Pajara: Difference between revisions

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{{interwiki

| en = Pajara

| de = Pajara

~~| en = Pajara~~

| es =

| ja =

}}

{{Infobox ~~Regtemp~~

{{Infobox regtemp

| Title = Pajara

| Subgroups = 2.3.5.7

| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.17

| Comma basis = [[50/49]], [[64/63]]

| Comma basis = [[50/49]], [[64/63]] (7-limit);<br>[[50/49]], [[64/63]], [[99/98]] (11-limit);<br>[[50/49]], [[64/63]], [[85/84]], [[99/98]]<br>(2.3.5.7.11.17)

| Edo join 1 = 12 | Edo join 2 = 22

| ~~Generator~~ = 3/2 | ~~Generator~~ tuning = 707.~~344~~ | Optimization method = CWE

| Mapping = 2; 1 -2 -2 -6 1

| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]

| Generators = 3/2 | Generators tuning = 707.4 | Optimization method = CWE

~~| Mapping = 2; 1 -2 1~~

| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]

| Pergen = (P8/2, P5)

| Odd limit 1 = 7 | Mistuning 1 = 17.5 | Complexity 1 = 8

| Odd limit 1 = 9 | Mistuning 1 = 17.5 | Complexity 1 = 10

| Odd limit 2 = (7~~-limit)~~ 21 | Mistuning 2 = 17.5 | Complexity 2 = 10

| 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.

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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]].

Pajara has [[mos scale]]s of 10, 12, and 22 notes. The 10-note mos, Pajara[10], is notable for sharing a number of desirable properties with [[5L 2s|diatonic]], while having fundamentally different categories; for example, the ~7/4 is a now major 8-step, rather than a minor 6-step. This mos and the LsssLsssss [[modmos]] are called the ''symmetric'' and ''pentachordal'' decatonic scales and were independently invented/discovered by [[Paul Erlich]] and [[Gene Ward Smith]]. They are often thought of as subsets of [[22edo]], without much loss of generality and accuracy.

Pajara has [[mos scale]]s of 10, 12, and 22 notes. The 10-note mos, Pajara[10], is notable for sharing a number of desirable properties with [[5L 2s|diatonic]], while having fundamentally different categories; for example, the ~7/4 is a now major 8-step, rather than a minor 6-step. This mos and the LsssLsssss [[modmos]] are called the ''symmetric'' and ''pentachordal'' decatonic scales and were independently invented/discovered by [[Paul Erlich]]<ref>Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]</ref> and [[Gene Ward Smith]]. They are often thought of as subsets of [[22edo]], without much loss of generality and accuracy.

As does all diaschismic temperaments, pajara has a natural extension to prime [[17/1|17]], obtained by tempering out [[136/135]], [[256/255]], and [[289/288]]. This extension notably also tempers out [[120/119]], which equates the 1/(12:10:8:7) utonal tetrad with the otonal [[10:12:15:17]].

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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, ~~loses some accuracy for 3 and 5. It is best tuned sharp of 22edo. However~~, it equates [[12/11]] with [[10/9]], and the only tuning equating [[11/10]] with both is 22edo.

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"

|+ Pajara (12 & 22)

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

|-

! rowspan="2" | #

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| 45/44, 56/55, 81/80

|}

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

|+ Pajarous (10 & 22)

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

|-

! rowspan="2" | #

Line 141:

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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 ==

=== 10-note (proper) ===

The true mos is called the ''symmetric'' decatonic scale, because it repeats exactly at the half-octave, so the symmetric scale starting from 7/5~10/7 is the same as the symmetric scale starting from 1/1. The near-mos, LsssLsssss, in which only the 5-step interval violates the rule of no more than 2 intervals per class, is called the ''pentachordal'' decatonic, because it consists of two identical [[pentachord]]s plus a split 9/8~8/7 whole tone to complete the octave.

The true mos is called the ''symmetric'' decatonic scale, because it repeats exactly at the half-octave, so the symmetric scale starting from {{nowrap|7/5~10/7}} is the same as the symmetric scale starting from 1/1. The near-mos, LsssLsssss, in which only the 5-step interval violates the rule of no more than 2 intervals per class, is called the ''pentachordal'' decatonic, because it consists of two identical [[pentachord]]s plus a split {{nowrap|9/8~8/7}} whole tone to complete the octave.

=== 12-note (proper) ===

=== Scala files ===

* [[Pajara12]]

* [[12-22h]]

== Tunings ==

As with [[archy]], there is a tradeoff in pajara between accuracy of 3 and accuracy of 7. Unlike tunings of archy which ~~often hover~~ around ~~710-712 cents~~, however, pajara is conventionally tuned flat of 22edo ~~– this is because~~ tunings sharp of about 710 ~~cents~~ lose a large degree of accuracy in 5/4 and especially 6/5.

As with [[archy]], there is a tradeoff in pajara between accuracy of 3 and accuracy of 7. Unlike tunings of archy which the fifth is around 710–712{{c}}, however, pajara is conventionally tuned flat of 22edo, since tunings sharp of about 710{{c}} lose a large degree of accuracy in 5/4 and especially 6/5.

=== Norm-based tunings ===

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

|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings

|-

! rowspan="2" |

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Tenney

| CTE: ~3/2 = 708.3557{{c}}

| CWE: ~3/2 = 707.3438{{c}}

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

|}

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

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

|-

! rowspan="2" |

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Tenney

| CTE: ~3/2 = 708.1993{{c}}

| 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

|}

=== Tuning spectrum ===

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| 5/3

| 705.214

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

| 5-odd-limit minimax

|-

| 20\34

Line 248:

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~~

|

|-

|

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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

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== References ==

* Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http:/~~/sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]~~

[[Category:Pajara| ]]

@@ Line 1: / Line 1: @@
 {{interwiki
+| en = Pajara
 | de = Pajara
-| en = Pajara
 | es =
 | ja =
 }}
-{{Infobox Regtemp
+{{Infobox regtemp
 | Title = Pajara
-| Subgroups = 2.3.5.7
+| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.17
-| Comma basis = [[50/49]], [[64/63]]
+| Comma basis = [[50/49]], [[64/63]] (7-limit);<br>[[50/49]], [[64/63]], [[99/98]] (11-limit);<br>[[50/49]], [[64/63]], [[85/84]], [[99/98]]<br>(2.3.5.7.11.17)
 | Edo join 1 = 12 | Edo join 2 = 22
-| Generator = 3/2 | Generator tuning = 707.344 | Optimization method = CWE
+| Mapping = 2; 1 -2 -2 -6 1
-| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
+| Generators = 3/2 | Generators tuning = 707.4 | Optimization method = CWE
-| Mapping = 2; 1 -2 1
+| MOS scales = [[2L&nbsp;8s]], [[10L&nbsp;2s]], [[12L&nbsp;10s]]
 | Pergen = (P8/2, P5)
-| Odd limit 1 = 7 | Mistuning 1 = 17.5 | Complexity 1 = 8
+| Odd limit 1 = 9 | Mistuning 1 = 17.5 | Complexity 1 = 10
-| Odd limit 2 = (7-limit) 21 | Mistuning 2 = 17.5 | Complexity 2 = 10
+| 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.
@@ Line 21: / Line 21: @@
 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]].
-Pajara has [[mos scale]]s of 10, 12, and 22 notes. The 10-note mos, Pajara[10], is notable for sharing a number of desirable properties with [[5L 2s|diatonic]], while having fundamentally different categories; for example, the ~7/4 is a now major 8-step, rather than a minor 6-step. This mos and the LsssLsssss [[modmos]] are called the ''symmetric'' and ''pentachordal'' decatonic scales and were independently invented/discovered by [[Paul Erlich]] and [[Gene Ward Smith]]. They are often thought of as subsets of [[22edo]], without much loss of generality and accuracy.
+Pajara has [[mos scale]]s of 10, 12, and 22 notes. The 10-note mos, Pajara[10], is notable for sharing a number of desirable properties with [[5L 2s|diatonic]], while having fundamentally different categories; for example, the ~7/4 is a now major 8-step, rather than a minor 6-step. This mos and the LsssLsssss [[modmos]] are called the ''symmetric'' and ''pentachordal'' decatonic scales and were independently invented/discovered by [[Paul Erlich]]<ref>Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]</ref> and [[Gene Ward Smith]]. They are often thought of as subsets of [[22edo]], without much loss of generality and accuracy.
 As does all diaschismic temperaments, pajara has a natural extension to prime [[17/1|17]], obtained by tempering out [[136/135]], [[256/255]], and [[289/288]]. This extension notably also tempers out [[120/119]], which equates the 1/(12:10:8:7) utonal tetrad with the otonal [[10:12:15:17]].
@@ Line 28: / 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, loses some accuracy for 3 and 5. It is best tuned sharp of 22edo. However, it equates [[12/11]] with [[10/9]], and the only tuning equating [[11/10]] with both is 22edo.
+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"
-|+ Pajara (12 & 22)
+|+ style="font-size: 105%;" | Pajara ({{nowrap| 12 & 22 }})
 |-
 ! rowspan="2" | #
@@ Line 86: / Line 86: @@
 | 45/44, 56/55, 81/80
 |}
 {| class="wikitable center-1 right-2 right-4"
-|+ Pajarous (10 & 22)
+|+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }})
 |-
 ! rowspan="2" | #
@@ Line 141: / 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 ==
 === 10-note (proper) ===
-{{Main| 2L 8s }}
+{{Main| 2L&nbsp;8s }}
-The true mos is called the ''symmetric'' decatonic scale, because it repeats exactly at the half-octave, so the symmetric scale starting from 7/5~10/7 is the same as the symmetric scale starting from 1/1. The near-mos, LsssLsssss, in which only the 5-step interval violates the rule of no more than 2 intervals per class, is called the ''pentachordal'' decatonic, because it consists of two identical [[pentachord]]s plus a split 9/8~8/7 whole tone to complete the octave.
+The true mos is called the ''symmetric'' decatonic scale, because it repeats exactly at the half-octave, so the symmetric scale starting from {{nowrap|7/5~10/7}} is the same as the symmetric scale starting from 1/1. The near-mos, LsssLsssss, in which only the 5-step interval violates the rule of no more than 2 intervals per class, is called the ''pentachordal'' decatonic, because it consists of two identical [[pentachord]]s plus a split {{nowrap|9/8~8/7}} whole tone to complete the octave.
 === 12-note (proper) ===
-{{Main| 10L 2s }}
+{{Main| 10L&nbsp;2s }}
 === Scala files ===
+* [[Pajara12]]
 * [[12-22h]]
 == Tunings ==
-As with [[archy]], there is a tradeoff in pajara between accuracy of 3 and accuracy of 7. Unlike tunings of archy which often hover around 710-712 cents, however, pajara is conventionally tuned flat of 22edo – this is because tunings sharp of about 710 cents lose a large degree of accuracy in 5/4 and especially 6/5.
+As with [[archy]], there is a tradeoff in pajara between accuracy of 3 and accuracy of 7. Unlike tunings of archy which the fifth is around 710–712{{c}}, however, pajara is conventionally tuned flat of 22edo, since tunings sharp of about 710{{c}} lose a large degree of accuracy in 5/4 and especially 6/5.
+=== Norm-based tunings ===
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
+|-
+! rowspan="2" |
+! colspan="3" | Euclidean
+|-
+! Constrained
+! Constrained & skewed
+! Destretched
+|-
+! Tenney
+| CTE: ~3/2 = 708.3557{{c}}
+| CWE: ~3/2 = 707.3438{{c}}
+| POTE: ~3/2 = 707.0477{{c}}
+|}
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
+|-
+! rowspan="2" |
+! colspan="3" | Euclidean
+|-
+! Constrained
+! Constrained & skewed
+! Destretched
+|-
+! Tenney
+| CTE: ~3/2 = 708.1993{{c}}
+| 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
+|}
 === Tuning spectrum ===
@@ Line 203: / Line 269: @@
 | 5/3
 | 705.214
-| 5- and 15-odd-limit minimax
+| 5-odd-limit minimax
 |-
 | 20\34
@@ Line 248: / 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 264: / 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
@@ Line 313: / Line 389: @@
 == References ==
-* Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]
+<references/>
 [[Category:Pajara| ]] <!-- main article -->