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[edit]
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.
Basic 7-limit pajara[edit]
| 771.81 | 878.86 | 985.90 | 1092.95 | 0. | 107.05 | 214.10 | 321.14 | 428.19 |
| 14/9 | 5/3 | 7/4~16/9 | 1/1 | 9/8~8/7 | 6/5 | 9/7 | ||
| 171.81 | 278.86 | 385.90 | 492.95 | 600. | 707.05 | 814.10 | 921.14 | 1028.19 |
| 10/9 | 7/6 | 5/4 | 4/3 | 7/5~10/7 | 3/2 | 8/5 | 12/7 | 9/5 |
11-limit pajara[edit]
| 344.92 | 451.80 | 558.69 | 665.57 | 772.46 | 879.34 | 986.23 | 1093.11 | 0. | 106.89 | 213.77 | 320.66 | 427.54 | 534.43 | 641.31 | 748.20 | 855.08 |
| 11/9 | 11/8 | 14/9~11/7 | 5/3 | 7/4~16/9 | 1/1 | 9/8~8/7 | 6/5 | 14/9~9/7 | 16/11 | 18/11 | ||||||
| 944.92 | 1051.80 | 1158.69 | 65.57 | 172.46 | 279.34 | 386.23 | 493.11 | 600. | 706.89 | 813.77 | 920.66 | 1027.54 | 1134.43 | 41.31 | 148.20 | 255.08 |
| 11/6 | 11/10~10/9 | 7/6 | 5/4 | 4/3 | 7/5~10/7 | 3/2 | 8/5 | 12/7 | 9/5 | 12/11 |
Pajarous[edit]
| 432.96 | 542.54 | 652.11 | 761.69 | 871.27 | 980.85 | 1090.42 | 0. | 109.58 | 219.15 | 328.73 | 438.31 | 547.89 | 657.46 | 767.04 |
| 14/11 | 16/11 | 14/9 | 18/11~5/3 | 7/4~16/9 | 1/1 | 9/8~8/7 | 6/5~11/9 | 9/7 | 11/8 | 11/7 | ||||
| 1032.96 | 1142.54 | 52.11 | 161.69 | 271.27 | 380.85 | 490.42 | 600. | 709.58 | 819.15 | 928.73 | 1038.31 | 1147.89 | 57.46 | 167.04 |
| 20/11 | 12/11~10/9 | 7/6 | 5/4 | 4/3 | 7/5~10/7 | 3/2 | 8/5 | 12/7 | 9/5~11/6 | 11/10 |
MOSes[edit]
10-note (proper)[edit]
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)[edit]
See 10L 2s.
Spectrum of Pajara Tunings by Eigenmonzos[edit]
| EDO degree | Eigenmonzo | Decatonic seventh |
|---|---|---|
| 7\12 | 700.000 | |
| 3/2 | 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 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 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 limit minimax) | |
| 32\54 | 711.111 | |
| 13/11 | 711.151 (13 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[edit]
- Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf
Music[edit]
12-22hexachordal Sonatina both by Joel Grant Taylor, in the hexachordal dodecatonic MODMOS.
Smoke Filled Bar by Chris Vaisvil, also in 12-22h.
Chord Sequence in Paul Erlich's Decatonic Major by Jake Freivald