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 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 marked ares. Finally, chords that require any two of 100/99, 245/243 and 540/539 to vanish are marked 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 generates MOS scales of 5, 7, 12, 17, 22, and 27 notes. The highest complexity of any chord on this list is 18 generators, and would thus require the 22-note MOS. That being said, even the 5- and 7-note MOSes contain some chords in the 2.3.7-subgroup, though the 12- and 17-note MOSes are needed to explore full 7- and 11-limit harmonies. Superpyth has hardly been explored in the 11-limit, and full 7-limit superpyth hasn't been explored much either, so these MOS scales are 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 | |
| 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 | |
| 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) |