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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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{{interwiki
{{interwiki
| en = Pajara
| de = Pajara
| de = Pajara
| en = Pajara
| es =  
| es =  
| ja =  
| ja =  
}}
}}
{{Infobox Regtemp
{{Infobox regtemp
| Title = Pajara
| 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
| 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)
| Pergen = (P8/2, P5)
| Odd limit 1 = 7 | Mistuning 1 = 17.5 | Complexity 1 = 22
| Odd limit 1 = 9 | Mistuning 1 = 17.5 | Complexity 1 = 10
| Odd limit 2 = (7-limit) 21 | Mistuning 2 = 17.5 | Complexity 2 = 22
| 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. This means that [[49/48]] and [[25/24]] are tempered to the same interval, and allows for a simple alteration to produce a "minor" harmonic chord with 6/5 and 12/7 by flattening the third and seventh the same amount.  
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]]<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.


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


== Interval chains ==
== Interval chains ==
In the following table, odd harmonics 1–11 and their inverses are in '''bold'''.  
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"
{| class="wikitable center-1 right-2 right-4"
|+ style="font-size: 105%;" | Pajara ({{nowrap| 12 & 22 }})
|-
|-
! rowspan="2" | #
! rowspan="2" | #
Line 79: Line 84:
| '''16/11'''
| '''16/11'''
| 43.1
| 43.1
| 45/44, 81/80
| 45/44, 56/55, 81/80
|}
|}
<nowiki/>* In 11-limit CWE tuning, octave-reduced


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.
{| class="wikitable center-1 right-2 right-4"
 
|+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }})
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Intervals of pajara (12 &amp; 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
| 709.6
! 1
| '''3/2'''
! 2
| 109.6
! 3
| 15/14, 16/15, 21/20
! 4
! 5
|-
|-
! Cents*
| 2
| 0.00
| 219.1
| 106.89
| '''8/7''', '''9/8'''
| 213.77
| 819.1
| 320.66
| '''8/5'''
| 427.54
| 534.43
|-
|-
! Ratios
| 3
| 1/1
| 928.7
| 16/15, 15/14
| 12/7
| 9/8, 8/7
| 328.7
| 6/5
| 6/5, 11/9
| 14/11, 9/7
| 15/11
|-
|-
! Generator
| 4
! 6
| 438.2
! 7
| 9/7
! 8
| 1038.2
! 9
| 9/5, 11/6
! 10
! 11
|-
|-
! Cents*
| 5
| 41.31
| 1147.8
| 148.20
| 27/14, 48/25, 55/28, <br>88/45, 96/49
| 255.08
| 547.8
| 361.97
| '''11/8''', 27/20
| 468.85
| 575.74
|-
|-
! Ratios
| 6
|  
| 657.3
| 12/11
| 22/15
|
| 57.3
|
| 22/21, 33/32, 81/80
|  
|  
|}
|}
<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}}
 
== Scales ==
=== 10-note (proper) ===
{{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 {{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"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Intervals of pajarous (10 &amp; 22)
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
|-
! Generator
! rowspan="2" |
!  
! colspan="3" | Euclidean
! −10
|-
! −9
! Constrained
! −8
! Constrained & skewed
! −7
! Destretched
! −6
|-
|-
! Cents*
! Tenney
|  
| CTE: ~3/2 = 708.3557{{c}}
| 104.22
| CWE: ~3/2 = 707.3438{{c}}
| 213.80
| POTE: ~3/2 = 707.0477{{c}}
| 323.38
|}
| 432.96
{| class="wikitable mw-collapsible mw-collapsed"
| 542.53
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
|-
! Ratios
! rowspan="2" |  
|
! colspan="3" | Euclidean
|
|
|
| 14/11
| 15/11
|-
|-
! Generator
! Constrained
! −5
! Constrained & skewed
! −4
! Destretched
! −3
! −2
! −1
! 0
|-
|-
! Cents*
! Tenney
| 52.11
| CTE: ~3/2 = 708.1993{{c}}
| 161.69
| CWE: ~3/2 = 707.1826{{c}}
| 271.27
| POTE: ~3/2 = 706.8851{{c}}
| 380.84
|}
| 490.42
 
| 600.00
=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
|-
|-
! Ratios
! rowspan="2" | Target
|
! colspan="2" | Minimax
| 12/11, 10/9
| 7/6
| 5/4
| 4/3
| 7/5, 10/7
|-
|-
! Generator
! Generator
! 0
! Eigenmonzo*
! 1
! 2
! 3
! 4
! 5
|-
! Cents*
| 0.00
| 109.58
| 219.16
| 328.73
| 438.31
| 547.89
|-
|-
! Ratios
| 7-odd-limit
| 1/1
| ~3/2 = 709.363{{c}}
| 16/15, 15/14
| 35/24
| 9/8, 8/7
| 6/5, 11/9
| 9/7
| 11/8
|-
|-
! Generator
| 9-odd-limit
! 6
| ~3/2 = 708.128{{c}}
! 7
| 35/18
! 8
! 9
! 10
!
|-
|-
! Cents*
| 11-odd-limit
| 57.47
| ~3/2 = 708.128{{c}}
| 167.04
| 35/18
| 276.62
| 386.20
| 495.78
|
|-
! Ratios
|
| 11/10
|
|
|
|
|}
|}
<nowiki>*</nowiki> In 11-limit POTE tuning
== Scales ==
=== 10-note (proper) ===
{{Main| 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 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.
=== 12-note (proper) ===
{{Main| 10L 2s }}
=== Scala files ===
* [[12-22h]]


== Tunings ==
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
Line 346: Line 269:
| 5/3
| 5/3
| 705.214
| 705.214
| 5- and 15-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 20\34
| 20\34
Line 391: Line 314:
| 11/8
| 11/8
| 708.114
| 708.114
|  
| 11- and 15-odd-limit minimax
|-
|
|36/35
|708.128
|9-odd-limit minimax
|-
|-
|  
|  
| 11/10
| 11/10
| 708.749
| 708.749
| 11-odd-limit minimax
|
|-
|-
|  
|  
Line 407: Line 335:
| 709.091
| 709.091
| Upper bound of 11-odd-limit diamond monotone
| Upper bound of 11-odd-limit diamond monotone
|-
|
|48/35
|709.363
|7-odd-limit minimax
|-
|-
|  
|  
| 7/6
| 7/6
| 711.043
| 711.043
| 7-odd-limit minimax
|
|-
|-
| 32\54
| 32\54
Line 456: Line 389:


== References ==
== 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 -->
[[Category:Pajara| ]] <!-- main article -->
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