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 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.

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 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.

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.

See Diaschismic family#Pajara for technical data.

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.

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*		104.22	213.80	323.38	432.96	542.53
Ratios					14/11	15/11
Generator	−5	−4	−3	−2	−1	0
Cents*	52.11	161.69	271.27	380.84	490.42	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	109.58	219.16	328.73	438.31	547.89
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*	57.47	167.04	276.62	386.20	495.78
Ratios		11/10

* In 11-limit POTE tuning

Theoretical properties

Pajara is unique in that it introduces a well-defined "minor" analogue to the harmonic tetrad, because

MOSes

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 "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.

12-note (proper)

Main article: 10L 2s

Tuning spectrum

Gencom: [7/5 3/2; 50/49 64/63 65/63 99/98]

Gencom mapping: [⟨2 2 7 8 14 5], ⟨0 1 -2 -2 -6 2]]

ET generator	Eigenmonzo (unchanged-interval)	decatonic seventh (¢)	comments
7\12		700.000
	4/3	701.955
41\70		702.857
34\58		703.448
61\104		703.846
27\46		704.348
	14/11	704.377
	10/9	704.399
74\126		704.762
47\80		705.000
114\194		705.155
	6/5	705.214	5 and 15-odd-limit minimax
67\114		705.263
87\148		705.405
20\34		705.882
93\158		706.329
73\124		706.452
126\214		706.542
	11/9	706.574
53\90		706.667
139\236		706.780
	5/4	706.843	7 and 11-limit POTT
86\146		706.849
119\202		706.931
33\56		707.143
	12/11	707.234
112\190		707.368
	15/11	707.390
79\134		707.463
125\212		707.547
46\78		707.692
105\178		707.865
59\100		708.000
	11/8	708.114
72\122		708.196
	11/10	708.749	11-odd-limit minimax
	9/7	708.771
13\22		709.091
58\98		710.204
45\76		710.526
122\206		710.680
77\130		710.769
109\184		710.870
	7/6	711.043	7-odd-limit minimax
32\54		711.111
	13/11	711.151	13-odd-limit minimax
83\140		711.429
51\86		711.628
	16/15	711.731
70\118		711.864
19\32		712.500
44\74		713.5135
	13/10	713.553
25\42		714.286
31\52		715.385
	8/7	715.587
6\10		720.000

References

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

Music

Pieces by Joel Grant Taylor, in the hexachordal dodecatonic MODMOS:
- 12-22hexachordal Dirge
- 12-22hexachordal Sonatina
Smoke Filled Bar by Chris Vaisvil, also in 12-22h.
Chord Sequence in Paul Erlich's Decatonic Major by Jake Freivald