Pajara

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 tempering 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 tempering of 50/49 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. This minor chord is, in fact, the utonal tetrad, having subharmonics in the pattern 12/10/8/7, which is an inversion of 7/6/5/4.

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 and Gene Ward Smith. They are often thought of as subsets of 22edo, without much loss of generality and accuracy.

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

Interval chains

In the following table, odd harmonics 1–11 and their inverses are in bold.

#	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, 81/80

* 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.

Intervals of pajara (12 & 22)

Generator	−11	−10	−9	−8	−7	−6
Cents*	24.26	131.15	238.03	344.92	451.80	558.69
Ratios				11/9		11/8
Generator	−5	−4	−3	−2	−1	0
Cents*	65.57	172.46	279.34	386.23	493.11	600.00
Ratios		11/10, 10/9	7/6	5/4	4/3	7/5, 10/7
Generator	0	1	2	3	4	5
Cents*	0.00	106.89	213.77	320.66	427.54	534.43
Ratios	1/1	16/15, 15/14	9/8, 8/7	6/5	14/11, 9/7	15/11
Generator	6	7	8	9	10	11
Cents*	41.31	148.20	255.08	361.97	468.85	575.74
Ratios		12/11

* In 11-limit POTE tuning

Intervals of pajarous (10 & 22)

Generator		−10	−9	−8	−7	−6
Cents*		109.09	218.18	327.27	436.36	545.45
Ratios					14/11	15/11
Generator	−5	−4	−3	−2	−1	0
Cents*	54.55	163.64	272.73	381.82	490.91	600.00
Ratios		12/11, 10/9	7/6	5/4	4/3	7/5, 10/7
Generator	0	1	2	3	4	5
Cents*	0.00	109.09	218.18	327.27	436.36	545.45
Ratios	1/1	16/15, 15/14	9/8, 8/7	6/5, 11/9	9/7	11/8
Generator	6	7	8	9	10
Cents*	54.55	163.64	272.73	381.82	491.91
Ratios		11/10

* In 22edo tuning

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

• 12-22h

Tunings

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- and 15-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
	11/10	708.749	11-odd-limit minimax
	9/7	708.771
13\22		709.091	Upper bound of 11-odd-limit diamond monotone
	7/6	711.043	7-odd-limit minimax
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

• Chord Sequence in Paul Erlich's Decatonic Major (2014) – in Pajara[10], 22edo tuning

Joel Grant Taylor

• Dirge – in the hexachordal dodecatonic modmos, 12-22h
• Sonatina – ditto

Chris Vaisvil

• Smoke Filled Bar (2012) – blog | play – in 12-22h.

References

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