Chords of pajara: Difference between revisions
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This page lists all [[11-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[pajara]] temperament. Each chord listed has multiple {{W|Chord inversion|inversions}}; only one is listed, that being the inversion where all notes are a nonnegative number of perfect fifth [[generator]]s above the root or semioctave, which may not be the optimal {{W|Voicing (music)|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. | This page lists all [[11-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[pajara]] temperament. Each chord listed has multiple {{W|Chord inversion|inversions}}; only one is listed, that being the inversion where all notes are a nonnegative number of perfect fifth [[generator]]s above the root or semioctave, which may not be the optimal {{W|Voicing (music)|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 [[dyadic chord #Essentially tempered dyadic chord|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 [[dyadic chord #Essentially tempered dyadic chord|essentially tempered]], it is classified based on which [[comma]]s are needed to define the chord. Chords essentially tempered by [[ | If a chord is [[dyadic chord #Essentially tempered dyadic chord|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 [[dyadic chord #Essentially tempered dyadic chord|essentially tempered]], it is classified based on which [[comma]]s are needed to define the chord. Chords essentially tempered by [[64/63]] are labeled [[archytas chords|archytas]], by [[99/98]] [[mothwellsmic chords|mothwellsmic]], by [[100/99]] [[ptolemismic chords|ptolemismic]], by [[176/175]] [[valinorsmic chords|valinorsmic]], by [[225/224]] [[marvel chords|marvel]], and by [[896/891]] [[pentacircle chords|pentacircle]]. Chords that require any two of 50/49, 64/63, and 225/224 to vanish are labeled [[pajara chords|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|jubilee]] [not known to exist]. Chords that require any two of 64/63, 99/98, and 896/891 to vanish are labeled [[supra chords|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|ares]]. Chords that require any two of 99/98, 176/175, and 225/224 to vanish are labeled [[minerva chords|minerva]], and chords that require any two of 100/99, 225/224, and 896/891 to vanish are labeled [[apollo chords|apollo]]. Finally, chords that require any three independent commas listed above to vanish are labeled [[undecimal pajara chords|pajara11]] [not known to exist]. | ||
== Triads == | == Triads == | ||
Revision as of 06:54, 24 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 64/63 are labeled 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 [not known to exist]. 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 | |
| 2 | 0–1–2 | 1–9/8–3/2 | Ambitonal | |
| 3 | 0–0'–2' | 1–7/5–8/5 | oops it's just a rotation of #1 | |
| 4 | 0–2–2' | 1–8/7–8/5 | ||
| 5 | 0–0'–3 | 1–7/5–12/7 | ||
| 6 | 0–1–3 | 1–3/2–12/7 | ||
| 7 | 0–2–3 | 1–8/7–12/7 | ||
| 8 | 0–0'–3' | 1–6/5–7/5 | ||
| 9 | 0–1–3' | 1–6/5–3/2 | ||
| 10 | 0–2'–3' | 1–6/5–8/5 | ||
| 11 | 0–3–3' | 1–6/5–12/7 | ||
| 12 | 0–0'–4 | 1–9/7–7/5 | ||
| 13 | 0–1–4 | 1–9/7–3/2 | ||
| 14 | 0–2–4 | 1–8/7–9/7 | ||
| 15 | 0–2'–4 | 1–9/7–8/5 | ||
| 16 | 0–3–4 | 1–9/7–12/7 | ||
| 17 | 0–0'–4' | 1–7/5–9/5 | ||
| 18 | 0–1–4' | 1–3/2–9/5 | ||
| 19 | 0–2–4' | 1–8/7–9/5 | ||
| 20 | 0–2'–4' | 1–8/5–9/5 | ||
| 21 | 0–3'–4' | 1–6/5–9/5 | ||
| 22 | 0–4–4' | 1–9/7–9/5 | ||
| 23 | 0–2–6 | 1–8/7–16/11 | ||
| 24 | 0–2'–6 | 1–16/11–8/5 | ||
| 25 | 0–3–6 | 1–16/11–12/7 | ||
| 26 | 0–3'–6 | 1–6/5–16/11 | ||
| 27 | 0–4–6 | 1–9/7–16/11 | ||
| 28 | 0–4'–6 | 1–16/11–9/5 | ||
| 29 | 0–1–7 | 1–12/11–3/2 | ||
| 30 | 0–3–7 | 1–12/11–12/7 | ||
| 31 | 0–3'–7 | 1–12/11–6/5 | ||
| 32 | 0–4–7 | 1–12/11–9/7 | ||
| 33 | 0–4'–7 | 1–12/11–9/5 | ||
| 34 | 0–6–7 | 1–12/11–16/11 | ||
| 35 | 0–1–8 | 1–3/2–18/11 | ||
| 36 | 0–2–8 | 1–8/7–18/11 | ||
| 37 | 0–4–8 | 1–9/7–18/11 | ||
| 38 | 0–4'–8 | 1–18/11–9/5 | ||
| 39 | 0–6–8 | 1–16/11–18/11 | ||
| 40 | 0–7–8 | 1–12/11–18/11 |
Tetrads
| # | Generators | Transversal | Type | Comments |
|---|---|---|---|---|
| 1 | 0–0'–2–2' | 1–8/7–7/5–8/5 | ||
| 2 | 0–0'–2–3 | 1–8/7–7/5–12/7 | ||
| 3 | 0–1–2–3 | 1–8/7–3/2–12/7 | ||
| 4 | 0–0'–2'–3' | 1–6/5–7/5–8/5 | ||
| 5 | 0–0'–3–3' | 1–6/5–7/5–12/7 | ||
| 6 | 0–1–3–3' | 1–6/5–3/2–12/7 | ||
| 7 | 0–0'–2–4 | 1–8/7–9/7–7/5 | ||
| 8 | 0–1–2–4 | 1–8/7–9/7–3/2 | ||
| 9 | 0–0'–2'–4 | 1–9/7–7/5–8/5 | ||
| 10 | 0–2–2'–4 | 1–8/7–9/7–8/5 | ||
| 11 | 0–0'–3–4 | 1–9/7–7/5–12/7 | ||
| 12 | 0–1–3–4 | 1–9/7–3/2–12/7 | ||
| 13 | 0–2–3–4 | 1–8/7–9/7–12/7 | ||
| 14 | 0–0'–2–4' | 1–8/7–7/5–9/5 | ||
| 15 | 0–1–2–4' | 1–8/7–3/2–9/5 | ||
| 16 | 0–0'–2'–4' | 1–7/5–8/5–9/5 | ||
| 17 | 0–2–2'–4' | 1–8/7–8/5–9/5 | ||
| 18 | 0–0'–3'–4' | 1–6/5–7/5–9/5 | ||
| 19 | 0–1–3'–4' | 1–6/5–3/2–9/5 | ||
| 20 | 0–2'–3'–4' | 1–6/5–8/5–9/5 | ||
| 21 | 0–0'–4–4' | 1–9/7–7/5–9/5 | ||
| 22 | 0–1–4–4' | 1–9/7–3/2–9/5 | ||
| 23 | 0–2–4–4' | 1–8/7–9/7–9/5 | ||
| 24 | 0–2'–4–4' | 1–9/7–8/5–9/5 | ||
| 25 | 0–2–2'–6 | 1–8/7–16/11–8/5 | ||
| 26 | 0–2–3–6 | 1–8/7–16/11–12/7 | ||
| 27 | 0–2'–3'–6 | 1–6/5–16/11–8/5 | ||
| 28 | 0–3–3'–6 | 1–6/5–16/11–12/7 | ||
| 29 | 0–2–4–6 | 1–8/7–9/7–16/11 | ||
| 30 | 0–2'–4–6 | 1–9/7–16/11–8/5 | ||
| 31 | 0–3–4–6 | 1–9/7–16/11–12/7 | ||
| 32 | 0–2–4'–6 | 1–8/7–16/11–9/5 | ||
| 33 | 0–2'–4'–6 | 1–16/11–8/5–9/5 | ||
| 34 | 0–3'–4'–6 | 1–6/5–16/11–9/5 | ||
| 35 | 0–4–4'–6 | 1–9/7–16/11–9/5 | ||
| 36 | 0–1–3–7 | 1–12/11–3/2–12/7 | ||
| 37 | 0–1–3'–7 | 1–12/11–6/5–3/2 | ||
| 38 | 0–3–3'–7 | 1–12/11–6/5–12/7 | ||
| 39 | 0–1–4–7 | 1–12/11–9/7–3/2 | ||
| 40 | 0–3–4–7 | 1–12/11–9/7–12/7 | ||
| 41 | 0–1–4'–7 | 1–12/11–3/2–9/5 | ||
| 42 | 0–3'–4'–7 | 1–12/11–6/5–9/5 | ||
| 43 | 0–4–4'–7 | 1–12/11–9/7–9/5 | ||
| 44 | 0–3–6–7 | 1–12/11–16/11–12/7 | ||
| 45 | 0–3'–6–7 | 1–12/11–6/5–16/11 | ||
| 46 | 0–4–6–7 | 1–12/11–9/7–16/11 | ||
| 47 | 0–4'–6–7 | 1–12/11–16/11–9/5 | ||
| 48 | 0–1–2–8 | 1–8/7–3/2–18/11 | ||
| 49 | 0–1–4–8 | 1–9/7–3/2–18/11 | ||
| 50 | 0–2–4–8 | 1–8/7–9/7–18/11 | ||
| 51 | 0–1–4'–8 | 1–3/2–18/11–9/5 | ||
| 52 | 0–2–4'–8 | 1–8/7–18/11–9/5 | ||
| 53 | 0–4–4'–8 | 1–9/7–18/11–9/5 | ||
| 54 | 0–2–6–8 | 1–8/7–16/11–18/11 | ||
| 55 | 0–4–6–8 | 1–9/7–16/11–18/11 | ||
| 56 | 0–4'–6–8 | 1–16/11–18/11–9/5 | ||
| 57 | 0–1–7–8 | 1–12/11–3/2–18/11 | ||
| 58 | 0–4–7–8 | 1–12/11–9/7–18/11 | ||
| 59 | 0–4'–7–8 | 1–12/11–18/11–9/5 | ||
| 60 | 0–6–7–8 | 1–12/11–16/11–18/11 |
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'–2–2'–4' | 1–8/7–7/5–8/5–9/5 | ||
| 5 | 0–0'–2'–3'–4' | 1–6/5–7/5–8/5–9/5 | ||
| 6 | 0–0'–2–4–4' | 1–8/7–9/7–7/5–9/5 | ||
| 7 | 0–1–2–4–4' | 1–8/7–9/7–3/2–9/5 | ||
| 8 | 0–0'–2'–4–4' | 1–9/7–7/5–8/5–9/5 | ||
| 9 | 0–2–2'–4–4' | 1–8/7–9/7–8/5–9/5 | ||
| 10 | 0–2–2'–4–6 | 1–8/7–9/7–16/11–8/5 | ||
| 11 | 0–2–3–4–6 | 1–8/7–9/7–16/11–12/7 | ||
| 12 | 0–2–2'–4'–6 | 1–8/7–16/11–8/5–9/5 | ||
| 13 | 0–2'–3'–4'–6 | 1–6/5–16/11–8/5–9/5 | ||
| 14 | 0–2–4–4'–6 | 1–8/7–9/7–16/11–9/5 | ||
| 15 | 0–2'–4–4'–6 | 1–9/7–16/11–8/5–9/5 | ||
| 16 | 0–1–3–3'–7 | 1–12/11–6/5–3/2–12/7 | ||
| 17 | 0–1–3–4–7 | 1–12/11–9/7–3/2–12/7 | ||
| 18 | 0–1–3'–4'–7 | 1–12/11–6/5–3/2–9/5 | ||
| 19 | 0–1–4–4'–7 | 1–12/11–9/7–3/2–9/5 | ||
| 20 | 0–3–3'–6–7 | 1–12/11–6/5–16/11–12/7 | ||
| 21 | 0–3–4–6–7 | 1–12/11–9/7–16/11–12/7 | ||
| 22 | 0–3'–4'–6–7 | 1–12/11–6/5–16/11–9/5 | ||
| 23 | 0–4–4'–6–7 | 1–12/11–9/7–16/11–9/5 | ||
| 24 | 0–1–2–4–8 | 1–8/7–9/7–3/2–18/11 | ||
| 25 | 0–1–2–4'–8 | 1–8/7–3/2–18/11–9/5 | ||
| 26 | 0–1–4–4'–8 | 1–9/7–3/2–18/11–9/5 | ||
| 27 | 0–2–4–4'–8 | 1–8/7–9/7–18/11–9/5 | ||
| 28 | 0–2–4–6–8 | 1–8/7–9/7–16/11–18/11 | ||
| 29 | 0–2–4'–6–8 | 1–8/7–16/11–18/11–9/5 | ||
| 30 | 0–4–4'–6–8 | 1–9/7–16/11–18/11–9/5 | ||
| 31 | 0–1–4–7–8 | 1–12/11–9/7–3/2–18/11 | ||
| 32 | 0–1–4'–7–8 | 1–12/11–3/2–18/11–9/5 | ||
| 33 | 0–4–4'–7–8 | 1–12/11–9/7–18/11–9/5 | ||
| 34 | 0–4–6–7–8 | 1–12/11–9/7–16/11–18/11 | ||
| 35 | 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 |