Pajara: Difference between revisions

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| Odd limit 2 = 2.3.5.7.11.17 21 | Mistuning 2 = 22.4 | Complexity 2 = 22

}}

'''Pajara''' (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a [[regular temperament|temperament]] with a half-octave [[period]] that represents both [[7/5]] and [[10/7]], so [[50/49]] is [[tempering out|tempered out]] and it is in the [[jubilismic clan]]. The [[generator]] is a [[3/2|perfect fifth]] in the neighborhood of 707–711 [[cent]]s, or that minus a half-octave period, which is a semitone representing [[15/14]] and [[16/15]]. One period minus 2 such semitones is [[~]][[5/4]], which, if you work it out, implies that [[2048/2025]] is tempered out, so pajara is also in the [[diaschismic family]]. In fact, it shares the same structure as 5-limit [[diaschismic]]. Finally, two 4/3's (or an octave minus two semitones) represents 7/4 as well as 16/9, so [[64/63]] is tempered out and pajara is in the [[archytas clan]]. Tempering out any two of these commas (among others) produces the unique temperament pajara.

'''Pajara''' (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a [[regular temperament|temperament]] with a half-octave [[period]] that represents both [[7/5]] and [[10/7]], so [[50/49]] is [[tempering out|tempered out]] and it is in the [[jubilismic clan]]. The [[generator]] is a [[3/2|perfect fifth]] in the neighborhood of 707–711~~ ~~[[cent]]s, or that minus a half-octave period, which is a semitone representing [[15/14]] and [[16/15]]. One period minus 2 such semitones is [[~]][[5/4]], which, if you work it out, implies that [[2048/2025]] is tempered out, so pajara is also in the [[diaschismic family]]. In fact, it shares the same structure as 5-limit [[diaschismic]]. Finally, two 4/3's (or an octave minus two semitones) represents 7/4 as well as 16/9, so [[64/63]] is tempered out and pajara is in the [[archytas clan]]. Tempering out any two of these commas (among others) produces the unique temperament pajara.

Pajara has fairly low accuracy overall, due to the ~5/4 and ~7/4 necessarily being separated by 600 cents via vanishing of [[50/49]]. However, if one accepts the accuracy of [[12edo]] in the 5-limit, they would probably accept the accuracy of pajara as well. The vanishing of [[50/49]] means that [[49/48]] and [[25/24]] are tempered to the same interval, and allows for a simple alteration to produce the subharmonic sixth chord [[70:84:105:120|1/(12:10:8:7)]] with 6/5 and 12/7 by flattening the third and seventh the same amount from the harmonic seventh chord, [[4:5:6:7]].

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== Interval chains ==

There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but ~~22edo is the only~~ [[11~~-odd-limit~~]] [[~~diamond monotone~~]] ~~tuning~~, ~~where primes [[3/1|3]]~~ and [[5/~~1|5~~]] ~~are less accurate than in optimal tunings of canonical 11-limit pajara~~.

There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo, with the optimum at around 707-708 cents. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but it equates [[12/11]] with [[10/9]], and the only tuning equating [[11/10]] with both is 22edo.

In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.

{| class="wikitable center-1 right-2 right-4"

|+ style="font-size: 105%;" | Pajara (12&~~nbsp;& ~~22)

|+ style="font-size: 105%;" | Pajara ({{nowrap| 12 & 22 }})

|-

! rowspan="2" | #

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{| class="wikitable center-1 right-2 right-4"

|+ style="font-size: 105%;" | Pajarous (10&~~nbsp;& ~~22)

|+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }})

|-

! rowspan="2" | #

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|}

<nowiki/>* In 11-limit CWE tuning, octave-reduced

== Chords and harmony ==

In pajara, a decatonic system of interval classification based on the [[2L 8s]] (jaric) [[mos scale]] is preferred over the [[diatonic]] interval classification system traditionally used in western music, which is used in [[meantone]]. If we count scale degrees similarly to diatonic, then [[2/1]] is a "hendecave" (11ve), as there are 10 scale degrees, and we repeat at 2/1 at the 11th. In this system, [[3/2]] is a perfect 7th, and [[4/3]] is a perfect 5th. The intervals [[5/4]] and [[6/5]] are major and minor decatonic 4ths respectively, rather than being major and minor 3rds by diatonic interval classification in meantone. Importantly, [[7/4]] is now a major decatonic 9th, with [[12/7]] being its minor counterpart. This is in contrast to diatonic, where 7/4 is considered a subminor 7th, and 12/7 a supermajor 6th.

By decatonic interval classification, the [[4:5:6:7]] tetrad is written as P1–M4–P7–M9. It can be considered the ''major tetrad'', since the non-perfect intervals, those being the decatonic 4th and 9th, are both major intervals. If we instead use a minor interval for the 4th and 9th; that is, a P1–m4–P7–m9 chord, then we get a tetrad approximating [[70:84:105:120|1/(12:10:8:7)]], which can be considered the ''minor tetrad''.

== Scales ==

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=== Scala files ===

* [[Pajara12]]

* [[12-22h]]

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| POTE: ~3/2 = 707.0477{{c}}

|}

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings

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| CWE: ~3/2 = 707.1826{{c}}

| POTE: ~3/2 = 706.8851{{c}}

|}

=== Target tunings ===

{| class="wikitable center-all mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings

|-

! rowspan="2" | Target

! colspan="2" | Minimax

|-

! Generator

! Eigenmonzo*

|-

| 7-odd-limit

| ~3/2 = 709.363{{c}}

| 35/24

|-

| 9-odd-limit

| ~3/2 = 708.128{{c}}

| 35/18

|-

| 11-odd-limit

| ~3/2 = 708.128{{c}}

| 35/18

|}

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| 5/3

| 705.214

| 5~~- and 15~~-odd-limit minimax

| 5-odd-limit minimax

|-

| 20\34

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| 11/8

| 708.114

|

| 11- and 15-odd-limit minimax

|-

|

|36/35

|708.128

|9-odd-limit minimax

|-

|

| 11/10

| 708.749

| ~~11-odd-limit minimax~~

|

|-

|

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| 709.091

| Upper bound of 11-odd-limit diamond monotone

|-

|

|48/35

|709.363

|7-odd-limit minimax

|-

|

| 7/6

| 711.043

| ~~7-odd-limit minimax~~

|

|-

| 32\54

@@ Line 17: / Line 17: @@
 | Odd limit 2 = 2.3.5.7.11.17 21 | Mistuning 2 = 22.4 | Complexity 2 = 22
 }}
+'''Pajara''' (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a [[regular temperament|temperament]] with a half-octave [[period]] that represents both [[7/5]] and [[10/7]], so [[50/49]] is [[tempering out|tempered out]] and it is in the [[jubilismic clan]]. The [[generator]] is a [[3/2|perfect fifth]] in the neighborhood of 707–711 [[cent]]s, or that minus a half-octave period, which is a semitone representing [[15/14]] and [[16/15]]. One period minus 2 such semitones is [[~]][[5/4]], which, if you work it out, implies that [[2048/2025]] is tempered out, so pajara is also in the [[diaschismic family]]. In fact, it shares the same structure as 5-limit [[diaschismic]]. Finally, two 4/3's (or an octave minus two semitones) represents 7/4 as well as 16/9, so [[64/63]] is tempered out and pajara is in the [[archytas clan]]. Tempering out any two of these commas (among others) produces the unique temperament pajara.
-'''Pajara''' (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a [[regular temperament|temperament]] with a half-octave [[period]] that represents both [[7/5]] and [[10/7]], so [[50/49]] is [[tempering out|tempered out]] and it is in the [[jubilismic clan]]. The [[generator]] is a [[3/2|perfect fifth]] in the neighborhood of 707–711&nbsp;[[cent]]s, or that minus a half-octave period, which is a semitone representing [[15/14]] and [[16/15]]. One period minus 2 such semitones is [[~]][[5/4]], which, if you work it out, implies that [[2048/2025]] is tempered out, so pajara is also in the [[diaschismic family]]. In fact, it shares the same structure as 5-limit [[diaschismic]]. Finally, two 4/3's (or an octave minus two semitones) represents 7/4 as well as 16/9, so [[64/63]] is tempered out and pajara is in the [[archytas clan]]. Tempering out any two of these commas (among others) produces the unique temperament pajara.
 Pajara has fairly low accuracy overall, due to the ~5/4 and ~7/4 necessarily being separated by 600 cents via vanishing of [[50/49]]. However, if one accepts the accuracy of [[12edo]] in the 5-limit, they would probably accept the accuracy of pajara as well. The vanishing of [[50/49]] means that [[49/48]] and [[25/24]] are tempered to the same interval, and allows for a simple alteration to produce the subharmonic sixth chord [[70:84:105:120|1/(12:10:8:7)]] with 6/5 and 12/7 by flattening the third and seventh the same amount from the harmonic seventh chord, [[4:5:6:7]].
@@ Line 29: / Line 28: @@
 == Interval chains ==
-There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but 22edo is the only [[11-odd-limit]] [[diamond monotone]] tuning, where primes [[3/1|3]] and [[5/1|5]] are less accurate than in optimal tunings of canonical 11-limit pajara.
+There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo, with the optimum at around 707-708 cents. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but it equates [[12/11]] with [[10/9]], and the only tuning equating [[11/10]] with both is 22edo.
 In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.
 {| class="wikitable center-1 right-2 right-4"
-|+ style="font-size: 105%;" | Pajara (12&nbsp;&amp;&nbsp;22)
+|+ style="font-size: 105%;" | Pajara ({{nowrap| 12 & 22 }})
 |-
 ! rowspan="2" | #
@@ Line 89: / Line 88: @@
 {| class="wikitable center-1 right-2 right-4"
-|+ style="font-size: 105%;" | Pajarous (10&nbsp;&amp;&nbsp;22)
+|+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }})
 |-
 ! rowspan="2" | #
@@ Line 143: / Line 142: @@
 |}
 <nowiki/>* In 11-limit CWE tuning, octave-reduced
+== Chords and harmony ==
+{{See also| Chords of pajara }}
+In pajara, a decatonic system of interval classification based on the [[2L 8s]] (jaric) [[mos scale]] is preferred over the [[diatonic]] interval classification system traditionally used in western music, which is used in [[meantone]]. If we count scale degrees similarly to diatonic, then [[2/1]] is a "hendecave" (11ve), as there are 10 scale degrees, and we repeat at 2/1 at the 11th. In this system, [[3/2]] is a perfect 7th, and [[4/3]] is a perfect 5th. The intervals [[5/4]] and [[6/5]] are major and minor decatonic 4ths respectively, rather than being major and minor 3rds by diatonic interval classification in meantone. Importantly, [[7/4]] is now a major decatonic 9th, with [[12/7]] being its minor counterpart. This is in contrast to diatonic, where 7/4 is considered a subminor 7th, and 12/7 a supermajor 6th.
+By decatonic interval classification, the [[4:5:6:7]] tetrad is written as P1–M4–P7–M9. It can be considered the ''major tetrad'', since the non-perfect intervals, those being the decatonic 4th and 9th, are both major intervals. If we instead use a minor interval for the 4th and 9th; that is, a P1–m4–P7–m9 chord, then we get a tetrad approximating [[70:84:105:120|1/(12:10:8:7)]], which can be considered the ''minor tetrad''.
+{{Todo|complete section}}
 == Scales ==
@@ Line 154: / Line 162: @@
 === Scala files ===
+* [[Pajara12]]
 * [[12-22h]]
@@ Line 175: / Line 184: @@
 | POTE: ~3/2 = 707.0477{{c}}
 |}
 {| class="wikitable mw-collapsible mw-collapsed"
 |+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
@@ Line 190: / Line 198: @@
 | CWE: ~3/2 = 707.1826{{c}}
 | POTE: ~3/2 = 706.8851{{c}}
+|}
+=== Target tunings ===
+{| class="wikitable center-all mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
+|-
+! rowspan="2" | Target
+! colspan="2" | Minimax
+|-
+! Generator
+! Eigenmonzo*
+|-
+| 7-odd-limit
+| ~3/2 = 709.363{{c}}
+| 35/24
+|-
+| 9-odd-limit
+| ~3/2 = 708.128{{c}}
+| 35/18
+|-
+| 11-odd-limit
+| ~3/2 = 708.128{{c}}
+| 35/18
 |}
@@ Line 238: / Line 269: @@
 | 5/3
 | 705.214
-| 5- and 15-odd-limit minimax
+| 5-odd-limit minimax
 |-
 | 20\34
@@ Line 283: / Line 314: @@
 | 11/8
 | 708.114
-|
+| 11- and 15-odd-limit minimax
+|-
+|
+|36/35
+|708.128
+|9-odd-limit minimax
 |-
 |
 | 11/10
 | 708.749
-| 11-odd-limit minimax
+|
 |-
 |
@@ Line 299: / Line 335: @@
 | 709.091
 | Upper bound of 11-odd-limit diamond monotone
+|-
+|
+|48/35
+|709.363
+|7-odd-limit minimax
 |-
 |
 | 7/6
 | 711.043
-| 7-odd-limit minimax
+|
 |-
 | 32\54