Chords of superpyth: Difference between revisions
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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 or 10/9 above the root. | 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 or 10/9 above the root. | ||
Superpyth is generated by a sharp [[~]][[3/2]] between [[22edo|13\22]] (709.{{Overline|09}}[[{{c}}]]) and [[27edo|16\27]] (711.{{Overline|11}}{{c}}), and generates [[mos]] scales of the patterns [[2L 3s]] ( | Superpyth is generated by a sharp [[~]][[3/2]] between [[22edo|13\22]] (709.{{Overline|09}}[[{{c}}]]) and [[27edo|16\27]] (711.{{Overline|11}}{{c}}), and generates [[mos]] scales of the patterns [[2L 3s]] (pentic), [[5L 2s]] (diatonic), [[5L 7s]] (p-chromatic), [[5L 12s]], [[5L 17s]], and [[22L 5s]]. The pentic and diatonic scales contain some chords in the [[2.3.7 subgroup|2.3.7]] [[subgroup]], though the 12-note chromatic scale is needed to properly utilize intervals of [[5/1|5]], and intervals of [[11/1|11]] don't become common until the 17- and 22-note scales. 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 == | == Triads == | ||
| Line 319: | Line 319: | ||
| 1–9/7–3/2–12/7 | | 1–9/7–3/2–12/7 | ||
| Ambitonal | | Ambitonal | ||
| [[12:14:18:21]], [[14:18:21:24]]<br> 9-odd-limit [[ASS]] | | [[12:14:18:21]], [[14:18:21:24]]<br>[[9-odd-limit]] [[ASS]] | ||
|- | |- | ||
| 4 | | 4 | ||
| Line 534: | Line 534: | ||
| 0–7–9–16 | | 0–7–9–16 | ||
| 1–10/9–5/4–11/8 | | 1–10/9–5/4–11/8 | ||
| Ptolemismic | | Ptolemismic/valinorsmic | ||
| | | | ||
|- | |- | ||
Latest revision as of 01:22, 1 February 2026
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 nonnegative 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 or 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 (pentic), 5L 2s (diatonic), 5L 7s (p-chromatic), 5L 12s, 5L 17s, and 22L 5s. The pentic and diatonic scales contain some chords in the 2.3.7 subgroup, though the 12-note chromatic scale is needed to properly utilize intervals of 5, and intervals of 11 don't become common until the 17- and 22-note scales. 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/(10:8:7) |
| 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/valinorsmic | |
| 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 | 4:5:6:9:11 |
| 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 | 1/(24:20:16:11:9) |
| 19 | 0–2–3–4–18 | 1–8/7–9/7–11/7–12/7 | Otonal | 4:6:7:9:11 |
| 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 | 4:5:6:7:11 |
| 22 | 0–2–4–11–18 | 1–8/7–9/7–10/7–11/7 | Otonal | 4:5:7:9:11 |
| 23 | 0–3–4–11–18 | 1–9/7–10/7–11/7–12/7 | Otonal | 5:6:7:9:11 |
| 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 | 1/(24:20:14:11:9) |
| 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 | 1/(20:16:14:11:9) |
| 39 | 0–7–15–16–18 | 1–11/10–11/8–11/7–11/6 | Utonal | 1/(24:20:16:14:11) |
| 40 | 0–14–15–16–18 | 1–11/9–11/8–11/7–11/6 | Utonal | 1/(24:16:14:11:9) |
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) |