400/231 = 950.043, ~5/4 = 384.858
* [[CTE]]: ~2 = 1200.000{{c}}, ~400/231 = 950.043{{c}}, ~5/4 = 384.858{{c}}
* [[CWE]]: ~2 = 1\1, ~400/231 = 950.121, ~5/4 = 383.953
* [[CWE]]: ~2 = 1200.000{{c}}, ~400/231 = 950.121{{c}}, ~5/4 = 383.953{{c}}


{{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce }}
{{Optimal ET sequence|legend=1| 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce }}


[[Badness]]: 1.54 × 10<sup>-3</sup>
[[Badness]] (Smith): 1.54 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 664: Line 660:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~26/15 = 950.131, ~5/4 = 385.245
* CTE: ~2 = 1200.000{{c}}, ~26/15 = 950.131{{c}}, ~5/4 = 385.245{{c}}
* CWE: ~2 = 1\1, ~26/15 = 950.211, ~5/4 = 383.949
* CWE: ~2 = 1200.000{{c}}, ~26/15 = 950.211{{c}}, ~5/4 = 383.949{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}
{{Optimal ET sequence|legend=0| 19, 29, 43, 53, 72, 125f, 197ef, 269cef }}


Badness: 1.062 × 10<sup>-3</sup>
Badness (Smith): 1.062 × 10<sup>-3</sup>


== Morfil ==
== Morfil ==
Line 677: Line 673:


{{Mapping|legend=1| 1 0 1 -3 -2 | 0 1 2 6 5 | 0 0 -3 -6 -4 }}
{{Mapping|legend=1| 1 0 1 -3 -2 | 0 1 2 6 5 | 0 0 -3 -6 -4 }}
: mapping generators: ~2, ~3, ~84/55
: mapping generators: ~2, ~3, ~84/55


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.459, ~84/55 = 739.747
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.459{{c}}, ~84/55 = 739.747{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.111, ~84/55 = 739.451
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.111{{c}}, ~84/55 = 739.451{{c}}


{{Optimal ET sequence|legend=1| 29, 31, 60e, 91e, 94, 125 }}
{{Optimal ET sequence|legend=1| 29, 31, 60e, 91e, 94, 125 }}


[[Badness]]: 1.152 × 10<sup>-3</sup>
[[Badness]] (Smith): 1.152 × 10<sup>-3</sup>


== Catakleismoid ==
== Catakleismoid ==
Line 696: Line 691:


{{Mapping|legend=1| 1 0 1 -3 0 | 0 6 5 22 0 | 0 0 0 0 1 }}
{{Mapping|legend=1| 1 0 1 -3 0 | 0 6 5 22 0 | 0 0 0 0 1 }}
: mapping generators: ~2, ~6/5, ~11
: mapping generators: ~2, ~6/5, ~11


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~6/5 = 316.834, ~11/8 = 551.318
* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 316.834{{c}}, ~11/8 = 551.318{{c}}
* [[CWE]]: ~2 = 1\1, ~6/5 = 316.771, ~11/8 = 550.034
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 316.771{{c}}, ~11/8 = 550.034{{c}}


{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce }}
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce }}


[[Badness]]: 1.275 × 10<sup>-3</sup>
[[Badness]] (Smith): 1.275 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 715: Line 709:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 316.886, ~11/8 = 551.318
* CTE: ~2 = 1200.000{{c}}, ~6/5 = 316.886{{c}}, ~11/8 = 551.318{{c}}
* CWE: ~2 = 1\1, ~6/5 = 316.794, ~11/8 = 549.590
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 316.794{{c}}, ~11/8 = 549.590{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19, 34d, 53, 72, 125f, 197ef, 269cef }}
Optimal ET sequence: {{Optimal ET sequence| 19, 34d, 53, 72, 125f, 197ef, 269cef }}


Badness: 0.916 × 10<sup>-3</sup>
Badness (Smith): 0.916 × 10<sup>-3</sup>


== Subgroup extensions ==
== Subgroup extensions ==
Line 730: Line 724:
Mapping: {{mapping| 1 0 0 -5 8 | 0 1 0 2 -1 | 0 0 1 2 -1 }}
Mapping: {{mapping| 1 0 0 -5 8 | 0 1 0 2 -1 | 0 0 1 2 -1 }}


Optimal tuning:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.244, ~5/4 = 384.789
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.244{{c}}, ~5/4 = 384.789{{c}}


{{Optimal ET sequence| legend=1 | 9, 10, 12, 19, 22, 31, 41, 53 }}
{{Optimal ET sequence|legend=0| 9, 10, 12, 19, 22, 31, 41, 53 }}


=== Devimarvel ===
=== Devimarvel ===
Line 742: Line 736:
Mapping: {{mapping| 1 0 0 -5 9 | 0 1 0 2 -3 | 0 0 1 2 0 }}
Mapping: {{mapping| 1 0 0 -5 9 | 0 1 0 2 -3 | 0 0 1 2 0 }}


Optimal tuning:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.904, ~5/4 = 384.260
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.904{{c}}, ~5/4 = 384.260{{c}}


{{Optimal ET sequence| legend=1 | 10, 12, 19, 22, 31, 41, 53, 72, 94 }}
{{Optimal ET sequence|legend=0| 10, 12, 19, 22, 31, 41, 53, 72, 94 }}


== Notes ==
== References ==
<references />


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Marvel family| ]] <!-- main article -->
[[Category:Marvel family| ]] <!-- main article -->
[[Category:Marvel| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
<references />

Revision as of 12:45, 12 December 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The marvel family of rank-3 temperaments tempers out the 7-limit marvel comma (ratio: 225/224, monzo: [-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 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 (when using the 34d val) 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 generator list 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, with which it shares the optimal patent val. Another temperament which does little damage to tuning accuracy is compton, for which 240edo may be used. See marvel temperaments for some other rank-2 temperaments.

Subgroup: 2.3.5.7

Comma list: 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:

~15/14 length = 1.256, ~3/2 length = 1.369
Angle (~15/14, ~3/2) = 106.958 degrees

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 700.974 ¢, ~5/4 = 384.208 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 700.622 ¢, ~5/4 = 383.854 ¢

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⟩]
unchanged-interval (eigenmonzo) basis: 2.5/3.7
  • 9-odd-limit: 3 1/6-comma flat, 5 1/3-comma flat, 7 just
[[1 0 0 0⟩, [5/6 2/3 -1/3 1/6⟩, [5/3 -2/3 1/3 1/3⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5.7

Optimal ET sequence: 9, 10, 12, 19, 31, 41, 53, 72, 197, 269c

Badness (Smith): 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

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

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

Overview to 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. These and many others considered below use the same generators as marvel.

Temperaments discussed elsewhere include

Undecimal marvel (unimarv)

Main article: Marvel

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]]

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

Lattice basis:

~15/14 length = 1.0364, ~5/4 length = 1.0759
angle (~15/14, ~5/4) = 104.028 degrees

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.373 ¢, ~5/4 = 383.146 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 700.605 ¢, ~5/4 = 383.454 ¢

Minimax tuning:

[[1 0 0 0 0⟩, [4/3 8/9 -1/3 0 -1/9⟩, [8/3 -2/9 1/3 0 -2/9⟩, [3 4/3 0 0 -2/3⟩, [8/3 -2/9 -2/3 0 7/9⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5.11/9

Optimal ET sequence: 9, 10, 12e, 19, 22, 31, 41, 53, 72, 166, 197e, 238c, 269ce, 341ce

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 700.182 ¢, ~5/4 = 384.400 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 699.811 ¢, ~5/4 = 384.118 ¢

Minimax tuning:

  • 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9
  • 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.15/11.15/13

Optimal ET sequence: 19, 22, 31, 50, 53, 72, 103, 175f, 300ceff, 403bceeff, 578bbccdeeefff

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.563 ¢, ~5/4 = 383.015 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.100 ¢, ~5/4 = 383.132 ¢

Minimax tuning:

  • 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.13/5
  • 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13

Optimal ET sequence: 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf

Badness (Smith): 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

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

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

Subgroup-val mapping: [⟨1 0 0 -5 12 2 9], ⟨0 1 0 2 -1 4 -3], ⟨0 0 1 2 -3 -2 0]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.424 ¢, ~5/4 = 383.029 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.200 ¢, ~5/4 = 383.114 ¢

Optimal ET sequence: 41, 53, 72, 94, 113, 166

Badness (Smith): 0.773 × 10-3

Apotropaia

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.586 ¢, ~5/4 = 382.733 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.088 ¢, ~5/4 = 382.971 ¢

Optimal ET sequence: 41, 53g, 72, 166g, 238cfg

Badness (Smith): 0.869 × 10-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.443 ¢, ~5/4 = 382.739 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.201 ¢, ~5/4 = 382.940 ¢

Optimal ET sequence: 41, 53g, 72, 94, 113, 166g

Badness (Smith): 0.978 × 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 -13], ⟨0 1 0 2 -1 4 2], ⟨0 0 1 2 -3 -2 6]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.627 ¢, ~5/4 = 383.346 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.095 ¢, ~5/4 = 383.314 ¢

Optimal ET sequence: 41g, 53, 72, 166g, 238cfg

Badness (Smith): 0.917 × 10-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 325/324, 375/374, 385/384, 400/399

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.479 ¢, ~5/4 = 383.379 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.211 ¢, ~5/4 = 383.305 ¢

Optimal ET sequence: 41g, 53, 72, 94, 125f, 166g

Badness (Smith): 1.03 × 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.385 ¢, ~5/4 = 383.208 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 700.611 ¢, ~5/4 = 383.493 ¢

Minimax tuning:

  • 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.11/9
  • 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7.15/13

Optimal ET sequence: 9, 22f, 31, 63, 72, 103, 166, 238cf, 269ce, 507bcceff, 610bcceeff

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.997 ¢, ~5/4 = 383.134 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.928 ¢, ~5/4 = 383.228 ¢

Optimal ET sequence: 10, 19f, 22, 31, 41, 53, 94

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.702 ¢, ~5/4 = 382.074 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 700.718 ¢, ~5/4 = 382.661 ¢

Minimax tuning:

  • 13-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.9/5.13/9
  • 15-odd-limit eigenmonzo subgroup (unchanged-interval basis): 2.3.13/5

Optimal ET sequence: 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 700.028 ¢, ~5/4 = 382.694 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 699.404 ¢, ~5/4 = 382.917 ¢

Minimax tuning:

  • 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/11
  • 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/11

Optimal ET sequence: 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df

Badness (Smith): 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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~26/15 = 950.961 ¢, ~5/4 = 383.088 ¢
  • CWE: ~2 = 1200.000 ¢, ~26/15 = 950.426 ¢, ~5/4 = 383.479 ¢

Optimal ET sequence: 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff

Badness (Smith): 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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 700.379 ¢, ~5/4 = 386.617 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 700.301 ¢, ~5/4 = 386.579 ¢

Minimax tuning: 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/5.11/9

Optimal ET sequence: 9, 12, 19e, 22, 31, 53, 84e, 96, 127

Badness (Smith): 0.381 × 10-3

Projection pairs: 7 225/32 11 5625/512

Scales: 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.179 ¢, ~5/4 = 385.888 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.214 ¢, ~5/4 = 385.934 ¢

Minimax tuning:

  • 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7
  • 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/7

Optimal ET sequence: 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def

Badness (Smith): 0.818 × 10-3

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

Apollo

See also: Ptolemismic clan § 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 703.418 ¢, ~5/4 = 381.331 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 703.461 ¢, ~5/4 = 381.507 ¢

Minimax tuning: 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/5.11/9

Optimal ET sequence: 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce

Projection pairs: 7 225/32 11 100/9

Associated temperament: magic

Scales: apollo wholetone, indigo17

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 703.959 ¢, ~5/4 = 381.001 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 703.985 ¢, ~5/4 = 381.358 ¢

Minimax tuning: 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.11/9.13/9

Optimal ET sequence: 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 695.937 ¢, ~5/4 = 384.836 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 696.059 ¢, ~5/4 = 384.947 ¢

Minimax tuning: 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.11

Optimal ET sequence: 7d, 9, 10, 12, 19, 31e, 50e

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 695.894 ¢, ~5/4 = 384.602 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 695.948 ¢, ~5/4 = 384.637 ¢

Minimax tuning:

  • 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9
  • 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7.13/9

Optimal ET sequence: 9, 10, 12f, 19, 31e, 50e

Badness (Smith): 0.733 × 10-3

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

Malcolm

"Malcolm" redirects here. For Alexander Malcolm's JI scale, see Malcolm (scale).

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.322 ¢, ~5/4 = 382.976 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.054 ¢, ~5/4 = 382.622 ¢

Optimal ET sequence: 12e, 19e, 34d, 41, 53, 60e, 94, 229c, 248ce, 289cce, 342ccee, 383cce

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.235 ¢, ~5/4 = 383.205 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.088 ¢, ~5/4 = 383.063 ¢

Optimal ET sequence: 12e, 19e, 34d, 41, 53, 94, 429ccdeef, 523ccdeef

Badness (Smith): 1.075 × 10-3

Scales: malco

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. 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:

~15/14 length = 0.9111, ~3/2 length = 0.9477
angle (~15/14, ~3/2) = 65.933

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 699.995 ¢, ~5/4 = 384.330 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 699.861 ¢, ~5/4 = 383.772 ¢

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⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5.11

Optimal ET sequence: 10, 12, 19e, 29, 31, 41, 60e, 72, 247c, 319bcde, 391bcde, 463bccde

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 700.617 ¢, ~5/4 = 382.244 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 700.469 ¢, ~5/4 = 381.669 ¢

Optimal ET sequence: 10, 12f, 19e, 29, 31, 41, 60e, 72f, 91e, 101cd, 132def, 233ccddeef, 274ccddeff, 305ccddeeff

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 700.305 ¢, ~5/4 = 384.075 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 700.330 ¢, ~5/4 = 383.503 ¢

Optimal ET sequence: 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 699.698 ¢, ~5/4 = 384.884 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 699.554 ¢, ~5/4 = 384.350 ¢

Optimal ET sequence: 12f, 19e, 31, 53e, 60eff, 72, 103, 175f

Badness (Smith): 0.889 × 10-3

Protannic

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 441/440, 1001/1000

Mapping: [⟨1 0 0 -5 -13 21], ⟨0 1 0 2 6 -8], ⟨0 0 1 2 3 -2]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 698.947 ¢, ~5/4 = 385.480 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 699.422 ¢, ~5/4 = 383.983 ¢

Optimal ET sequence: 29, 31, 43, 60e, 72, 103, 175f, 482bccddeefff, 554bbccddeeeffff

Badness (Smith): 0.953 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 375/374, 441/440

Mapping: [⟨1 0 0 -5 -13 21 12], ⟨0 1 0 2 6 -8 -5], ⟨0 0 1 2 3 -2 0]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 698.969 ¢, ~5/4 = 385.440 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 699.485 ¢, ~5/4 = 383.874 ¢

Optimal ET sequence: 29g, 31, 43, 60e, 72, 103, 175f, 307bcdeeffg, 379bccdeeffgg, 482bccddeefffgg, 554bbccddeeeffffgg

Badness (Smith): 0.772 × 10-3

Fantastic

Besides 4375/4356, fantastic also tempers out 9801/9800 and splits the octave in two.

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

Optimal tunings:

  • CTE: ~99/70 = 600.000 ¢, ~3/2 = 701.132 ¢, ~5/4 = 383.926 ¢
  • CWE: ~99/70 = 600.000 ¢, ~3/2 = 700.816 ¢, ~5/4 = 383.535 ¢

Optimal ET sequence: 12, 22, 34d, 50, 60e, 72, 166, 238c, 310c

Badness (Smith): 0.743 × 10-3

Hestia

Named by Graham Breed in 2011, hestia was found to be locally efficient in the higher limits among all rank-3 extensions of marvel[1], although it is a weak extension.

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~400/231 = 950.043 ¢, ~5/4 = 384.858 ¢
  • CWE: ~2 = 1200.000 ¢, ~400/231 = 950.121 ¢, ~5/4 = 383.953 ¢

Optimal ET sequence: 19, 29, 43, 53, 72, 197e, 269ce, 341ce, 610bce

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~26/15 = 950.131 ¢, ~5/4 = 385.245 ¢
  • CWE: ~2 = 1200.000 ¢, ~26/15 = 950.211 ¢, ~5/4 = 383.949 ¢

Optimal ET sequence: 19, 29, 43, 53, 72, 125f, 197ef, 269cef

Badness (Smith): 1.062 × 10-3

Morfil

Subgroup: 2.3.5.7.11

Comma list: 225/224, 1331/1323

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

mapping generators: ~2, ~3, ~84/55

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.459 ¢, ~84/55 = 739.747 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.111 ¢, ~84/55 = 739.451 ¢

Optimal ET sequence: 29, 31, 60e, 91e, 94, 125

Badness (Smith): 1.152 × 10-3

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~6/5 = 316.834 ¢, ~11/8 = 551.318 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 316.771 ¢, ~11/8 = 550.034 ¢

Optimal ET sequence: 19, 34d, 53, 72, 197e, 269ce

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~6/5 = 316.886 ¢, ~11/8 = 551.318 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 316.794 ¢, ~11/8 = 549.590 ¢

Optimal ET sequence: 19, 34d, 53, 72, 125f, 197ef, 269cef

Badness (Smith): 0.916 × 10-3

Subgroup extensions

Char

Subgroup: 2.3.5.7.17

Comma list: 120/119, 225/224

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.244 ¢, ~5/4 = 384.789 ¢

Optimal ET sequence: 9, 10, 12, 19, 22, 31, 41, 53

Devimarvel

Subgroup: 2.3.5.7.19

Comma list: 225/224, 400/399

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.904 ¢, ~5/4 = 384.260 ¢

Optimal ET sequence: 10, 12, 19, 22, 31, 41, 53, 72, 94

References

  1. ↑ Yahoo! Tuning Group | Artemis and friends
Retrieved from "https://en.xen.wiki/index.php?title=Marvel_family&oldid=218839"