Chords of wizard

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Below are listed the dyadic chords of 11-limit wizard temperament. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 540/539 are swetismic, by 385/384 keenanismic, and by 225/224 marvel. Those requiring tempering by any two of 540/539, 385/384 or 225/224 are labeled unimarvel.

The normal mapping for wizard is wiz = [<2 1 5 2 8|, <0 6 -1 10 -3|]. From this we may derive a val v = wiz[1] + 100 wiz[2] = <2 601 -95 1002 -292|. Since v is approximately 100 times the generator chain val, <0 6 -1 10 -3|, it will sort elements of the 11-limit tonality diamond in the same order, but it uniquely encodes each interval. Since it is a val of wizard, we may take a chord, find the smallest element under the mapping, and use it to translate the whole chord starting at zero. The val v is what might be called a quasi higher rank val; it has a comma comma basis 225/224, 385/384, 4000/3993, 1001255211106304, which are the commas of wizard plus a complex interval which can play no role in ordinary circumstances, so that v can be used in place of the mapping for wizard.

Under "Chord" is listed the chord, normalized to start from zero, in the mapping by v. If we look at the highest, rightmost, element of the chord, divide that by 100, round, and multiply by 2, we get the Graham complexity of the chord. Redundantly for the sake of convenience, the Graham complexity is listed in the last column.

Wizard has MOS of size 6, 10, 16, 22, 28, 50 and 72. It may be seen that even ten notes supplies enough chords to be interesting, and that by the time we reach 22 we might feel we are pretty well supplied.

Triads

Number Chord Transversal Type Complexity
1 0-101-200 1-8/5-9/7 marvel 4
2 0-101-300 1-8/5-16/11 utonal 6
3 0-201-300 1-20/11-16/11 otonal 6
4 0-101-399 1-8/5-7/6 keenanismic 8
5 0-200-399 1-9/7-7/6 swetismic 8
6 0-201-399 1-20/11-7/6 swetismic 8
7 0-300-399 1-16/11-7/6 keenanismic 8
8 0-200-599 1-9/7-3/2 utonal 12
9 0-399-599 1-7/6-3/2 otonal 12
10 0-101-698 1-8/5-6/5 otonal 14
11 0-599-698 1-3/2-6/5 utonal 14
12 0-201-897 1-20/11-12/11 otonal 18
13 0-300-897 1-16/11-12/11 otonal 18
14 0-599-897 1-3/2-12/11 utonal 18
15 0-698-897 1-6/5-12/11 utonal 18
16 0-101-998 1-8/5-7/4 keenanismic 20
17 0-300-998 1-16/11-7/4 keenanismic 20
18 0-399-998 1-7/6-7/4 utonal 20
19 0-599-998 1-3/2-7/4 otonal 20
20 0-698-998 1-6/5-7/4 keenanismic 20
21 0-897-998 1-12/11-7/4 keenanismic 20
22 0-101-1097 1-8/5-7/5 otonal 22
23 0-200-1097 1-9/7-7/5 swetismic 22
24 0-399-1097 1-7/6-7/5 utonal 22
25 0-698-1097 1-6/5-7/5 otonal 22
26 0-897-1097 1-12/11-7/5 swetismic 22
27 0-998-1097 1-7/4-7/5 utonal 22
28 0-101-1196 1-8/5-9/8 marvel 24
29 0-200-1196 1-9/7-9/8 utonal 24
30 0-599-1196 1-3/2-9/8 ambitonal 24
31 0-998-1196 1-7/4-9/8 otonal 24
32 0-1097-1196 1-7/5-9/8 marvel 24
33 0-201-1296 1-20/11-14/11 otonal 26
34 0-300-1296 1-16/11-14/11 otonal 26
35 0-399-1296 1-7/6-14/11 utonal 26
36 0-897-1296 1-12/11-14/11 otonal 26
37 0-998-1296 1-7/4-14/11 utonal 26
38 0-1097-1296 1-7/5-14/11 utonal 26
39 0-101-1297 1-8/5-9/5 otonal 26
40 0-200-1297 1-9/7-9/5 utonal 26
41 0-599-1297 1-3/2-9/5 utonal 26
42 0-698-1297 1-6/5-9/5 otonal 26
43 0-1097-1297 1-7/5-9/5 otonal 26
44 0-1196-1297 1-9/8-9/5 utonal 26
45 0-200-1496 1-9/7-18/11 utonal 30
46 0-201-1496 1-20/11-18/11 otonal 30
47 0-300-1496 1-16/11-18/11 otonal 30
48 0-399-1496 1-7/6-18/11 swetismic 30
49 0-599-1496 1-3/2-18/11 utonal 30
50 0-897-1496 1-12/11-18/11 otonal 30
51 0-1097-1496 1-7/5-18/11 swetismic 30
52 0-1196-1496 1-9/8-18/11 utonal 30
53 0-1296-1496 1-14/11-18/11 otonal 30
54 0-1297-1496 1-9/5-18/11 utonal 30

Tetrads

Number Chord Transversal Type Complexity
1 0-101-200-399 1-8/5-9/7-7/6 unimarvel 8
2 0-101-300-399 1-8/5-16/11-7/6 keenanismic 8
3 0-201-300-399 1-20/11-16/11-7/6 unimarvel 8
4 0-200-399-599 1-9/7-7/6-3/2 swetismic 12
5 0-201-300-897 1-20/11-16/11-12/11 otonal 18
6 0-599-698-897 1-3/2-6/5-12/11 utonal 18
7 0-101-300-998 1-8/5-16/11-7/4 keenanismic 20
8 0-101-399-998 1-8/5-7/6-7/4 keenanismic 20
9 0-300-399-998 1-16/11-7/6-7/4 keenanismic 20
10 0-399-599-998 1-7/6-3/2-7/4 ambitonal 20
11 0-101-698-998 1-8/5-6/5-7/4 keenanismic 20
12 0-599-698-998 1-3/2-6/5-7/4 keenanismic 20
13 0-300-897-998 1-16/11-12/11-7/4 keenanismic 20
14 0-599-897-998 1-3/2-12/11-7/4 keenanismic 20
15 0-698-897-998 1-6/5-12/11-7/4 keenanismic 20
16 0-101-200-1097 1-8/5-9/7-7/5 unimarvel 22
17 0-101-399-1097 1-8/5-7/6-7/5 keenanismic 22
18 0-200-399-1097 1-9/7-7/6-7/5 swetismic 22
19 0-101-698-1097 1-8/5-6/5-7/5 otonal 22
20 0-698-897-1097 1-6/5-12/11-7/5 swetismic 22
21 0-101-998-1097 1-8/5-7/4-7/5 keenanismic 22
22 0-399-998-1097 1-7/6-7/4-7/5 utonal 22
23 0-698-998-1097 1-6/5-7/4-7/5 keenanismic 22
24 0-897-998-1097 1-12/11-7/4-7/5 unimarvel 22
25 0-101-200-1196 1-8/5-9/7-9/8 marvel 24
26 0-200-599-1196 1-9/7-3/2-9/8 utonal 24
27 0-101-998-1196 1-8/5-7/4-9/8 unimarvel 24
28 0-599-998-1196 1-3/2-7/4-9/8 otonal 24
29 0-101-1097-1196 1-8/5-7/5-9/8 marvel 24
30 0-200-1097-1196 1-9/7-7/5-9/8 unimarvel 24
31 0-998-1097-1196 1-7/4-7/5-9/8 marvel 24
32 0-201-300-1296 1-20/11-16/11-14/11 otonal 26
33 0-201-399-1296 1-20/11-7/6-14/11 swetismic 26
34 0-300-399-1296 1-16/11-7/6-14/11 keenanismic 26
35 0-201-897-1296 1-20/11-12/11-14/11 otonal 26
36 0-300-897-1296 1-16/11-12/11-14/11 otonal 26
37 0-300-998-1296 1-16/11-7/4-14/11 keenanismic 26
38 0-399-998-1296 1-7/6-7/4-14/11 utonal 26
39 0-897-998-1296 1-12/11-7/4-14/11 keenanismic 26
40 0-399-1097-1296 1-7/6-7/5-14/11 utonal 26
41 0-897-1097-1296 1-12/11-7/5-14/11 swetismic 26
42 0-998-1097-1296 1-7/4-7/5-14/11 utonal 26
43 0-101-200-1297 1-8/5-9/7-9/5 marvel 26
44 0-200-599-1297 1-9/7-3/2-9/5 utonal 26
45 0-101-698-1297 1-8/5-6/5-9/5 otonal 26
46 0-599-698-1297 1-3/2-6/5-9/5 ambitonal 26
47 0-101-1097-1297 1-8/5-7/5-9/5 otonal 26
48 0-200-1097-1297 1-9/7-7/5-9/5 swetismic 26
49 0-698-1097-1297 1-6/5-7/5-9/5 otonal 26
50 0-101-1196-1297 1-8/5-9/8-9/5 marvel 26
51 0-200-1196-1297 1-9/7-9/8-9/5 utonal 26
52 0-599-1196-1297 1-3/2-9/8-9/5 utonal 26
53 0-1097-1196-1297 1-7/5-9/8-9/5 marvel 26
54 0-201-300-1496 1-20/11-16/11-18/11 otonal 30
55 0-200-399-1496 1-9/7-7/6-18/11 swetismic 30
56 0-201-399-1496 1-20/11-7/6-18/11 swetismic 30
57 0-300-399-1496 1-16/11-7/6-18/11 unimarvel 30
58 0-200-599-1496 1-9/7-3/2-18/11 utonal 30
59 0-399-599-1496 1-7/6-3/2-18/11 swetismic 30
60 0-201-897-1496 1-20/11-12/11-18/11 otonal 30
61 0-300-897-1496 1-16/11-12/11-18/11 otonal 30
62 0-599-897-1496 1-3/2-12/11-18/11 ambitonal 30
63 0-200-1097-1496 1-9/7-7/5-18/11 swetismic 30
64 0-399-1097-1496 1-7/6-7/5-18/11 swetismic 30
65 0-897-1097-1496 1-12/11-7/5-18/11 swetismic 30
66 0-200-1196-1496 1-9/7-9/8-18/11 utonal 30
67 0-599-1196-1496 1-3/2-9/8-18/11 utonal 30
68 0-1097-1196-1496 1-7/5-9/8-18/11 unimarvel 30
69 0-201-1296-1496 1-20/11-14/11-18/11 otonal 30
70 0-300-1296-1496 1-16/11-14/11-18/11 otonal 30
71 0-399-1296-1496 1-7/6-14/11-18/11 swetismic 30
72 0-897-1296-1496 1-12/11-14/11-18/11 otonal 30
73 0-1097-1296-1496 1-7/5-14/11-18/11 swetismic 30
74 0-200-1297-1496 1-9/7-9/5-18/11 utonal 30
75 0-599-1297-1496 1-3/2-9/5-18/11 utonal 30
76 0-1097-1297-1496 1-7/5-9/5-18/11 swetismic 30
77 0-1196-1297-1496 1-9/8-9/5-18/11 utonal 30

Pentads

Number Chord Transversal Type Complexity
1 0-101-300-399-998 1-8/5-16/11-7/6-7/4 keenanismic 20
2 0-599-698-897-998 1-3/2-6/5-12/11-7/4 keenanismic 20
3 0-101-200-399-1097 1-8/5-9/7-7/6-7/5 unimarvel 22
4 0-101-399-998-1097 1-8/5-7/6-7/4-7/5 keenanismic 22
5 0-101-698-998-1097 1-8/5-6/5-7/4-7/5 keenanismic 22
6 0-698-897-998-1097 1-6/5-12/11-7/4-7/5 unimarvel 22
7 0-101-200-1097-1196 1-8/5-9/7-7/5-9/8 unimarvel 24
8 0-101-998-1097-1196 1-8/5-7/4-7/5-9/8 unimarvel 24
9 0-201-300-399-1296 1-20/11-16/11-7/6-14/11 unimarvel 26
10 0-201-300-897-1296 1-20/11-16/11-12/11-14/11 otonal 26
11 0-300-399-998-1296 1-16/11-7/6-7/4-14/11 keenanismic 26
12 0-300-897-998-1296 1-16/11-12/11-7/4-14/11 keenanismic 26
13 0-399-998-1097-1296 1-7/6-7/4-7/5-14/11 utonal 26
14 0-897-998-1097-1296 1-12/11-7/4-7/5-14/11 unimarvel 26
15 0-101-200-1097-1297 1-8/5-9/7-7/5-9/5 unimarvel 26
16 0-101-698-1097-1297 1-8/5-6/5-7/5-9/5 otonal 26
17 0-101-200-1196-1297 1-8/5-9/7-9/8-9/5 marvel 26
18 0-200-599-1196-1297 1-9/7-3/2-9/8-9/5 utonal 26
19 0-101-1097-1196-1297 1-8/5-7/5-9/8-9/5 marvel 26
20 0-200-1097-1196-1297 1-9/7-7/5-9/8-9/5 unimarvel 26
21 0-201-300-399-1496 1-20/11-16/11-7/6-18/11 unimarvel 30
22 0-200-399-599-1496 1-9/7-7/6-3/2-18/11 swetismic 30
23 0-201-300-897-1496 1-20/11-16/11-12/11-18/11 otonal 30
24 0-200-399-1097-1496 1-9/7-7/6-7/5-18/11 swetismic 30
25 0-200-599-1196-1496 1-9/7-3/2-9/8-18/11 utonal 30
26 0-200-1097-1196-1496 1-9/7-7/5-9/8-18/11 unimarvel 30
27 0-201-300-1296-1496 1-20/11-16/11-14/11-18/11 otonal 30
28 0-201-399-1296-1496 1-20/11-7/6-14/11-18/11 swetismic 30
29 0-300-399-1296-1496 1-16/11-7/6-14/11-18/11 unimarvel 30
30 0-201-897-1296-1496 1-20/11-12/11-14/11-18/11 otonal 30
31 0-300-897-1296-1496 1-16/11-12/11-14/11-18/11 otonal 30
32 0-399-1097-1296-1496 1-7/6-7/5-14/11-18/11 swetismic 30
33 0-897-1097-1296-1496 1-12/11-7/5-14/11-18/11 swetismic 30
34 0-200-599-1297-1496 1-9/7-3/2-9/5-18/11 utonal 30
35 0-200-1097-1297-1496 1-9/7-7/5-9/5-18/11 swetismic 30
36 0-200-1196-1297-1496 1-9/7-9/8-9/5-18/11 utonal 30
37 0-599-1196-1297-1496 1-3/2-9/8-9/5-18/11 utonal 30
38 0-1097-1196-1297-1496 1-7/5-9/8-9/5-18/11 unimarvel 30

Hexads

Number Chord Transversal Type Complexity
1 0-101-200-1097-1196-1297 1-8/5-9/7-7/5-9/8-9/5 unimarvel 26
2 0-201-300-399-1296-1496 1-20/11-16/11-7/6-14/11 18/11 unimarvel 30
3 0-201-300-897-1296-1496 1-20/11-16/11-12/11-14/11-18/11 otonal 30
4 0-200-599-1196-1297-1496 1-9/7-3/2-9/8-9/5-18/11 utonal 30
5 0-200-1097-1196-1297-1496 1-9/7-7/5-9/8-9/5-18/11 unimarvel 30