Marveltwin: Difference between revisions

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m Text replacement - "eigenmonzo (unchanged-interval) " to "unchanged-interval (eigenmonzo) "
- complexity spectra (rank-5 outlier & unnamed rank-4 temps)
 
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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.  
: 3 pure; 5, 7, 11 and 13 all flat by (325/324)<sup>1/3</sup>, which is 1.778 cents.  
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) 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 }}
{{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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{{Optimal ET sequence|legend=1| 12, 19, 41, 53, 72, 166 }}
{{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>
[[Badness]]: 3.668 × 10<sup>-6</sup>
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[[Comma list]]: 325/324, 364/363
[[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 }}
{{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
[[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 }}
{{Optimal ET sequence|legend=1| 12, 15, 26, 41, 46, 72, 87, 159 }}
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[[Comma list]]: 169/168, 325/324
[[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 }}
{{Optimal ET sequence|legend=1| 7, 19, 26, 46, 53, 72, 152 }}
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[[Comma list]]: 325/324, 540/539
[[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 }}
{{Optimal ET sequence|legend=1| 19, 41, 53, 72, 121, 166, 193 }}
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[[Comma list]]: 325/324, 352/351
[[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 }}
{{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
[[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 }}
{{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]]
* [[Marvel|Marvel (hecate)]] → [[Marvel family #Hecate|Marvel family]]
: +225/224, 385/384
: +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
: +385/384, 625/624


Latest revision as of 14:12, 26 September 2026

Todo: 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.

Rank-5 temperaments

Subgroup: 2.3.5.7.11.13

Comma list: 325/324

Mapping:

[⟨ 1 0 0 0 0 2 ],
⟨ 0 1 0 0 0 4 ],
⟨ 0 0 1 0 0 -2 ],
⟨ 0 0 0 1 0 0 ],
⟨ 0 0 0 0 1 0 ]]

Minimax tunings:

  • 13- and 15-odd-limit
[[ 1 0 0 0 0 0 ⟩
[ 0 1 0 0 0 0 ⟩
[ 2/3 4/3 1/3 0 0 -1/3 ⟩
[ 2/3 4/3 -2/3 1 0 -1/3 ⟩
[ 2/3 4/3 -2/3 0 1 -1/3 ⟩
[ 2/3 4/3 -2/3 0 0 2/3 ⟩]
3 pure; 5, 7, 11 and 13 all flat by (325/324)1/3, which is 1.778 cents.
unchanged-interval (eigenmonzo) basis: 2.3.7/5.11/5.13/5

Optimal ET sequence: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333

Rank-4 temperaments

225/224

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324

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

Minimax tunings:

  • 13-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5.13/5
  • 15-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11.15/13

Optimal ET sequence: 12, 19, 41, 53, 72, 166

Badness: 3.668 × 10-6

385/384

See Keenanismic family #Martwin.

364/363

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363

Optimal ET sequence: 15, 26, 41, 46, 72, 87, 121, 159, 193, 239, 280

Badness: 3.011 × 10-6

441/440

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 441/440

Optimal ET sequence: 12, 15, 26, 41, 46, 72, 87, 159

Badness: 3.037 × 10-6

169/168

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324

Optimal ET sequence: 7, 19, 26, 46, 53, 72, 152

Badness: 2.975 × 10-6

540/539

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 540/539

Optimal ET sequence: 19, 41, 53, 72, 121, 166, 193

Badness: 3.281 × 10-6

352/351

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351

Optimal ET sequence: 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401

Badness: 3.434 × 10-6

625/624

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624

Optimal ET sequence: 15, 19, 34, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333

Badness: 3.563 × 10-6

Rank-3 temperaments

Notable rank-3 temperaments of marveltwin include:

+385/384, 441/440
+225/224, 385/384
+385/384, 625/624
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