Chords of superpyth: Difference between revisions

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+ generators, replacing complexity as the former covers the latter. Finish for triads and hexads
Finish types for triads
Line 15: Line 15:
| 0–1–2
| 0–1–2
| 1–9/8–3/2
| 1–9/8–3/2
| ambitonal
| Ambitonal
|-
|-
| 2
| 2
| 0–1–3
| 0–1–3
| 1–3/2–12/7
| 1–3/2–12/7
| utonal
| Utonal
|-
|-
| 3
| 3
| 0–2–3
| 0–2–3
| 1–8/7–12/7
| 1–8/7–12/7
| otonal
| Otonal
|-
|-
| 4
| 4
| 0–1–4
| 0–1–4
| 1–9/7–3/2
| 1–9/7–3/2
| utonal
| Utonal
|-
|-
| 5
| 5
| 0–2–4
| 0–2–4
| 1–8/7–9/7
| 1–8/7–9/7
| ambitonal
| Ambitonal
|-
|-
| 6
| 6
| 0–3–4
| 0–3–4
| 1–9/7–12/7
| 1–9/7–12/7
| otonal
| Otonal
|-
|-
| 7
| 7
| 0–3–7
| 0–3–7
| 1–10/9–12/7
| 1–10/9–12/7
| sensamagic
| Sensamagic
|-
|-
| 8
| 8
| 0–4–7
| 0–4–7
| 1–10/9–9/7
| 1–10/9–9/7
| sensamagic
| Sensamagic
|-
|-
| 9
| 9
| 0–1–8
| 0–1–8
| 1–3/2–5/3
| 1–3/2–5/3
| otonal
| Otonal
|-
|-
| 10
| 10
| 0–4–8
| 0–4–8
| 1–9/7–5/3
| 1–9/7–5/3
| sensamagic
| Sensamagic
|-
|-
| 11
| 11
| 0–7–8
| 0–7–8
| 1–10/9–5/3
| 1–10/9–5/3
| utonal
| Utonal
|-
|-
| 12
| 12
| 0–1–9
| 0–1–9
| 1–5/4–3/2
| 1–5/4–3/2
| otonal
| Otonal
|-
|-
| 13
| 13
| 0–2–9
| 0–2–9
| 1–9/8–5/4
| 1–9/8–5/4
| otonal
| Otonal
|-
|-
| 14
| 14
| 0–7–9
| 0–7–9
| 1–10/9–5/4
| 1–10/9–5/4
| utonal
| Utonal
|-
|-
| 15
| 15
| 0–8–9
| 0–8–9
| 1–5/4–5/3
| 1–5/4–5/3
| utonal
| Utonal
|-
|-
| 16
| 16
| 0–2–11
| 0–2–11
| 1–8/7–10/7
| 1–8/7–10/7
|  
| Otonal
|-
|-
| 17
| 17
| 0–3–11
| 0–3–11
| 1–12/7–10/7
| 1–12/7–10/7
|  
| Otonal
|-
|-
| 18
| 18
| 0–4–11
| 0–4–11
| 1–9/7–10/7
| 1–9/7–10/7
|  
| Otonal
|-
|-
| 19
| 19
| 0–7–11
| 0–7–11
| 1–10/9–10/7
| 1–10/9–10/7
|  
| Utonal
|-
|-
| 20
| 20
| 0–8–11
| 0–8–11
| 1–5/3–10/7
| 1–5/3–10/7
|  
| Utonal
|-
|-
| 21
| 21
| 0–9–11
| 0–9–11
| 1–5/4–10/7
| 1–5/4–10/7
|  
| Utonal
|-
|-
| 22
| 22
| 0–3–14
| 0–3–14
| 1–12/7–11/9
| 1–12/7–11/9
|  
| Swetismic
|-
|-
| 23
| 23
| 0–7–14
| 0–7–14
| 1–10/9–11/9
| 1–10/9–11/9
|  
| Otonal/utonal
|-
|-
| 24
| 24
| 0–11–14
| 0–11–14
| 1–10/7–11/9
| 1–10/7–11/9
|  
| Swetismic
|-
|-
| 25
| 25
| 0–1–15
| 0–1–15
| 1–3/2–11/6
| 1–3/2–11/6
|  
| Otonal
|-
|-
| 26
| 26
| 0–4–15
| 0–4–15
| 1–9/7–11/6
| 1–9/7–11/6
|  
| Swetismic
|-
|-
| 27
| 27
| 0–7–15
| 0–7–15
| 1–10/9–11/6
| 1–11/10–11/6
|  
| Utonal
|-
|-
| 28
| 28
| 0–8–15
| 0–8–15
| 1–5/3–11/6
| 1–5/3–11/6
|  
| Otonal
|-
|-
| 29
| 29
| 0–11–15
| 0–11–15
| 1–10/7–11/6
| 1–10/7–11/6
|  
| Swetismic
|-
|-
| 30
| 30
| 0–14–15
| 0–14–15
| 1–11/9–11/6
| 1–11/9–11/6
|  
| Utonal
|-
|-
| 31
| 31
| 0–1–16
| 0–1–16
| 1–3/2–11/8
| 1–3/2–11/8
| otonal
| Otonal
|-
|-
| 32
| 32
| 0–2–16
| 0–2–16
| 1–9/8–11/8
| 1–9/8–11/8
| otonal
| Otonal
|-
|-
| 33
| 33
| 0–7–16
| 0–7–16
| 1–11/10–11/8
| 1–11/10–11/8
| utonal
| Utonal
|-
|-
| 34
| 34
| 0–8–16
| 0–8–16
| 1–5/3–11/8
| 1–5/3–11/8
| ptolemismic
| Ptolemismic
|-
|-
| 35
| 35
| 0–9–16
| 0–9–16
| 1–5/4–11/8
| 1–5/4–11/8
| otonal
| Otonal
|-
|-
| 36
| 36
| 0–14–16
| 0–14–16
| 1–11/9–11/8
| 1–11/9–11/8
| utonal
| Utonal
|-
|-
| 37
| 37
| 0–15–16
| 0–15–16
| 1–11/6–11/8
| 1–11/6–11/8
| utonal
| Utonal
|-
|-
| 38
| 38
| 0–2–18
| 0–2–18
| 1–8/7–11/7
| 1–8/7–11/7
|  
| Otonal
|-
|-
| 39
| 39
| 0–3–18
| 0–3–18
| 1–12/7–11/7
| 1–12/7–11/7
|  
| Otonal
|-
|-
| 40
| 40
| 0–4–18
| 0–4–18
| 1–9/7–11/7
| 1–9/7–11/7
|  
| Otonal
|-
|-
| 41
| 41
| 0–7–18
| 0–7–18
| 1–10/9–11/7
| 1–11/10–11/7
|  
| Utonal
|-
|-
| 42
| 42
| 0–9–18
| 0–9–18
| 1–5/4–11/7
| 1–5/4–11/7
|  
| Valinorsmic
|-
|-
| 43
| 43
| 0–11–18
| 0–11–18
| 1–10/7–11/7
| 1–10/7–11/7
|  
| Otonal
|-
|-
| 44
| 44
| 0–14–18
| 0–14–18
| 1–11/9–11/7
| 1–11/9–11/7
|  
| Utonal
|-
|-
| 45
| 45
| 0–15–18
| 0–15–18
| 1–11/6–11/7
| 1–11/6–11/7
|  
| Utonal
|-
|-
| 46
| 46
| 0–16–18
| 0–16–18
| 1–11/8–11/7
| 1–11/8–11/7
|  
| Utonal
|}
|}


Line 822: Line 822:
| 0–2–3–4–11–18
| 0–2–3–4–11–18
| 1–8/7–9/7–10/7–11/7–12/7
| 1–8/7–9/7–10/7–11/7–12/7
| otonal
| Otonal
|-
|-
| 3
| 3
Line 847: Line 847:
| 0–7–14–15–16–18
| 0–7–14–15–16–18
| 1–11/10–11/9–11/8–11/7–11/6
| 1–11/10–11/9–11/8–11/7–11/6
| utonal
| Utonal
|}
|}


[[Category:Todo:expand]]
[[Category:Todo:expand]]

Revision as of 07:33, 13 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.

Chords essentially tempered by 64/63 are labeled archytas, by 100/99 ptolemismic, by 176/175 valinorsmic, by 245/243 sensamagic, 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 Ambitonal
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 1–12/7–10/7 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 1–5/3–10/7 Utonal
21 0–9–11 1–5/4–10/7 Utonal
22 0–3–14 1–12/7–11/9 Swetismic
23 0–7–14 1–10/9–11/9 Otonal/utonal
24 0–11–14 1–10/7–11/9 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 1–3/2–11/8 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 1–5/3–11/8 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 1–11/6–11/8 Utonal
38 0–2–18 1–8/7–11/7 Otonal
39 0–3–18 1–12/7–11/7 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 1–11/6–11/7 Utonal
46 0–16–18 1–11/8–11/7 Utonal

Tetrads

# Generators Transversal Type
1 1–3/2–8/7–12/7
2 1–3/2–8/7–9/7
3 1–3/2–12/7–9/7
4 1–8/7–12/7–9/7
5 1–12/7–9/7–10/9
6 1–3/2–9/7–5/3
7 1–9/7–10/9–5/3
8 1–3/2–8/7–5/4
9 1–3/2–5/3–5/4
10 1–10/9–5/3–5/4
11 1–8/7–12/7–10/7
12 1–8/7–9/7–10/7
13 1–12/7–9/7–10/7
14 1–12/7–10/9–10/7
15 1–9/7–10/9–10/7
16 1–9/7–5/3–10/7
17 1–10/9–5/3–10/7
18 1–8/7–5/4–10/7
19 1–10/9–5/4–10/7
20 1–5/3–5/4–10/7
21 1–12/7–10/9–11/9
22 1–12/7–10/7–11/9
23 1–10/9–10/7–11/9
24 1–3/2–9/7–11/6
25 1–9/7–10/9–11/6
26 1–3/2–5/3–11/6
27 1–9/7–5/3–11/6
28 1–10/9–5/3–11/6
29 1–9/7–10/7–11/6
30 1–10/9–10/7–11/6
31 1–5/3–10/7–11/6
32 1–10/9–11/9–11/6
33 1–10/7–11/9–11/6
34 1–3/2–8/7–11/8
35 1–3/2–5/3–11/8
36 1–10/9–5/3–11/8
37 1–3/2–5/4–11/8
38 1–8/7–5/4–11/8
39 1–10/9–5/4–11/8
40 1–5/3–5/4–11/8
41 1–10/9–11/9–11/8
42 1–3/2–11/6–11/8
43 1–10/9–11/6–11/8
44 1–5/3–11/6–11/8
45 1–11/9–11/6–11/8
46 1–8/7–12/7–11/7
47 1–8/7–9/7–11/7
48 1–12/7–9/7–11/7
49 1–12/7–10/9–11/7
50 1–9/7–10/9–11/7
51 1–8/7–5/4–11/7
52 1–10/9–5/4–11/7
53 1–8/7–10/7–11/7
54 1–12/7–10/7–11/7
55 1–9/7–10/7–11/7
56 1–10/9–10/7–11/7
57 1–5/4–10/7–11/7
58 1–12/7–11/9–11/7
59 1–10/9–11/9–11/7
60 1–10/7–11/9–11/7
61 1–9/7–11/6–11/7
62 1–10/9–11/6–11/7
63 1–10/7–11/6–11/7
64 1–11/9–11/6–11/7
65 1–8/7–11/8–11/7
66 1–10/9–11/8–11/7
67 1–5/4–11/8–11/7
68 1–11/9–11/8–11/7
69 1–11/6–11/8–11/7

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 1–9/7–10/9–5/3–10/7–11/6
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 1–12/7–9/7–10/9–10/7–11/7
4 0–3–7–11–14–18 1–12/7–10/9–10/7–11/9–11/7
5 0–4–7–11–15–18 1–9/7–10/9–10/7–11/6–11/7
6 0–7–11–14–15–18 1–10/9–10/7–11/9–11/6–11/7
7 0–7–14–15–16–18 1–11/10–11/9–11/8–11/7–11/6 Utonal
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