Marveltwin: Difference between revisions

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== Marveltwin and marvel ==

The marveltwin comma, [[325/324]], bears a curiously close analogy to the marvel comma, [[225/224]]. 325/324 can be added to the [[11-limit]] version of marvel, which tempers out 225/224 and 385/384, to get [[13-limit]] marvel, aka hecate. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.

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: 3 pure; 5, 7, 11 and 13 all flat by (325/324)1/3, which is 1.778 cents.

: [[eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.3.7/5.11/5.13/5

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5.13/5

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

{{Optimal ET sequence|legend=1| 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 }}

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[[Minimax tuning]]s:

* 13-limit ~~eigenmonzo (~~unchanged-interval) basis: 2.7.11/5.13/5

* 13-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5.13/5

* 15-limit ~~eigenmonzo (~~unchanged-interval) basis: 2.7.15/11.15/13

* 15-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11.15/13

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

[[Badness]]: 3.668 × 10-6

=== 385/384 ===

[[~~Subgroup~~]]~~: 2~~.~~3.5.7.11.13~~

See [[Keenanismic family #Martwin]].

~~[[Comma list]]: 325/324, 385/384~~

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

~~{{Optimal ET sequence|legend=1| 7, 15, 19, 26, 34, 41, 46, 53, 72, 87, 140, 159, 212, 299 }}~~

~~[[Badness]]: 2.206 × 10-6~~

=== 364/363 ===

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[[Comma list]]: 325/324, 364/363

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

{{Optimal ET sequence|legend=1| 15, 26, 41, 46, 72, 87, 121, 159, 193, 239, 280 }}

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[[Comma list]]: 325/324, 441/440

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

{{Optimal ET sequence|legend=1| 12, 15, 26, 41, 46, 72, 87, 159 }}

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[[Comma list]]: 169/168, 325/324

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

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[[Comma list]]: 325/324, 540/539

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

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[[Comma list]]: 325/324, 352/351

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

{{Optimal ET sequence|legend=1| 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401 }}

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[[Comma list]]: 325/324, 625/624

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

{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 }}

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* [[Marvel|Marvel (hecate)]] → [[Marvel family #Hecate|Marvel family]]

: +225/224, 385/384

* [[Enlil|Enlil a.k.a. sumatra]] → [[Kleismic rank ~~three~~ family #Enlil|Kleismic rank-3 family]]

* [[Enlil|Enlil a.k.a. sumatra]] → [[Kleismic rank-3 family #Enlil|Kleismic rank-3 family]]

: +385/384, 625/624

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[[Category:Commatic realms]]

[[Category:Marveltwin]]

~~{{Todo| cleanup }}~~

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-__FORCETOC__
+{{Todo|inline=1| intro }}
 == Marveltwin and marvel ==
 The marveltwin comma, [[325/324]], bears a curiously close analogy to the marvel comma, [[225/224]]. 325/324 can be added to the [[11-limit]] version of marvel, which tempers out 225/224 and 385/384, to get [[13-limit]] marvel, aka hecate. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.
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 |}
 : 3 pure; 5, 7, 11 and 13 all flat by (325/324)<sup>1/3</sup>, which is 1.778 cents.
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3.7/5.11/5.13/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5.13/5
-[[Complexity spectrum]]: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 16/15, 18/13, 13/12, 7/6, 16/13, 7/5, 11/8, 9/7, 12/11, 15/14, 11/10, 11/9, 13/10, 14/13, 15/13, 14/11, 15/11, 13/11
 {{Optimal ET sequence|legend=1| 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 }}
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 [[Minimax tuning]]s:
-* 13-limit eigenmonzo (unchanged-interval) basis: 2.7.11/5.13/5
+* 13-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5.13/5
-* 15-limit eigenmonzo (unchanged-interval) basis: 2.7.15/11.15/13
+* 15-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11.15/13
 {{Optimal ET sequence|legend=1| 12, 19, 41, 53, 72, 166 }}
-[[Complexity spectrum]]: 4/3, 5/4, 6/5, 16/15, 15/14, 9/8, 10/9, 7/5, 9/7, 7/6, 18/13, 8/7, 13/12, 16/13, 11/8, 12/11, 15/13, 11/10, 15/11, 11/9, 13/10, 14/13, 14/11, 13/11
 [[Badness]]: 3.668 × 10<sup>-6</sup>
 === 385/384 ===
-[[Subgroup]]: 2.3.5.7.11.13
+See [[Keenanismic family #Martwin]].
-[[Comma list]]: 325/324, 385/384
-[[Complexity spectrum]]: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 11/8, 7/6, 12/11, 16/15, 18/13, 13/12, 9/7, 7/5, 16/13, 11/9, 11/10, 13/11, 14/13, 15/14, 15/11, 13/10, 15/13, 14/11
-{{Optimal ET sequence|legend=1| 7, 15, 19, 26, 34, 41, 46, 53, 72, 87, 140, 159, 212, 299 }}
-[[Badness]]: 2.206 × 10<sup>-6</sup>
 === 364/363 ===
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 [[Comma list]]: 325/324, 364/363
-[[Complexity spectrum]]: 4/3, 6/5, 10/9, 11/8, 5/4, 12/11, 14/11, 9/8, 13/11, 11/9, 8/7, 13/12, 18/13, 16/15, 7/6, 11/10, 7/5, 16/13, 9/7, 15/11, 15/14, 13/10, 15/13, 14/13
 {{Optimal ET sequence|legend=1| 15, 26, 41, 46, 72, 87, 121, 159, 193, 239, 280 }}
@@ Line 89: / Line 76: @@
 [[Comma list]]: 325/324, 441/440
-[[Complexity spectrum]]: 4/3, 6/5, 8/7, 10/9, 14/11, 5/4, 7/6, 9/8, 7/5, 16/15, 13/12, 18/13, 9/7, 11/8, 12/11, 15/14, 16/13, 11/9, 11/10, 13/11, 15/11, 14/13, 13/10, 15/13
 {{Optimal ET sequence|legend=1| 12, 15, 26, 41, 46, 72, 87, 159 }}
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 [[Comma list]]: 169/168, 325/324
-[[Complexity spectrum]]: 10/9, 6/5, 4/3, 14/13, 13/12, 16/13, 5/4, 18/13, 9/8, 13/10, 8/7, 7/6, 15/13, 16/15, 9/7, 7/5, 11/8, 12/11, 13/11, 11/10, 15/14, 11/9, 14/11, 15/11
 {{Optimal ET sequence|legend=1| 7, 19, 26, 46, 53, 72, 152 }}
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 [[Comma list]]: 325/324, 540/539
-[[Complexity spectrum]]: 4/3, 6/5, 7/6, 9/7, 10/9, 5/4, 7/5, 9/8, 15/14, 8/7, 16/15, 18/13, 13/12, 12/11, 11/9, 11/10, 11/8, 15/11, 16/13, 14/13, 15/13, 13/10, 13/11, 14/11
 {{Optimal ET sequence|legend=1| 19, 41, 53, 72, 121, 166, 193 }}
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 [[Comma list]]: 325/324, 352/351
-[[Complexity spectrum]]: 4/3, 10/9, 6/5, 9/8, 5/4, 13/11, 16/13, 11/9, 13/12, 12/11, 16/15, 18/13, 8/7, 7/6, 11/8, 9/7, 15/11, 11/10, 7/5, 14/13, 13/10, 15/14, 14/11, 15/13
 {{Optimal ET sequence|legend=1| 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401 }}
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 [[Comma list]]: 325/324, 625/624
-[[Complexity spectrum]]: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 16/13, 8/7, 7/5, 7/6, 15/14, 11/8, 9/7, 11/10, 12/11, 14/13, 15/11, 11/9, 13/11, 14/11
 {{Optimal ET sequence|legend=1| 15, 19, 34, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 }}
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 * [[Marvel|Marvel (hecate)]] → [[Marvel family #Hecate|Marvel family]]
 : +225/224, 385/384
-* [[Enlil|Enlil a.k.a. sumatra]] → [[Kleismic rank three family #Enlil|Kleismic rank-3 family]]
+* [[Enlil|Enlil a.k.a. sumatra]] → [[Kleismic rank-3 family #Enlil|Kleismic rank-3 family]]
 : +385/384, 625/624
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 [[Category:Commatic realms]]
 [[Category:Marveltwin]]
-{{Todo| cleanup }}