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'''Pajara''' (pronounced /pəˈd͡ʒɑːrə/, 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]]. The generator is in the neighborhood of 107-111 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]]. In fact, it shares the same structure as 5-limit [[srutal]]. 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]]. Tempering out any two of these commas (among others) produces the unique temperament pajara.
{{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 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 loses some accuracy overall, due to 5/4 and 7/4 necessarily being separated by 600 cents. This means that [[49/48]] and [[25/24]] are tempered to the same interval. As such, pajara has fundamentally different categories, as a conventional semifourth (~250 cents) is now a neutral interval of some variety. As a result, a unique feature of pajara is a well-defined "minor" version of the [[4:5:6:7|harmonic tetrad]] (as the 7th harmonic is now a major interval), consisting of [1/1 6/5 3/2 12/7], as an essentially tempered alteration to [1/1 5/4 3/2 7/4] where both the third and the harmonic seventh are flattened by a chroma.
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.


The 10-note MOS and LsssLsssss almost-MOS 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.
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]].


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


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


{| class="wikitable mw-collapsible mw-collapsed"
In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.
|+ style="font-size: 105%; white-space: nowrap;" | Intervals of pajara (12 &amp; 22)
 
{| class="wikitable center-1 right-2 right-4"
|+ style="font-size: 105%;" | Pajara ({{nowrap| 12 & 22 }})
|-
|-
! Generator
! rowspan="2" | #
! −11
! colspan="2" | Period 0
! −10
! colspan="2" | Period 1
! −9
! −8
! −7
! −6
|-
|-
! Cents*
! Cents*
| 24.26
! Approximate ratios
| 131.15
| 238.03
| 344.92
| 451.80
| 558.69
|-
! Ratios
|
|
|
| 11/9
|
| 11/8
|-
! Generator
! −5
! −4
! −3
! −2
! −1
! 0
|-
! Cents*
! Cents*
| 65.57
! Approximate ratios
| 172.46
| 279.34
| 386.23
| 493.11
| 600.00
|-
|-
! Ratios
| 0
|  
| 0.0
| 11/10, 10/9
| '''1/1'''
| 7/6
| 600.0
| 5/4
| 4/3
| 7/5, 10/7
| 7/5, 10/7
|-
|-
! Generator
| 1
! 0
| 707.2
! 1
| '''3/2'''
! 2
| 107.2
! 3
| 15/14, 16/15, 21/20
! 4
! 5
|-
|-
! Cents*
| 2
| 0.00
| 214.4
| 106.89
| '''8/7''', '''9/8'''
| 213.77
| 814.4
| 320.66
| '''8/5'''
| 427.54
| 534.43
|-
|-
! Ratios
| 3
| 1/1
| 921.5
| 16/15, 15/14
| 12/7
| 9/8, 8/7
| 321.5
| 6/5
| 6/5
| 14/11, 9/7
| 15/11
|-
|-
! Generator
| 4
! 6
| 428.7
! 7
| 9/7, 14/11
! 8
| 1028.7
! 9
| 9/5, 20/11
! 10
! 11
|-
|-
! Cents*
| 5
| 41.31
| 1135.9
| 148.20
| 21/11, 27/14, 48/25, <br>64/33, 96/49
| 255.08
| 535.9
| 361.97
| 15/11, 27/20
| 468.85
| 575.74
|-
|-
! Ratios
| 6
|  
| 643.1
| 12/11
| '''16/11'''
|  
| 43.1
|  
| 45/44, 56/55, 81/80
|
|
|}
|}
<nowiki />* In 11-limit POTE tuning


{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable center-1 right-2 right-4"
|+ style="font-size: 105%; white-space: nowrap;" | Intervals of pajarous (10 &amp; 22)
|+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }})
|-
|-
! Generator
! rowspan="2" | #
!  
! colspan="2" | Period 0
! −10
! colspan="2" | Period 1
! −9
! −8
! −7
! −6
|-
|-
! Cents*
! Cents*
|
! Approximate ratios
| 104.22
| 213.80
| 323.38
| 432.96
| 542.53
|-
! Ratios
|
|
|
|
| 14/11
| 15/11
|-
! Generator
! −5
! −4
! −3
! −2
! −1
! 0
|-
! Cents*
! Cents*
| 52.11
! Approximate ratios
| 161.69
| 271.27
| 380.84
| 490.42
| 600.00
|-
|-
! Ratios
| 0
|  
| 0.0
| 12/11, 10/9
| '''1/1'''
| 7/6
| 600.0
| 5/4
| 4/3
| 7/5, 10/7
| 7/5, 10/7
|-
|-
! Generator
| 1
! 0
| 709.6
! 1
| '''3/2'''
! 2
| 109.6
! 3
| 15/14, 16/15, 21/20
! 4
! 5
|-
|-
! Cents*
| 2
| 0.00
| 219.1
| 109.58
| '''8/7''', '''9/8'''
| 219.16
| 819.1
| 328.73
| '''8/5'''
| 438.31
| 547.89
|-
|-
! Ratios
| 3
| 1/1
| 928.7
| 16/15, 15/14
| 12/7
| 9/8, 8/7
| 328.7
| 6/5, 11/9
| 6/5, 11/9
|-
| 4
| 438.2
| 9/7
| 9/7
| 11/8
| 1038.2
|-
| 9/5, 11/6
! Generator
! 6
! 7
! 8
! 9
! 10
!
|-
|-
! Cents*
| 5
| 57.47
| 1147.8
| 167.04
| 27/14, 48/25, 55/28, <br>88/45, 96/49
| 276.62
| 547.8
| 386.20
| '''11/8''', 27/20
| 495.78
|
|-
|-
! Ratios
| 6
|  
| 657.3
| 11/10
| 22/15
|
| 57.3
|
| 22/21, 33/32, 81/80
|  
|  
|}
|}
<nowiki>*</nowiki> In 11-limit POTE tuning=
<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}}


== MOSes ==
== Scales ==
=== 10-note (proper) ===
=== 10-note (proper) ===
{{Main|2L&nbsp;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 "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.
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) ===
=== 12-note (proper) ===
{{main|10L&nbsp;2s}}
{{Main| 10L&nbsp;2s }}


== Tuning spectrum ==
=== Scala files ===
Gencom: [7/5 3/2; 50/49 64/63 65/63 99/98]
* [[Pajara12]]
* [[12-22h]]


Gencom mapping: [{{val| 2 2 7 8 14 5 }}, {{val| 0 1 -2 -2 -6 2 }}]
== 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.


{| class="wikitable center-all"
=== 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
|-
|-
! ET<br />generator
! Constrained
! [[eigenmonzo|Eigenmonzo<br />(unchanged-interval]])
! Constrained & skewed
! decatonic<br />seventh (¢)
! Destretched
! comments
|-
! 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 ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval]])
! Generator (¢)
! Comments
|-
|-
| 7\12
| 7\12
|  
|  
| 700.000
| 700.000
|  
| Lower bound of 9- and 11-odd-limit diamond monotone
|-
|-
|  
|  
| 4/3
| 3/2
| 701.955
| 701.955
|
|-
| 41\70
|
| 702.857
|  
|  
|-
|-
Line 257: Line 244:
|  
|  
| 703.448
| 703.448
|  
| 58ddee val
|-
| 61\104
|
| 703.846
|
|-
|-
| 27\46
| 27\46
|  
|  
| 704.348
| 704.348
|  
| 46de val
|-
|-
|  
|  
| 14/11
| 11/7
| 704.377
| 704.377
|  
|  
|-
|-
|  
|  
| 10/9
| 9/5
| 704.399
| 704.399
|
|-
| 74\126
|
| 704.762
|  
|  
|-
|-
Line 287: Line 264:
|  
|  
| 705.000
| 705.000
|  
| 80ddee val
|-
|-
| 114\194
|  
|  
| 705.155
| 5/3
|
|-
|
| 6/5
| 705.214
| 705.214
| 5 and 15-odd-limit minimax
| 5-odd-limit minimax
|-
| 67\114
|
| 705.263
|
|-
| 87\148
|
| 705.405
|
|-
|-
| 20\34
| 20\34
|  
|  
| 705.882
| 705.882
|  
| 34d val
|-
| 93\158
|
| 706.329
|
|-
| 73\124
|
| 706.452
|
|-
| 126\214
|
| 706.542
|
|-
|-
|  
|  
Line 337: Line 284:
|  
|  
| 706.667
| 706.667
|  
| 90dde val
|-
| 139\236
|
| 706.780
|
|-
|-
|  
|  
| 5/4
| 5/4
| 706.843
| 706.843
| 7 and 11-limit POTT
| 7- and 11-limit POTT
|-
| 86\146
|
| 706.849
|
|-
| 119\202
|
| 706.931
|
|-
|-
| 33\56
| 33\56
|  
|  
| 707.143
| 707.143
|  
| 56d val
|-
|-
|  
|  
| 12/11
| 11/6
| 707.234
| 707.234
|
|-
| 112\190
|
| 707.368
|  
|  
|-
|-
Line 377: Line 304:
| 15/11
| 15/11
| 707.390
| 707.390
|
|-
| 79\134
|
| 707.463
|
|-
| 125\212
|
| 707.547
|  
|  
|-
|-
Line 392: Line 309:
|  
|  
| 707.692
| 707.692
|  
| 78dd val
|-
| 105\178
|
| 707.865
|
|-
| 59\100
|
| 708.000
|
|-
|-
|  
|  
| 11/8
| 11/8
| 708.114
| 708.114
|  
| 11- and 15-odd-limit minimax
|-
|-
| 72\122
|
|  
|36/35
| 708.196
|708.128
|  
|9-odd-limit minimax
|-
|-
|  
|  
| 11/10
| 11/10
| 708.749
| 708.749
| 11-odd-limit minimax
|
|-
|-
|  
|  
Line 427: Line 334:
|  
|  
| 709.091
| 709.091
|  
| Upper bound of 11-odd-limit diamond monotone
|-
| 58\98
|
| 710.204
|
|-
| 45\76
|
| 710.526
|
|-
| 122\206
|
| 710.680
|
|-
| 77\130
|
| 710.769
|
|-
|-
| 109\184
|
|  
|48/35
| 710.870
|709.363
|  
|7-odd-limit minimax
|-
|-
|  
|  
| 7/6
| 7/6
| 711.043
| 711.043
| 7-odd-limit minimax
|
|-
|-
| 32\54
| 32\54
|  
|  
| 711.111
| 711.111
|  
| 54e val
|-
|
| 13/11
| 711.151
| 13-odd-limit minimax
|-
|-
| 83\140
|
| 711.429
|  
|  
|-
| 15/8
| 51\86
|
| 711.628
|
|-
|
| 16/15
| 711.731
| 711.731
|
|-
| 70\118
|
| 711.864
|  
|  
|-
|-
Line 492: Line 359:
|  
|  
| 712.500
| 712.500
|  
| 32e val
|-
| 44\74
|
| 713.5135
|
|-
|
| 13/10
| 713.553
|
|-
|-
| 25\42
| 25\42
|  
|  
| 714.286
| 714.286
|  
| 42cee val
|-
| 31\52
|
| 715.385
|
|-
|-
|  
|  
| 8/7
| 7/4
| 715.587
| 715.587
|  
|  
Line 522: Line 374:
|  
|  
| 720.000
| 720.000
|  
| 10e val, upper bound of 9-odd-limit diamond monotone
|}
|}


== References ==
== Music ==
* 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]
; [[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
 
; [[Joel Grant Taylor]]
* [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


== Music ==
; [[Chris Vaisvil]]
* Pieces by [[Joel Grant Taylor]], in the hexachordal dodecatonic [[MODMOS]]:
* ''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/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge]
** [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina]
* [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.
* [https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs Chord Sequence in Paul Erlich's Decatonic Major] by [[Jake Freivald]]


== See also ==
== References ==
* [[Pajara extensions]] - 13-limit extensions for pajara
<references/>
* [[Lumatone mapping for diaschismic]]


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