Chords of superpyth: Difference between revisions

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


Revision as of 15:05, 14 December 2025

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Below are listed the 11-odd-limit dyadic chords of 11-limit superpyth temperament. 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. If the chord is essentially tempered, it is analyzed in terms of the transversal that employs 8/7 and 10/9.

Superpyth has MOS scales of 5, 7, 12, 17, 22, and 27. The highest complexity of any chord on this list is 18, and would thus require the 22-note mos, but even the 5- and 7-note MOSes contain enough chords to be interesting, though the 12- and 17-note MOSes are needed to properly explore full 7- and 11-limit harmonies.

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 tempering are marked ares. Chords that require any two of 100/99, 245/243 and 540/539 tempering are marked octarod. Chords that require 176/175 and 540/539 tempering are marked guanyin.

Triads

# Generators Transversal Type
1 0–1–2 1–9/8–3/2 Ambitonal
2 0–1–3 1–3/2–12/7 Utonal
3 0–2–3 1–8/7–12/7 Otonal
4 0–1–4 1–9/7–3/2 Utonal
5 0–2–4 1–8/7–9/7 Otonal/utonal
6 0–3–4 1–9/7–12/7 Otonal
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
10 0–4–8 1–9/7–5/3 Sensamagic
11 0–7–8 1–10/9–5/3 Utonal
12 0–1–9 1–5/4–3/2 Otonal
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
16 0–2–11 1–8/7–10/7 Otonal
17 0–3–11 1–10/7–12/7 Otonal
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
21 0–9–11 1–5/4–10/7 Utonal
22 0–3–14 1–11/9–12/7 Swetismic
23 0–7–14 1–10/9–11/9 Otonal/utonal
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
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
3 0–1–3–4 1–9/7–3/2–12/7 Ambitonal
4 0–2–3–4 1–8/7–9/7–12/7 Otonal
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
9 0–1–8–9 1–5/4–3/2–5/3 Ambitonal
10 0–7–8–9 1–10/9–5/4–5/3 Utonal
11 0–2–3–11 1–8/7–10/7–12/7 Otonal
12 0–2–4–11 1–8/7–9/7–10/7 Otonal
13 0–3–4–11 1–9/7–10/7–12/7 Otonal
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
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
20 0–8–9–11 1–5/4–10/7–5/3 Utonal
21 0–3–7–14 1–10/9–11/9–12/7 Swetismic
22 0–3–11–14 1–11/9–10/7–12/7 Swetismic
23 0–7–11–14 1–10/9–11/9–10/7 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
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 1–10/9–9/7–11/7 Swetismic
51 0–2–9–18 1–8/7–5/4–11/7 Valinorsmic
52 0–7–9–18 1–10/9–5/4–11/7 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
1 1–3/2–8/7–12/7–9/7
2 1–8/7–12/7–9/7–10/7
3 1–12/7–9/7–10/9–10/7
4 1–9/7–10/9–5/3–10/7
5 1–10/9–5/3–5/4–10/7
6 1–12/7–10/9–10/7–11/9
7 1–3/2–9/7–5/3–11/6
8 1–9/7–10/9–5/3–11/6
9 1–9/7–10/9–10/7–11/6
10 1–9/7–5/3–10/7–11/6
11 1–10/9–5/3–10/7–11/6
12 1–10/9–10/7–11/9–11/6
13 1–3/2–8/7–5/4–11/8
14 1–3/2–5/3–5/4–11/8
15 1–10/9–5/3–5/4–11/8
16 1–3/2–5/3–11/6–11/8
17 1–10/9–5/3–11/6–11/8
18 1–10/9–11/9–11/6–11/8
19 1–8/7–12/7–9/7–11/7
20 1–12/7–9/7–10/9–11/7
21 1–8/7–12/7–10/7–11/7
22 1–8/7–9/7–10/7–11/7
23 1–12/7–9/7–10/7–11/7
24 1–12/7–10/9–10/7–11/7
25 1–9/7–10/9–10/7–11/7
26 1–8/7–5/4–10/7–11/7
27 1–10/9–5/4–10/7–11/7
28 1–12/7–10/9–11/9–11/7
29 1–12/7–10/7–11/9–11/7
30 1–10/9–10/7–11/9–11/7
31 1–9/7–10/9–11/6–11/7
32 1–9/7–10/7–11/6–11/7
33 1–10/9–10/7–11/6–11/7
34 1–10/9–11/9–11/6–11/7
35 1–10/7–11/9–11/6–11/7
36 1–8/7–5/4–11/8–11/7
37 1–10/9–5/4–11/8–11/7
38 1–10/9–11/9–11/8–11/7
39 1–10/9–11/6–11/8–11/7
40 1–11/9–11/6–11/8–11/7

Hexads

# Generators Transversal Type
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
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
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