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. This 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.
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.
| 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
| 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
Scales
10-note (proper)
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)
Scala files
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
- Chord Sequence in Paul Erlich's Decatonic Major (2014) – in Pajara[10], 22edo tuning
References
- Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf