Pajara: Difference between revisions

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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 [[70:84:105:120|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 scale]]s of 10, 12, and 22 notes. The 10-note mos, Pajara[10], is notable for sharing a number of desirable properties with [[5L 2s|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.

Pajara has [[mos scale]]s of 10, 12, and 22 notes. The 10-note mos, Pajara[10], is notable for sharing a number of desirable properties with [[5L 2s|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]]<ref>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]</ref> 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 [[17/1|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]].

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== 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]~~

[[Category:Pajara| ]]

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 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 [[70:84:105:120|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 scale]]s of 10, 12, and 22 notes. The 10-note mos, Pajara[10], is notable for sharing a number of desirable properties with [[5L 2s|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.
+Pajara has [[mos scale]]s of 10, 12, and 22 notes. The 10-note mos, Pajara[10], is notable for sharing a number of desirable properties with [[5L 2s|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]]<ref>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]</ref> 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 [[17/1|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]].
@@ Line 346: / Line 346: @@
 == 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]
+<references/>
 [[Category:Pajara| ]] <!-- main article -->