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== 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. It is best tuned flat of 22edo. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but 22edo is the only [[11-odd-limit]] [[diamond monotone]] tuning, where primes [[3/1|3]] and [[5/1|5]] are less accurate than in optimal tunings of canonical 11-limit pajara.
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. It is best tuned flat of 22edo, with the optimum at around 707-708 cents. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but it equates [[12/11]] with [[10/9]], and the only tuning equating [[11/10]] with both is 22edo.


In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.  
In the following tables, odd harmonics 1–21 and their inverses are in '''bold'''.  


{| class="wikitable center-1 right-2 right-4"
{| class="wikitable center-1 right-2 right-4"
Line 48: Line 48:
| '''1/1'''
| '''1/1'''
| 600.0
| 600.0
| 7/5, 10/7
| 7/5, 10/7, 17/12, 24/17
|-
|-
| 1
| 1
| 707.2
| 707.4
| '''3/2'''
| '''3/2''', '''32/21'''
| 107.2
| 107.4
| 15/14, 16/15, 21/20
| 15/14, '''16/15''', '''17/16''',<br>18/17, 21/20
|-
|-
| 2
| 2
| 214.4
| 214.7
| '''8/7''', '''9/8'''
| '''8/7''', '''9/8''', 17/15
| 814.4
| 814.7
| '''8/5'''
| '''8/5''', 34/21
|-
|-
| 3
| 3
| 921.5
| 922.1
| 12/7
| 12/7, 17/10
| 321.5
| 322.1
| 6/5
| 6/5, 17/14
|-
|-
| 4
| 4
| 428.7
| 429.5
| 9/7, 14/11
| 9/7, 14/11
| 1028.7
| 1029.5
| 9/5, 20/11
| 9/5, 20/11
|-
|-
| 5
| 5
| 1135.9
| 1136.9
| 21/11, 27/14, 48/25, <br>64/33, 96/49
| 21/11, 27/14, 48/25, <br>64/33, 96/49
| 535.9
| 536.9
| 15/11, 27/20
| 15/11, 27/20
|-
|-
| 6
| 6
| 643.1
| 644.2
| '''16/11'''
| '''16/11''', 36/25, 72/49
| 43.1
| 44.2
| 45/44, 56/55, 81/80
| 45/44, 56/55, 81/80
|}
|}
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| '''1/1'''
| '''1/1'''
| 600.0
| 600.0
| 7/5, 10/7
| 7/5, 10/7, 17/12, 24/17
|-
|-
| 1
| 1
| 709.6
| 709.5
| '''3/2'''
| '''3/2''', '''32/21'''
| 109.6
| 109.5
| 15/14, 16/15, 21/20
| 15/14, '''16/15''', '''17/16''',<br>18/17, 21/20
|-
|-
| 2
| 2
| 219.1
| 219.1
| '''8/7''', '''9/8'''
| '''8/7''', '''9/8''', 17/15
| 819.1
| 819.1
| '''8/5'''
| '''8/5''', 34/21
|-
|-
| 3
| 3
| 928.7
| 928.6
| 12/7
| 12/7, 17/10
| 328.7
| 328.6
| 6/5, 11/9
| 6/5, 11/9, 17/14
|-
|-
| 4
| 4
| 438.2
| 438.2
| 9/7
| 9/7, 22/17
| 1038.2
| 1038.2
| 9/5, 11/6
| 9/5, 11/6
|-
|-
| 5
| 5
| 1147.8
| 1147.7
| 27/14, 48/25, 55/28, <br>88/45, 96/49
| 27/14, 48/25, 55/28, <br>88/45, 96/49
| 547.8
| 547.7
| '''11/8''', 27/20
| '''11/8''', 27/20
|-
|-
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| 22/21, 33/32, 81/80
| 22/21, 33/32, 81/80
|}
|}
<nowiki/>* In 11-limit CWE tuning, octave-reduced
<nowiki/>* In 2.3.5.7.11.17-subgroup CWE tuning, octave-reduced
 
== Chords and harmony ==
{{See also| Chords of pajara }}
 
In pajara, a decatonic system of interval classification based on the [[2L 8s]] (jaric) [[mos scale]] is preferred over the [[diatonic]] interval classification system traditionally used in western music, which is used in [[meantone]]. If we count scale degrees similarly to diatonic, then [[2/1]] is a "hendecave" (11ve), as there are 10 scale degrees, and we repeat at 2/1 at the 11th. In this system, [[3/2]] is a perfect 7th, and [[4/3]] is a perfect 5th. The intervals [[5/4]] and [[6/5]] are major and minor decatonic 4ths respectively, rather than being major and minor 3rds by diatonic interval classification in meantone. Importantly, [[7/4]] is now a major decatonic 9th, with [[12/7]] being its minor counterpart. This is in contrast to diatonic, where 7/4 is considered a subminor 7th, and 12/7 a supermajor 6th.
 
By decatonic interval classification, the [[4:5:6:7]] tetrad is written as P1–M4–P7–M9. It can be considered the ''major tetrad'', since the non-perfect intervals, those being the decatonic 4th and 9th, are both major intervals. If we instead use a minor interval for the 4th and 9th; that is, a P1–m4–P7–m9 chord, then we get a tetrad approximating [[70:84:105:120|1/(12:10:8:7)]], which can be considered the ''minor tetrad''.
 
{{Todo|complete section}}


== Scales ==
== Scales ==
Line 153: Line 162:


=== Scala files ===
=== Scala files ===
* [[Pajara12]]
* [[12-22h]]
* [[12-22h]]


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| POTE: ~3/2 = 707.0477{{c}}
| POTE: ~3/2 = 707.0477{{c}}
|}
|}
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
Line 189: Line 198:
| CWE: ~3/2 = 707.1826{{c}}
| CWE: ~3/2 = 707.1826{{c}}
| POTE: ~3/2 = 706.8851{{c}}
| POTE: ~3/2 = 706.8851{{c}}
|}
=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 7-odd-limit
| ~3/2 = 709.363{{c}}
| 35/24
|-
| 9-odd-limit
| ~3/2 = 708.128{{c}}
| 35/18
|-
| 11-odd-limit
| ~3/2 = 708.128{{c}}
| 35/18
|}
|}


Line 237: Line 269:
| 5/3
| 5/3
| 705.214
| 705.214
| 5- and 15-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 20\34
| 20\34
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| 11/8
| 11/8
| 708.114
| 708.114
|  
| 11- and 15-odd-limit minimax
|-
|
|36/35
|708.128
|9-odd-limit minimax
|-
|-
|  
|  
| 11/10
| 11/10
| 708.749
| 708.749
| 11-odd-limit minimax
|
|-
|-
|  
|  
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| 709.091
| 709.091
| Upper bound of 11-odd-limit diamond monotone
| Upper bound of 11-odd-limit diamond monotone
|-
|
|48/35
|709.363
|7-odd-limit minimax
|-
|-
|  
|  
| 7/6
| 7/6
| 711.043
| 711.043
| 7-odd-limit minimax
|
|-
|-
| 32\54
| 32\54
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