Pajara: Difference between revisions
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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 [[Paul Erlich]] and [[Gene Ward Smith]]. They are often thought of as subsets of [[22edo]], without much loss of generality and accuracy. | ||
==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 22edo, which is the intersection of both systems. | 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 === | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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|} | |} | ||
===11-limit pajara=== | === 11-limit pajara === | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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|} | |} | ||
===Pajarous=== | === Pajarous === | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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|} | |} | ||
==MOSes== | == MOSes == | ||
===10-note (proper)=== | === 10-note (proper) === | ||
See [[2L 8s]]. | 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 [[10L 2s]]. | See [[10L 2s]]. | ||
==Tuning spectrum== | == Tuning spectrum == | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||
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|- | |- | ||
| | | | | | ||
| | 3 | | | 4/3 | ||
| | 701.955 | | | 701.955 | ||
| | | | | | ||
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| | 6/5 | | | 6/5 | ||
| | 705.214 | | | 705.214 | ||
| | 5 and 15 limit minimax | | | 5 and 15-odd-limit minimax | ||
|- | |- | ||
| | 67\114 | | | 67\114 | ||
| Line 330: | Line 326: | ||
| | 5/4 | | | 5/4 | ||
| | 706.843 | | | 706.843 | ||
| | 7 and 11 limit POTT | | | 7 and 11-limit POTT | ||
|- | |- | ||
| | 86\146 | | | 86\146 | ||
| Line 400: | Line 396: | ||
| | 11/10 | | | 11/10 | ||
| | 708.749 | | | 708.749 | ||
| | 11 limit minimax | | | 11-odd-limit minimax | ||
|- | |- | ||
| | | | | | ||
| Line 440: | Line 436: | ||
| | 7/6 | | | 7/6 | ||
| | 711.043 | | | 711.043 | ||
| | 7 limit minimax | | | 7-odd-limit minimax | ||
|- | |- | ||
| | 32\54 | | | 32\54 | ||
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| | 13/11 | | | 13/11 | ||
| | 711.151 | | | 711.151 | ||
| | 13 limit minimax | | | 13-odd-limit minimax | ||
|- | |- | ||
| | 83\140 | | | 83\140 | ||
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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] | * 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 == | ||
* Pieces by [[Joel Grant Taylor]], in the hexachordal dodecatonic [[MODMOS]]: | * Pieces by [[Joel Grant Taylor]], in the hexachordal dodecatonic [[MODMOS]]: | ||
** [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge] | ** [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge] | ||
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* [http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] by [http://chrisvaisvil.com/?p=2403 Chris Vaisvil], also in 12-22h. | * [http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] by [http://chrisvaisvil.com/?p=2403 Chris Vaisvil], also in 12-22h. | ||
* [https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs Chord Sequence in Paul Erlich's Decatonic Major] by [[Jake Freivald]] | * [https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs Chord Sequence in Paul Erlich's Decatonic Major] by [[Jake Freivald]] | ||
== See also == | |||
* [[Pajara extensions]] - 13-limit extensions for pajara | |||
[[Category:Erlich]] | [[Category:Erlich]] | ||