Xenharmonic Wiki

Chords of magic

Revision as of 15:13, 14 November 2024 by ArrowHead294 (talk | contribs) (→Pentads: Capitalisation)

Below are listed the dyadic chords of 11-limit magic temperament. Typing the chords requires consideration of the fact that magic conflates 10/9 and 11/10 and so also 9/5 and 20/11. If a transversal can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If the chord is essentially tempered, it is analyzed in terms of the transversal which employs 10/9 and 9/5.

Chords requiring tempering only by 225/224 are labeled marvel, by 245/243 sensamagic, by 100/99 ptolemismic, by 896/891 pentacircle, by 385/384 keenanismic, and by 540/539 swetismic. Those requiring any two of 225/224, 100/99 or 896/891 are labeled apollo, any two of 100/99, 245/243 or 540/539 octarod, any two of 245/243, 896/891 or 385/384 sensamagic11, any two of 225/224, 385/384 or 540/539 unimarvel. Chords requiring both 100/99 and 385/384 are labeled supermagic. Finally, anything requiring three independent commas among those discussed above is labeled magic.

Magic has mos scales of sizes 7, 10, 13, 16, 19 and 22 notes. It may be seen that even the seven-note mos is not without a few harmonic resources, and the larger ones do much better.

The chord names use ups and downs as described on the pergens page. The generator is vM3 and the enharmonic interval is ^5dd2. To simplify the names, lifts and drops are also used. One lift is 41 generators, octave-reduced. (41 because 41edo is a near-optimal tuning of magic.) Thus /1 = ^d2 = 20¢ - 8.2c, i.e. an up minus a tempered pythagorean comma. /C = ^Dbb and \C = vB#. ^C = /B# and vC = \Dbb.

Genspan 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Cents (41edo) 0 380 761 1141 322 702 1083 263 644 1024 205 585 966 146
Ratios 1/1 5/4 14/9 27/14 6/5 3/2 15/8 7/6 (16/11) 9/5 9/8 7/5 7/4 (12/11)
Interval P1 vM3 vvA5

v\m6

^^d8

^/M7

^m3 P5 vM7 vvA2

v\m3

^^d5

^/A4

^m7 M2 vA4

\d5

vvA6

v\m7

^^m2

^/A1

TODO: complete the tables

Triads

# Chord Transversal Type Name Comments
1 0-1-2 1-5/4-14/9 Marvel v(vv#5)
2 0-2-4 1-14/9-6/5 Sensamagic ^m(vv#5)
3 0-1-5 1-5/4-3/2 Otonal v
4 0-4-5 1-6/5-3/2 Utonal ^m
5 0-2-7 1-14/9-7/6 Utonal ^/ 1-9/7-3/2
6 0-5-7 1-3/2-7/6 Otonal v\m
7 0-1-8 1-5/4-16/11 Keenanismic v(^^b5)
8 0-4-8 1-6/5-16/11 Ptolemismic ^m(^^b5)
9 0-7-8 1-7/6-16/11 Keenanismic v\m(^^b5)
10 0-1-9 1-5/4-20/11 Utonal v^7no5
11 0-2-9 1-14/9-9/5 Sensamagic ^m7(vv#5)no3
12 0-4-9 1-6/5-9/5 Otonal ^m7no5 or 1-3/2-5/3 = v6no3
13 0-5-9 1-3/2-9/5 Utonal ^m7no3
14 0-7-9 1-7/6-9/5 Sensamagic v\m,v7no5
15 0-8-9 1-16/11-20/11 Otonal v(^^^bb5) 1-5/4-11/8
16 0-1-10 1-5/4-9/8 Otonal v,9no5
17 0-2-10 1-14/9-9/8 Pentacircle 2(vv#5)
18 0-5-10 1-3/2-9/8 Ambitonal 2
19 0-8-10 1-16/11-9/8 Pentacircle 2(^^b5)
20 0-9-10 1-9/5-9/8 Utonal ^9no35 or ^7sus2no5
21 0-1-11 1-5/4-7/5 Marvel v(\b5)
22 0-2-11 1-14/9-7/5 Utonal ^/,^7no5 1-9/7-9/5
23 0-4-11 1-6/5-7/5 Otonal ^m(\b5)
24 0-7-11 1-7/6-7/5 Utonal v\m(\b5)
25 0-9-11 1-9/5-7/5 Otonal ^/(^b5) 1-9/7-10/7
26 0-10-11 1-9/8-7/5 Marvel v,7no5 1-5/4-16/9
27 0-1-12 1-5/4-7/4 Otonal v,v/7
28 0-2-12 1-14/9-7/4 Utonal ^/,9no5 1-9/8-9/7
29 0-4-12 1-6/5-7/4 Keenanismic ^m,v\7
30 0-5-12 1-3/2-7/4 Otonal v\m7no3
31 0-7-12 1-7/6-7/4 Utonal v\m7no5
32 0-8-12 1-16/11-7/4 Keenanismic ^m(v\b5) 1-6/5-11/8
33 0-10-12 1-9/8-7/4 Otonal v\m7sus2
34 0-11-12 1-7/5-7/4 Utonal v\7(\b5)no3
35 0-1-13 1-5/4-12/11 Keenanismic
36 0-2-13 1-14/9-12/11 Swetismic 1-9/7-7/5
37 0-4-13 1-6/5-12/11 Utonal
38 0-5-13 1-3/2-12/11 Utonal
39 0-8-13 1-16/11-12/11 Otonal
40 0-9-13 1-20/11-12/11 Otonal
41 0-11-13 1-7/5-12/11 Swetismic
42 0-12-13 1-7/4-12/11 Keenanismic
43 0-5-18 1-3/2-18/11 Utonal
44 0-7-18 1-7/6-18/11 Swetismic
45 0-8-18 1-16/11-18/11 Otonal
46 0-9-18 1-9/5-18/11 Utonal
47 0-10-18 1-9/8-18/11 Utonal
48 0-11-18 1-7/5-18/11 Swetismic
49 0-13-18 1-12/11-18/11 Otonal
50 0-2-20 1-14/9-14/11 Utonal
51 0-7-20 1-7/6-14/11 Utonal
52 0-8-20 1-16/11-14/11 Otonal
53 0-9-20 1-20/11-14/11 Otonal
54 0-10-20 1-9/8-14/11 Pentacircle
55 0-11-20 1-7/5-14/11 Utonal
56 0-12-20 1-7/4-14/11 Utonal
57 0-13-20 1-12/11-14/11 Otonal
58 0-18-20 1-18/11-14/11 Otonal

Tetrads

# Chord Transversal Type Name Comments
1 0-1-2-9 1-5/4-14/9-9/5 Magic v^7(vv#5)
2 0-2-4-9 1-14/9-6/5-9/5 Sensamagic ^m7(vv#5)
3 0-1-5-9 1-5/4-3/2-9/5 Ptolemismic v^7
4 0-4-5-9 1-6/5-3/2-9/5 Ambitonal ^m7 or 1-5/4-3/2-5/3 = v6
5 0-2-7-9 1-14/9-7/6-9/5 Sensamagic ^/,vv#9 1-9/7-3/2-7/3
6 0-5-7-9 1-3/2-7/6-9/5 Sensamagic v\m^7
7 0-1-8-9 1-5/4-16/11-9/5 Supermagic v^7(^^b5)
8 0-4-8-9 1-6/5-16/11-9/5 Ptolemismic ^m7(^^b5)
9 0-7-8-9 1-7/6-16/11-9/5 Magic v\m,^7(^^b5)
10 0-1-2-10 1-5/4-14/9-9/8 Apollo v,9(vv#5)
11 0-1-5-10 1-5/4-3/2-9/8 Otonal v,9
12 0-1-8-10 1-5/4-16/11-9/8 Sensamagic11 v,9(^^b5)
13 0-1-9-10 1-5/4-9/5-9/8 Ptolemismic v^7,9no5 or v9(^7)no5
14 0-2-9-10 1-14/9-9/5-9/8 Sensamagic11 ^9(vv#5)no3 or ^7(vv#5)sus2
15 0-5-9-10 1-3/2-9/5-9/8 Utonal ^9no3 or ^7sus2 or 2,^7
16 0-8-9-10 1-16/11-9/5-9/8 Apollo
17 0-1-2-11 1-5/4-14/9-7/5 Marvel
18 0-2-4-11 1-14/9-6/5-7/5 Sensamagic
19 0-2-7-11 1-14/9-7/6-7/5 Utonal
20 0-1-9-11 1-5/4-9/5-7/5 Apollo
21 0-2-9-11 1-14/9-9/5-7/5 Sensamagic
22 0-4-9-11 1-6/5-9/5-7/5 Otonal ^m7(\b5)
23 0-7-9-11 1-7/6-9/5-7/5 Sensamagic
24 0-1-10-11 1-5/4-9/8-7/5 Marvel
25 0-2-10-11 1-14/9-9/8-7/5 Apollo
26 0-9-10-11 1-9/5-9/8-7/5 Marvel
27 0-1-2-12 1-5/4-14/9-7/4 Marvel
28 0-2-4-12 1-14/9-6/5-7/4 Sensamagic11
29 0-1-5-12 1-5/4-3/2-7/4 Otonal v,v\7
30 0-4-5-12 1-6/5-3/2-7/4 Keenanismic ^mv\7
31 0-2-7-12 1-14/9-7/6-7/4 Utonal v\m7(vv#5)
32 0-5-7-12 1-3/2-7/6-7/4 Ambitonal v\m7
33 0-1-8-12 1-5/4-16/11-7/4 Keenanismic
34 0-4-8-12 1-6/5-16/11-7/4 Supermagic
35 0-7-8-12 1-7/6-16/11-7/4 Keenanismic v\m7(^^b5)
36 0-1-10-12 1-5/4-9/8-7/4 Otonal
37 0-2-10-12 1-14/9-9/8-7/4 Pentacircle
38 0-5-10-12 1-3/2-9/8-7/4 Otonal v\7sus2 or 2,v\7 or v\9no3
39 0-8-10-12 1-16/11-9/8-7/4 Sensamagic11
40 0-1-11-12 1-5/4-7/5-7/4 Marvel
41 0-2-11-12 1-14/9-7/5-7/4 Utonal
42 0-4-11-12 1-6/5-7/5-7/4 Keenanismic
43 0-7-11-12 1-7/6-7/5-7/4 Utonal
44 0-10-11-12 1-9/8-7/5-7/4 Marvel
45 0-1-2-13 1-5/4-14/9-12/11 Unimarvel
46 0-2-4-13 1-14/9-6/5-12/11 Octarod
47 0-1-5-13 1-5/4-3/2-12/11 Keenanismic
48 0-4-5-13 1-6/5-3/2-12/11 Utonal
49 0-1-8-13 1-5/4-16/11-12/11 Keenanismic
50 0-4-8-13 1-6/5-16/11-12/11 Ptolemismic
51 0-1-9-13 1-5/4-9/5-12/11 Supermagic
52 0-2-9-13 1-14/9-9/5-12/11 Octarod
53 0-4-9-13 1-6/5-9/5-12/11 Ptolemismic
54 0-5-9-13 1-3/2-9/5-12/11 Ptolemismic
55 0-8-9-13 1-16/11-20/11-12/11 Otonal
56 0-1-11-13 1-5/4-7/5-12/11 Unimarvel
57 0-2-11-13 1-14/9-7/5-12/11 Swetismic
58 0-4-11-13 1-6/5-7/5-12/11 Octarod
59 0-9-11-13 1-9/5-7/5-12/11 Octarod
60 0-1-12-13 1-5/4-7/4-12/11 Keenanismic
61 0-2-12-13 1-14/9-7/4-12/11 Unimarvel
62 0-4-12-13 1-6/5-7/4-12/11 Supermagic
63 0-5-12-13 1-3/2-7/4-12/11 Keenanismic
64 0-8-12-13 1-16/11-7/4-12/11 Keenanismic
65 0-11-12-13 1-7/5-7/4-12/11 Unimarvel
66 0-5-7-18 1-3/2-7/6-18/11 Swetismic
67 0-7-8-18 1-7/6-16/11-18/11 Unimarvel
68 0-5-9-18 1-3/2-9/5-18/11 Utonal
69 0-7-9-18 1-7/6-9/5-18/11 Octarod
70 0-8-9-18 1-16/11-20/11-18/11 Otonal
71 0-5-10-18 1-3/2-9/8-18/11 Utonal
72 0-8-10-18 1-16/11-9/8-18/11 Pentacircle
73 0-9-10-18 1-9/5-9/8-18/11 Utonal
74 0-7-11-18 1-7/6-7/5-18/11 Swetismic
75 0-9-11-18 1-9/5-7/5-18/11 Octarod
76 0-10-11-18 1-9/8-7/5-18/11 Unimarvel
77 0-5-13-18 1-3/2-12/11-18/11 Ambitonal
78 0-8-13-18 1-16/11-12/11-18/11 Otonal
79 0-9-13-18 1-20/11-12/11-18/11 Otonal
80 0-11-13-18 1-7/5-12/11-18/11 Swetismic
81 0-2-7-20 1-14/9-7/6-14/11 Utonal
82 0-7-8-20 1-7/6-16/11-14/11 Keenanismic
83 0-2-9-20 1-14/9-9/5-14/11 Octarod
84 0-7-9-20 1-7/6-9/5-14/11 Octarod
85 0-8-9-20 1-16/11-20/11-14/11 Otonal
86 0-2-10-20 1-14/9-9/8-14/11 Pentacircle
87 0-8-10-20 1-16/11-9/8-14/11 Pentacircle
88 0-9-10-20 1-9/5-9/8-14/11 Apollo
89 0-2-11-20 1-14/9-7/5-14/11 Utonal
90 0-7-11-20 1-7/6-7/5-14/11 Utonal
91 0-9-11-20 1-9/5-7/5-14/11 Ptolemismic
92 0-10-11-20 1-9/8-7/5-14/11 Apollo
93 0-2-12-20 1-14/9-7/4-14/11 Utonal
94 0-7-12-20 1-7/6-7/4-14/11 Utonal
95 0-8-12-20 1-16/11-7/4-14/11 Keenanismic
96 0-10-12-20 1-9/8-7/4-14/11 Pentacircle
97 0-11-12-20 1-7/5-7/4-14/11 Utonal
98 0-2-13-20 1-14/9-12/11-14/11 Swetismic
99 0-8-13-20 1-16/11-12/11-14/11 Otonal
100 0-9-13-20 1-20/11-12/11-14/11 Otonal
101 0-11-13-20 1-7/5-12/11-14/11 Octarod
102 0-12-13-20 1-7/4-12/11-14/11 Keenanismic
103 0-7-18-20 1-7/6-18/11-14/11 Swetismic
104 0-8-18-20 1-16/11-18/11-14/11 Otonal
105 0-9-18-20 1-20/11-18/11-14/11 Otonal
106 0-10-18-20 1-9/8-18/11-14/11 Pentacircle
107 0-11-18-20 1-7/5-18/11-14/11 Octarod
108 0-13-18-20 1-12/11-18/11-14/11 Otonal

Pentads

# Chord Transversal Type
1 0-1-2-9-10 1-5/4-14/9-9/5-9/8 Magic
2 0-1-5-9-10 1-5/4-3/2-9/5-9/8 Ptolemismic
3 0-1-8-9-10 1-5/4-16/11-9/5-9/8 Magic
4 0-1-2-9-11 1-5/4-14/9-9/5-7/5 Magic
5 0-2-4-9-11 1-14/9-6/5-9/5-7/5 Sensamagic
6 0-2-7-9-11 1-14/9-7/6-9/5-7/5 Sensamagic
7 0-1-2-10-11 1-5/4-14/9-9/8-7/5 Apollo
8 0-1-9-10-11 1-5/4-9/5-9/8-7/5 Apollo
9 0-2-9-10-11 1-14/9-9/5-9/8-7/5 Magic
10 0-1-2-10-12 1-5/4-14/9-9/8-7/4 Apollo
11 0-1-5-10-12 1-5/4-3/2-9/8-7/4 Otonal
12 0-1-8-10-12 1-5/4-16/11-9/8-7/4 Sensamagic11
13 0-1-2-11-12 1-5/4-14/9-7/5-7/4 Marvel
14 0-2-4-11-12 1-14/9-6/5-7/5-7/4 Sensamagic11
15 0-2-7-11-12 1-14/9-7/6-7/5-7/4 Utonal
16 0-1-10-11-12 1-5/4-9/8-7/5-7/4 Marvel
17 0-2-10-11-12 1-14/9-9/8-7/5-7/4 Apollo
18 0-1-2-9-13 1-5/4-14/9-9/5-12/11 Magic
19 0-2-4-9-13 1-14/9-6/5-9/5-12/11 Octarod
20 0-1-5-9-13 1-5/4-3/2-9/5-12/11 Supermagic
21 0-4-5-9-13 1-6/5-3/2-9/5-12/11 Ptolemismic
22 0-1-8-9-13 1-5/4-16/11-9/5-12/11 supermagic
23 0-4-8-9-13 1-6/5-16/11-9/5-12/11 Ptolemismic
24 0-1-2-11-13 1-5/4-14/9-7/5-12/11 Unimarvel
25 0-2-4-11-13 1-14/9-6/5-7/5-12/11 Octarod
26 0-1-9-11-13 1-5/4-9/5-7/5-12/11 Magic
27 0-2-9-11-13 1-14/9-9/5-7/5-12/11 Octarod
28 0-4-9-11-13 1-6/5-9/5-7/5-12/11 Octarod
29 0-1-2-12-13 1-5/4-14/9-7/4-12/11 Unimarvel
30 0-2-4-12-13 1-14/9-6/5-7/4-12/11 Magic
31 0-1-5-12-13 1-5/4-3/2-7/4-12/11 Keenanismic
32 0-4-5-12-13 1-6/5-3/2-7/4-12/11 Supermagic
33 0-1-8-12-13 1-5/4-16/11-7/4-12/11 Keenanismic
34 0-4-8-12-13 1-6/5-16/11-7/4-12/11 Supermagic
35 0-1-11-12-13 1-5/4-7/5-7/4-12/11 Unimarvel
36 0-2-11-12-13 1-14/9-7/5-7/4-12/11 Unimarvel
37 0-4-11-12-13 1-6/5-7/5-7/4-12/11 Magic
38 0-5-7-9-18 1-3/2-7/6-9/5-18/11 Octarod
39 0-7-8-9-18 1-7/6-16/11-9/5-18/11 Magic
40 0-5-9-10-18 1-3/2-9/5-9/8-18/11 Utonal
41 0-8-9-10-18 1-16/11-9/5-9/8-18/11 Apollo
42 0-7-9-11-18 1-7/6-9/5-7/5-18/11 Octarod
43 0-9-10-11-18 1-9/5-9/8-7/5-18/11 Magic
44 0-5-9-13-18 1-3/2-9/5-12/11-18/11 Ptolemismic
45 0-8-9-13-18 1-16/11-20/11-12/11-18/11 Otonal
46 0-9-11-13-18 1-9/5-7/5-12/11-18/11 Octarod
47 0-2-7-9-20 1-14/9-7/6-9/5-14/11 Octarod
48 0-7-8-9-20 1-7/6-16/11-9/5-14/11 Magic
49 0-2-9-10-20 1-14/9-9/5-9/8-14/11 Magic
50 0-8-9-10-20 1-16/11-9/5-9/8-14/11 Apollo
51 0-2-7-11-20 1-14/9-7/6-7/5-14/11 Utonal
52 0-2-9-11-20 1-14/9-9/5-7/5-14/11 Octarod
53 0-7-9-11-20 1-7/6-9/5-7/5-14/11 Octarod
54 0-2-10-11-20 1-14/9-9/8-7/5-14/11 Apollo
55 0-9-10-11-20 1-9/5-9/8-7/5-14/11 Apollo
56 0-2-7-12-20 1-14/9-7/6-7/4-14/11 Utonal
57 0-7-8-12-20 1-7/6-16/11-7/4-14/11 Keenanismic
58 0-2-10-12-20 1-14/9-9/8-7/4-14/11 Pentacircle
59 0-8-10-12-20 1-16/11-9/8-7/4-14/11 Sensamagic11
60 0-2-11-12-20 1-14/9-7/5-7/4-14/11 Utonal
61 0-7-11-12-20 1-7/6-7/5-7/4-14/11 Utonal
62 0-10-11-12-20 1-9/8-7/5-7/4-14/11 Apollo
63 0-2-9-13-20 1-14/9-9/5-12/11-14/11 Octarod
64 0-8-9-13-20 1-16/11-20/11-12/11-14/11 Otonal
65 0-2-11-13-20 1-14/9-7/5-12/11-14/11 Octarod
66 0-9-11-13-20 1-9/5-7/5-12/11-14/11 Octarod
67 0-2-12-13-20 1-14/9-7/4-12/11-14/11 Unimarvel
68 0-8-12-13-20 1-16/11-7/4-12/11-14/11 Keenanismic
69 0-11-12-13-20 1-7/5-7/4-12/11-14/11 Magic
70 0-7-8-18-20 1-7/6-16/11-18/11-14/11 Unimarvel
71 0-7-9-18-20 1-7/6-9/5-18/11-14/11 Octarod
72 0-8-9-18-20 1-16/11-20/11-18/11-14/11 Otonal
73 0-8-10-18-20 1-16/11-9/8-18/11-14/11 Pentacircle
74 0-9-10-18-20 1-9/5-9/8-18/11-14/11 Apollo
75 0-7-11-18-20 1-7/6-7/5-18/11-14/11 Octarod
76 0-9-11-18-20 1-9/5-7/5-18/11-14/11 Octarod
77 0-10-11-18-20 1-9/8-7/5-18/11-14/11 Magic
78 0-8-13-18-20 1-16/11-12/11-18/11-14/11 Otonal
79 0-9-13-18-20 1-20/11-12/11-18/11-14/11 Otonal
80 0-11-13-18-20 1-7/5-12/11-18/11-14/11 Octarod

Hexads

# Chord Transversal Type
1 0-1-2-9-10-11 1-5/4-14/9-9/5-9/8-7/5 magic
2 0-1-2-10-11-12 1-5/4-14/9-9/8-7/5-7/4 apollo
3 0-1-2-9-11-13 1-5/4-14/9-9/5-7/5-12/11 magic
4 0-2-4-9-11-13 1-14/9-6/5-9/5-7/5-12/11 octarod
5 0-1-2-11-12-13 1-5/4-14/9-7/5-7/4-12/11 unimarvel
6 0-2-4-11-12-13 1-14/9-6/5-7/5-7/4-12/11 magic
7 0-2-7-9-11-20 1-14/9-7/6-9/5-7/5-14/11 octarod
8 0-2-9-10-11-20 1-14/9-9/5-9/8-7/5-14/11 magic
9 0-2-7-11-12-20 1-14/9-7/6-7/5-7/4-14/11 utonal
10 0-2-10-11-12-20 1-14/9-9/8-7/5-7/4-14/11 apollo
11 0-2-9-11-13-20 1-14/9-9/5-7/5-12/11-14/11 octarod
12 0-2-11-12-13-20 1-14/9-7/5-7/4-12/11-14/11 magic
13 0-7-8-9-18-20 1-7/6-16/11-9/5-18/11-14/11 magic
14 0-8-9-10-18-20 1-16/11-9/5-9/8-18/11-14/11 apollo
15 0-7-9-11-18-20 1-7/6-9/5-7/5-18/11-14/11 octarod
16 0-9-10-11-18-20 1-9/5-9/8-7/5-18/11-14/11 magic
17 0-8-9-13-18-20 1-16/11-20/11-12/11-18/11-14/11 otonal
18 0-9-11-13-18-20 1-9/5-7/5-12/11-18/11-14/11 octarod
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