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Pajara: Difference between revisions

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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.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{| class="wikitable mw-collapsible mw-collapsed"
<div style="line-height:1.6;">'''Intervals of pajara (12&amp;22)'''</div>
|+ style="font-size: 105%; white-space: nowrap;" | Intervals of pajara (12 &amp; 22)
<div class="mw-collapsible-content">
{| class="wikitable"
|-
|-
! | Generator
! Generator
! | -11
! −11
! | -10
! −10
! | -9
! −9
! | -8
! −8
! | -7
! −7
! | -6
! −6
|-
|-
! | Cents*
! Cents*
| | 24.26
| 24.26
| | 131.15
| 131.15
| | 238.03
| 238.03
| | 344.92
| 344.92
| | 451.80
| 451.80
| | 558.69
| 558.69
|-
|-
! | Ratios
! Ratios
| |  
|  
| |  
|  
| |  
|  
| | 11/9
| 11/9
| |  
|  
| | 11/8
| 11/8
|-
|-
! | Generator
! Generator
! | -5
! −5
! | -4
! −4
! | -3
! −3
! | -2
! −2
! | -1
! −1
! | 0
! 0
|-
|-
! | Cents*
! Cents*
| | 65.57
| 65.57
| | 172.46
| 172.46
| | 279.34
| 279.34
| | 386.23
| 386.23
| | 493.11
| 493.11
| | 600.00
| 600.00
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 11/10, 10/9
| 11/10, 10/9
| | 7/6
| 7/6
| | 5/4
| 5/4
| | 4/3
| 4/3
| | 7/5, 10/7
| 7/5, 10/7
|-
|-
! | Generator
! Generator
! | 0
! 0
! | 1
! 1
! | 2
! 2
! | 3
! 3
! | 4
! 4
! | 5
! 5
|-
|-
! | Cents*
! Cents*
| | 0.00
| 0.00
| | 106.89
| 106.89
| | 213.77
| 213.77
| | 320.66
| 320.66
| | 427.54
| 427.54
| | 534.43
| 534.43
|-
|-
! | Ratios
! Ratios
| | 1/1
| 1/1
| | 16/15, 15/14
| 16/15, 15/14
| | 9/8, 8/7
| 9/8, 8/7
| | 6/5
| 6/5
| | 14/11, 9/7
| 14/11, 9/7
| | 15/11
| 15/11
|-
|-
! | Generator
! Generator
! | 6
! 6
! | 7
! 7
! | 8
! 8
! | 9
! 9
! | 10
! 10
! | 11
! 11
|-
|-
! | Cents*
! Cents*
| | 41.31
| 41.31
| | 148.20
| 148.20
| | 255.08
| 255.08
| | 361.97
| 361.97
| | 468.85
| 468.85
| | 575.74
| 575.74
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 12/11
| 12/11
| |  
|  
| |  
|  
| |  
|  
| |  
|  
|}
|}
<nowiki>*</nowiki> in 11-limit POTE tuning
<nowiki />* In 11-limit POTE tuning
</div></div>


 
{| class="wikitable mw-collapsible mw-collapsed"
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
|+ style="font-size: 105%; white-space: nowrap;" | Intervals of pajarous (10 &amp; 22)
<div style="line-height:1.6;">'''Intervals of pajarous (10&amp;22)'''</div>
<div class="mw-collapsible-content">
{| class="wikitable"
|-
|-
! | Generator
! Generator
! |
!  
! | -10
! −10
! | -9
! −9
! | -8
! −8
! | -7
! −7
! | -6
! −6
|-
|-
! | Cents*
! Cents*
| |  
|  
| | 104.22
| 104.22
| | 213.80
| 213.80
| | 323.38
| 323.38
| | 432.96
| 432.96
| | 542.53
| 542.53
|-
|-
! | Ratios
! Ratios
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| | 14/11
| 14/11
| | 15/11
| 15/11
|-
|-
! | Generator
! Generator
! | -5
! −5
! | -4
! −4
! | -3
! −3
! | -2
! −2
! | -1
! −1
! | 0
! 0
|-
|-
! | Cents*
! Cents*
| | 52.11
| 52.11
| | 161.69
| 161.69
| | 271.27
| 271.27
| | 380.84
| 380.84
| | 490.42
| 490.42
| | 600.00
| 600.00
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 12/11, 10/9
| 12/11, 10/9
| | 7/6
| 7/6
| | 5/4
| 5/4
| | 4/3
| 4/3
| | 7/5, 10/7
| 7/5, 10/7
|-
|-
! | Generator
! Generator
! | 0
! 0
! | 1
! 1
! | 2
! 2
! | 3
! 3
! | 4
! 4
! | 5
! 5
|-
|-
! | Cents*
! Cents*
| | 0.00
| 0.00
| | 109.58
| 109.58
| | 219.16
| 219.16
| | 328.73
| 328.73
| | 438.31
| 438.31
| | 547.89
| 547.89
|-
|-
! | Ratios
! Ratios
| | 1/1
| 1/1
| | 16/15, 15/14
| 16/15, 15/14
| | 9/8, 8/7
| 9/8, 8/7
| | 6/5, 11/9
| 6/5, 11/9
| | 9/7
| 9/7
| | 11/8
| 11/8
|-
|-
! | Generator
! Generator
! | 6
! 6
! | 7
! 7
! | 8
! 8
! | 9
! 9
! | 10
! 10
! |
!  
|-
|-
! | Cents*
! Cents*
| | 57.47
| 57.47
| | 167.04
| 167.04
| | 276.62
| 276.62
| | 386.20
| 386.20
| | 495.78
| 495.78
| |  
|  
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 11/10
| 11/10
| |  
|  
| |  
|  
| |  
|  
| |  
|  
|}
|}
<nowiki>*</nowiki> in 11-limit POTE tuning
<nowiki />* In 11-limit POTE tuning
</div></div>


== MOSes ==
== MOSes ==
=== 10-note (proper) ===
=== 10-note (proper) ===
See [[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 "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]].
{{main|10L&nbsp;2s}}


== Tuning spectrum ==
== Tuning spectrum ==
Line 238: Line 230:
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! | ET<br>generator
! ET<br />generator
! | [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
! [[eigenmonzo|Eigenmonzo<br />(unchanged-interval]])
! | decatonic<br>seventh (¢)
! decatonic<br />seventh (¢)
! | comments
! comments
|-
|-
| | 7\12
| 7\12
| |  
|  
| | 700.000
| 700.000
| |  
|  
|-
|-
| |  
|  
| | 4/3
| 4/3
| | 701.955
| 701.955
| |  
|  
|-
|-
| | 41\70
| 41\70
| |  
|  
| | 702.857
| 702.857
| |  
|  
|-
|-
| | 34\58
| 34\58
| |  
|  
| | 703.448
| 703.448
| |  
|  
|-
|-
| | 61\104
| 61\104
| |  
|  
| | 703.846
| 703.846
| |  
|  
|-
|-
| | 27\46
| 27\46
| |  
|  
| | 704.348
| 704.348
| |  
|  
|-
|-
| |  
|  
| | 14/11
| 14/11
| | 704.377
| 704.377
| |  
|  
|-
|-
| |  
|  
| | 10/9
| 10/9
| | 704.399
| 704.399
| |  
|  
|-
|-
| | 74\126
| 74\126
| |  
|  
| | 704.762
| 704.762
| |  
|  
|-
|-
| | 47\80
| 47\80
| |  
|  
| | 705.000
| 705.000
| |  
|  
|-
|-
| | 114\194
| 114\194
| |  
|  
| | 705.155
| 705.155
| |  
|  
|-
|-
| |  
|  
| | 6/5
| 6/5
| | 705.214
| 705.214
| | 5 and 15-odd-limit minimax
| 5 and 15-odd-limit minimax
|-
|-
| | 67\114
| 67\114
| |  
|  
| | 705.263
| 705.263
| |  
|  
|-
|-
| | 87\148
| 87\148
| |  
|  
| | 705.405
| 705.405
| |  
|  
|-
|-
| | 20\34
| 20\34
| |  
|  
| | 705.882
| 705.882
| |  
|  
|-
|-
| | 93\158
| 93\158
| |  
|  
| | 706.329
| 706.329
| |  
|  
|-
|-
| | 73\124
| 73\124
| |  
|  
| | 706.452
| 706.452
| |  
|  
|-
|-
| | 126\214
| 126\214
| |  
|  
| | 706.542
| 706.542
| |  
|  
|-
|-
| |  
|  
| | 11/9
| 11/9
| | 706.574
| 706.574
| |  
|  
|-
|-
| | 53\90
| 53\90
| |  
|  
| | 706.667
| 706.667
| |  
|  
|-
|-
| | 139\236
| 139\236
| |  
|  
| | 706.780
| 706.780
| |  
|  
|-
|-
| |  
|  
| | 5/4
| 5/4
| | 706.843
| 706.843
| | 7 and 11-limit POTT
| 7 and 11-limit POTT
|-
|-
| | 86\146
| 86\146
| |  
|  
| | 706.849
| 706.849
| |  
|  
|-
|-
| | 119\202
| 119\202
| |  
|  
| | 706.931
| 706.931
| |  
|  
|-
|-
| | 33\56
| 33\56
| |  
|  
| | 707.143
| 707.143
| |  
|  
|-
|-
| |  
|  
| | 12/11
| 12/11
| | 707.234
| 707.234
| |  
|  
|-
|-
| | 112\190
| 112\190
| |  
|  
| | 707.368
| 707.368
| |  
|  
|-
|-
| |  
|  
| | 15/11
| 15/11
| | 707.390
| 707.390
| |  
|  
|-
|-
| | 79\134
| 79\134
| |  
|  
| | 707.463
| 707.463
| |  
|  
|-
|-
| | 125\212
| 125\212
| |  
|  
| | 707.547
| 707.547
| |  
|  
|-
|-
| | 46\78
| 46\78
| |  
|  
| | 707.692
| 707.692
| |  
|  
|-
|-
| | 105\178
| 105\178
| |  
|  
| | 707.865
| 707.865
| |  
|  
|-
|-
| | 59\100
| 59\100
| |  
|  
| | 708.000
| 708.000
| |  
|  
|-
|-
| |  
|  
| | 11/8
| 11/8
| | 708.114
| 708.114
| |  
|  
|-
|-
| | 72\122
| 72\122
| |  
|  
| | 708.196
| 708.196
| |  
|  
|-
|-
| |  
|  
| | 11/10
| 11/10
| | 708.749
| 708.749
| | 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
| |  
|  
| | 9/7
| 9/7
| | 708.771
| 708.771
| |  
|  
|-
|-
| | 13\22
| 13\22
| |  
|  
| | 709.091
| 709.091
| |  
|  
|-
|-
| | 58\98
| 58\98
| |  
|  
| | 710.204
| 710.204
| |  
|  
|-
|-
| | 45\76
| 45\76
| |  
|  
| | 710.526
| 710.526
| |  
|  
|-
|-
| | 122\206
| 122\206
| |  
|  
| | 710.680
| 710.680
| |  
|  
|-
|-
| | 77\130
| 77\130
| |  
|  
| | 710.769
| 710.769
| |  
|  
|-
|-
| | 109\184
| 109\184
| |  
|  
| | 710.870
| 710.870
| |  
|  
|-
|-
| |  
|  
| | 7/6
| 7/6
| | 711.043
| 711.043
| | 7-odd-limit minimax
| 7-odd-limit minimax
|-
|-
| | 32\54
| 32\54
| |  
|  
| | 711.111
| 711.111
| |  
|  
|-
|-
| |  
|  
| | 13/11
| 13/11
| | 711.151
| 711.151
| | 13-odd-limit minimax
| 13-odd-limit minimax
|-
|-
| | 83\140
| 83\140
| |  
|  
| | 711.429
| 711.429
| |  
|  
|-
|-
| | 51\86
| 51\86
| |  
|  
| | 711.628
| 711.628
| |  
|  
|-
|-
| |  
|  
| | 16/15
| 16/15
| | 711.731
| 711.731
| |  
|  
|-
|-
| | 70\118
| 70\118
| |  
|  
| | 711.864
| 711.864
| |  
|  
|-
|-
| | 19\32
| 19\32
| |  
|  
| | 712.500
| 712.500
| |  
|  
|-
|-
| | 44\74
| 44\74
| |  
|  
| | 713.5135
| 713.5135
| |  
|  
|-
|-
| |  
|  
| | 13/10
| 13/10
| | 713.553
| 713.553
| |  
|  
|-
|-
| | 25\42
| 25\42
| |  
|  
| | 714.286
| 714.286
| |  
|  
|-
|-
| | 31\52
| 31\52
| |  
|  
| | 715.385
| 715.385
| |  
|  
|-
|-
| |  
|  
| | 8/7
| 8/7
| | 715.587
| 715.587
| |  
|  
|-
|-
| | 6\10
| 6\10
| |  
|  
| | 720.000
| 720.000
| |  
|  
|}
|}


Retrieved from "https://en.xen.wiki/w/Pajara"