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→Tunings: Added the minimax tunings according to the power limit method proposed by Dave & Doug and independently by myself.
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<span style="display: block; text-align: right;">Other languages: [[:de:Pajara Deutsch]]</span>
{{interwiki
| en = Pajara
| de = Pajara
| es =
| ja =
}}
{{Infobox regtemp
| Title = Pajara
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.17
| 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
| Mapping = 2; 1 -2 -2 -6 1
| Generators = 3/2 | Generators tuning = 707.4 | Optimization method = CWE
| MOS scales = [[2L&nbsp;8s]], [[10L&nbsp;2s]], [[12L&nbsp;10s]]
| Pergen = (P8/2, P5)
| Odd limit 1 = 9 | Mistuning 1 = 17.5 | Complexity 1 = 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.


Pajara (pronounced /p<span style="">əˈd͡ʒɑːr</span>ə/, with the J as in "jar") is a temperament with a half-octave period that represents both 7/5 and 10/7, so 50/49 is tempered out and it is in the [[Jubilismic_clan|jubilismic clan]]. The generator is in the neighborhood of 105-110 cents, so that period + generator represents 3/2. Period minus 2 generators is 5/4, which, if you work it out, implies that 2048/2025 is tempered out, so pajara is also in the [[Diaschismic_family|diaschismic family]]. Finally, two 4/3s (or a 2/1 minus two generators) represents 7/4 as well as 16/9, so 64/63 is tempered out and pajara is in the [[Archytas_clan|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]].  


The 10-note MOS and LsssLsssss almost-MOS are called the symmetric and pentachordal decatonic scales and were independently invented/discovered by [[Paul_Erlich|Paul Erlich]] and [[Gene_Ward_Smith|Gene Ward Smith]]. They are often thought of as subsets of [[22edo|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.


==Interval chains==
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]].
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. The other, called "pajarous" to avoid confusion, loses some accuracy and there's little reason to use it unless you're using [[22edo|22edo]], which is the intersection of both systems.


===Basic 7-limit pajara===
See [[Diaschismic family #Pajara]] for technical data. See [[Pajara extensions]] for a discussion on the 11-limit extensions.


{| class="wikitable"
== 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, 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 ({{nowrap| 12 & 22 }})
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
|-
! Cents*
! Approximate ratios
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
| 600.0
| 7/5, 10/7
|-
| 1
| 707.2
| '''3/2'''
| 107.2
| 15/14, 16/15, 21/20
|-
|-
| | 771.81
| 2
| | 878.86
| 214.4
| | 985.90
| '''8/7''', '''9/8'''
| | 1092.95
| 814.4
| | 0.
| '''8/5'''
| | 107.05
| | 214.10
| | 321.14
| | 428.19
|-
|-
| | 14/9
| 3
| | 5/3
| 921.5
| | 7/4~16/9
| 12/7
| |
| 321.5
| | 1/1
| 6/5
| |
| | 9/8~8/7
| | 6/5
| | 9/7
|-
|-
| | 171.81
| 4
| | 278.86
| 428.7
| | 385.90
| 9/7, 14/11
| | 492.95
| 1028.7
| | 600.
| 9/5, 20/11
| | 707.05
| | 814.10
| | 921.14
| | 1028.19
|-
|-
| | 10/9
| 5
| | 7/6
| 1135.9
| | 5/4
| 21/11, 27/14, 48/25, <br>64/33, 96/49
| | 4/3
| 535.9
| | 7/5~10/7
| 15/11, 27/20
| | 3/2
|-
| | 8/5
| 6
| | 12/7
| 643.1
| | 9/5
| '''16/11'''
| 43.1
| 45/44, 56/55, 81/80
|}
|}


===11-limit pajara===
{| class="wikitable center-1 right-2 right-4"
 
|+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }})
{| class="wikitable"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
|-
|-
| | 344.92
! Cents*
| | 451.80
! Approximate ratios
| | 558.69
! Cents*
| | 665.57
! Approximate ratios
| | 772.46
| | 879.34
| | 986.23
| | 1093.11
| | 0.
| | 106.89
| | 213.77
| | 320.66
| | 427.54
| | 534.43
| | 641.31
| | 748.20
| | 855.08
|-
|-
| | 11/9
| 0
| |
| 0.0
| | 11/8
| '''1/1'''
| |
| 600.0
| | 14/9~11/7
| 7/5, 10/7
| | 5/3
| | 7/4~16/9
| |
| | 1/1
| |
| | 9/8~8/7
| | 6/5
| | 14/9~9/7
| |
| | 16/11
| |
| | 18/11
|-
|-
| | 944.92
| 1
| | 1051.80
| 709.6
| | 1158.69
| '''3/2'''
| | 65.57
| 109.6
| | 172.46
| 15/14, 16/15, 21/20
| | 279.34
| | 386.23
| | 493.11
| | 600.
| | 706.89
| | 813.77
| | 920.66
| | 1027.54
| | 1134.43
| | 41.31
| | 148.20
| | 255.08
|-
|-
| |
| 2
| | 11/6
| 219.1
| |
| '''8/7''', '''9/8'''
| |
| 819.1
| | 11/10~10/9
| '''8/5'''
| | 7/6
| | 5/4
| | 4/3
| | 7/5~10/7
| | 3/2
| | 8/5
| | 12/7
| | 9/5
| |
| |
| | 12/11
| |
|}
 
===Pajarous===
 
{| class="wikitable"
|-
|-
| | 432.96
| 3
| | 542.54
| 928.7
| | 652.11
| 12/7
| | 761.69
| 328.7
| | 871.27
| 6/5, 11/9
| | 980.85
| | 1090.42
| | 0.
| | 109.58
| | 219.15
| | 328.73
| | 438.31
| | 547.89
| | 657.46
| | 767.04
|-
|-
| | 14/11
| 4
| |
| 438.2
| | 16/11
| 9/7
| | 14/9
| 1038.2
| | 18/11~5/3
| 9/5, 11/6
| | 7/4~16/9
| |
| | 1/1
| |
| | 9/8~8/7
| | 6/5~11/9
| | 9/7
| | 11/8
| |
| | 11/7
|-
|-
| | 1032.96
| 5
| | 1142.54
| 1147.8
| | 52.11
| 27/14, 48/25, 55/28, <br>88/45, 96/49
| | 161.69
| 547.8
| | 271.27
| '''11/8''', 27/20
| | 380.85
| | 490.42
| | 600.
| | 709.58
| | 819.15
| | 928.73
| | 1038.31
| | 1147.89
| | 57.46
| | 167.04
|-
|-
| | 20/11
| 6
| |
| 657.3
| |
| 22/15
| | 12/11~10/9
| 57.3
| | 7/6
| 22/21, 33/32, 81/80
| | 5/4
| | 4/3
| | 7/5~10/7
| | 3/2
| | 8/5
| | 12/7
| | 9/5~11/6
| |
| |
| | 11/10
|}
|}
<nowiki/>* In 11-limit CWE tuning, octave-reduced


==MOSes==
== Chords and harmony ==
{{See also| Chords of pajara }}


===10-note (proper)===
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.
See [[2L_8s|2L 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 "no more than 2 intervals per class" rule, is called the "pentachordal" decatonic, because it consists of two identical "pentachords" plus a split 9/8~8/7 whole tone to complete the octave.
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''.


===12-note (proper)===
{{Todo|complete section}}
See [[10L_2s|10L 2s]].


==Spectrum of Pajara Tunings by Eigenmonzos==
== Scales ==
=== 10-note (proper) ===
{{Main| 2L&nbsp;8s }}


{| class="wikitable"
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&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 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
|-
|-
! | EDO degree
! rowspan="2" |  
! | Eigenmonzo
! colspan="3" | Euclidean
! | Decatonic seventh
|-
|-
| | 7\12
! Constrained
| |
! Constrained & skewed
| | 700.000
! Destretched
|-
|-
| |  
! Tenney
| | 3/2
| CTE: ~3/2 = 708.3557{{c}}
| | 701.955
| 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
|-
|-
| | 41\70
! rowspan="2" |  
| |
! colspan="3" | Euclidean
| | 702.857
|-
|-
| | 34\58
! Constrained
| |
! Constrained & skewed
| | 703.448
! Destretched
|-
|-
| | 61\104
! Tenney
| |  
| CTE: ~3/2 = 708.1993{{c}}
| | 703.846
| 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
|-
|-
| | 27\46
! rowspan="2" | Target
| |
! colspan="2" | Minimax
| | 704.348
|-
|-
| |
! Generator
| | 14/11
! Eigenmonzo*
| | 704.377
|-
|-
| |
| 7-odd-limit
| | 10/9
| ~3/2 = 709.363{{c}}
| | 704.399
| 35/24
|-
|-
| | 74\126
| 9-odd-limit
| |
| ~3/2 = 708.128{{c}}
| | 704.762
| 35/18
|-
|-
| | 47\80
| 11-odd-limit
| |
| ~3/2 = 708.128{{c}}
| | 705.000
| 35/18
|-
|}
| | 114\194
 
| |
=== Tuning spectrum ===
| | 705.155
{| class="wikitable center-all left-4"
|-
| |
| | 6/5
| | 705.214 (5 and 15 limit minimax)
|-
| | 67\114
| |
| | 705.263
|-
| | 87\148
| |
| | 705.405
|-
| | 20\34
| |
| | 705.882
|-
| | 93\158
| |
| | 706.329
|-
| | 73\124
| |
| | 706.452
|-
| | 126\214
| |
| | 706.542
|-
| |
| | 11/9
| | 706.574
|-
| | 53\90
| |
| | 706.667
|-
| | 139\236
| |
| | 706.780
|-
| |
| | 5/4
| | 706.843 (7 and 11 limit POTT)
|-
| | 86\146
| |
| | 706.849
|-
| | 119\202
| |
| | 706.931
|-
| | 33\56
| |
| | 707.143
|-
| |
| | 12/11
| | 707.234
|-
| | 112\190
| |
| | 707.368
|-
|-
| |
! Edo<br>generator
| | 15/11
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval]])
| | 707.390
! Generator (¢)
! Comments
|-
|-
| | 79\134
| 7\12
| |  
|  
| | 707.463
| 700.000
| Lower bound of 9- and 11-odd-limit diamond monotone
|-
|-
| | 125\212
|  
| |
| 3/2
| | 707.547
| 701.955
|  
|-
|-
| | 46\78
| 34\58
| |  
|  
| | 707.692
| 703.448
| 58ddee val
|-
|-
| | 105\178
| 27\46
| |  
|  
| | 707.865
| 704.348
| 46de val
|-
|-
| | 59\100
|  
| |
| 11/7
| | 708.000
| 704.377
|  
|-
|-
| |  
|  
| | 11/8
| 9/5
| | 708.114
| 704.399
|  
|-
|-
| | 72\122
| 47\80
| |  
|  
| | 708.196
| 705.000
| 80ddee val
|-
|-
| |  
|  
| | 11/10
| 5/3
| | 708.749 (11 limit minimax)
| 705.214
| 5-odd-limit minimax
|-
|-
| |  
| 20\34
| | 9/7
|  
| | 708.771
| 705.882
| 34d val
|-
|-
| | 13\22
|  
| |
| 11/9
| | 709.091
| 706.574
|  
|-
|-
| | 58\98
| 53\90
| |  
|  
| | 710.204
| 706.667
| 90dde val
|-
|-
| | 45\76
|  
| |
| 5/4
| | 710.526
| 706.843
| 7- and 11-limit POTT
|-
|-
| | 122\206
| 33\56
| |  
|  
| | 710.680
| 707.143
| 56d val
|-
|-
| | 77\130
|  
| |
| 11/6
| | 710.769
| 707.234
|  
|-
|-
| | 109\184
|  
| |
| 15/11
| | 710.870
| 707.390
|  
|-
|-
| |  
| 46\78
| | 7/6
|  
| | 711.043 (7 limit minimax)
| 707.692
| 78dd val
|-
|-
| | 32\54
|  
| |
| 11/8
| | 711.111
| 708.114
| 11- and 15-odd-limit minimax
|-
|-
| |  
|
| | 13/11
|36/35
| | 711.151 (13 limit minimax)
|708.128
|9-odd-limit minimax
|-
|-
| | 83\140
|  
| |
| 11/10
| | 711.429
| 708.749
|
|-
|-
| | 51\86
|  
| |
| 9/7
| | 711.628
| 708.771
|  
|-
|-
| |  
| 13\22
| | 16/15
|  
| | 711.731
| 709.091
| Upper bound of 11-odd-limit diamond monotone
|-
|-
| | 70\118
|
| |
|48/35
| | 711.864
|709.363
|7-odd-limit minimax
|-
|-
| | 19\32
|  
| |
| 7/6
| | 712.500
| 711.043
|
|-
|-
| | 44\74
| 32\54
| |  
|  
| | 713.5135
| 711.111
| 54e val
|-
|-
| |  
|  
| | 13/10
| 15/8
| | 713.553
| 711.731
|  
|-
|-
| | 25\42
| 19\32
| |  
|  
| | 714.286
| 712.500
| 32e val
|-
|-
| | 31\52
| 25\42
| |  
|  
| | 715.385
| 714.286
| 42cee val
|-
|-
| |  
|  
| | 8/7
| 7/4
| | 715.587
| 715.587
|
|-
|-
| | 6\10
| 6\10
| |  
|  
| | 720.000
| 720.000
| 10e val, upper bound of 9-odd-limit diamond monotone
|}
|}


==References==
== Music ==
<ul><li>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]</li></ul>
; [[Jake Freivald]]
* [https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs ''Chord Sequence in Paul Erlich's Decatonic Major''] (2014) – in Pajara[10], 22edo tuning


=Music=
; [[Joel Grant Taylor]]
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge] and
* [https://web.archive.org/web/20201127012345/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 ''Dirge''] – in the hexachordal dodecatonic modmos, [[12-22h]]
* [https://web.archive.org/web/20201127012408/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 ''Sonatina''] – ditto


[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina] both by [[Joel_Grant_Taylor|Joel Grant Taylor]], in the hexachordal dodecatonic MODMOS.
; [[Chris Vaisvil]]
* ''Smoke Filled Bar'' (2012) – [https://www.chrisvaisvil.com/smoke-filled-bar/ blog] | [https://web.archive.org/web/20230530093324/http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 play] – in 12-22h.


[http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] by [http://chrisvaisvil.com/?p=2403 Chris Vaisvil], also in 12-22h.
== References ==
<references/>


[https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs Chord Sequence in Paul Erlich's Decatonic Major] by Jake Freivald
[[Category:Pajara| ]] <!-- main article -->
[[Category:erlich]]
[[Category:Rank-2 temperaments]]
[[Category:pajara]]
[[Category:Archytas clan]]
[[Category:temperament]]
[[Category:Diaschismic family]]
[[Category:Jubilismic clan]]
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