Chords of pajara: Difference between revisions

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Overthink (talk | contribs)
autopatrolled
2,982 edits
remove duplicate tetrads; classify a few
Overthink (talk | contribs)
autopatrolled
2,982 edits
remove the remaining duplicates
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== Pentads ==
== Pentads ==


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Revision as of 23:51, 26 January 2026

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This page lists all 11-odd-limit dyadic chords of 11-limit pajara temperament. Each chord listed has multiple inversions; only one is listed, that being the inversion where all notes are a nonnegative number of perfect fifth generators above the root or semioctave, which may not be the optimal voicing of the chord. Note that there are many common chords, such as the classical major seventh chord with ratios 8:10:12:15, which are not listed; in this case because 15/8 is not a ratio of the 11-odd-limit.

If a chord is essentially just, then 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 it is equally well analyzed with either. If a chord is essentially tempered, it is classified based on which commas are needed to define the chord. Chords essentially tempered by 50/49 are labeled jubilismic, by 64/63 archytas, by 99/98 mothwellsmic, by 100/99 ptolemismic, by 176/175 valinorsmic, by 225/224 marvel, and by 896/891 pentacircle. Chords that require any two of 50/49, 64/63, and 225/224 to vanish are labeled pajara [not known to exist], and chords that require any two of 50/49, 99/98, and 100/99 to vanish are labeled jubilee. Chords that require any two of 64/63, 99/98, and 896/891 to vanish are labeled supra [placeholder name, not known to exist], and chords that require any two of 64/63, 100/99, and 176/175 to vanish are labeled ares. Chords that require any two of 99/98, 176/175, and 225/224 to vanish are labeled minerva, and chords that require any two of 100/99, 225/224, and 896/891 to vanish are labeled apollo. Finally, chords that require any three independent commas listed above to vanish are labeled pajara11 [not known to exist].

Triads

# Generators Transversal Type Comments
1 0–0'–2 1–8/7–10/7 Otonal 4:5:7
2 0–1–2 1–9/8–3/2 Ambitonal 6:8:9, 8:9:12
3 0–2–2' 1–8/7–8/5 Utonal 1/(10:8:7)
4 0–0'–3 1–10/7–12/7 Otonal 5:6:7
5 0–1–3 1–3/2–12/7 Utonal 1/(12:8:7)
6 0–2–3 1–8/7–12/7 Otonal 4:6:7
7 0–1–3' 1–6/5–3/2 Utonal 1/(6:5:4)
8 0–2'–3' 1–6/5–8/5 Otonal 4:5:6
9 0–3–3' 1–6/5–12/7 Utonal 1/(7:6:5)
10 0–0'–4 1–9/7–10/7 Otonal
11 0–1–4 1–9/7–3/2 Utonal 1/(9:7:6)
12 0–2–4 1–8/7–9/7 Otonal/utonal 7:8:9~1/(9:8:7)
13 0–2'–4 1–9/7–8/5 Marvel/valinorsmic
14 0–3–4 1–9/7–12/7 Otonal 6:7:9
15 0–1–4' 1–3/2–9/5 Utonal 1/(9:6:5)
16 0–2–4' 1–9/8–9/5 Utonal
17 0–2'–4' 1–8/5–9/5 Otonal
18 0–3'–4' 1–6/5–9/5 Otonal 6:9:10
19 0–4–4' 1–9/7–9/5 Utonal
20 0–2–6 1–8/7–16/11 Utonal
21 0–2'–6 1–16/11–8/5 Utonal
22 0–3–6 1–16/11–12/7 Mothwellsmic
23 0–3'–6 1–6/5–16/11 Ptolemismic
24 0–4–6 1–14/11–16/11 Otonal
25 0–4'–6 1–16/11–20/11 Otonal
26 0–1–7 1–12/11–3/2 Utonal
27 0–3–7 1–12/11–12/7 Utonal
28 0–3'–7 1–12/11–6/5 Utonal
29 0–4–7 1–12/11–14/11 Otonal
30 0–4'–7 1–12/11–20/11 Otonal
31 0–6–7 1–12/11–16/11 Otonal
32 0–1–8 1–3/2–18/11 Utonal
33 0–2–8 1–9/8–18/11 Utonal
34 0–4–8 1–9/7–18/11 Otonal/utonal
35 0–4'–8 1–18/11–9/5 Otonal/utonal
36 0–6–8 1–16/11–18/11 Otonal
37 0–7–8 1–12/11–18/11 Otonal

Tetrads

# Generators Transversal Type Comments
1 0–0'–2–2' 1–8/7–7/5–8/5 Jubilismic
2 0–0'–2–3 1–8/7–10/7–12/7 Otonal 4:5:6:7
3 0–1–2–3 1–8/7–3/2–12/7 Archytas
4 0–0'–3–3' 1–6/5–7/5–12/7 Jubilismic
5 0–1–3–3' 1–6/5–3/2–12/7 Utonal 1/(12:10:8:7)
6 0–0'–2–4 1–8/7–9/7–10/7 Otonal
7 0–1–2–4 1–9/8–9/7–3/2 Utonal 1/(9:8:7:6)
8 0–0'–2'–4 1–9/7–7/5–8/5
9 0–2–2'–4 1–8/7–9/7–8/5
10 0–0'–3–4 1–9/7–10/7–12/7 Otonal 6:7:9:10
11 0–1–3–4 1–9/7–3/2–12/7
12 0–2–3–4 1–8/7–9/7–12/7
13 0–1–2–4' 1–8/7–3/2–9/5
14 0–2–2'–4' 1–8/7–8/5–9/5
15 0–1–3'–4' 1–6/5–3/2–9/5
16 0–2'–3'–4' 1–6/5–8/5–9/5
17 0–0'–4–4' 1–9/7–7/5–9/5
18 0–1–4–4' 1–9/7–3/2–9/5 Utonal 1/(9:7:6:5)
19 0–2–4–4' 1–8/7–9/7–9/5
20 0–2'–4–4' 1–9/7–8/5–9/5
21 0–2–2'–6 1–8/7–16/11–8/5
22 0–2–3–6 1–8/7–16/11–12/7
23 0–2'–3'–6 1–6/5–16/11–8/5
24 0–3–3'–6 1–6/5–16/11–12/7 Jubilee
25 0–2–4–6 1–8/7–9/7–16/11
26 0–2'–4–6 1–9/7–16/11–8/5
27 0–3–4–6 1–9/7–16/11–12/7
28 0–2–4'–6 1–8/7–16/11–9/5
29 0–2'–4'–6 1–16/11–8/5–9/5
30 0–3'–4'–6 1–6/5–16/11–9/5
31 0–4–4'–6 1–9/7–16/11–9/5
32 0–1–3–7 1–12/11–3/2–12/7
33 0–1–3'–7 1–12/11–6/5–3/2
34 0–3–3'–7 1–12/11–6/5–12/7
35 0–1–4–7 1–12/11–9/7–3/2
36 0–3–4–7 1–12/11–9/7–12/7
37 0–1–4'–7 1–12/11–3/2–9/5
38 0–3'–4'–7 1–12/11–6/5–9/5
39 0–4–4'–7 1–12/11–9/7–9/5
40 0–3–6–7 1–12/11–16/11–12/7
41 0–3'–6–7 1–12/11–6/5–16/11
42 0–4–6–7 1–12/11–9/7–16/11
43 0–4'–6–7 1–12/11–16/11–9/5
44 0–1–2–8 1–8/7–3/2–18/11
45 0–1–4–8 1–9/7–3/2–18/11
46 0–2–4–8 1–8/7–9/7–18/11
47 0–1–4'–8 1–3/2–18/11–9/5
48 0–2–4'–8 1–8/7–18/11–9/5
49 0–4–4'–8 1–9/7–18/11–9/5 Otonal/utonal 7:9:10:11~1/(14:11:10:9)
50 0–2–6–8 1–8/7–16/11–18/11
51 0–4–6–8 1–9/7–16/11–18/11
52 0–4'–6–8 1–16/11–18/11–9/5
53 0–1–7–8 1–12/11–3/2–18/11
54 0–4–7–8 1–12/11–9/7–18/11
55 0–4'–7–8 1–12/11–18/11–20/11 Otonal
56 0–6–7–8 1–12/11–16/11–18/11 Otonal

Pentads

# Generators Transversal Type Comments
1 0–0'–2–2'–4 1–8/7–9/7–7/5–8/5
2 0–0'–2–3–4 1–8/7–9/7–7/5–12/7
3 0–1–2–3–4 1–8/7–9/7–3/2–12/7
4 0–0'–2–4–4' 1–8/7–9/7–7/5–9/5
5 0–1–2–4–4' 1–8/7–9/7–3/2–9/5
6 0–2–2'–4–4' 1–8/7–9/7–8/5–9/5
7 0–2–2'–4–6 1–8/7–9/7–16/11–8/5
8 0–2–3–4–6 1–8/7–9/7–16/11–12/7
9 0–2–2'–4'–6 1–8/7–16/11–8/5–9/5
10 0–2'–3'–4'–6 1–6/5–16/11–8/5–9/5
11 0–2–4–4'–6 1–8/7–9/7–16/11–9/5
12 0–2'–4–4'–6 1–9/7–16/11–8/5–9/5
13 0–1–3–3'–7 1–12/11–6/5–3/2–12/7
14 0–1–3–4–7 1–12/11–9/7–3/2–12/7
15 0–1–3'–4'–7 1–12/11–6/5–3/2–9/5
16 0–1–4–4'–7 1–12/11–9/7–3/2–9/5
17 0–3–3'–6–7 1–12/11–6/5–16/11–12/7
18 0–3–4–6–7 1–12/11–9/7–16/11–12/7
19 0–3'–4'–6–7 1–12/11–6/5–16/11–9/5
20 0–4–4'–6–7 1–12/11–9/7–16/11–9/5
21 0–1–2–4–8 1–8/7–9/7–3/2–18/11
22 0–1–2–4'–8 1–8/7–3/2–18/11–9/5
23 0–1–4–4'–8 1–9/7–3/2–18/11–9/5
24 0–2–4–4'–8 1–8/7–9/7–18/11–9/5
25 0–2–4–6–8 1–8/7–9/7–16/11–18/11
26 0–2–4'–6–8 1–8/7–16/11–18/11–9/5
27 0–4–4'–6–8 1–9/7–16/11–18/11–9/5
28 0–1–4–7–8 1–12/11–9/7–3/2–18/11
29 0–1–4'–7–8 1–12/11–3/2–18/11–9/5
30 0–4–4'–7–8 1–12/11–9/7–18/11–9/5
31 0–4–6–7–8 1–12/11–9/7–16/11–18/11
32 0–4'–6–7–8 1–12/11–16/11–18/11–9/5

Hexads

# Generators Transversal Type Comments
1 0–0'–2–2'–4–4' 1–8/7–9/7–7/5–8/5–9/5
2 0–2–2'–4–4'–6 1–8/7–9/7–16/11–8/5–9/5
3 0–1–2–4–4'–8 1–8/7–9/7–3/2–18/11–9/5
4 0–2–4–4'–6–8 1–8/7–9/7–16/11–18/11–9/5
5 0–1–4–4'–7–8 1–12/11–9/7–3/2–18/11–9/5
6 0–4–4'–6–7–8 1–12/11–9/7–16/11–18/11–9/5
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