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Chords of superpyth

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Below is a complete list of all 11-odd-limit dyadic chords in 11-limit superpyth temperament. Note that there are many common chords, for example 8:10:12:15, which are not listed; in this case due to 15/8 not being in the 11-odd-limit. Every chord listed has multiple inversions; only one is listed, that being the inversion where all notes are a positive number of perfect fifth generators above the root.

If a chord is essentially just, it is classified as otonal if it is best analyzed in terms of the harmonic series, utonal if best analyzed in terms of the subharmonic series, and ambitonal if equally well analyzed by either. If a chord is essentially tempered, it is classified based on which commas are needed to define the chord. Chords essentially tempered by 64/63 are labeled archytas, by 100/99 ptolemismic, by 176/175 valinorsmic, by 245/243 sensamagic, and by 540/539 swetismic. Chords that require any two of 64/63, 100/99 and 176/175 to vanish are labeled ares. Finally, chords that require any two of 100/99, 245/243 and 540/539 to vanish are labeled octarod.

Typing the chords requires consideration of the fact that superpyth conflates 9/8 with 8/7, and 11/10 with 10/9. 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. However, sometimes multiple such transversals exist, in which case the chord is a plurichord, and the type is given for all possible interpretations. If the chord is essentially tempered, it is analyzed in terms of the transversal that requires the minimum amount of commas to be tempered out; if there is a tie between multiple transversals, it is analyzed in terms of the transversal which employs 8/7 and 10/9 above the root.

Superpyth is generated by a sharp ~3/2 between 13\22 (709.09 ¢) and 16\27 (711.11 ¢), and generates mos scales of the patterns 2L 3s (pentatonic), 5L 2s (diatonic), 5L 7s (p-chromatic), 5L 12s, 5L 17s, and 22L 5s. The highest complexity of any chord on this list is 18 generators, and would thus require the 22-note 5L 17s mos, though there are many chords of much lower complexity. Even the pentatonic and diatonic scales contain some chords in the 2.3.7 subgroup, though the 12- and 17-note scales are needed to properly utilize full 7- and 11-limit harmonies. Superpyth has hardly been explored in the 11-limit, and full 7-limit superpyth has not been explored much either, so these mos scales would be a great place to start such explorations.

Triads

# Generators Transversal Type Comments
1 0–1–2 1–9/8–3/2 Ambitonal 6:8:9, 8:9:12
2 0–1–3 1–3/2–12/7 Utonal 1/(12:8:7)
3 0–2–3 1–8/7–12/7 Otonal 4:6:7
4 0–1–4 1–9/7–3/2 Utonal 1/(9:7:6)
5 0–2–4 1–8/7–9/7 Otonal/utonal 7:8:9~1/(9:8:7)
6 0–3–4 1–9/7–12/7 Otonal 6:7:9
7 0–3–7 1–10/9–12/7 Sensamagic
8 0–4–7 1–10/9–9/7 Sensamagic
9 0–1–8 1–3/2–5/3 Otonal 6:9:10
10 0–4–8 1–9/7–5/3 Sensamagic
11 0–7–8 1–10/9–5/3 Utonal 1/(9:6:5)
12 0–1–9 1–5/4–3/2 Otonal 4:5:6
13 0–2–9 1–9/8–5/4 Otonal
14 0–7–9 1–10/9–5/4 Utonal
15 0–8–9 1–5/4–5/3 Utonal 1/(6:5:4)
16 0–2–11 1–8/7–10/7 Otonal 4:5:7
17 0–3–11 1–10/7–12/7 Otonal 5:6:7
18 0–4–11 1–9/7–10/7 Otonal
19 0–7–11 1–10/9–10/7 Utonal
20 0–8–11 1–10/7–5/3 Utonal 1/(7:6:5)
21 0–9–11 1–5/4–10/7 Utonal 1/(7:5:4)
22 0–3–14 1–11/9–12/7 Swetismic
23 0–7–14 1–10/9–11/9 Otonal/utonal 9:10:11~1/(11:10:9)
24 0–11–14 1–11/9–10/7 Swetismic
25 0–1–15 1–3/2–11/6 Otonal
26 0–4–15 1–9/7–11/6 Swetismic
27 0–7–15 1–11/10–11/6 Utonal
28 0–8–15 1–5/3–11/6 Otonal
29 0–11–15 1–10/7–11/6 Swetismic
30 0–14–15 1–11/9–11/6 Utonal
31 0–1–16 1–11/8–3/2 Otonal
32 0–2–16 1–9/8–11/8 Otonal
33 0–7–16 1–11/10–11/8 Utonal
34 0–8–16 1–11/8–5/3 Ptolemismic
35 0–9–16 1–5/4–11/8 Otonal
36 0–14–16 1–11/9–11/8 Utonal
37 0–15–16 1–11/8–11/6 Utonal
38 0–2–18 1–8/7–11/7 Otonal
39 0–3–18 1–11/7–12/7 Otonal
40 0–4–18 1–9/7–11/7 Otonal
41 0–7–18 1–11/10–11/7 Utonal
42 0–9–18 1–5/4–11/7 Valinorsmic
43 0–11–18 1–10/7–11/7 Otonal
44 0–14–18 1–11/9–11/7 Utonal
45 0–15–18 1–11/7–11/6 Utonal
46 0–16–18 1–11/8–11/7 Utonal

Tetrads

# Generators Transversal Type Comments
1 0–1–2–3 1–8/7–3/2–12/7 Archytas
2 0–1–2–4 1–9/8–9/7–3/2 Utonal 1/(9:7:6:4)
3 0–1–3–4 1–9/7–3/2–12/7 Ambitonal 12:14:18:21, 14:18:21:24
9-odd-limit ASS
4 0–2–3–4 1–8/7–9/7–12/7 Otonal 4:6:7:9
5 0–3–4–7 1–10/9–9/7–12/7 Sensamagic
6 0–1–4–8 1–9/7–3/2–5/3 Sensamagic
7 0–4–7–8 1–9/7–10/9–5/3 Sensamagic
8 0–1–2–9 1–9/8–5/4–3/2 Otonal 4:5:6:9
9 0–1–8–9 1–5/4–3/2–5/3 Ambitonal 10:12:15:18, 12:15:18:20
9-odd-limit ASS
10 0–7–8–9 1–10/9–5/4–5/3 Utonal 1/(9:6:5:4)
11 0–2–3–11 1–8/7–10/7–12/7 Otonal 4:5:6:7
12 0–2–4–11 1–8/7–9/7–10/7 Otonal 4:5:7:9
13 0–3–4–11 1–9/7–10/7–12/7 Otonal 6:7:9:10
14 0–3–7–11 1–10/9–10/7–12/7 Sensamagic
15 0–4–7–11 1–10/9–9/7–10/7 Sensamagic
16 0–4–8–11 1–9/7–10/7–5/3 Sensamagic
17 0–7–8–11 1–10/9–10/7–5/3 Utonal 1/(9:7:6:5)
18 0–2–9–11 1–8/7–5/4–10/7 Archytas/valinorsmic
19 0–7–9–11 1–10/9–5/4–10/7 Utonal 1/(9:7:5:4)
20 0–8–9–11 1–5/4–10/7–5/3 Utonal 1/(12:10:8:7)
21 0–3–7–14 1–11/10–11/9–12/7 Swetismic
22 0–3–11–14 1–11/9–10/7–12/7 Swetismic
23 0–7–11–14 1–11/10–11/9–10/7 Swetismic
24 0–1–4–15 1–9/7–3/2–11/6 Swetismic
25 0–4–7–15 1–10/9–9/7–11/6 Octarod
26 0–1–8–15 1–3/2–5/3–11/6 Otonal
27 0–4–8–15 1–9/7–5/3–11/6 Octarod
28 0–7–8–15 1–10/9–5/3–11/6 Ptolemismic
29 0–4–11–15 1–9/7–10/7–11/6 Swetismic
30 0–7–11–15 1–10/9–10/7–11/6 Octarod
31 0–8–11–15 1–10/7–5/3–11/6 Octarod
32 0–7–14–15 1–11/10–11/9–11/6 Utonal
33 0–11–14–15 1–11/9–10/7–11/6 Swetismic
34 0–1–2–16 1–9/8–11/8–3/2 Otonal
35 0–1–8–16 1–11/8–3/2–5/3 Ptolemismic
36 0–7–8–16 1–10/9–5/3–11/8 Ptolemismic
37 0–1–9–16 1–5/4–11/8–3/2 Otonal
38 0–2–9–16 1–9/8–5/4–11/8 Otonal
39 0–7–9–16 1–10/9–5/4–11/8 Ptolemismic
40 0–8–9–16 1–5/4–11/8–5/3 Ptolemismic
41 0–9–14–16 1–11/10–11/9–11/8 Utonal
42 0–1–15–16 1–11/8–3/2–11/6 Ambitonal 11-odd-limit ASS
43 0–7–15–16 1–11/10–11/8–11/6 Utonal
44 0–8–15–16 1–11/8–5/3–11/6 Ptolemismic
45 0–14–15–16 1–11/9–11/8–11/6 Utonal
46 0–2–3–18 1–8/7–11/7–12/7 Otonal
47 0–2–4–18 1–8/7–9/7–11/7 Otonal
48 0–3–4–18 1–9/7–11/7–12/7 Otonal
49 0–3–7–18 1–10/9–11/7–12/7 Octarod
50 0–4–7–18 1–11/10–9/7–11/7 Swetismic
51 0–2–9–18 1–8/7–5/4–11/7 Valinorsmic
52 0–7–9–18 1–11/10–5/4–11/7 Valinorsmic
53 0–2–11–18 1–8/7–10/7–11/7 Otonal
54 0–3–11–18 1–10/7–11/7–12/7 Otonal
55 0–4–11–18 1–9/7–10/7–11/7 Otonal
56 0–7–11–18 1–10/9–10/7–11/7 Ptolemismic
57 0–9–11–18 1–5/4–10/7–11/7 Valinorsmic
58 0–3–14–18 1–11/9–11/7–12/7 Swetismic
59 0–7–14–18 1–11/10–11/9–11/7 Utonal
60 0–11–14–18 1–11/9–10/7–11/7 Swetismic
61 0–4–15–18 1–9/7–11/7–11/6 Swetismic
62 0–7–15–18 1–11/10–11/7–11/6 Utonal
63 0–11–15–18 1–10/7–11/7–11/6 Swetismic
64 0–14–15–18 1–11/9–11/7–11/6 Utonal
65 0–2–16–18 1–8/7–11/8–11/7 Archytas
66 0–7–16–18 1–11/10–11/8–11/7 Utonal
67 0–9–16–18 1–5/4–11/8–11/7 Valinorsmic
68 0–14–16–18 1–11/9–11/8–11/7 Utonal
69 0–15–16–18 1–11/8–11/7–11/6 Utonal

Pentads

# Generators Transversal Type Comments
1 0–1–2–3–4 1–8/7–9/7–3/2–12/7 Archytas
2 0–2–3–4–11 1–8/7–9/7–10/7–12/7 Otonal 4:5:6:7:9
3 0–3–4–7–11 1–10/9–9/7–10/7–12/7 Sensamagic
4 0–4–7–8–11 1–10/9–9/7–10/7–5/3 Sensamagic
5 0–7–8–9–11 1–10/9–5/4–10/7–5/3 Utonal 1/(24:20:16:14:9)
6 0–3–7–11–14 1–10/9–11/9–10/7–12/7 Octarod
7 0–1–4–8–15 1–9/7–3/2–5/3–11/6 Octarod
8 0–4–7–8–15 1–10/9–9/7–5/3–11/6 Octarod
9 0–4–7–11–15 1–9/7–10/9–10/7–11/6 Octarod
10 0–4–8–11–15 1–9/7–5/3–10/7–11/6 Octarod
11 0–7–8–11–15 1–10/9–5/3–10/7–11/6 Octarod
12 0–7–11–14–15 1–10/9–11/9–10/7–11/6 Octarod
13 0–1–2–9–16 1–9/8–5/4–11/8–3/2 Otonal
14 0–1–8–9–16 1–5/4–11/8–3/2–5/3 Ptolemismic
15 0–7–8–9–16 1–10/9–5/3–5/4–11/8 Ptolemismic
16 0–1–8–15–16 1–11/8–3/2–5/3–11/6 Ptolemismic
17 0–7–8–15–16 1–10/9–11/8–5/3–11/6 Ptolemismic
18 0–7–14–15–16 1–11/10–11/9–11/8–11/6 Utonal
19 0–2–3–4–18 1–8/7–9/7–11/7–12/7 Otonal
20 0–3–4–7–18 1–10/9–9/7–11/7–12/7 Octarod
21 0–2–3–11–18 1–8/7–10/7–11/7–12/7 Otonal
22 0–2–4–11–18 1–8/7–9/7–10/7–11/7 Otonal
23 0–3–4–11–18 1–9/7–10/7–11/7–12/7 Otonal
24 0–3–7–11–18 1–12/7–10/9–10/7–11/7 Octarod
25 0–4–7–11–18 1–10/9–9/7–10/7–11/7 Octarod
26 0–2–9–11–18 1–8/7–5/4–10/7–11/7 Valinorsmic
27 0–7–9–11–18 1–10/9–5/4–10/7–11/7 Ares
28 0–3–7–14–18 1–10/9–11/9–11/7–12/7 Octarod
29 0–3–11–14–18 1–11/9–10/7–11/7–12/7 Swetismic
30 0–7–11–14–18 1–10/9–11/9–10/7–11/7 Octarod
31 0–4–7–15–18 1–10/9–9/7–11/7–11/6 Octarod
32 0–4–11–15–18 1–9/7–10/7–11/7–11/6 Octarod
33 0–7–11–15–18 1–10/9–10/7–11/7–11/6 Octarod
34 0–7–14–15–18 1–11/10–11/9–11/7–11/6 Utonal
35 0–11–14–15–18 1–11/9–10/7–11/7–11/6 Octarod
36 0–2–9–16–18 1–8/7–5/4–11/8–11/7 Ares
37 0–7–9–16–18 1–11/10–5/4–11/8–11/7 Valinorsmic
38 0–7–14–16–18 1–11/10–11/9–11/8–11/7 Utonal
39 0–7–15–16–18 1–11/10–11/8–11/7–11/6 Utonal
40 0–14–15–16–18 1–11/9–11/8–11/7–11/6 Utonal

Hexads

# Generators Transversal Type Comments
1 0–4–7–8–11–15 1–10/9–9/7–10/7–5/3–11/6 Octarod
2 0–2–3–4–11–18 1–8/7–9/7–10/7–11/7–12/7 Otonal 4:5:6:7:9:11
3 0–3–4–7–11–18 1–10/9–9/7–10/7–11/7–12/7 Octarod
4 0–3–7–11–14–18 1–10/9–11/9–10/7–11/7–12/7 Octarod
5 0–4–7–11–15–18 1–10/9–9/7–10/7–11/7–11/6 Octarod
6 0–7–11–14–15–18 1–10/9–10/7–11/9–11/6–11/7 Octarod
7 0–7–14–15–16–18 1–11/10–11/9–11/8–11/7–11/6 Utonal 1/(24:20:16:14:11:9)
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