Pajara: Difference between revisions

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| Mapping = 2; 1 -2 -2 -6 1

| Generators = 3/2 | Generators tuning = 707.4 | Optimization method = CWE

| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]

| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]

| Pergen = (P8/2, P5)

| Odd limit 1 = 9 | Mistuning 1 = 17.5 | Complexity 1 = 10

| Odd limit 2 = 2.3.5.7.11.17 21 | Mistuning 2 = 22.4 | Complexity 2 = 22

}}

'''Pajara''' (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a [[regular temperament|temperament]] with a half-octave [[period]] that represents both [[7/5]] and [[10/7]], so [[50/49]] is [[tempering out|tempered out]] and it is in the [[jubilismic clan]]. The [[generator]] is a [[3/2|perfect fifth]] in the neighborhood of 707–711 [[cent]]s, 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''' (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a [[regular temperament|temperament]] with a half-octave [[period]] that represents both [[7/5]] and [[10/7]], so [[50/49]] is [[tempering out|tempered out]] and it is in the [[jubilismic clan]]. The [[generator]] is a [[3/2|perfect fifth]] in the neighborhood of 707–711 [[cent]]s, 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 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]].

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{| class="wikitable center-1 right-2 right-4"

|+ Pajara (12 & 22)

|+ style="font-size: 105%;" | Pajara (12 & 22)

|-

! rowspan="2" | #

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| 45/44, 56/55, 81/80

|}

{| class="wikitable center-1 right-2 right-4"

|+ Pajarous (10 & 22)

|+ style="font-size: 105%;" | Pajarous (10 & 22)

|-

! rowspan="2" | #

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== 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 [[pentachord]]s 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 {{nowrap|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 [[pentachord]]s plus a split {{nowrap|9/8~8/7}} whole tone to complete the octave.

=== 12-note (proper) ===

=== Scala files ===

@@ Line 12: / Line 12: @@
 | Mapping = 2; 1 -2 -2 -6 1
 | Generators = 3/2 | Generators tuning = 707.4 | Optimization method = CWE
-| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
+| MOS scales = [[2L&nbsp;8s]], [[10L&nbsp;2s]], [[12L&nbsp;10s]]
 | Pergen = (P8/2, P5)
 | Odd limit 1 = 9 | Mistuning 1 = 17.5 | Complexity 1 = 10
 | Odd limit 2 = 2.3.5.7.11.17 21 | Mistuning 2 = 22.4 | Complexity 2 = 22
 }}
-'''Pajara''' (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a [[regular temperament|temperament]] with a half-octave [[period]] that represents both [[7/5]] and [[10/7]], so [[50/49]] is [[tempering out|tempered out]] and it is in the [[jubilismic clan]]. The [[generator]] is a [[3/2|perfect fifth]] in the neighborhood of 707–711 [[cent]]s, 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''' (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a [[regular temperament|temperament]] with a half-octave [[period]] that represents both [[7/5]] and [[10/7]], so [[50/49]] is [[tempering out|tempered out]] and it is in the [[jubilismic clan]]. The [[generator]] is a [[3/2|perfect fifth]] in the neighborhood of 707–711&nbsp;[[cent]]s, 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 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]].
@@ Line 33: / Line 34: @@
 {| class="wikitable center-1 right-2 right-4"
-|+ Pajara (12 & 22)
+|+ style="font-size: 105%;" | Pajara (12&nbsp;&amp;&nbsp;22)
 |-
 ! rowspan="2" | #
@@ Line 86: / Line 87: @@
 | 45/44, 56/55, 81/80
 |}
 {| class="wikitable center-1 right-2 right-4"
-|+ Pajarous (10 & 22)
+|+ style="font-size: 105%;" | Pajarous (10&nbsp;&amp;&nbsp;22)
 |-
 ! rowspan="2" | #
@@ Line 144: / Line 146: @@
 == Scales ==
 === 10-note (proper) ===
-{{Main| 2L 8s }}
+{{Main| 2L&nbsp;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 rule of no more than 2 intervals per class, is called the ''pentachordal'' decatonic, because it consists of two identical [[pentachord]]s 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 {{nowrap|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 [[pentachord]]s plus a split {{nowrap|9/8~8/7}} whole tone to complete the octave.
 === 12-note (proper) ===
-{{Main| 10L 2s }}
+{{Main| 10L&nbsp;2s }}
 === Scala files ===