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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 {{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 [[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].
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 the same 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.


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 9/5, 8/7, 9/7, and 7/5 above the root; if there's still a tie, 7/5, 8/7, 9/5, and 9/7 are prioritized in that order.
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, 99/98, and 100/99 to vanish are labeled [[undecimal jubilismic chords|jubilismic11]]. Chords that require any two of 64/63, 100/99, and 176/175 to vanish are labeled [[ares chords|ares]]. Finally, chords that require any two of 100/99, 225/224, and 896/891 to vanish are labeled [[apollo chords|apollo]].<ref group="note" name="nominerva">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.</ref>
 
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 ==
== Triads ==
Line 90: Line 93:
| 0–2'–4
| 0–2'–4
| 1–9/7–8/5
| 1–9/7–8/5
| Marvel/valinorsmic
| Marvel
|  
|  
|-
|-
Line 250: Line 253:
| 0–0'–2–2'
| 0–0'–2–2'
| 1–8/7–7/5–8/5
| 1–8/7–7/5–8/5
| Jubilismic/marvel
| Jubilismic
|  
|  
|-
|-
Line 292: Line 295:
| 0–0'–2'–4
| 0–0'–2'–4
| 1–9/7–10/7–8/5
| 1–9/7–10/7–8/5
| Marvel/valinorsmic
| Marvel
|  
|  
|-
|-
Line 328: Line 331:
| 0–2–2'–4'
| 0–2–2'–4'
| 1–8/7–8/5–9/5
| 1–8/7–8/5–9/5
| Archytas/marvel/valinorsmic
| Archytas
|  
|  
|-
|-
Line 346: Line 349:
| 0–0'–4–4'
| 0–0'–4–4'
| 1–9/7–7/5–9/5
| 1–9/7–7/5–9/5
| Jubilismic/ptolemismic
| Jubilismic
|  
|  
|-
|-
Line 364: Line 367:
| 0–2'–4–4'
| 0–2'–4–4'
| 1–9/7–8/5–9/5
| 1–9/7–8/5–9/5
| Marvel/valinorsmic
| Marvel
|  
|  
|-
|-
Line 418: Line 421:
| 0–2'–4'–6
| 0–2'–4'–6
| 1–16/11–8/5–9/5
| 1–16/11–8/5–9/5
| Ptolemismic/valinorsmic
| Ptolemismic
|  
|  
|-
|-
Line 585: Line 588:


== Pentads ==
== Pentads ==
{| class="wikitable sortable center-1"
{| class="wikitable sortable center-1"
|-
|-
Line 596: Line 598:
| 1
| 1
| 0–0'–2–2'–4
| 0–0'–2–2'–4
| 1–8/7–9/7–7/5–8/5
| 1–8/7–9/7–10/7–8/5
|  
| Marvel
|  
|  
|-
|-
| 2
| 2
| 0–0'–2–3–4
| 0–0'–2–3–4
| 1–8/7–9/7–7/5–12/7
| 1–8/7–9/7–10/7–12/7
|  
| Otonal
|  
| [[4:5:6:7:9]]
|-
|-
| 3
| 3
| 0–1–2–3–4
| 0–1–2–3–4
| 1–8/7–9/7–3/2–12/7
| 1–8/7–9/7–3/2–12/7
|  
| Archytas
|  
|  
|-
|-
| 4
| 4
| 0–0'–2–4–4'
| 0–0'–2–4–4'
| 1–8/7–9/7–7/5–9/5
| 1–8/7–9/7–10/7–9/5
|  
| Ares
|  
|  
|-
|-
| 5
| 5
| 0–1–2–4–4'
| 0–1–2–4–4'
| 1–8/7–9/7–3/2–9/5
| 1–9/8–9/7–3/2–9/5
|  
| Utonal
|  
| [[210:252:315:360:560|1/(24:20:16:14:9)]]
|-
|-
| 6
| 6
| 0–2–2'–4–4'
| 0–2–2'–4–4'
| 1–8/7–9/7–8/5–9/5
| 1–9/8–9/7–8/5–9/5
|  
| Marvel
|  
|  
|-
|-
| 7
| 7
| 0–2–2'–4–6
| 0–2–2'–4–6
| 1–8/7–9/7–16/11–8/5
| 1–8/7–14/11–16/11–8/5
|  
| Apollo
|  
|  
|-
|-
Line 639: Line 641:
| 0–2–3–4–6
| 0–2–3–4–6
| 1–8/7–9/7–16/11–12/7
| 1–8/7–9/7–16/11–12/7
|  
| Mothwellsmic
|  
|  
|-
|-
| 9
| 9
| 0–2–2'–4'–6
| 0–2–2'–4'–6
| 1–8/7–16/11–8/5–9/5
| 1–8/7–16/11–8/5–20/11
|  
| Valinorsmic
|  
|  
|-
|-
Line 651: Line 653:
| 0–2'–3'–4'–6
| 0–2'–3'–4'–6
| 1–6/5–16/11–8/5–9/5
| 1–6/5–16/11–8/5–9/5
|  
| Ptolemismic
|  
|  
|-
|-
Line 657: Line 659:
| 0–2–4–4'–6
| 0–2–4–4'–6
| 1–8/7–9/7–16/11–9/5
| 1–8/7–9/7–16/11–9/5
|  
| Apollo
|  
|  
|-
|-
| 12
| 12
| 0–2'–4–4'–6
| 0–2'–4–4'–6
| 1–9/7–16/11–8/5–9/5
| 1–14/11–16/11–8/5–20/11
|  
| Valinorsmic
|  
|  
|-
|-
Line 669: Line 671:
| 0–1–3–3'–7
| 0–1–3–3'–7
| 1–12/11–6/5–3/2–12/7
| 1–12/11–6/5–3/2–12/7
|  
| Utonal
|  
| [[770:924:1155:1320:1680|1/(24:20:16:14:11)]]
|-
|-
| 14
| 14
| 0–1–3–4–7
| 0–1–3–4–7
| 1–12/11–9/7–3/2–12/7
| 1–12/11–9/7–3/2–12/7
|  
| Mothwellsmic
|  
|  
|-
|-
Line 681: Line 683:
| 0–1–3'–4'–7
| 0–1–3'–4'–7
| 1–12/11–6/5–3/2–9/5
| 1–12/11–6/5–3/2–9/5
|  
| Ptolemismic
|  
|  
|-
|-
Line 687: Line 689:
| 0–1–4–4'–7
| 0–1–4–4'–7
| 1–12/11–9/7–3/2–9/5
| 1–12/11–9/7–3/2–9/5
|  
| Jubilismic11
|  
|  
|-
|-
Line 693: Line 695:
| 0–3–3'–6–7
| 0–3–3'–6–7
| 1–12/11–6/5–16/11–12/7
| 1–12/11–6/5–16/11–12/7
|  
| Jubilismic11
|  
|  
|-
|-
Line 699: Line 701:
| 0–3–4–6–7
| 0–3–4–6–7
| 1–12/11–9/7–16/11–12/7
| 1–12/11–9/7–16/11–12/7
|  
| Mothwellsmic
|  
|  
|-
|-
Line 705: Line 707:
| 0–3'–4'–6–7
| 0–3'–4'–6–7
| 1–12/11–6/5–16/11–9/5
| 1–12/11–6/5–16/11–9/5
|  
| Ptolemismic
|  
|  
|-
|-
| 20
| 20
| 0–4–4'–6–7
| 0–4–4'–6–7
| 1–12/11–9/7–16/11–9/5
| 1–12/11–14/11–16/11–20/11
|  
| Otonal
|  
| [[4:5:6:7:11]]
|-
|-
| 21
| 21
| 0–1–2–4–8
| 0–1–2–4–8
| 1–8/7–9/7–3/2–18/11
| 1–9/8–9/7–3/2–18/11
|  
| Utonal
|  
| [[462:693:792:1008:1232|1/(24:16:14:11:9)]]
|-
|-
| 22
| 22
| 0–1–2–4'–8
| 0–1–2–4'–8
| 1–8/7–3/2–18/11–9/5
| 1–9/8–3/2–18/11–9/5
|  
| Utonal
|  
| [[330:396:495:720:880|1/(24:20:16:11:9)]]
|-
|-
| 23
| 23
| 0–1–4–4'–8
| 0–1–4–4'–8
| 1–9/7–3/2–18/11–9/5
| 1–9/7–3/2–18/11–9/5
|  
| Utonal
|  
| [[1155:1386:1980:2520:3080|1/(24:20:14:11:9)]]
|-
|-
| 24
| 24
| 0–2–4–4'–8
| 0–2–4–4'–8
| 1–8/7–9/7–18/11–9/5
| 1–9/8–9/7–18/11–9/5
|  
| Utonal
|  
| [[924:1155:1320:2016:2464|1/(20:16:14:11:9)]]
|-
|-
| 25
| 25
| 0–2–4–6–8
| 0–2–4–6–8
| 1–8/7–9/7–16/11–18/11
| 1–9/8–9/7–16/11–18/11
|  
| Pentacircle
|  
|  
|-
|-
Line 747: Line 749:
| 0–2–4'–6–8
| 0–2–4'–6–8
| 1–8/7–16/11–18/11–9/5
| 1–8/7–16/11–18/11–9/5
|  
| Ares
|  
|  
|-
|-
| 27
| 27
| 0–4–4'–6–8
| 0–4–4'–6–8
| 1–9/7–16/11–18/11–9/5
| 1–14/11–16/11–18/11–20/11
|  
| Otonal
|  
| [[4:5:7:9:11]]
|-
|-
| 28
| 28
| 0–1–4–7–8
| 0–1–4–7–8
| 1–12/11–9/7–3/2–18/11
| 1–12/11–9/7–3/2–18/11
|  
| Mothwellsmic
|  
|  
|-
|-
Line 765: Line 767:
| 0–1–4'–7–8
| 0–1–4'–7–8
| 1–12/11–3/2–18/11–9/5
| 1–12/11–3/2–18/11–9/5
|  
| Ptolemismic
|  
|  
|-
|-
| 30
| 30
| 0–4–4'–7–8
| 0–4–4'–7–8
| 1–12/11–9/7–18/11–9/5
| 1–12/11–14/11–18/11–20/11
|  
| Otonal
|  
| [[5:6:7:9:11]]
|-
|-
| 31
| 31
| 0–4–6–7–8
| 0–4–6–7–8
| 1–12/11–9/7–16/11–18/11
| 1–12/11–14/11–16/11–18/11
|  
| Otonal
|  
| [[4:6:7:9:11]]
|-
|-
| 32
| 32
| 0–4'–6–7–8
| 0–4'–6–7–8
| 1–12/11–16/11–18/11–9/5
| 1–12/11–16/11–18/11–20/11
|  
| Otonal
|  
| [[4:5:6:9:11]]
|}
|}


== Hexads ==
== Hexads ==
{| class="wikitable sortable center-1"
{| class="wikitable sortable center-1"
|-
|-
Line 799: Line 800:
| 1
| 1
| 0–0'–2–2'–4–4'
| 0–0'–2–2'–4–4'
| 1–8/7–9/7–7/5–8/5–9/5
| 1–9/8–9/7–7/5–8/5–9/5
| Pajara
| Apollo
|  
|  
|-
|-
| 2
| 2
| 0–2–2'–4–4'–6
| 0–2–2'–4–4'–6
| 1–8/7–9/7–16/11–8/5–9/5
| 1–9/8–9/7–16/11–8/5–9/5
|  
| Apollo
|  
|  
|-
|-
Line 817: Line 818:
| 4
| 4
| 0–2–4–4'–6–8
| 0–2–4–4'–6–8
| 1–8/7–9/7–16/11–18/11–9/5
| 1–9/8–9/7–16/11–18/11–9/5
|  
| Apollo
|  
|  
|-
|-
Line 824: Line 825:
| 0–1–4–4'–7–8
| 0–1–4–4'–7–8
| 1–12/11–9/7–3/2–18/11–9/5
| 1–12/11–9/7–3/2–18/11–9/5
|  
| Jubilismic11
|  
|  
|-
|-
Line 833: Line 834:
| [[4:5:6:7:9:11]]
| [[4:5:6:7:9:11]]
|}
|}
== Notes ==
<references group="note"/>
[[Category:Pajara]]
[[Category:Lists of chords]]
[[Category:Dyadic chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
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