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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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| Mapping = 2; 1 -2 -2 -6 1
| Mapping = 2; 1 -2 -2 -6 1
| Generators = 3/2 | Generators tuning = 707.4 | Optimization method = CWE
| Generators = 3/2 | Generators tuning = 707.4 | Optimization method = CWE
| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
| Pergen = (P8/2, P5)
| Pergen = (P8/2, P5)
| Odd limit 1 = 9 | Mistuning 1 = 17.5 | Complexity 1 = 10
| Odd limit 1 = 9 | Mistuning 1 = 17.5 | Complexity 1 = 10
Line 28: Line 28:


== 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. 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'''.  
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"
|+ Pajara (12 & 22)
|+ style="font-size: 105%;" | Pajara ({{nowrap| 12 & 22 }})
|-
|-
! rowspan="2" | #
! rowspan="2" | #
Line 86: Line 86:
| 45/44, 56/55, 81/80
| 45/44, 56/55, 81/80
|}
|}
{| class="wikitable center-1 right-2 right-4"
{| class="wikitable center-1 right-2 right-4"
|+ Pajarous (10 & 22)
|+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }})
|-
|-
! rowspan="2" | #
! rowspan="2" | #
Line 141: Line 142:
|}
|}
<nowiki/>* In 11-limit CWE tuning, octave-reduced
<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 ==
== Scales ==
=== 10-note (proper) ===
=== 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) ===
=== 12-note (proper) ===
{{Main| 10L 2s }}
{{Main| 10L&nbsp;2s }}


=== Scala files ===
=== Scala files ===
* [[Pajara12]]
* [[12-22h]]
* [[12-22h]]


Line 173: Line 184:
| POTE: ~3/2 = 707.0477{{c}}
| POTE: ~3/2 = 707.0477{{c}}
|}
|}
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
Line 188: Line 198:
| CWE: ~3/2 = 707.1826{{c}}
| CWE: ~3/2 = 707.1826{{c}}
| POTE: ~3/2 = 706.8851{{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 236: Line 269:
| 5/3
| 5/3
| 705.214
| 705.214
| 5- and 15-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 20\34
| 20\34
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| 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
|
|-
|-
|  
|  
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| 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

Latest revision as of 17:44, 14 April 2026

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);
50/49, 64/63, 99/98 (11-limit);
50/49, 64/63, 85/84, 99/98
(2.3.5.7.11.17)
Reduced mapping ⟨2; 1 -2 -2 -6 1]
ET join 12 & 22
Generators (CWE) ~3/2 = 707.4 ¢
MOS scales 2L 8s, 10L 2s, 12L 10s
Ploidacot diploid monocot
Pergen (P8/2, P5)
Minimax error 9-odd-limit: 17.5 ¢;
2.3.5.7.11.17 21-odd-limit: 22.4 ¢
Target scale size 9-odd-limit: 10 notes;
2.3.5.7.11.17 21-odd-limit: 22 notes

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 a perfect fifth in the neighborhood of 707–711 cents, 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 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 scales of 10, 12, and 22 notes. The 10-note mos, Pajara[10], is notable for sharing a number of desirable properties with 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[1] 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, 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.

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.

Pajara (12 & 22)
# Period 0 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
2 214.4 8/7, 9/8 814.4 8/5
3 921.5 12/7 321.5 6/5
4 428.7 9/7, 14/11 1028.7 9/5, 20/11
5 1135.9 21/11, 27/14, 48/25,
64/33, 96/49 		535.9 15/11, 27/20
6 643.1 16/11 43.1 45/44, 56/55, 81/80
Pajarous (10 & 22)
# Period 0 Period 1
Cents* Approximate ratios Cents* Approximate ratios
0 0.0 1/1 600.0 7/5, 10/7
1 709.6 3/2 109.6 15/14, 16/15, 21/20
2 219.1 8/7, 9/8 819.1 8/5
3 928.7 12/7 328.7 6/5, 11/9
4 438.2 9/7 1038.2 9/5, 11/6
5 1147.8 27/14, 48/25, 55/28,
88/45, 96/49 		547.8 11/8, 27/20
6 657.3 22/15 57.3 22/21, 33/32, 81/80

* 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 1/(12:10:8:7), which can be considered the minor tetrad.

Scales

10-note (proper)

Main article: 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 pentachords plus a split 9/8~8/7 whole tone to complete the octave.

12-note (proper)

Main article: 10L 2s

Scala files

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 ¢, however, pajara is conventionally tuned flat of 22edo, since tunings sharp of about 710 ¢ lose a large degree of accuracy in 5/4 and especially 6/5.

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 708.3557 ¢ CWE: ~3/2 = 707.3438 ¢ POTE: ~3/2 = 707.0477 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 708.1993 ¢ CWE: ~3/2 = 707.1826 ¢ POTE: ~3/2 = 706.8851 ¢

Target tunings

Odd-limit-based target tunings
Target Minimax
Generator Eigenmonzo*
7-odd-limit ~3/2 = 709.363 ¢ 35/24
9-odd-limit ~3/2 = 708.128 ¢ 35/18
11-odd-limit ~3/2 = 708.128 ¢ 35/18

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
7\12 700.000 Lower bound of 9- and 11-odd-limit diamond monotone
3/2 701.955
34\58 703.448 58ddee val
27\46 704.348 46de val
11/7 704.377
9/5 704.399
47\80 705.000 80ddee val
5/3 705.214 5-odd-limit minimax
20\34 705.882 34d val
11/9 706.574
53\90 706.667 90dde val
5/4 706.843 7- and 11-limit POTT
33\56 707.143 56d val
11/6 707.234
15/11 707.390
46\78 707.692 78dd val
11/8 708.114 11- and 15-odd-limit minimax
36/35 708.128 9-odd-limit minimax
11/10 708.749
9/7 708.771
13\22 709.091 Upper bound of 11-odd-limit diamond monotone
48/35 709.363 7-odd-limit minimax
7/6 711.043
32\54 711.111 54e val
15/8 711.731
19\32 712.500 32e val
25\42 714.286 42cee val
7/4 715.587
6\10 720.000 10e val, upper bound of 9-odd-limit diamond monotone

Music

Jake Freivald
Joel Grant Taylor
Chris Vaisvil
  • Smoke Filled Bar (2012) – blog | play – in 12-22h.

References

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