Chords of pajara
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[note 1], which may not be the optimal root position 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, 99/98, and 100/99 to vanish are labeled jubilismic11. 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, 225/224, and 896/891 to vanish are labeled apollo.[note 2]
Typing the chords requires consideration of the fact that pajara conflates several pairs of consonances: 11/10~10/9, 9/8~8/7, 14/11~9/7, 7/5~10/7, and their octave complements. 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, then the types are prioritized as follows: jubilismic > archytas > marvel > mothwellsmic > ptolemismic > valinorsmic > pentacircle, and jubilismic11 > ares > apollo.
In the "Generators" column of the tables below, each number corresponds to a note of the chord, with the number shown being the number of perfect fifths which need to be stacked above the root or semioctave to reach that note. If an apostrophe follows the number, then the note is reached by stacking fifths above the semioctave; if no apostrophe follows the number, then the note is reached by stacking fifths above the root. For example, the note 3 is reached by stacking three fifths above the root, which in pajara is equal to 12/7 when octave reduced. The note 3' is reached by stacking 3 fifths above the semioctave, which in pajara is 6/5 when octave reduced.
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 | |
| 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 | 4:5:7:9 |
| 7 | 0–1–2–4 | 1–9/8–9/7–3/2 | Utonal | 1/(9:7:6:4) |
| 8 | 0–0'–2'–4 | 1–9/7–10/7–8/5 | Marvel | |
| 9 | 0–2–2'–4 | 1–8/7–9/7–8/5 | Marvel | |
| 10 | 0–0'–3–4 | 1–9/7–10/7–12/7 | Otonal | 5:6:7:9 |
| 11 | 0–1–3–4 | 1–9/7–3/2–12/7 | Ambitonal | 12:14:18:21, 14:18:21:24 9-odd-limit ASS |
| 12 | 0–2–3–4 | 1–8/7–9/7–12/7 | Otonal | 4:6:7:9 |
| 13 | 0–1–2–4' | 1–9/8–3/2–9/5 | Utonal | 1/(9:6:5:4) |
| 14 | 0–2–2'–4' | 1–8/7–8/5–9/5 | Archytas | |
| 15 | 0–1–3'–4' | 1–6/5–3/2–9/5 | Ambitonal | 10:12:15:18, 12:15:18:20 9-odd-limit ASS |
| 16 | 0–2'–3'–4' | 1–6/5–8/5–9/5 | Otonal | 4:5:6:9 |
| 17 | 0–0'–4–4' | 1–9/7–7/5–9/5 | Jubilismic | |
| 18 | 0–1–4–4' | 1–9/7–3/2–9/5 | Utonal | 1/(9:7:6:5) |
| 19 | 0–2–4–4' | 1–9/8–9/7–9/5 | Utonal | 1/(9:7:5:4) |
| 20 | 0–2'–4–4' | 1–9/7–8/5–9/5 | Marvel | |
| 21 | 0–2–2'–6 | 1–8/7–16/11–8/5 | Utonal | |
| 22 | 0–2–3–6 | 1–8/7–16/11–12/7 | Mothwellsmic | |
| 23 | 0–2'–3'–6 | 1–6/5–16/11–8/5 | Ptolemismic | |
| 24 | 0–3–3'–6 | 1–6/5–16/11–12/7 | Jubilismic11 | |
| 25 | 0–2–4–6 | 1–8/7–9/7–16/11 | Mothwellsmic | |
| 26 | 0–2'–4–6 | 1–14/11–16/11–8/5 | Valinorsmic | |
| 27 | 0–3–4–6 | 1–9/7–16/11–12/7 | Mothwellsmic | |
| 28 | 0–2–4'–6 | 1–8/7–16/11–20/11 | Valinorsmic | |
| 29 | 0–2'–4'–6 | 1–16/11–8/5–9/5 | Ptolemismic | |
| 30 | 0–3'–4'–6 | 1–6/5–16/11–9/5 | Ptolemismic | |
| 31 | 0–4–4'–6 | 1–14/11–16/11–20/11 | Otonal | |
| 32 | 0–1–3–7 | 1–12/11–3/2–12/7 | Utonal | |
| 33 | 0–1–3'–7 | 1–12/11–6/5–3/2 | Utonal | |
| 34 | 0–3–3'–7 | 1–12/11–6/5–12/7 | Utonal | |
| 35 | 0–1–4–7 | 1–12/11–9/7–3/2 | Mothwellsmic | |
| 36 | 0–3–4–7 | 1–12/11–9/7–12/7 | Mothwellsmic | |
| 37 | 0–1–4'–7 | 1–12/11–3/2–9/5 | Ptolemismic | |
| 38 | 0–3'–4'–7 | 1–12/11–6/5–9/5 | Ptolemismic | |
| 39 | 0–4–4'–7 | 1–12/11–14/11–20/11 | Otonal | |
| 40 | 0–3–6–7 | 1–12/11–16/11–12/7 | Mothwellsmic | |
| 41 | 0–3'–6–7 | 1–12/11–6/5–16/11 | Ptolemismic | |
| 42 | 0–4–6–7 | 1–12/11–14/11–16/11 | Otonal | |
| 43 | 0–4'–6–7 | 1–12/11–16/11–20/11 | Otonal | |
| 44 | 0–1–2–8 | 1–9/8–3/2–18/11 | Utonal | |
| 45 | 0–1–4–8 | 1–9/7–3/2–18/11 | Utonal | |
| 46 | 0–2–4–8 | 1–9/8–9/7–18/11 | Utonal | |
| 47 | 0–1–4'–8 | 1–3/2–18/11–9/5 | Utonal | |
| 48 | 0–2–4'–8 | 1–9/8–18/11–9/5 | Utonal | |
| 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 | Archytas | |
| 51 | 0–4–6–8 | 1–14/11–16/11–18/11 | Otonal | |
| 52 | 0–4'–6–8 | 1–16/11–18/11–20/11 | Otonal | |
| 53 | 0–1–7–8 | 1–12/11–3/2–18/11 | Ambitonal | 11-odd-limit ASS |
| 54 | 0–4–7–8 | 1–12/11–14/11–18/11 | Otonal | |
| 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 | 1–8/7–9/7–10/7–8/5 | Marvel | |
| 2 | 0–0'–2–3–4 | 1–8/7–9/7–10/7–12/7 | Otonal | 4:5:6:7:9 |
| 3 | 0–1–2–3–4 | 1–8/7–9/7–3/2–12/7 | Archytas | |
| 4 | 0–0'–2–4–4' | 1–8/7–9/7–10/7–9/5 | Ares | |
| 5 | 0–1–2–4–4' | 1–9/8–9/7–3/2–9/5 | Utonal | 1/(24:20:16:14:9) |
| 6 | 0–2–2'–4–4' | 1–9/8–9/7–8/5–9/5 | Marvel | |
| 7 | 0–2–2'–4–6 | 1–8/7–14/11–16/11–8/5 | Apollo | |
| 8 | 0–2–3–4–6 | 1–8/7–9/7–16/11–12/7 | Mothwellsmic | |
| 9 | 0–2–2'–4'–6 | 1–8/7–16/11–8/5–20/11 | Valinorsmic | |
| 10 | 0–2'–3'–4'–6 | 1–6/5–16/11–8/5–9/5 | Ptolemismic | |
| 11 | 0–2–4–4'–6 | 1–8/7–9/7–16/11–9/5 | Apollo | |
| 12 | 0–2'–4–4'–6 | 1–14/11–16/11–8/5–20/11 | Valinorsmic | |
| 13 | 0–1–3–3'–7 | 1–12/11–6/5–3/2–12/7 | Utonal | 1/(24:20:16:14:11) |
| 14 | 0–1–3–4–7 | 1–12/11–9/7–3/2–12/7 | Mothwellsmic | |
| 15 | 0–1–3'–4'–7 | 1–12/11–6/5–3/2–9/5 | Ptolemismic | |
| 16 | 0–1–4–4'–7 | 1–12/11–9/7–3/2–9/5 | Jubilismic11 | |
| 17 | 0–3–3'–6–7 | 1–12/11–6/5–16/11–12/7 | Jubilismic11 | |
| 18 | 0–3–4–6–7 | 1–12/11–9/7–16/11–12/7 | Mothwellsmic | |
| 19 | 0–3'–4'–6–7 | 1–12/11–6/5–16/11–9/5 | Ptolemismic | |
| 20 | 0–4–4'–6–7 | 1–12/11–14/11–16/11–20/11 | Otonal | 4:5:6:7:11 |
| 21 | 0–1–2–4–8 | 1–9/8–9/7–3/2–18/11 | Utonal | 1/(24:16:14:11:9) |
| 22 | 0–1–2–4'–8 | 1–9/8–3/2–18/11–9/5 | Utonal | 1/(24:20:16:11:9) |
| 23 | 0–1–4–4'–8 | 1–9/7–3/2–18/11–9/5 | Utonal | 1/(24:20:14:11:9) |
| 24 | 0–2–4–4'–8 | 1–9/8–9/7–18/11–9/5 | Utonal | 1/(20:16:14:11:9) |
| 25 | 0–2–4–6–8 | 1–9/8–9/7–16/11–18/11 | Pentacircle | |
| 26 | 0–2–4'–6–8 | 1–8/7–16/11–18/11–9/5 | Ares | |
| 27 | 0–4–4'–6–8 | 1–14/11–16/11–18/11–20/11 | Otonal | 4:5:7:9:11 |
| 28 | 0–1–4–7–8 | 1–12/11–9/7–3/2–18/11 | Mothwellsmic | |
| 29 | 0–1–4'–7–8 | 1–12/11–3/2–18/11–9/5 | Ptolemismic | |
| 30 | 0–4–4'–7–8 | 1–12/11–14/11–18/11–20/11 | Otonal | 5:6:7:9:11 |
| 31 | 0–4–6–7–8 | 1–12/11–14/11–16/11–18/11 | Otonal | 4:6:7:9:11 |
| 32 | 0–4'–6–7–8 | 1–12/11–16/11–18/11–20/11 | Otonal | 4:5:6:9:11 |
Hexads
| # | Generators | Transversal | Type | Comments |
|---|---|---|---|---|
| 1 | 0–0'–2–2'–4–4' | 1–9/8–9/7–7/5–8/5–9/5 | Apollo | |
| 2 | 0–2–2'–4–4'–6 | 1–9/8–9/7–16/11–8/5–9/5 | Apollo | |
| 3 | 0–1–2–4–4'–8 | 1–9/8–9/7–3/2–18/11–9/5 | Utonal | 1/(24:20:16:14:11:9) |
| 4 | 0–2–4–4'–6–8 | 1–9/8–9/7–16/11–18/11–9/5 | Apollo | |
| 5 | 0–1–4–4'–7–8 | 1–12/11–9/7–3/2–18/11–9/5 | Jubilismic11 | |
| 6 | 0–4–4'–6–7–8 | 1–12/11–14/11–16/11–18/11–20/11 | Otonal | 4:5:6:7:9:11 |
Notes
- ↑ Sometimes there are two such inversions, in which case the one which appears first in generation order is kept. Note that a note reached by stacking fifths above the root (e.g. 3) comes before the same note displaced by a semioctave (e.g. 3').
- ↑ Pajara temperament tempers out 99/98 and 176/175, thus supporting minerva. While minerva chords exist, in pajara all of them can be tempered via apollo instead, so no chords are labeled as minerva in the tables.