# 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 five

Comma: 325/324

13 and 15 limit minimax tuning

 [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>

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

Eigenmonzo subgroup: 2.3.7/5.11/5.13/5

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:

 <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]

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: 2.7.11/5.13/5

15-limit eigenmonzo subgroup: 2.7.15/11.15/13

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:

 <1 0 0 -5 0 2] <0 1 0 2 0 4] <0 0 1 2 0 -2] <0 0 0 0 1 0]

Edos: 12, 19, 41, 53, 72, 166

## 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

Edos: 7, 15, 19, 26, 34, 41, 46, 53, 72, 87, 140, 159, 212, 299

## 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

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

## 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

Edos: 12, 15, 26, 41, 46, 72, 87, 159

## 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

Edos: 7, 19, 26, 46, 53, 72, 152

## 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

Edos: 19, 41, 53, 72, 121, 166, 193

## 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

Edos: 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401

## 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

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

# Rank three

## Portending

Commas: 325/324, 364/363, 441/440

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

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

## Marvel (Hecate)

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

Edos: 19, 41, 53, 72, 166