Transformations of glumma

Here are the 12 versions of glumma; the first being a mode of the
original. "recanbm" means in the cubic lattice description of the scale,
the "n" values range from -1 to 1, and the "m" values from 0 to 1;
"recancm" means instead the "m" values range from -1 to 0 instead of 0 to
1. Hence "reca1b2", which is glumma, has chords [i,j,0] where
-1<=i<=1 and 0<=j<=1. However glumma does not seem the most interesting
of these scales; recta3c1 not only has nice, low-ratio note values, it is
the most regular in terms of the variation in step size. The presence of
both 15/14 and 16/15 among the step sizes suggests tempering by
225/224~1.

reca1b2 := [1, 21/20, 35/32, 6/5, 5/4, 21/16, 35/24, 3/2, 49/32, 12/7,
7/4,
9/5];
reca1b3 := [1, 15/14, 35/32, 6/5, 5/4, 9/7, 35/24, 3/2, 25/16, 12/7, 7/4,
15
/8];
reca2b1 := [1, 25/24, 21/20, 35/32, 5/4, 21/16, 10/7, 35/24, 3/2, 49/32,
5/3
, 7/4];
reca2b3 := [1, 21/20, 15/14, 9/8, 5/4, 21/16, 10/7, 3/2, 5/3, 7/4, 25/14,
15
/8];
reca3b1 := [1, 49/48, 15/14, 35/32, 7/6, 5/4, 7/5, 35/24, 3/2, 25/16,
7/4,
15/8];
reca3b2 := [1, 21/20, 15/14, 9/8, 7/6, 49/40, 5/4, 21/16, 7/5, 3/2, 7/4,
15/
8];
reca1c2 := [1, 25/24, 35/32, 8/7, 6/5, 5/4, 10/7, 35/24, 3/2, 5/3, 12/7,
7/4
];
reca1c3 := [1, 49/48, 35/32, 7/6, 6/5, 5/4, 7/5, 35/24, 3/2, 8/5, 12/7,
7/4]
;
reca2c1 := [1, 21/20, 8/7, 6/5, 5/4, 21/16, 10/7, 3/2, 5/3, 12/7, 7/4,
9/5];
reca2c3 := [1, 21/20, 7/6, 49/40, 5/4, 21/16, 4/3, 7/5, 10/7, 3/2, 5/3,
7/4]
;
reca3c1 := [1, 15/14, 7/6, 6/5, 5/4, 9/7, 7/5, 3/2, 8/5, 12/7, 7/4,
15/8];
reca3c2 := [1, 15/14, 7/6, 5/4, 4/3, 7/5, 10/7, 3/2, 5/3, 7/4, 25/14,
15/8];

I checked each of these 12 scales for intervals which were close to a
7-limit consonance by a comma less than 15 cents. All of them had at
least one such comma, and the 31-et covers all of the commas.

> reca1b2 := [1, 21/20, 35/32, 6/5, 5/4, 21/16, 35/24, 3/2, 49/32,
> 12/7,
> 7/4,
> 9/5];

126/125 ^ 1728/1715 = [10, 9, 7, -9, 17, -9]

"Small diesic" temperament covered by 27, 31 and 58

> reca1b3 := [1, 15/14, 35/32, 6/5, 5/4, 9/7, 35/24, 3/2, 25/16, 12/7,
> 7/4,
> 15
> /8];

1728/1715

> reca2b1 := [1, 25/24, 21/20, 35/32, 5/4, 21/16, 10/7, 35/24, 3/2,
> 49/32,
> 5/3
> , 7/4];

126/125

> reca2b3 := [1, 21/20, 15/14, 9/8, 5/4, 21/16, 10/7, 3/2, 5/3, 7/4,
> 25/14,
> 15
> /8];

126/125

> reca3b1 := [1, 49/48, 15/14, 35/32, 7/6, 5/4, 7/5, 35/24, 3/2, 25/16,
> 7/4,
> 15/8];

225/224

> reca3b2 := [1, 21/20, 15/14, 9/8, 7/6, 49/40, 5/4, 21/16, 7/5, 3/2,
> 7/4,
> 15/
> 8];

126/125 ^ 225/224 ^ 2401/2400 = 31-et

> reca1c2 := [1, 25/24, 35/32, 8/7, 6/5, 5/4, 10/7, 35/24, 3/2, 5/3,
> 12/7,
> 7/4
> ];

126/125 ^ 1728/1715, small diesic

> reca1c3 := [1, 49/48, 35/32, 7/6, 6/5, 5/4, 7/5, 35/24, 3/2, 8/5,
> 12/7,
> 7/4]
> ;

1728/1715

> reca2c1 := [1, 21/20, 8/7, 6/5, 5/4, 21/16, 10/7, 3/2, 5/3, 12/7,
> 7/4,
> 9/5];

1029/1024^126/125 = [9, 5, -3, -21, 30, -13]

"quartaminorthirds" covered by 31, 46 and 77

> reca2c3 := [1, 21/20, 7/6, 49/40, 5/4, 21/16, 4/3, 7/5, 10/7, 3/2,
> 5/3,
> 7/4]
> ;

2401/2400^126/125 = 126/125^1728/1715 = small diesic

> reca3c1 := [1, 15/14, 7/6, 6/5, 5/4, 9/7, 7/5, 3/2, 8/5, 12/7, 7/4,
> 15/8];

225/224

> reca3c2 := [1, 15/14, 7/6, 5/4, 4/3, 7/5, 10/7, 3/2, 5/3, 7/4, 25/14,
> 15/8];

126/125^225/224 = [1, 4, 10, 12 -13, 4]

meantone, covered by 19, 31, 43, 50, 74