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 [[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 | This page lists all [[11-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[pajara]] temperament. Each chord listed has multiple [[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<ref group="note>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 that note displaced by a semioctave (e.g. 3').</ref>, 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 [[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 [[50/49]] are labeled [[jubilismic chords|jubilismic]], by [[64/63]] [[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]], and chords that require any two of 50/49, 99/98, and 100/99 to vanish are labeled [[undecimal jubilismic chords|jubilismic11]]. 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]. | 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 [[50/49]] are labeled [[jubilismic chords|jubilismic]], by [[64/63]] [[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]], and chords that require any two of 50/49, 99/98, and 100/99 to vanish are labeled [[undecimal jubilismic chords|jubilismic11]]. 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]. | ||
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 9/7, 9/5, 8/7, and 7/5 are | 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, it is analyzed in terms of the transversal which employs as many as possible of 9/7, 9/5, 8/7, and 7/5 above the root; if there's still a tie, then 9/7, 9/5, 8/7, and 7/5 are prioritized in that order. | ||
== Triads == | == Triads == | ||
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|- | |- | ||
! # | ! # | ||
! class="unsortable" | Generators | ! class="unsortable" | Generators<ref group="note" name="generators>Each number corresponds to a note of the chord, with the number being the number of perfect fifths which need to be stacked to reach that note. If an apostrophe follows the number, then the note is displaced by a semioctave.</ref> | ||
! class="unsortable" | Transversal | ! class="unsortable" | Transversal | ||
! Type | ! Type | ||
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|- | |- | ||
! # | ! # | ||
! class="unsortable" | Generators | ! class="unsortable" | Generators<ref group="note" name="generators"/> | ||
! class="unsortable" | Transversal | ! class="unsortable" | Transversal | ||
! Type | ! Type | ||
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|- | |- | ||
! # | ! # | ||
! class="unsortable" | Generators | ! class="unsortable" | Generators<ref group="note" name="generators"/> | ||
! class="unsortable" | Transversal | ! class="unsortable" | Transversal | ||
! Type | ! Type | ||
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|- | |- | ||
! # | ! # | ||
! class="unsortable" | Generators | ! class="unsortable" | Generators<ref group="note" name="generators"/> | ||
! class="unsortable" | Transversal | ! class="unsortable" | Transversal | ||
! Type | ! Type | ||
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| [[4:5:6:7:9:11]] | | [[4:5:6:7:9:11]] | ||
|} | |} | ||
== Notes == | |||
<references group="note"/> | |||