Chords of diaschismic

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Below are listed the dyadic chords of 11-limit diaschismic temperament. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 441/440 are labeled werckismic, by 176/175 valinorsmic, by 896/891 pentacircle, and by 126/125 starling. Those requiring tempering by any two of 126/125, 176/175 or 441/440 are labeled thrush, and by both 896/891 and 441/440 pele.

The normal mapping for diaschismic is dia = [<2 0 11 31 45|, <0 1 -2 -8 -12|]. From this we may derive a val v = dia[1] - 100 dia[2] = <2 -100 211 831 1245| which we may use to sort and normalize the chords of diaschismic. 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.

Diaschsmic has MOS of size 10, 12, 22, 34, 46 and 58. It may be seen that even ten notes are not without resources, mostly essentially just. By the time we get to 22 notes, we are well-supplied.

Triads

Number Chord Transversal Type Complexity
1 0-104-208 1-4/3-16/9 ambitonal 4
2 0-104-311 1-4/3-5/3 otonal 6
3 0-207-311 1-5/4-5/3 utonal 6
4 0-104-413 1-4/3-10/9 otonal 8
5 0-207-413 1-5/4-10/9 utonal 8
6 0-208-413 1-16/9-10/9 otonal 8
7 0-311-413 1-5/3-10/9 utonal 8
8 0-207-414 1-5/4-11/7 valinorsmic 8
9 0-208-414 1-16/9-11/7 pentacircle 8
10 0-207-620 1-5/4-7/5 starling 12
11 0-208-620 1-16/9-7/5 werckismic 12
12 0-311-620 1-5/3-7/5 starling 12
13 0-413-620 1-10/9-7/5 starling 12
14 0-414-620 1-11/7-7/5 werckismic 12
15 0-207-827 1-5/4-7/4 otonal 16
16 0-413-827 1-10/9-7/4 werckismic 16
17 0-414-827 1-11/7-7/4 werckismic 16
18 0-620-827 1-7/5-7/4 utonal 16
19 0-104-929 1-4/3-7/6 otonal 18
20 0-311-929 1-5/3-7/6 otonal 18
21 0-620-929 1-7/5-7/6 utonal 18
22 0-827-929 1-7/4-7/6 utonal 18
23 0-207-1032 1-5/4-11/10 valinorsmic 20
24 0-414-1032 1-11/7-11/10 utonal 20
25 0-620-1032 1-7/5-11/10 otonal 20
26 0-827-1032 1-7/4-11/10 valinorsmic 20
27 0-104-1033 1-4/3-14/9 otonal 20
28 0-208-1033 1-16/9-14/9 otonal 20
29 0-413-1033 1-10/9-14/9 otonal 20
30 0-620-1033 1-7/5-14/9 utonal 20
31 0-827-1033 1-7/4-14/9 utonal 20
32 0-929-1033 1-7/6-14/9 utonal 20
33 0-207-1239 1-5/4-11/8 otonal 24
34 0-208-1239 1-16/9-11/8 pentacircle 24
35 0-414-1239 1-11/7-11/8 utonal 24
36 0-827-1239 1-7/4-11/8 otonal 24
37 0-1032-1239 1-11/10-11/8 utonal 24
38 0-1033-1239 1-14/9-11/8 pentacircle 24
39 0-104-1343 1-4/3-11/6 otonal 26
40 0-311-1343 1-5/3-11/6 otonal 26
41 0-414-1343 1-11/7-11/6 utonal 26
42 0-929-1343 1-7/6-11/6 otonal 26
43 0-1032-1343 1-11/10-11/6 utonal 26
44 0-1239-1343 1-11/8-11/6 utonal 26
45 0-104-1445 1-4/3-11/9 otonal 28
46 0-208-1445 1-16/9-11/9 otonal 28
47 0-413-1445 1-10/9-11/9 otonal 28
48 0-414-1445 1-11/7-11/9 utonal 28
49 0-620-1445 1-7/5-11/9 werckismic 28
50 0-827-1445 1-7/4-11/9 werckismic 28
51 0-1032-1445 1-11/10-11/9 utonal 28
52 0-1033-1445 1-14/9-11/9 otonal 28
53 0-1239-1445 1-11/8-11/9 utonal 28
54 0-1343-1445 1-11/6-11/9 utonal 28

Tetrads

Number Chord Transversal Type Complexity
1 0-104-208-413 1-4/3-16/9-10/9 otonal 8
2 0-104-311-413 1-4/3-5/3-10/9 ambitonal 8
3 0-207-311-413 1-5/4-5/3-10/9 utonal 8
4 0-207-311-620 1-5/4-5/3-7/5 starling 12
5 0-207-413-620 1-5/4-10/9-7/5 starling 12
6 0-208-413-620 1-16/9-10/9-7/5 thrush 12
7 0-311-413-620 1-5/3-10/9-7/5 starling 12
8 0-207-414-620 1-5/4-11/7-7/5 thrush 12
9 0-208-414-620 1-16/9-11/7-7/5 pele 12
10 0-207-413-827 1-5/4-10/9-7/4 werckismic 16
11 0-207-414-827 1-5/4-11/7-7/4 thrush 16
12 0-207-620-827 1-5/4-7/5-7/4 starling 16
13 0-413-620-827 1-10/9-7/5-7/4 thrush 16
14 0-414-620-827 1-11/7-7/5-7/4 werckismic 16
15 0-104-311-929 1-4/3-5/3-7/6 otonal 18
16 0-311-620-929 1-5/3-7/5-7/6 starling 18
17 0-620-827-929 1-7/5-7/4-7/6 utonal 18
18 0-207-414-1032 1-5/4-11/7-11/10 valinorsmic 20
19 0-207-620-1032 1-5/4-7/5-11/10 thrush 20
20 0-414-620-1032 1-11/7-7/5-11/10 werckismic 20
21 0-207-827-1032 1-5/4-7/4-11/10 valinorsmic 20
22 0-414-827-1032 1-11/7-7/4-11/10 thrush 20
23 0-620-827-1032 1-7/5-7/4-11/10 valinorsmic 20
24 0-104-208-1033 1-4/3-16/9-14/9 otonal 20
25 0-104-413-1033 1-4/3-10/9-14/9 otonal 20
26 0-208-413-1033 1-16/9-10/9-14/9 otonal 20
27 0-208-620-1033 1-16/9-7/5-14/9 werckismic 20
28 0-413-620-1033 1-10/9-7/5-14/9 starling 20
29 0-413-827-1033 1-10/9-7/4-14/9 werckismic 20
30 0-620-827-1033 1-7/5-7/4-14/9 utonal 20
31 0-104-929-1033 1-4/3-7/6-14/9 ambitonal 20
32 0-620-929-1033 1-7/5-7/6-14/9 utonal 20
33 0-827-929-1033 1-7/4-7/6-14/9 utonal 20
34 0-207-414-1239 1-5/4-11/7-11/8 valinorsmic 24
35 0-208-414-1239 1-16/9-11/7-11/8 pentacircle 24
36 0-207-827-1239 1-5/4-7/4-11/8 otonal 24
37 0-414-827-1239 1-11/7-7/4-11/8 werckismic 24
38 0-207-1032-1239 1-5/4-11/10-11/8 valinorsmic 24
39 0-414-1032-1239 1-11/7-11/10-11/8 utonal 24
40 0-827-1032-1239 1-7/4-11/10-11/8 valinorsmic 24
41 0-208-1033-1239 1-16/9-14/9-11/8 pentacircle 24
42 0-827-1033-1239 1-7/4-14/9-11/8 pentacircle 24
43 0-104-311-1343 1-4/3-5/3-11/6 otonal 26
44 0-104-929-1343 1-4/3-7/6-11/6 otonal 26
45 0-311-929-1343 1-5/3-7/6-11/6 otonal 26
46 0-414-1032-1343 1-11/7-11/10-11/6 utonal 26
47 0-414-1239-1343 1-11/7-11/8-11/6 utonal 26
48 0-1032-1239-1343 1-11/10-11/8-11/6 utonal 26
49 0-104-208-1445 1-4/3-16/9-11/9 otonal 28
50 0-104-413-1445 1-4/3-10/9-11/9 otonal 28
51 0-208-413-1445 1-16/9-10/9-11/9 otonal 28
52 0-208-414-1445 1-16/9-11/7-11/9 pentacircle 28
53 0-208-620-1445 1-16/9-7/5-11/9 werckismic 28
54 0-413-620-1445 1-10/9-7/5-11/9 thrush 28
55 0-414-620-1445 1-11/7-7/5-11/9 werckismic 28
56 0-413-827-1445 1-10/9-7/4-11/9 werckismic 28
57 0-414-827-1445 1-11/7-7/4-11/9 werckismic 28
58 0-620-827-1445 1-7/5-7/4-11/9 werckismic 28
59 0-414-1032-1445 1-11/7-11/10-11/9 utonal 28
60 0-620-1032-1445 1-7/5-11/10-11/9 werckismic 28
61 0-827-1032-1445 1-7/4-11/10-11/9 thrush 28
62 0-104-1033-1445 1-4/3-14/9-11/9 otonal 28
63 0-208-1033-1445 1-16/9-14/9-11/9 otonal 28
64 0-413-1033-1445 1-10/9-14/9-11/9 otonal 28
65 0-620-1033-1445 1-7/5-14/9-11/9 werckismic 28
66 0-827-1033-1445 1-7/4-14/9-11/9 werckismic 28
67 0-208-1239-1445 1-16/9-11/8-11/9 pentacircle 28
68 0-414-1239-1445 1-11/7-11/8-11/9 utonal 28
69 0-827-1239-1445 1-7/4-11/8-11/9 werckismic 28
70 0-1032-1239-1445 1-11/10-11/8-11/9 utonal 28
71 0-1033-1239-1445 1-14/9-11/8-11/9 pentacircle 28
72 0-104-1343-1445 1-4/3-11/6-11/9 ambitonal 28
73 0-414-1343-1445 1-11/7-11/6-11/9 utonal 28
74 0-1032-1343-1445 1-11/10-11/6-11/9 utonal 28
75 0-1239-1343-1445 1-11/8-11/6-11/9 utonal 28

Pentads

Number Chord Transversal Type Complexity
1 0-207-311-413-620 1-5/4-5/3-10/9-7/5 starling 12
2 0-207-413-620-827 1-5/4-10/9-7/5-7/4 thrush 16
3 0-207-414-620-827 1-5/4-11/7-7/5-7/4 thrush 16
4 0-207-414-620-1032 1-5/4-11/7-7/5-11/10 thrush 20
5 0-207-414-827-1032 1-5/4-11/7-7/4-11/10 thrush 20
6 0-207-620-827-1032 1-5/4-7/5-7/4-11/10 thrush 20
7 0-414-620-827-1032 1-11/7-7/5-7/4-11/10 thrush 20
8 0-104-208-413-1033 1-4/3-16/9-10/9-14/9 otonal 20
9 0-208-413-620-1033 1-16/9-10/9-7/5-14/9 thrush 20
10 0-413-620-827-1033 1-10/9-7/5-7/4-14/9 thrush 20
11 0-620-827-929-1033 1-7/5-7/4-7/6-14/9 utonal 20
12 0-207-414-827-1239 1-5/4-11/7-7/4-11/8 thrush 24
13 0-207-414-1032-1239 1-5/4-11/7-11/10-11/8 valinorsmic 24
14 0-207-827-1032-1239 1-5/4-7/4-11/10-11/8 valinorsmic 24
15 0-414-827-1032-1239 1-11/7-7/4-11/10-11/8 thrush 24
16 0-104-311-929-1343 1-4/3-5/3-7/6-11/6 otonal 26
17 0-414-1032-1239-1343 1-11/7-11/10-11/8-11/6 utonal 26
18 0-104-208-413-1445 1-4/3-16/9-10/9-11/9 otonal 28
19 0-208-413-620-1445 1-16/9-10/9-7/5-11/9 thrush 28
20 0-208-414-620-1445 1-16/9-11/7-7/5-11/9 pele 28
21 0-413-620-827-1445 1-10/9-7/5-7/4-11/9 thrush 28
22 0-414-620-827-1445 1-11/7-7/5-7/4-11/9 werckismic 28
23 0-414-620-1032-1445 1-11/7-7/5-11/10-11/9 werckismic 28
24 0-414-827-1032-1445 1-11/7-7/4-11/10-11/9 thrush 28
25 0-620-827-1032-1445 1-7/5-7/4-11/10-11/9 thrush 28
26 0-104-208-1033-1445 1-4/3-16/9-14/9-11/9 otonal 28
27 0-104-413-1033-1445 1-4/3-10/9-14/9-11/9 otonal 28
28 0-208-413-1033-1445 1-16/9-10/9-14/9-11/9 otonal 28
29 0-208-620-1033-1445 1-16/9-7/5-14/9-11/9 werckismic 28
30 0-413-620-1033-1445 1-10/9-7/5-14/9-11/9 thrush 28
31 0-413-827-1033-1445 1-10/9-7/4-14/9-11/9 werckismic 28
32 0-620-827-1033-1445 1-7/5-7/4-14/9-11/9 werckismic 28
33 0-208-414-1239-1445 1-16/9-11/7-11/8-11/9 pentacircle 28
34 0-414-827-1239-1445 1-11/7-7/4-11/8-11/9 werckismic 28
35 0-414-1032-1239-1445 1-11/7-11/10-11/8-11/9 utonal 28
36 0-827-1032-1239-1445 1-7/4-11/10-11/8-11/9 thrush 28
37 0-208-1033-1239-1445 1-16/9-14/9-11/8-11/9 pentacircle 28
38 0-827-1033-1239-1445 1-7/4-14/9-11/8-11/9 pele 28
39 0-414-1032-1343-1445 1-11/7-11/10-11/6-11/9 utonal 28
40 0-414-1239-1343-1445 1-11/7-11/8-11/6-11/9 utonal 28
41 0-1032-1239-1343-1445 1-11/10-11/8-11/6-11/9 utonal 28

Hexads

Number Chord Transversal Type Complexity
1 0-207-414-620-827-1032 1-5/4-11/7-7/5-7/4-11/10 thrush 20
2 0-207-414-827-1032-1239 1-5/4-11/7-7/4-11/10-11/8 thrush 24
3 0-414-620-827-1032-1445 1-11/7-7/5-7/4-11/10-11/9 thrush 28
4 0-104-208-413-1033-1445 1-4/3-16/9-10/9-14/9-11/9 otonal 28
5 0-208-413-620-1033-1445 1-16/9-10/9-7/5-14/9-11/9 thrush 28
6 0-413-620-827-1033-1445 1-10/9-7/5-7/4-14/9-11/9 thrush 28
7 0-414-827-1032-1239-1445 1-11/7-7/4-11/10-11/8-11/9 thrush 28
8 0-414-1032-1239-1343-1445 1-11/7-11/10-11/8-11/6-11/9 utonal 28