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 |
50/49, 64/63, 99/98 (11-limit)
11-limit 15-odd-limit: 17.5 ¢
11-limit 15-odd-limit: 22 notes
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[1] 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. The other, called pajarous to avoid confusion, loses some accuracy for 3 and 5. It is best tuned sharp of 22edo. However, 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.
| # | 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 |
| # | 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
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
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
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 708.3557 ¢ | CWE: ~3/2 = 707.3438 ¢ | POTE: ~3/2 = 707.0477 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 708.1993 ¢ | CWE: ~3/2 = 707.1826 ¢ | POTE: ~3/2 = 706.8851 ¢ |
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