Pajara: Difference between revisions
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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. | |||
The 10-note MOS and LsssLsssss almost-MOS are called the symmetric and pentachordal decatonic scales and were independently invented/discovered by [[ | |||
==Interval chains== | ==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 | 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=== | ===Basic 7-limit pajara=== | ||
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===10-note (proper)=== | ===10-note (proper)=== | ||
See [[ | 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. | 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)=== | ===12-note (proper)=== | ||
See [[ | See [[10L 2s]]. | ||
==Spectrum of Pajara Tunings by Eigenmonzos== | ==Spectrum of Pajara Tunings by Eigenmonzos== | ||
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==References== | ==References== | ||
* Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf] | |||
==Music== | ==Music== | ||