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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
 
[[Comma list]]: 325/324


13 and 15 limit minimax tuning
13 and 15 limit minimax tuning
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Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333
Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333


=Rank four=
== Rank-4 temperaments ==
 
=== 225/224 ===
==225/224==


13-limit eigenmonzo subgroup (unchanged-interval basis): 2.7.11/5.13/5
13-limit eigenmonzo subgroup (unchanged-interval basis): 2.7.11/5.13/5
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10^6 * Badness: 3.668
10^6 * Badness: 3.668


==385/384==
=== 385/384 ===


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
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
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10^6 * Badness: 2.206
10^6 * Badness: 2.206


==364/363==
=== 364/363 ===


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
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
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10^6 * Badness: 3.011
10^6 * Badness: 3.011


==441/440==
=== 441/440 ===


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, 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
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10^6 * Badness: 3.037
10^6 * Badness: 3.037


==169/168==
=== 169/168 ===


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
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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10^6 * Badness: 2.975
10^6 * Badness: 2.975


==540/539==
=== 540/539 ===


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
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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10^6 * Badness: 3.281
10^6 * Badness: 3.281


==352/351==
=== 352/351 ===


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
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
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10^6 * Badness: 3.434
10^6 * Badness: 3.434


==625/624==
=== 625/624 ===


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
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
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10^6 * Badness: 3.563
10^6 * Badness: 3.563


=Rank three=
== Rank-3 temperaments ==
 
=== Portending ===
==Portending==
Commas: 325/324, 364/363, 441/440
Commas: 325/324, 364/363, 441/440


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10^5 * Badness: 62.715
10^5 * Badness: 62.715


==Marvel (Hecate)==
=== Marvel (hecate) ===
Commas: 225/224, 325/324, 385/384
Commas: 225/224, 325/324, 385/384


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10^5 * Badness: 72.113
10^5 * Badness: 72.113


==Sumatra==
=== Sumatra ===
Commas: 325/324, 385/384, 625/624
Commas: 325/324, 385/384, 625/624


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