Pajara

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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 105-110 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. 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.

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.

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

MOSes

10-note (proper)

See 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)

See 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

Music

See also

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