Marvel family: Difference between revisions

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m Units & misc. cleanup
Switch to Sintel's badness, WE & CWE tunings
Line 28: Line 28:
Lattice basis:  
Lattice basis:  
: ~15/14 length = 1.256, ~3/2 length = 1.369
: ~15/14 length = 1.256, ~3/2 length = 1.369
: Angle (~15/14, ~3/2) = 106.958 degrees
: angle (~15/14, ~3/2) = 106.958°


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.974{{c}}, ~5/4 = 384.208{{c}}
* [[WE]]: ~2 = 1200.5971{{c}}, ~3/2 = 700.7560{{c}}, ~5/4 = 383.8285{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.622{{c}}, ~5/4 = 383.854{{c}}
: [[error map]]: {{val| +0.597 -0.602 -1.291 +0.940 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.6222{{c}}, ~5/4 = 383.8540{{c}}
: error map: {{val| 0.000 -1.333 -2.460 +0.127 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.974{{c}}, ~5/4 = 384.208{{c}} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 44: Line 47:
{{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]] (Smith): 0.0365 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.161


[[Projection pair]]s: <code>7 225/32</code>
[[Projection pair]]s: <code>7 225/32</code>
Line 91: Line 94:
Lattice basis:  
Lattice basis:  
: ~15/14 length = 1.0364, ~5/4 length = 1.0759
: ~15/14 length = 1.0364, ~5/4 length = 1.0759
: angle (~15/14, ~5/4) = 104.028 degrees
: angle (~15/14, ~5/4) = 104.028°


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.373{{c}}, ~5/4 = 383.146{{c}}
* [[WE]]: ~2 = 1200.6395{{c}}, ~3/2 = 700.7619{{c}}, ~5/4 = 383.7447{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.605{{c}}, ~5/4 = 383.454{{c}}
: [[error map]]: {{val| +0.639 -0.554 -1.290 +0.827 -0.117 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.6048{{c}}, ~5/4 = 383.4538{{c}}
: error map: {{val| 0.000 -1.350 -2.860 -0.709 -2.284 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.373{{c}}, ~5/4 = 383.146{{c}} -->


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 8/9 -1/3 0 -1/9 }}, {{monzo| 8/3 -2/9 1/3 0 -2/9 }}, {{monzo| 3 4/3 0 0 -2/3 }}, {{monzo| 8/3 -2/9 -2/3 0 7/9 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 8/9 -1/3 0 -1/9 }}, {{monzo| 8/3 -2/9 1/3 0 -2/9 }}, {{monzo| 3 4/3 0 0 -2/3 }}, {{monzo| 8/3 -2/9 -2/3 0 7/9 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9


{{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]] (Smith): 0.255 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.306


[[Projection pair]]s: <code>7 225/32 11 4096/375</code>
[[Projection pair]]s: <code>7 225/32 11 4096/375</code>
Line 132: Line 138:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.182{{c}}, ~5/4 = 384.400{{c}}
* WE: ~2 = 1200.8043{{c}}, ~3/2 = 700.2057{{c}}, ~5/4 = 384.3188{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 699.811{{c}}, ~5/4 = 384.118{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.8109{{c}}, ~5/4 = 384.1177{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.182{{c}}, ~5/4 = 384.400{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 141: Line 148:
{{Optimal ET sequence|legend=0| 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 (Smith): 0.690 × 10<sup>-3</sup>
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
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 ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 153: Line 161:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.563{{c}}, ~5/4 = 383.015{{c}}
* WE: ~2 = 1200.5788{{c}}, ~3/2 = 701.3161{{c}}, ~5/4 = 383.3471{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.100{{c}}, ~5/4 = 383.132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1003{{c}}, ~5/4 = 383.1315{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.563{{c}}, ~5/4 = 383.015{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 162: Line 171:
{{Optimal ET sequence|legend=0| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}
{{Optimal ET sequence|legend=0| 19, 22f, 31f, 41, 53, 72, 125f, 166, 238cf }}


Badness (Smith): 0.721 × 10<sup>-3</sup>
Badness (Sintel): 0.674


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>
Line 168: Line 177:
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


==== 2.3.5.7.11.13.19 subgroup ====
===== 2.3.5.7.11.13.19 subgroup =====
Subgroup: 2.3.5.7.11.13.19
Subgroup: 2.3.5.7.11.13.19


Line 176: Line 185:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.424{{c}}, ~5/4 = 383.029{{c}}
* WE: ~2 = 1200.4716{{c}}, ~3/2 = 701.4395{{c}}, ~5/4 = 383.2779{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.200{{c}}, ~5/4 = 383.114{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2002{{c}}, ~5/4 = 383.1136{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.424{{c}}, ~5/4 = 383.029{{c}} -->


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


Badness (Smith): 0.773 × 10<sup>-3</sup>
Badness (Sintel): 0.756


==== Apotropaia ====
==== Apotropaia ====
Line 191: Line 201:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.586{{c}}, ~5/4 = 382.733{{c}}
* WE: ~2 = 1200.6057{{c}}, ~3/2 = 701.3157{{c}}, ~5/4 = 383.2243{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.088{{c}}, ~5/4 = 382.971{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0877{{c}}, ~5/4 = 382.9709{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.586{{c}}, ~5/4 = 382.733{{c}} -->


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


Badness (Smith): 0.869 × 10<sup>-3</sup>
Badness (Sintel): 0.827


===== 19-limit =====
===== 19-limit =====
Line 206: Line 217:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.443{{c}}, ~5/4 = 382.739{{c}}
* WE: ~2 = 1200.4927{{c}}, ~3/2 = 701.4506{{c}}, ~5/4 = 383.1291{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.201{{c}}, ~5/4 = 382.940{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2007{{c}}, ~5/4 = 382.9404{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.443{{c}}, ~5/4 = 382.739{{c}} -->


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


Badness (Smith): 0.978 × 10<sup>-3</sup>
Badness (Sintel): 1.01


==== Enodia ====
==== Enodia ====
Line 221: Line 233:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.627{{c}}, ~5/4 = 383.346{{c}}
* WE: ~2 = 1200.6165{{c}}, ~3/2 = 701.3259{{c}}, ~5/4 = 383.5032{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.095{{c}}, ~5/4 = 383.314{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0949{{c}}, ~5/4 = 383.3140{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.627{{c}}, ~5/4 = 383.346{{c}} -->


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


Badness (Smith): 0.917 × 10<sup>-3</sup>
Badness (Sintel): 0.872


===== 19-limit =====
===== 19-limit =====
Line 236: Line 249:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.479{{c}}, ~5/4 = 383.379{{c}}
* WE: ~2 = 1200.5038{{c}}, ~3/2 = 701.4654{{c}}, ~5/4 = 383.4549{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.211{{c}}, ~5/4 = 383.305{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2113{{c}}, ~5/4 = 383.3052{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.479{{c}}, ~5/4 = 383.379{{c}} -->


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


Badness (Smith): 1.03 × 10<sup>-3</sup>
Badness (Sintel): 1.05


=== Marvell ===
=== Marvell ===
Line 251: Line 265:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.385{{c}}, ~5/4 = 383.208{{c}}
* WE: ~2 = 1200.6404{{c}}, ~3/2 = 700.7675{{c}}, ~5/4 = 383.7772{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 700.611{{c}}, ~5/4 = 383.493{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6113{{c}}, ~5/4 = 383.4925{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.385{{c}}, ~5/4 = 383.208{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 258: Line 273:
* 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|legend=0| 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, 341cef }}


Badness (Smith): 0.862 × 10<sup>-3</sup>
Badness (Sintel): 0.806


=== Isis ===
=== Isis ===
Line 270: Line 285:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.997{{c}}, ~5/4 = 383.134{{c}}
* WE: ~2 = 1200.1824{{c}}, ~3/2 = 702.0223{{c}}, ~5/4 = 383.3028{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.928{{c}}, ~5/4 = 383.228{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9276{{c}}, ~5/4 = 383.2283{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.997{{c}}, ~5/4 = 383.134{{c}} -->


{{Optimal ET sequence|legend=0| 10, 19f, 22, 31, 41, 53, 94 }}
{{Optimal ET sequence|legend=0| 10, 19f, 22, 31, 41, 53, 84e, 94 }}


Badness (Smith): 0.866 × 10<sup>-3</sup>
Badness (Sintel): 0.810


Projection pairs: <code>7 225/32 11 4096/375 13 131072/10125</code>
Projection pairs: <code>7 225/32 11 4096/375 13 131072/10125</code>
Line 287: Line 303:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.702{{c}}, ~5/4 = 382.074{{c}}
* WE: ~2 = 1200.7533{{c}}, ~3/2 = 700.8957{{c}}, ~5/4 = 383.0580{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 700.718{{c}}, ~5/4 = 382.661{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.7184{{c}}, ~5/4 = 382.6611{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.702{{c}}, ~5/4 = 382.074{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 296: Line 313:
{{Optimal ET sequence|legend=0| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}
{{Optimal ET sequence|legend=0| 9, 19f, 22, 31f, 41, 63, 72, 185cf, 257cff }}


Badness (Smith): 0.920 × 10<sup>-3</sup>
Badness (Sintel): 0.861


Projection pairs: <code>7 225/32 11 4096/375 13 134217728/10546875</code>
Projection pairs: <code>7 225/32 11 4096/375 13 134217728/10546875</code>
Line 308: Line 325:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.028{{c}}, ~5/4 = 382.694{{c}}
* WE: ~2 = 1200.4789{{c}}, ~3/2 = 699.5125{{c}}, ~5/4 = 383.1304{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 699.404{{c}}, ~5/4 = 382.917{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4041{{c}}, ~5/4 = 382.9166{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.028{{c}}, ~5/4 = 382.694{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 315: Line 333:
* 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|legend=0| 9, 10, 19, 22f, 31, 41, 72f, 91, 122f, 163df }}
{{Optimal ET sequence|legend=0| 9, 10, 19, 22f, 31, 41, 72f, 91 }}


Badness (Smith): 0.745 × 10<sup>-3</sup>
Badness (Sintel): 0.697


Projection pairs: <code>7 225/32 11 4096/375 13 3375/256</code>
Projection pairs: <code>7 225/32 11 4096/375 13 3375/256</code>
Line 330: Line 348:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~26/15 = 950.961{{c}}, ~5/4 = 383.088{{c}}
* WE: ~2 = 1200.7720{{c}}, ~3/2 = 950.8976{{c}}, ~5/4 = 383.8283{{c}}
* CWE: ~2 = 1200.000{{c}}, ~26/15 = 950.426{{c}}, ~5/4 = 383.479{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.4265{{c}}, ~5/4 = 383.4790{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~26/15 = 950.961{{c}}, ~5/4 = 383.088{{c}} -->


{{Optimal ET sequence|legend=0| 9, 10, 19, 44, 53, 72, 125f, 197ef, 269ceff }}
{{Optimal ET sequence|legend=0| 9, 10, 19, 43e, 44, 53, 72, 125f, 197ef }}


Badness (Smith): 0.9997 × 10<sup>-3</sup>
Badness (Sintel): 0.935


== Minerva ==
== Minerva ==
Line 348: Line 367:
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°


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.379{{c}}, ~5/4 = 386.617{{c}}
* [[WE]]: ~2 = 1200.1086{{c}}, ~3/2 = 700.3226{{c}}, ~5/4 = 386.5931{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.301{{c}}, ~5/4 = 386.579{{c}}
: [[error map]]: {{val| +0.109 -1.524 +0.497 +5.114 -4.192 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.3006{{c}}, ~5/4 = 386.5785{{c}}
: error map: {{val| 0.000 -1.654 +0.265 +4.932 -4.403 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.379{{c}}, ~5/4 = 386.617{{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 358: Line 380:
{{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]] (Smith): 0.381 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.458


[[Projection pair]]s: <code>7 225/32 11 5625/512</code>
[[Projection pair]]s: <code>7 225/32 11 5625/512</code>
Line 374: Line 396:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.179{{c}}, ~5/4 = 385.888{{c}}
* WE: ~2 = 1199.9127{{c}}, ~3/2 = 701.1832{{c}}, ~5/4 = 385.9313{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.214{{c}}, ~5/4 = 385.934{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2143{{c}}, ~5/4 = 385.9336{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.179{{c}}, ~5/4 = 385.888{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 381: Line 404:
* 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|legend=0| 12f, 19e, 22, 31, 53, 84e, 118d, 171de, 202def }}
{{Optimal ET sequence|legend=0| 12f, 19e, 22, 31, 53, 84e, 118d }}


Badness (Smith): 0.818 × 10<sup>-3</sup>
Badness (Sintel): 0.765


Projection pairs: <code>7 225/32 11 5625/512 13 625/48</code>
Projection pairs: <code>7 225/32 11 5625/512 13 625/48</code>
Line 397: Line 420:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.418{{c}}, ~5/4 = 381.331{{c}}
* [[WE]]: ~2 = 1199.8250{{c}}, ~3/2 = 703.3820{{c}}, ~5/4 = 381.5476{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.461{{c}}, ~5/4 = 381.507{{c}}
: [[error map]]: {{val| -0.175 +1.252 -5.116 +0.858 +4.313 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4612{{c}}, ~5/4 = 381.5071{{c}}
: error map: {{val| 0.000 +1.506 -4.807 +1.111 +4.774 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.418{{c}}, ~5/4 = 381.331{{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 }}
 
[[Badness]] (Sintel): 0.637


[[Projection pair]]s: <code>7 225/32 11 100/9</code>
[[Projection pair]]s: <code>7 225/32 11 100/9</code>
Line 418: Line 446:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.959{{c}}, ~5/4 = 381.001{{c}}
* WE: ~2 = 1199.6919{{c}}, ~3/2 = 703.8176{{c}}, ~5/4 = 381.4372{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 703.985{{c}}, ~5/4 = 381.358{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.9853{{c}}, ~5/4 = 381.3579{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.959{{c}}, ~5/4 = 381.001{{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|legend=0| 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 }}
 
Badness (Sintel): 0.964


Projection pairs: <code>7 225/32 11 100/9 13 3200/243</code>
Projection pairs: <code>7 225/32 11 100/9 13 3200/243</code>
Line 435: Line 466:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 695.937{{c}}, ~5/4 = 384.836{{c}}
* [[WE]]: ~2 = 1199.7106{{c}}, ~3/2 = 696.0036{{c}}, ~5/4 = 384.9572{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 696.059{{c}}, ~5/4 = 384.947{{c}}
: [[error map]]: {{val| -0.289 -6.241 -1.935 -7.194 +25.068 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.0586{{c}}, ~5/4 = 384.9472{{c}}
: error map: {{val| 0.000 -5.896 -1.367 -6.814 +25.746 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 695.937{{c}}, ~5/4 = 384.836{{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 }}


[[Badness]] (Smith): 0.464 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.557


[[Projection pair]]s: <code>7 225/32 11 45/4</code>
[[Projection pair]]s: <code>7 225/32 11 45/4</code>
Line 454: Line 488:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.894{{c}}, ~5/4 = 384.602{{c}}
* WE: ~2 = 1199.8192{{c}}, ~3/2 = 695.9054{{c}}, ~5/4 = 384.6205{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 695.948{{c}}, ~5/4 = 384.637{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.9480{{c}}, ~5/4 = 384.6372{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.894{{c}}, ~5/4 = 384.602{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 461: Line 496:
* 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|legend=0| 9, 10, 12f, 19, 31e, 50e }}
{{Optimal ET sequence|legend=0| 9, 10, 12f, 19, 31e }}


Badness (Smith): 0.733 × 10<sup>-3</sup>
Badness (Sintel): 0.686


Projection pairs: <code>7 225/32 11 45/4 13 3375/256</code>
Projection pairs: <code>7 225/32 11 45/4 13 3375/256</code>
Line 477: Line 512:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.322{{c}}, ~5/4 = 382.976{{c}}
* [[WE]]: ~2 = 1199.5261{{c}}, ~3/2 = 702.1990{{c}}, ~5/4 = 382.5759{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.054{{c}}, ~5/4 = 382.622{{c}}
: [[error map]]: {{val| +0.526 +0.770 -2.686 +1.250 -1.077 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0535{{c}}, ~5/4 = 382.6222{{c}}
: error map: {{val| 0.000 +0.098 -3.692 +0.525 -2.188 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.322{{c}}, ~5/4 = 382.976{{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 }}


[[Badness]] (Smith): 1.250 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.50


=== 13-limit ===
=== 13-limit ===
Line 492: Line 530:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.235{{c}}, ~5/4 = 383.205{{c}}
* WE: ~2 = 1200.2427{{c}}, ~3/2 = 702.1575{{c}}, ~5/4 = 383.0704{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.088{{c}}, ~5/4 = 383.063{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0882{{c}}, ~5/4 = 383.0630{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.235{{c}}, ~5/4 = 383.205{{c}} -->


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


Badness (Smith): 1.075 × 10<sup>-3</sup>
Badness (Sintel): 1.00


Scales: [[malco]]
Scales: [[malco]]
Line 514: Line 553:
Lattice basis:  
Lattice basis:  
: ~15/14 length = 0.9111, ~3/2 length = 0.9477
: ~15/14 length = 0.9111, ~3/2 length = 0.9477
: angle (~15/14, ~3/2) = 65.933
: angle (~15/14, ~3/2) = 65.933°


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 699.995{{c}}, ~5/4 = 384.330{{c}}
* [[WE]]: ~2 = 1200.7854{{c}}, ~3/2 = 700.2562{{c}}, ~5/4 = 383.7624{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 699.861{{c}}, ~5/4 = 383.772{{c}}
: [[error map]]: {{val| +0.785 -0.913 -0.980 -0.003 +0.721 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.8610{{c}}, ~5/4 = 383.7724{{c}}
: error map: {{val| 0.000 -2.094 -2.541 -1.559 -0.835 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 699.995{{c}}, ~5/4 = 384.330{{c}} -->


[[Minimax tuning]]:
[[Minimax tuning]]:
Line 527: Line 569:
{{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]] (Smith): 0.344 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.402


[[Projection pair]]s: <code>7 225/32 11 91125/8192</code>
[[Projection pair]]s: <code>7 225/32 11 91125/8192</code>
Line 549: Line 591:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.617{{c}}, ~5/4 = 382.244{{c}}
* WE: ~2 = 1200.8252{{c}}, ~3/2 = 700.8823{{c}}, ~5/4 = 381.6647{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 700.469{{c}}, ~5/4 = 381.669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.4689{{c}}, ~5/4 = 381.6687{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.617{{c}}, ~5/4 = 382.244{{c}} -->


{{Optimal ET sequence|legend=0| 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, 101cd }}


Badness (Smith): 0.736 × 10<sup>-3</sup>
Badness (Sintel): 0.689


=== Prodigious ===
=== Prodigious ===
Line 564: Line 607:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.305{{c}}, ~5/4 = 384.075{{c}}
* WE: ~2 = 1200.6284{{c}}, ~3/2 = 700.7075{{c}}, ~5/4 = 383.4599{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 700.330{{c}}, ~5/4 = 383.503{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.3302{{c}}, ~5/4 = 383.5030{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 700.305{{c}}, ~5/4 = 384.075{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 29, 31f, 41, 72, 185cf, 341cf, 413bcff, 526bccdff }}
{{Optimal ET sequence|legend=0| 12f, 29, 31f, 41, 72, 185cf, 257cff }}


Badness (Smith): 0.900 × 10<sup>-3</sup>
Badness (Sintel): 0.841


=== Prodigal ===
=== Prodigal ===
Line 579: Line 623:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 699.698{{c}}, ~5/4 = 384.884{{c}}
* WE: ~2 = 1200.7798{{c}}, ~3/2 = 699.9410{{c}}, ~5/4 = 384.3494{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 699.554{{c}}, ~5/4 = 384.350{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.5538{{c}}, ~5/4 = 384.3496{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 699.698{{c}}, ~5/4 = 384.884{{c}} -->


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


Badness (Smith): 0.889 × 10<sup>-3</sup>
Badness (Sintel): 0.831


=== Protannic ===
=== Protannic ===
Line 594: Line 639:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 698.947{{c}}, ~5/4 = 385.480{{c}}
* WE: ~2 = 1200.9450{{c}}, ~3/2 = 700.1045{{c}}, ~5/4 = 383.8716{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 699.422{{c}}, ~5/4 = 383.983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4224{{c}}, ~5/4 = 383.9828{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 698.947{{c}}, ~5/4 = 385.480{{c}} -->


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


Badness (Smith): 0.953 × 10<sup>-3</sup>
Badness (Sintel): 0.891


==== 17-limit ====
==== 17-limit ====
Line 609: Line 655:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 698.969{{c}}, ~5/4 = 385.440{{c}}
* WE: ~2 = 1200.9342{{c}}, ~3/2 = 700.1708{{c}}, ~5/4 = 383.7444{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 699.485{{c}}, ~5/4 = 383.874{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.4849{{c}}, ~5/4 = 383.8744{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 698.969{{c}}, ~5/4 = 385.440{{c}} -->


{{Optimal ET sequence|legend=0| 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 (Smith): 0.772 × 10<sup>-3</sup>
Badness (Sintel): 0.734


== Fantastic ==
== Fantastic ==
Line 627: Line 674:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~99/70 = 600.000{{c}}, ~3/2 = 701.132{{c}}, ~5/4 = 383.926{{c}}
* [[WE]]: ~99/70 = 600.2896{{c}}, ~3/2 = 700.9624{{c}}, ~5/4 = 383.4829{{c}}
* [[CWE]]: ~99/70 = 600.000{{c}}, ~3/2 = 700.816{{c}}, ~5/4 = 383.535{{c}}
: [[error map]]: {{val| +0.579 -0.413 -1.672 +0.644 +0.579 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 700.8160{{c}}, ~5/4 = 383.5350{{c}}
: error map: {{val| 0.000 -1.139 -2.779 -0.124 -0.713 }}
<!-- * [[CTE]]: ~99/70 = 600.000{{c}}, ~3/2 = 701.132{{c}}, ~5/4 = 383.926{{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]] (Smith): 0.743 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.893


== Hestia ==
== Hestia ==
Line 645: Line 695:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~400/231 = 950.043{{c}}, ~5/4 = 384.858{{c}}
* [[WE]]: ~2 = 1200.6417{{c}}, ~400/231 = 950.6555{{c}}, ~5/4 = 383.8518{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~400/231 = 950.121{{c}}, ~5/4 = 383.953{{c}}
: [[error map]]: {{val| +0.642 -0.644 -1.179 +0.858 -0.131 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 950.1211{{c}}, ~5/4 = 383.9530{{c}}
: error map: {{val| 0.000 -1.712 -2.361 -0.436 -2.165 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~400/231 = 950.043{{c}}, ~5/4 = 384.858{{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 }}


[[Badness]] (Smith): 1.54 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.845


=== 13-limit ===
=== 13-limit ===
Line 660: Line 713:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~26/15 = 950.131{{c}}, ~5/4 = 385.245{{c}}
* WE: ~2 = 1200.8238{{c}}, ~26/15 = 950.8873{{c}}, ~5/4 = 383.8191{{c}}
* CWE: ~2 = 1200.000{{c}}, ~26/15 = 950.211{{c}}, ~5/4 = 383.949{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.2108{{c}}, ~5/4 = 383.9486{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~26/15 = 950.131{{c}}, ~5/4 = 385.245{{c}} -->


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


Badness (Smith): 1.062 × 10<sup>-3</sup>
Badness (Sintel): 0.993


== Morfil ==
== Morfil ==
Line 672: Line 726:
[[Comma list]]: 225/224, 1331/1323
[[Comma list]]: 225/224, 1331/1323


{{Mapping|legend=1| 1 0 1 -3 -2 | 0 1 2 6 5 | 0 0 -3 -6 -4 }}
{{Mapping|legend=1| 1 0 -2 -9 -6 | 0 1 2 6 5 | 0 0 3 6 4 }}
: mapping generators: ~2, ~3, ~84/55
: mapping generators: ~2, ~3, ~55/42


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.459{{c}}, ~84/55 = 739.747{{c}}
* [[WE]]: ~2 = 1200.5931{{c}}, ~3/2 = 701.2447{{c}}, ~55/42 = 460.8465{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.111{{c}}, ~84/55 = 739.451{{c}}
: [[error map]]: {{val| +0.593 -0.117 -1.285 +1.942 -2.302 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.1108{{c}}, ~55/42 = 460.5488{{c}}
: error map: {{val| 0.000 -0.844 -2.446 +1.131 -3.569 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.459{{c}}, ~84/55 = 739.747{{c}} -->


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


[[Badness]] (Smith): 1.152 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.38


== Catakleismoid ==
== Catakleismoid ==
Line 694: Line 751:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 316.834{{c}}, ~11/8 = 551.318{{c}}
* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}, ~11/8 = 549.5258{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 316.771{{c}}, ~11/8 = 550.034{{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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 316.834{{c}}, ~11/8 = 551.318{{c}} -->


{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce }}
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197e, 269ce, 341ce }}


[[Badness]] (Smith): 1.275 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.53


=== 13-limit ===
=== 13-limit ===
Line 709: Line 769:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~6/5 = 316.886{{c}}, ~11/8 = 551.318{{c}}
* WE: ~2 = 1200.8238{{c}}, ~6/5 = 316.9478{{c}}, ~11/8 = 548.9611{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 316.794{{c}}, ~11/8 = 549.590{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}, ~11/8 = 549.5903{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 316.886{{c}}, ~11/8 = 551.318{{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 }}


Badness (Smith): 0.916 × 10<sup>-3</sup>
Badness (Sintel): 0.853


== Subgroup extensions ==
== Subgroup extensions ==
Line 725: Line 786:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.244{{c}}, ~5/4 = 384.789{{c}}
* WE: ~2 = 1199.3681{{c}}, ~3/2 = 701.3669{{c}}, ~5/4 = 384.9641{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6286{{c}}, ~5/4 = 385.0841{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.244{{c}}, ~5/4 = 384.789{{c}} -->


{{Optimal ET sequence|legend=0| 9, 10, 12, 19, 22, 31, 41, 53 }}
{{Optimal ET sequence|legend=0| 9, 10, 12, 19, 22, 31, 41, 53 }}
Badness (Sintel): 0.381


=== Devimarvel ===
=== Devimarvel ===
Line 737: Line 802:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.904{{c}}, ~5/4 = 384.260{{c}}
* WE: ~2 = 1200.3454{{c}}, ~3/2 = 701.1728{{c}}, ~5/4 = 383.7499{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.9546{{c}}, ~5/4 = 383.7912{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.904{{c}}, ~5/4 = 384.260{{c}} -->
 
{{Optimal ET sequence|legend=0| 10, 12, 19, 22, 31, 41, 53, 72, 94, 113, 125, 291c }}


{{Optimal ET sequence|legend=0| 10, 12, 19, 22, 31, 41, 53, 72, 94 }}
Badness (Sintel): 0.310


== References ==
== References ==

Revision as of 11:42, 13 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°

Optimal tunings:

  • WE: ~2 = 1200.5971 ¢, ~3/2 = 700.7560 ¢, ~5/4 = 383.8285 ¢
error map: ⟨+0.597 -0.602 -1.291 +0.940]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.6222 ¢, ~5/4 = 383.8540 ¢
error map: ⟨0.000 -1.333 -2.460 +0.127]

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 (Sintel): 0.161

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°

Optimal tunings:

  • WE: ~2 = 1200.6395 ¢, ~3/2 = 700.7619 ¢, ~5/4 = 383.7447 ¢
error map: ⟨+0.639 -0.554 -1.290 +0.827 -0.117]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.6048 ¢, ~5/4 = 383.4538 ¢
error map: ⟨0.000 -1.350 -2.860 -0.709 -2.284]

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 (Sintel): 0.306

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:

  • WE: ~2 = 1200.8043 ¢, ~3/2 = 700.2057 ¢, ~5/4 = 384.3188 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.8109 ¢, ~5/4 = 384.1177 ¢

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 (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

13-limit

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:

  • WE: ~2 = 1200.5788 ¢, ~3/2 = 701.3161 ¢, ~5/4 = 383.3471 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.1003 ¢, ~5/4 = 383.1315 ¢

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 (Sintel): 0.674

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:

  • WE: ~2 = 1200.4716 ¢, ~3/2 = 701.4395 ¢, ~5/4 = 383.2779 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2002 ¢, ~5/4 = 383.1136 ¢

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

Badness (Sintel): 0.756

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:

  • WE: ~2 = 1200.6057 ¢, ~3/2 = 701.3157 ¢, ~5/4 = 383.2243 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.0877 ¢, ~5/4 = 382.9709 ¢

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

Badness (Sintel): 0.827

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:

  • WE: ~2 = 1200.4927 ¢, ~3/2 = 701.4506 ¢, ~5/4 = 383.1291 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2007 ¢, ~5/4 = 382.9404 ¢

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

Badness (Sintel): 1.01

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:

  • WE: ~2 = 1200.6165 ¢, ~3/2 = 701.3259 ¢, ~5/4 = 383.5032 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.0949 ¢, ~5/4 = 383.3140 ¢

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

Badness (Sintel): 0.872

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:

  • WE: ~2 = 1200.5038 ¢, ~3/2 = 701.4654 ¢, ~5/4 = 383.4549 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2113 ¢, ~5/4 = 383.3052 ¢

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

Badness (Sintel): 1.05

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:

  • WE: ~2 = 1200.6404 ¢, ~3/2 = 700.7675 ¢, ~5/4 = 383.7772 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.6113 ¢, ~5/4 = 383.4925 ¢

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, 341cef

Badness (Sintel): 0.806

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:

  • WE: ~2 = 1200.1824 ¢, ~3/2 = 702.0223 ¢, ~5/4 = 383.3028 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9276 ¢, ~5/4 = 383.2283 ¢

Optimal ET sequence: 10, 19f, 22, 31, 41, 53, 84e, 94

Badness (Sintel): 0.810

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:

  • WE: ~2 = 1200.7533 ¢, ~3/2 = 700.8957 ¢, ~5/4 = 383.0580 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.7184 ¢, ~5/4 = 382.6611 ¢

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 (Sintel): 0.861

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:

  • WE: ~2 = 1200.4789 ¢, ~3/2 = 699.5125 ¢, ~5/4 = 383.1304 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.4041 ¢, ~5/4 = 382.9166 ¢

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

Badness (Sintel): 0.697

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:

  • WE: ~2 = 1200.7720 ¢, ~3/2 = 950.8976 ¢, ~5/4 = 383.8283 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.4265 ¢, ~5/4 = 383.4790 ¢

Optimal ET sequence: 9, 10, 19, 43e, 44, 53, 72, 125f, 197ef

Badness (Sintel): 0.935

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°

Optimal tunings:

  • WE: ~2 = 1200.1086 ¢, ~3/2 = 700.3226 ¢, ~5/4 = 386.5931 ¢
error map: ⟨+0.109 -1.524 +0.497 +5.114 -4.192]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.3006 ¢, ~5/4 = 386.5785 ¢
error map: ⟨0.000 -1.654 +0.265 +4.932 -4.403]

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 (Sintel): 0.458

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:

  • WE: ~2 = 1199.9127 ¢, ~3/2 = 701.1832 ¢, ~5/4 = 385.9313 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2143 ¢, ~5/4 = 385.9336 ¢

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

Badness (Sintel): 0.765

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:

  • WE: ~2 = 1199.8250 ¢, ~3/2 = 703.3820 ¢, ~5/4 = 381.5476 ¢
error map: ⟨-0.175 +1.252 -5.116 +0.858 +4.313]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.4612 ¢, ~5/4 = 381.5071 ¢
error map: ⟨0.000 +1.506 -4.807 +1.111 +4.774]

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

Optimal ET sequence: 12, 19, 22, 34d, 41, 104

Badness (Sintel): 0.637

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:

  • WE: ~2 = 1199.6919 ¢, ~3/2 = 703.8176 ¢, ~5/4 = 381.4372 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.9853 ¢, ~5/4 = 381.3579 ¢

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

Badness (Sintel): 0.964

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:

  • WE: ~2 = 1199.7106 ¢, ~3/2 = 696.0036 ¢, ~5/4 = 384.9572 ¢
error map: ⟨-0.289 -6.241 -1.935 -7.194 +25.068]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.0586 ¢, ~5/4 = 384.9472 ¢
error map: ⟨0.000 -5.896 -1.367 -6.814 +25.746]

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

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

Badness (Sintel): 0.557

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:

  • WE: ~2 = 1199.8192 ¢, ~3/2 = 695.9054 ¢, ~5/4 = 384.6205 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 695.9480 ¢, ~5/4 = 384.6372 ¢

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

Badness (Sintel): 0.686

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:

  • WE: ~2 = 1199.5261 ¢, ~3/2 = 702.1990 ¢, ~5/4 = 382.5759 ¢
error map: ⟨+0.526 +0.770 -2.686 +1.250 -1.077]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0535 ¢, ~5/4 = 382.6222 ¢
error map: ⟨0.000 +0.098 -3.692 +0.525 -2.188]

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

Badness (Sintel): 1.50

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:

  • WE: ~2 = 1200.2427 ¢, ~3/2 = 702.1575 ¢, ~5/4 = 383.0704 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0882 ¢, ~5/4 = 383.0630 ¢

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

Badness (Sintel): 1.00

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:

  • WE: ~2 = 1200.7854 ¢, ~3/2 = 700.2562 ¢, ~5/4 = 383.7624 ¢
error map: ⟨+0.785 -0.913 -0.980 -0.003 +0.721]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.8610 ¢, ~5/4 = 383.7724 ¢
error map: ⟨0.000 -2.094 -2.541 -1.559 -0.835]

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 (Sintel): 0.402

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:

  • WE: ~2 = 1200.8252 ¢, ~3/2 = 700.8823 ¢, ~5/4 = 381.6647 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.4689 ¢, ~5/4 = 381.6687 ¢

Optimal ET sequence: 10, 12f, 19e, 29, 31, 41, 60e, 72f, 101cd

Badness (Sintel): 0.689

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:

  • WE: ~2 = 1200.6284 ¢, ~3/2 = 700.7075 ¢, ~5/4 = 383.4599 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.3302 ¢, ~5/4 = 383.5030 ¢

Optimal ET sequence: 12f, 29, 31f, 41, 72, 185cf, 257cff

Badness (Sintel): 0.841

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:

  • WE: ~2 = 1200.7798 ¢, ~3/2 = 699.9410 ¢, ~5/4 = 384.3494 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.5538 ¢, ~5/4 = 384.3496 ¢

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

Badness (Sintel): 0.831

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:

  • WE: ~2 = 1200.9450 ¢, ~3/2 = 700.1045 ¢, ~5/4 = 383.8716 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.4224 ¢, ~5/4 = 383.9828 ¢

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

Badness (Sintel): 0.891

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:

  • WE: ~2 = 1200.9342 ¢, ~3/2 = 700.1708 ¢, ~5/4 = 383.7444 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.4849 ¢, ~5/4 = 383.8744 ¢

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

Badness (Sintel): 0.734

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:

  • WE: ~99/70 = 600.2896 ¢, ~3/2 = 700.9624 ¢, ~5/4 = 383.4829 ¢
error map: ⟨+0.579 -0.413 -1.672 +0.644 +0.579]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 700.8160 ¢, ~5/4 = 383.5350 ¢
error map: ⟨0.000 -1.139 -2.779 -0.124 -0.713]

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

Badness (Sintel): 0.893

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:

  • WE: ~2 = 1200.6417 ¢, ~400/231 = 950.6555 ¢, ~5/4 = 383.8518 ¢
error map: ⟨+0.642 -0.644 -1.179 +0.858 -0.131]
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 950.1211 ¢, ~5/4 = 383.9530 ¢
error map: ⟨0.000 -1.712 -2.361 -0.436 -2.165]

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

Badness (Sintel): 1.845

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:

  • WE: ~2 = 1200.8238 ¢, ~26/15 = 950.8873 ¢, ~5/4 = 383.8191 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.2108 ¢, ~5/4 = 383.9486 ¢

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

Badness (Sintel): 0.993

Morfil

Subgroup: 2.3.5.7.11

Comma list: 225/224, 1331/1323

Mapping: [⟨1 0 -2 -9 -6], ⟨0 1 2 6 5], ⟨0 0 3 6 4]]

mapping generators: ~2, ~3, ~55/42

Optimal tunings:

  • WE: ~2 = 1200.5931 ¢, ~3/2 = 701.2447 ¢, ~55/42 = 460.8465 ¢
error map: ⟨+0.593 -0.117 -1.285 +1.942 -2.302]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.1108 ¢, ~55/42 = 460.5488 ¢
error map: ⟨0.000 -0.844 -2.446 +1.131 -3.569]

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

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

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

Optimal tunings:

  • WE: ~2 = 1200.5965 ¢, ~6/5 = 316.8893 ¢, ~11/8 = 549.5258 ¢
error map: ⟨+0.596 -0.619 -1.271 +0.948 -0.003]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7705 ¢, ~11/8 = 550.0343 ¢
error map: ⟨0.000 -1.332 -2.461 +0.126 -1.284]

Optimal ET sequence: 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: [⟨1 0 1 -3 0 0], ⟨0 6 5 22 0 14], ⟨0 0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.8238 ¢, ~6/5 = 316.9478 ¢, ~11/8 = 548.9611 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7939 ¢, ~11/8 = 549.5903 ¢

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

Badness (Sintel): 0.853

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:

  • WE: ~2 = 1199.3681 ¢, ~3/2 = 701.3669 ¢, ~5/4 = 384.9641 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6286 ¢, ~5/4 = 385.0841 ¢

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

Badness (Sintel): 0.381

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:

  • WE: ~2 = 1200.3454 ¢, ~3/2 = 701.1728 ¢, ~5/4 = 383.7499 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.9546 ¢, ~5/4 = 383.7912 ¢

Optimal ET sequence: 10, 12, 19, 22, 31, 41, 53, 72, 94, 113, 125, 291c

Badness (Sintel): 0.310

References

  1. ↑ Yahoo! Tuning Group | Artemis and friends
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