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 Vdia = [⟨2 0 11 31 45], ⟨0 1 -2 -8 -12]]. From this we may derive a val V = Vdia[0] - 100 Vdia[1] = ⟨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 scales 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.

Chords are named with ups and downs. The comma (the actual punctuation mark) is pronounced "add", thus Cv,9 is "C down add 9". The pergen is (P8/2, P5), #2 in the notation guide for rank-2 pergens. The period is both vA4 and ^d5. The P5 generator in mingen form is reduced by one period to become both ^m2 and vA1, and hence M2/2. The enharmonic interval is ^^d2, thus ^^C = B#. One up is 6 mingen generators minus 1 period. ^1 = 6c, where c = the amount in cents the tempered fifth exceeds 700¢. For 46edo, c = 4.3¢ and ^1 = 1\46 = 26.1¢.

In Diaschismic, 5/4 = vM3, 7/4 = vm7 and 11/8 = vd5. Thus ^1 equals ~81/80 and ~64/63 and ~8192/8019 = [13 -6 0 0 -1⟩. This won't be true for most (P8, P5/2) temperaments. So although the ratios are specific to diaschismic, the chord names apply to any (P8, P5/2) temperament with A4 > d5. If A4 < d5, the arrows flip direction.

Diaschismic's two genchains, with cents in 46edo
1st genchain Genspan 2nd genchain
Cents Ratio Interval Cents Ratio Interval
261 ^^M2 14 861 18/11 ^A5
757 ^^5 13 157 12/11 ^A1
52 45/44 ^^1 12 652 16/11 ^A4
548 15/11 ^^4 11 1148 27/14, 64/33 ^M7
1043 20/11 ^^m7 10 443 9/7 ^M3
339 40/33 ^^m3 9 939 12/7 ^M6
835 45/28 A5, ^^m6 8 235 8/7 ^M2
130 15/14 A1, ^^m2 7 730 32/21 ^5
626 10/7 A4, ^^d5 6 26 81/80, 64/63 ^1
1122 40/21, 21/11 M7 5 522 27/20 ^4
417 14/11 M3 4 1017 9/5 ^m7
913 27/16, 56/33 M6 3 313 6/5 ^m3
209 9/8 M2 2 809 8/5 vA5, ^m6
704 3/2 P5 1 104 16/15 vA1, ^m2
0 1/1 P1 0 600 45/32, 64/45 vA4, ^d5
496 4/3 P4 -1 1096 15/8 vM7, vd8
991 16/9 m7 -2 391 5/4 vM3, ^d4
287 32/27, 33/28 m3 -3 887 5/3 vM6
783 11/7 m6 -4 183 10/9 vM2
78 22/21, 21/20 m2 -5 678 40/27 v5
574 7/5 d5, vvA4 -6 1174 160/81, 63/32 v8
1070 28/15 d8, vvM7 -7 470 21/16 v4
365 56/45 d4, vvM3 -8 965 7/4 vm7
861 33/20 vvM6 -9 261 7/6 vm3
157 11/10 vvM2 -10 757 14/9 vm6
652 22/15 vv5 -11 52 28/27, 33/32 vm2
1148 88/45 vv8 -12 548 11/8 vd5
443 vv4 -13 1043 11/6 vd8
939 vvm7 -14 339 11/9 vd4

In the genspans columns below, those on the 2nd genchain are marked with an apostrophe, thus 4 = M3 and 4' = ^m7.

TODO: Complete the tables

Triads

# Chord Transversal Type Ratios Genspans Name Complexity
1 0-104-208 1-4/3-16/9 ambitonal 1-4/3-3/2

1-9/8-3/2

0 -1 1

0 2 1

C4

C2

4
2 0-104-311 1-4/3-5/3 otonal 1-5/4-3/2 0 -2' 1 Cv 6
3 0-207-311 1-5/4-5/3 utonal 1-6/5-3/2 0 3' 1 C^m 6
4 0-104-413 1-4/3-10/9 otonal 1-6/5-9/5 0 3' 4' C^m7no5 8
5 0-207-413 1-5/4-10/9 utonal 1-5/4-20/9

1-9/5-9/4

0 -2' -4'

0 4' 2

Cv,v9no5

C^9no35

8
6 0-208-413 1-16/9-10/9 otonal 1-5/4-9/4 0 -2' 2 Cv,9no5 8
7 0-311-413 1-5/3-10/9 utonal 1-3/2-9/5 0 1 4' C^m7no3 8
8 0-207-414 1-5/4-11/7 valinorsmic 1-5/4-11/7

1-5/4-8/5

1-14/11-8/5

0 -2' -4

0 -2' 2'

0 4 -2'

Cv(vv#5)

Cv(v#5)

C(v#5)

8
9 0-208-414 1-16/9-11/7 pentacircle 1-9/8-14/11 0 2 4 C,9no5 8
10 0-207-620 1-5/4-7/5 starling 1-5/4-7/5 0 -2' -6 Cv(b5) 12
11 0-208-620 1-16/9-7/5 werckismic 1-14/11-10/7 0 4 6 C(^^b5) 12
12 0-311-620 1-5/3-7/5 starling 1-6/5-10/7 0 3' 6 C^m(^^d5) 12
13 0-413-620 1-10/9-7/5 starling 1-10/9-7/5 0 -4' -6 Cv2(b5) 12
14 0-414-620 1-11/7-7/5 werckismic 1-14/11-16/9 0 4 -2 C7no5 12
15 0-207-827 1-5/4-7/4 otonal 1-5/4-7/4 0 -2' -8' Cv7no5 16
16 0-413-827 1-10/9-7/4 werckismic 1-10/9-7/4

1-8/7-14/11

0 -4' -8'

0 8' 4

Cv7v9no35

C,^9no5

16
17 0-414-827 1-11/7-7/4 werckismic 1-9/5-16/7

1-10/9-14/11

0 4' 8'

0 -4' 4

C^7^9no35

C,v9no5

16
18 0-620-827 1-7/5-7/4 utonal 1-5/4-10/7 0 -2' 6 Cv(^^b5) 16
19 0-104-929 1-4/3-7/6 otonal 1-3/2-7/4 0 1 -8' Cv7no3 18
20 0-311-929 1-5/3-7/6 otonal 1-7/6-5/3

1-6/5-7/5

0 -9' -3'

0 3' -6

Cvm6no5

C^dim

18
21 0-620-929 1-7/5-7/6 utonal 1-7/6-7/5

1-6/5-12/7

0 -9' -6

0 3' 9'

Cvdim

C^m6no5

18
22 0-827-929 1-7/4-7/6 utonal 1-7/6-7/4 0 -9' -8' Cvm7no5 18
23 0-207-1032 1-5/4-11/10 valinorsmic 1-5/4-11/5

1-20/11-16/7

0 -2' -10

0 10 8'

Cv,vv9no5

C^^7^9no35

20
24 0-414-1032 1-11/7-11/10 utonal 1-14/11-7/5 0 4 -6 C(b5) 20
25 0-620-1032 1-7/5-11/10 otonal 1-11/10-7/5

1-14/11-20/11

0 -10 -6

0 4 10

Cvv2(b5)

C,^^7no5

20
26 0-827-1032 1-7/4-11/10 valinorsmic 1-7/4-11/5

1-5/4-45/28

0 -8' -10

0 -2' 8'

Cv7vv9no35

Cv,v9no5

20
27 0-104-1033 1-4/3-14/9 otonal 1-7/6-3/2 0 -9' 1 Cvm 20
28 0-208-1033 1-16/9-14/9 otonal 1-7/4-9/4

1-9/7-16/7

0 -8' 2

0 10' 8'

Cv9no35

C^,^9no5

20
29 0-413-1033 1-10/9-14/9 otonal 1-9/7-10/7 0 10' 6 C^(^^b5) 20
30 0-620-1033 1-7/5-14/9 utonal 1-9/7-9/5 0 10' 4' C^7no5 20
31 0-827-1033 1-7/4-14/9 utonal 1-9/7-9/4

1-8/7-16/9

0 10' 2

0 8' -2

C^,9no5

C^2,7no5

20
32 0-929-1033 1-7/6-14/9 utonal 1-9/7-3/2 0 10' 1 C^ 20
33 0-207-1239 1-5/4-11/8 otonal 1-5/4-11/8 0 -2' -12' Cv(vb5) 24
34 0-208-1239 1-16/9-11/8 pentacircle 1-9/7-16/11 0 10' 12' C^,^#11no5 24
35 0-414-1239 1-11/7-11/8 utonal 1-14/11-7/4 0 4 -8' C,v7no5 24
36 0-827-1239 1-7/4-11/8 otonal 1-14/11-16/11 0 4 12' C,^#11no5 24
37 0-1032-1239 1-11/10-11/8 utonal 1-11/10-11/8

1-5/4-20/11

0 -10 -12'

0 -2' 10

Cvv2(vb5)

Cv,^^7no5

24
38 0-1033-1239 1-14/9-11/8 pentacircle 1-9/7-16/9 0 10' -2 C^,7no5 24
39 0-104-1343 1-4/3-11/6 otonal 1-11/8-3/2 0 -12' 1 C,vd5no3 26
40 0-311-1343 1-5/3-11/6 otonal 1-6/5-11/5 0 3' -10 C^m,vv9no5 26
41 0-414-1343 1-11/7-11/6 utonal 1-7/6-14/11 0 -9' 4 C,vm10no5 26
42 0-929-1343 1-7/6-11/6 otonal 1-7/6-11/6 0 -9' -13' Cvm,vb8 26
43 0-1032-1343 1-11/10-11/6 utonal 1-9/7-12/5 0 3' 10' C^,^m10 26
44 0-1239-1343 1-11/8-11/6 utonal 1-4/3-16/11 0 -1 12' C4,^#11no5 26
45 0-104-1445 1-4/3-11/9 otonal 28
46 0-208-1445 1-16/9-11/9 otonal 1-11/9-16/9 0 -14' -2 Cvb4,7no5 28
47 0-413-1445 1-10/9-11/9 otonal 1-10/9-11/9

1-11/10-9/5

0 -4' -14'

0 -10 4'

Cv2,vb4no5

Cvv2,^7

28
48 0-414-1445 1-11/7-11/9 utonal 1-11/9-11/7

1-14/11-14/9

1-9/7-18/11

0 -14' -4

0 4 -10'

0 10' 14'

Cvb4,b6no5

C,vm6no5

C^(^#5)

28
49 0-620-1445 1-7/5-11/9 werckismic 1-11/9-7/5 0 -14' -6 Cvb4(b5) 28
50 0-827-1445 1-7/4-11/9 werckismic 1-11/9-7/4

1-8/7-7/5

0 -14' -8'

0 8' -6

Cvb4,v7no5

C^2(b5)

28
51 0-1032-1445 1-11/10-11/9 utonal 1-11/10-11/9

1-20/11-20/9

0 -10 -14'

0 10 -4'

Cvv2,vb4no5

C^^7,v9no35

28
52 0-1033-1445 1-14/9-11/9 otonal 1-14/11-18/11 0 4 14' C(^#5) 28
53 0-1239-1445 1-11/8-11/9 utonal 1-16/11-16/9 0 12' -2 C^#4,7no5 28
54 0-1343-1445 1-11/6-11/9 utonal 1-11/9-11/6 0 -14' -13' CvvvM7no5 28

Tetrads

# Chord Transversal Type Ratios Genspans Name 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

# Chord Transversal Type Ratios Genspans Name 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

# Chord Transversal Type Ratios Genspans Name Complexity
1 0-207-414-620-827-1032 1-5/4-11/7-7/5-7/4-11/10 thrush 1-5/4-7/5-7/4-11/5-22/7 0 -2' -6 -8' -10 -4 v7vv9b13(b5) 20
2 0-207-414-827-1032-1239 1-5/4-11/7-7/4-11/10-11/8 thrush 1-5/4-11/8-7/4-11/5-22/7 0 -2' -12' -8' -10 -4 v7vv9b13(vb5) 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
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