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=Marveltwin and Marvel=
 
== 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.
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 five=
== Rank-5 temperaments ==
Comma: 325/324
[[Subgroup]]: 2.3.5.7.11.13


13 and 15 limit minimax tuning
[[Comma list]]: 325/324


{| class="wikitable"
[[Mapping]]:
{| class="right-all"
|-
|-
| | [1 0 0 0 0 0>
| {{lbrack}}⟨ || 1 || 0 || 0 || 0 || 0 || 2 || {{rbrack}},
|-
|-
| | [0 1 0 0 0 0>
| ⟨ || 0 || 1 || 0 || 0 || 0 || 4 || {{rbrack}},
|-
|-
| | [2/3 4/3 1/3 0 0 -1/3>
| ⟨ || 0 || 0 || 1 || 0 || 0 || -2 || {{rbrack}},
|-
|-
| | [2/3 4/3 -2/3 1 0 -1/3>
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || {{rbrack}},
|-
|-
| | [2/3 4/3 -2/3 0 1 -1/3>
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || {{rbrack}}{{rbrack}}
|-
| | [2/3 4/3 -2/3 0 0 2/3>
|}
|}


Fifths are pure; 5, 7, 11 and 13 are all flat by (325/324)^(1/3), which is 1.778 cents.
[[Minimax tuning]]s:  
 
* 13- and 15-odd-limit
[[Eigenmonzo subgroup]] ([[unchanged-interval basis]]): 2.3.7/5.11/5.13/5
: {| class="right-all"
 
[[Spectrum_of_a_temperament|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
 
Mapping:  
 
{| class="wikitable"
|-
|-
| | <1 0 0 0 0 2]
| {{lbrack}}{{lbrack}} || 1 || 0 || 0 || 0 || 0 || 0 || ⟩
|-
|-
| | <0 1 0 0 0 4]
| {{lbrack}} || 0 || 1 || 0 || 0 || 0 || 0 || ⟩
|-
| | <0 0 1 0 0 -2]
|-
| | <0 0 0 1 0 0]
|-
| | <0 0 0 0 1 0]
|}
 
Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333
 
=Rank four=
 
==225/224==
 
13-limit eigenmonzo subgroup (unchanged-interval basis): 2.7.11/5.13/5
 
15-limit eigenmonzo subgroup (unchanged-interval basis): 2.7.15/11.15/13
 
[[Spectrum_of_a_temperament|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
 
Mapping:
 
{| class="wikitable"
|-
|-
| | <1 0 0 -5 0 2]
| {{lbrack}} || 2/3 || 4/3 || 1/3 || 0 || 0 || -1/3 || ⟩
|-
|-
| | <0 1 0 2 0 4]
| {{lbrack}} || 2/3 || 4/3 || -2/3 || 1 || 0 || -1/3 || ⟩
|-
|-
| | <0 0 1 2 0 -2]
| {{lbrack}} || 2/3 || 4/3 || -2/3 || 0 || 1 || -1/3 || ⟩
|-
|-
| | <0 0 0 0 1 0]
| {{lbrack}} || 2/3 || 4/3 || -2/3 || 0 || 0 || 2/3 || ⟩{{rbrack}}
|}
|}
Edos: 12, 19, 41, 53, 72, 166
: 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


10^6 * Badness: 3.668
[[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


==385/384==
{{Optimal ET sequence|legend=1| 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 }}


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
== Rank-4 temperaments ==
=== 225/224 ===
[[Subgroup]]: 2.3.5.7.11.13


Edos: 7, 15, 19, 26, 34, 41, 46, 53, 72, 87, 140, 159, 212, 299
[[Comma list]]: 225/224, 325/324


10^6 * Badness: 2.206
{{Mapping|legend=1| 1 0 0 -5 0 2 | 0 1 0 2 0 4 | 0 0 1 2 0 -2 | 0 0 0 0 1 0 }}


==364/363==
[[Minimax tuning]]s:
* 13-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5.13/5
* 15-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11.15/13


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| 12, 19, 41, 53, 72, 166 }}


Edos: 15, 26, 41, 46, 72, 87, 121, 159, 193, 239, 280
[[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


10^6 * Badness: 3.011
[[Badness]]: 3.668 × 10<sup>-6</sup>


==441/440==
=== 385/384 ===
See [[Keenanismic family #Martwin]].


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
=== 364/363 ===
[[Subgroup]]: 2.3.5.7.11.13


Edos: 12, 15, 26, 41, 46, 72, 87, 159
[[Comma list]]: 325/324, 364/363


10^6 * Badness: 3.037
[[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


==169/168==
{{Optimal ET sequence|legend=1| 15, 26, 41, 46, 72, 87, 121, 159, 193, 239, 280 }}


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
[[Badness]]: 3.011 × 10<sup>-6</sup>


Edos: 7, 19, 26, 46, 53, 72, 152
=== 441/440 ===
[[Subgroup]]: 2.3.5.7.11.13


10^6 * Badness: 2.975
[[Comma list]]: 325/324, 441/440


==540/539==
[[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


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| 12, 15, 26, 41, 46, 72, 87, 159 }}


Edos: 19, 41, 53, 72, 121, 166, 193
[[Badness]]: 3.037 × 10<sup>-6</sup>


10^6 * Badness: 3.281
=== 169/168 ===
[[Subgroup]]: 2.3.5.7.11.13


==352/351==
[[Comma list]]: 169/168, 325/324


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
[[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


Edos: 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401
{{Optimal ET sequence|legend=1| 7, 19, 26, 46, 53, 72, 152 }}


10^6 * Badness: 3.434
[[Badness]]: 2.975 × 10<sup>-6</sup>


==625/624==
=== 540/539 ===
[[Subgroup]]: 2.3.5.7.11.13


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


Edos: 15, 19, 34, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333
[[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


10^6 * Badness: 3.563
{{Optimal ET sequence|legend=1| 19, 41, 53, 72, 121, 166, 193 }}


=Rank three=
[[Badness]]: 3.281 × 10<sup>-6</sup>


==Portending==
=== 352/351 ===
Commas: 325/324, 364/363, 441/440
[[Subgroup]]: 2.3.5.7.11.13


Spectrum: 8/7, 4/3, 11/8, 6/5, 14/11, 7/6, 10/9, 12/11, 5/4, 13/11, 9/8, 7/5, 11/9, 9/7, 18/13, 13/12, 16/15, 11/10, 15/14, 16/13, 14/13, 15/11, 13/10, 15/13
[[Comma list]]: 325/324, 352/351


Edos: 15, 26, 41, 46, 72, 87, 159
[[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


10^5 * Badness: 62.715
{{Optimal ET sequence|legend=1| 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401 }}


==Marvel (Hecate)==
[[Badness]]: 3.434 × 10<sup>-6</sup>
Commas: 225/224, 325/324, 385/384


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
=== 625/624 ===
[[Subgroup]]: 2.3.5.7.11.13


Edos: 19, 41, 53, 72, 166
[[Comma list]]: 325/324, 625/624


10^5 * Badness: 72.113
[[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


==Sumatra==
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 }}
Commas: 325/324, 385/384, 625/624


Spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 8/7, 11/8, 11/10, 7/5, 12/11, 7/6, 16/13, 15/11, 15/14, 11/9, 9/7, 13/11, 14/13, 14/11
[[Badness]]: 3.563 × 10<sup>-6</sup>


Edos: 15, 19, 34, 53, 72, 87, 140, 159, 212, 299
== Rank-3 temperaments ==
Notable rank-3 temperaments of marveltwin include:  


Optimal patent val: [[299edo|299edo]]
* [[Portent|Portending]] → [[Gamelismic family #Portending|Gamelismic family]]
 
: +385/384, 441/440
10^5 * Badness: 68.005
* [[Marvel|Marvel (hecate)]] → [[Marvel family #Hecate|Marvel family]]
: +225/224, 385/384
* [[Enlil|Enlil a.k.a. sumatra]] → [[Kleismic rank-3 family #Enlil|Kleismic rank-3 family]]
: +385/384, 625/624


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]
[[Category:Commatic realms]]
[[Category:Commatic realms]]
[[Category:Marveltwin]]
[[Category:Marveltwin]]
{{Todo| cleanup }}

Latest revision as of 13:30, 27 August 2025

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

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: 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

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

See Keenanismic family #Martwin.

364/363

Subgroup: 2.3.5.7.11.13

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: 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

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: 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

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: 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

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: 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

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: 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

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: 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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