[tuning-math] The chord-generator approach to Miracle and Orwell
- From: Gene W Smith <genewardsmith@xxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxxxx
- Date: Fri, 12 Jul 2002 23:51:55 -0700
The simplest generators for a chain of chords, in terms of the cubic
lattice picture, are [1 0 0], [0 1 0] and [0 0 1]. If we take a chain of
two of these, and look at the corresponding note, or in other words at
the roots of the chords
[2 0 0], [0 2 0] and [0 0 2] we obtain 35/24, 21/20 and 15/14
respectively. From (15/14)^2/(8/7) = 225/224 and
(3/2)/(8/7)^3 = 1029/1024 we see that tempering a chain of chords with
generator [0 0 1] via Miracle is a likely plan.
Similarly, from (21/20)^2/(10/9) = (21/20)^3/(7/6) = 4000/3969, it would
seem the planar temperament defined by 4000/3969 works well with the
chords generated by [0 1 0]. One approach to this is to add 245/243 or
2401/2400 to the mix, and obtain the linear temperament with wedgie [8 18
11 -25 5 10], which is covered by 41 and 68, and has a generator of about
21/20. Finally, from (15/7)/(35/24)^2 = 1728/1715 and (28/9)/(35/24)^3 =
6144/6125, it would seem Orwell is a good choice for tempering the chain
of chords defined by [1 0 0].
Below I give scales derived from chains of generators of size 1 to 15, in
72-et (for [0 0 1]), 68-et (for [0 1 0]) and 84-et
(for [1 0 0]). It should be noted that eventually, these all produce
MOS--Blackjack for 72-et, the 22-note Orwell MOS, and a 27-note 5/68 MOS.
72-et scales
1 [0, 7, 23, 42, 58, 65] [7, 16, 19, 16, 7, 7]
2 [0, 7, 23, 30, 42, 49, 58, 65] [7, 16, 7, 12, 7, 9, 7, 7]
3 [0, 7, 14, 23, 30, 42, 49, 58, 65] [7, 7, 9, 7, 12, 7, 9, 7, 7]
4 [0, 7, 14, 23, 30, 37, 42, 49, 56, 58, 65]
[7, 7, 9, 7, 7, 5, 7, 7, 2, 7, 7]
5 [0, 7, 14, 21, 23, 30, 37, 42, 49, 56, 58, 65]
[7, 7, 7, 2, 7, 7, 5, 7, 7, 2, 7, 7]
6 [0, 7, 14, 21, 23, 30, 37, 42, 44, 49, 56, 58, 63, 65]
[7, 7, 7, 2, 7, 7, 5, 2, 5, 7, 2, 5, 2, 7]
7 [0, 7, 14, 21, 23, 28, 30, 37, 42, 44, 49, 56, 58, 63, 65]
[7, 7, 7, 2, 5, 2, 7, 5, 2, 5, 7, 2, 5, 2, 7]
8 [0, 7, 14, 14, 21, 23, 28, 30, 37, 42, 44, 49, 51, 56, 58, 63, 65,
70]
[7, 7, 7, 2, 5, 2, 7, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]
9 [0, 7, 14, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56, 58, 63, 65,
70]
[7, 7, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]
10 [0, 5, 7, 14, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56, 58, 63,
65, 70]
[5, 2, 7, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]
11 [0, 5, 7, 14, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56, 58, 63,
65, 70]
[5, 2, 7, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]
12 [0, 5, 7, 12, 14, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56, 58,
63, 65, 70]
[5, 2, 5, 2, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]
13 [0, 5, 7, 12, 14, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56, 58,
63, 65, 70]
[5, 2, 5, 2, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]
14 [0, 5, 7, 12, 14, 19, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56,
58, 63, 65, 70]
[5, 2, 5, 2, 5, 2, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]
15 [0, 5, 7, 12, 14, 19, 21, 23, 28, 30, 35, 37, 42, 44, 49, 51, 56,
58, 63, 65, 70]
[5, 2, 5, 2, 5, 2, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2]
84-et scales
1 [0, 11, 27, 46, 49, 68] [11, 16, 19, 3, 19, 16]
2 [0, 11, 27, 30, 46, 49, 68, 73] [11, 16, 3, 16, 3, 19, 5, 11]
3 [0, 8, 11, 27, 30, 46, 49, 57, 68, 73]
[8, 3, 16, 3, 16, 3, 8, 11, 5, 11]
4 [0, 8, 11, 27, 30, 35, 46, 49, 57, 68, 73, 76]
[8, 3, 16, 3, 5, 11, 3, 8, 11, 5, 3, 8]
5 [0, 8, 11, 19, 27, 30, 35, 46, 49, 54, 57, 68, 73, 76]
[8, 3, 8, 8, 3, 5, 11, 3, 5, 3, 11, 5, 3, 8]
6 [0, 8, 11, 19, 27, 30, 35, 38, 46, 49, 54, 57, 68, 73, 76, 81]
[8, 3, 8, 8, 3, 5, 3, 8, 3, 5, 3, 11, 5, 3, 5, 3]
7 [0, 8, 11, 16, 19, 27, 30, 35, 38, 46, 49, 54, 57, 65, 68, 73, 76,
81]
[8, 3, 5, 3, 8, 3, 5, 3, 8, 3, 5, 3, 8, 3, 5, 3, 5, 3]
8 [0, 8, 11, 16, 19, 27, 30, 35, 38, 43, 46, 49, 54, 57, 65, 68, 73,
76, 81]
[8, 3, 5, 3, 8, 3, 5, 3, 5, 3, 3, 5, 3, 8, 3, 5, 3, 5, 3]
9 [0, 8, 11, 16, 19, 27, 30, 35, 38, 43, 46, 49, 54, 57, 62, 65, 68,
73, 76, 81]
[8, 3, 5, 3, 8, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3]
10 [0, 5, 8, 11, 16, 19, 27, 30, 35, 38,
43, 46, 49, 54, 57, 62, 65, 68, 73, 76, 81]
[5, 3, 3, 5, 3, 8, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3]
11 [0, 5, 8, 11, 16, 19, 24, 27, 30, 35,
38, 43, 46, 49, 54, 57, 62, 65, 68, 73, 76, 81]
[5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3]
12 [0, 5, 8, 11, 16, 19, 24, 27, 30, 35, 38,
43, 46, 49, 51, 54, 57, 62, 65, 68, 73, 76, 81]
[5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 2, 3, 3, 5, 3, 3, 5, 3, 5, 3]
13 [0, 5, 8, 11, 16, 19, 24, 27, 30, 35, 38,
43, 46, 49, 51, 54, 57, 62, 65, 68, 70, 73, 76, 81]
[5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3, 5, 3]
14 [0, 5, 8, 11, 13, 16, 19, 24, 27, 30, 35, 38,
43, 46, 49, 51, 54, 57, 62, 65, 68, 70, 73, 76, 81]
[5, 3, 3, 2, 3, 3, 5, 3, 3, 5, 3, 5, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3, 5,
3]
15 [0, 5, 8, 11, 13, 16, 19, 24, 27, 30, 32, 35, 38,
43, 46, 49, 51, 54, 57, 62, 65, 68, 70, 73, 76, 81]
[5, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3,
5, 3]
68-et scales
1 [0, 5, 22, 27, 40, 55] [5, 17, 5, 13, 15, 13]
2 [0, 5, 22, 27, 40, 45, 55, 60] [5, 17, 5, 13, 5, 10, 5, 8]
3 [0, 5, 10, 22, 27, 32, 40, 45, 55, 60]
[5, 5, 12, 5, 5, 8, 5, 10, 5, 8]
4 [0, 5, 10, 22, 27, 32, 40, 45, 50, 55, 60, 65]
[5, 5, 12, 5, 5, 8, 5, 5, 5, 5, 5, 3]
5 [0, 5, 10, 15, 22, 27, 32, 37, 40, 45, 50, 55, 60, 65]
[5, 5, 5, 7, 5, 5, 5, 3, 5, 5, 5, 5, 5, 3]
6 [0, 2, 5, 10, 15, 22, 27, 32, 37, 40, 45, 50, 55, 60, 65]
[2, 3, 5, 5, 7, 5, 5, 5, 3, 5, 5, 5, 5, 5, 3]
7 [0, 2, 5, 10, 15, 20, 22, 27, 32, 37, 40, 42, 45, 50, 55, 60, 65]
[2, 3, 5, 5, 5, 2, 5, 5, 5, 3, 2, 3, 5, 5, 5, 5, 3]
8 [0, 2, 5, 7, 10, 15, 20, 22, 27, 32, 37, 40, 42, 45, 50, 55, 60, 65]
[2, 3, 2, 3, 5, 5, 2, 5, 5, 5, 3, 2, 3, 5, 5, 5, 5, 3]
9 [0, 2, 5, 7, 10, 15, 20, 22, 25, 27, 32, 37, 40, 42, 45, 47, 50, 55,
60, 65]
[2, 3, 2, 3, 5, 5, 2, 3, 2, 5, 5, 3, 2, 3, 2, 3, 5, 5, 5, 3]
10 [0, 2, 5, 7, 10, 12, 15, 20, 22, 25, 27, 32, 37, 40, 42, 45, 47, 50,
55, 60, 65]
[2, 3, 2, 3, 2, 3, 5, 2, 3, 2, 5, 5, 3, 2, 3, 2, 3, 5, 5, 5, 3]
11 [0, 2, 5, 7, 10, 12, 15, 20, 22, 25, 27, 30, 32, 37, 40, 42, 45, 47,
50, 52, 55, 60, 65]
[2, 3, 2, 3, 2, 3, 5, 2, 3, 2, 3, 2, 5, 3, 2, 3, 2, 3, 2, 3, 5, 5, 3]
12 [0, 2, 5, 7, 10, 12, 15, 17, 20, 22, 25, 27,
30, 32, 37, 40, 42, 45, 47, 50, 52, 55, 60, 65]
[2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 5, 3, 2, 3, 2, 3, 2, 3, 5, 5, 3]
13 [0, 2, 2, 5, 7, 10, 12, 15, 17, 20, 22, 25, 27,
30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 60, 65]
[2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3,
5, 3]
14 [0, 2, 2, 5, 7, 7, 10, 12, 15, 17, 20, 22, 22, 25, 27,
30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 60, 65]
[2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3,
5, 3]
15 [0, 2, 2, 5, 7, 7, 10, 12, 15, 17, 20, 22, 22, 25, 27,
30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 60, 62, 65]
[2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3,
2, 3, 3]
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