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<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.

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].

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! class="unsortable" | Generators<ref group="note" name="generators>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.</ref>

! class="unsortable" | Generators<ref group="note" name="generators>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.</ref>

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! Type

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 {{Editable user page}}
-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.
+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].
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-! class="unsortable" | Generators<ref group="note" name="generators>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.</ref>
+! class="unsortable" | Generators<ref group="note" name="generators>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.</ref>
 ! class="unsortable" | Transversal
 ! Type