Pajara

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 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

• 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 ¢, 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
	11/10	708.749
	9/7	708.771
13\22		709.091	Upper bound of 11-odd-limit diamond monotone
	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

• 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.