Pajara extensions: Difference between revisions

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=Tuning Spectra=
{{Breadcrumb|Pajara}}


==Spectrum of Pajara Tunings by Eigenmonzos==
The [[11-limit]] [[pajara]] temperament has various [[extension]]s to the [[13-limit]]. The following temperaments are discussed in this article:
* '''Tridecimal pajara''' ({{nowrap| 12 & 22 }}) – tempering out 50/49, 64/63, 65/63, 99/98;
* '''Pajarina''' ({{nowrap| 12f & 22 }}) – tempering out 50/49, 64/63, 78/77, 99/98;
* '''Pajarita''' ({{nowrap| 12f & 22f }}) – tempering out 40/39, 50/49, 64/63, 66/65;


[[Gencom|Gencom]]: [7/5 3/2; 50/49 64/63 65/63 99/98]
In addition, we also consider the 13-limit extensions of important alternative 11-limit mappings. These are:
* '''Tridecimal pajarous''' ({{nowrap| 10 & 22 }}) – tempering out 50/49, 55/54, 64/63, 65/63;  
* '''Pajaro''' ({{nowrap| 10 & 22f }}) – tempering out 40/39, 50/49, 55/54, 64/63.
* '''Pajaric''' ({{nowrap| 10 & 12f }}) – tempering out 40/39, 45/44, 50/49, 56/55.


[[Gencom_map|Gencom map]]: [<2 2 7 8 14 5|, <0 1 -2 -2 -6 2|]
== Tuning spectra ==
 
=== Pajara ===
{| class="wikitable"
{| class="wikitable center-all left-4"
|-
|-
! | Eigenmonzo
! Edo<br>generator
! | Fifth
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! Generator (¢)
! Comments
|-
|-
| | 4/3
|  
| | 701.955
| 4/3
| 701.955
|
|-
|-
| | 14/11
|  
| | 704.377
| 14/11
| 704.377
|
|-
|-
| | 10/9
|  
| | 704.399
| 10/9
| 704.399
|
|-
|-
| | 6/5
|  
| | 705.214 (5 limit minimax)
| 6/5
| 705.214
| 5 limit minimax
|-
|-
| | 3125/2592
|  
| | 705.447 (5 limit least squares)
| 3125/2592
| 705.447
| 5 limit least squares
|-
|-
| | 20\34
| 20\34
| | 705.882
|  
| 705.882
|
|-
|-
| | 11/9
|  
| | 706.574
| 11/9
| 706.574
|
|-
|-
| | 5/4
|  
| | 706.843
| 5/4
| 706.843
|
|-
|-
| | |0 -50 5 5 29&gt;
|  
| | 707.106 (11 limit least squares)
| {{vector|0 -50 5 5 29}}
| 707.106
| 11 limit least squares
|-
|-
| | 33\56
| 33\56
| | 707.143
|  
| 707.143
|
|-
|-
| | 12/11
|  
| | 707.234
| 12/11
| 707.234
|
|-
|-
| | 35/27
|  
| | 707.246 (9 limit least squares)
| 35/27
| 707.246
| 9 limit least squares
|-
|-
| | 15/11
|  
| | 707.390
| 15/11
| 707.390
|
|-
|-
| | 11/8
|  
| | 708.114
| 11/8
| 708.114
|
|-
|-
| | 11/10
|  
| | 708.749 (11 limit minimax)
| 11/10
| 708.749
| 11 limit minimax
|-
|-
| | 9/7
|  
| | 708.771 (9 limit minimax)
| 9/7
| 708.771
| 9 limit minimax
|-
|-
| | 71663616/52521875
|  
| | 708.814 (7 limit least squares)
| 71663616/52521875
| 708.814
| 7 limit least squares
|-
|-
| | |0 -49 9 9 37 -19&gt;
|  
| | 708.901 (13 limit least squares)
| {{vector|0 -49 9 9 37 -19}}
| 708.901
| 13 limit least squares
|-
|-
| | |0 -51 7 10 42 -22&gt;
|  
| | 709.038 (15 limit least squares)
| {{vector|0 -51 7 10 42 -22}}
| 709.038
| 15 limit least squares
|-
|-
| | 13\22
| 13\22
| | 709.091
|  
| 709.091
|
|-
|-
| | 7/6
|  
| | 711.043 (7 limit minimax)
| 7/6
| 711.043
| 7 limit minimax
|-
|-
| | 32\54
| 32\54
| | 711.111
|  
| 711.111
|
|-
|-
| | 13/11
|  
| | 711.151 (13 and 15 limit minimax)
| 13/11
| 711.151
| 13 and 15 limit minimax
|-
|-
| | 16/15
|  
| | 711.731
| 16/15
| 711.731
|
|-
|-
| | 19\32
| 19\32
| | 712.500
|  
| 712.500
|
|-
|-
| | 13/10
|  
| | 713.553
| 13/10
| 713.553
|
|-
|-
| | 8/7
|  
| | 715.587
| 8/7
| 715.587
|
|-
|-
| | 15/13
|  
| | 717.420
| 15/13
| 717.420
|
|-
|-
| | 14/13
|  
| | 717.925
| 14/13
| 717.925
|
|-
|-
| | 15/14
|  
| | 719.443
| 15/14
| 719.443
|
|-
|-
| | 16/13
|  
| | 720.264
| 16/13
| 720.264
|
|-
|-
| | 13/12
|  
| | 738.573
| 13/12
| 738.573
|
|}
|}


==Spectrum of Pajarous Tunings by Eigenmonzos==
=== Pajarous ===
 
{| class="wikitable center-all left-4"
[[Gencom|Gencom]]: [7/5 3/2; 50/49 55/54 64/63 65/63]
 
[[Gencom_map|Gencom map]]: [&lt;2 2 7 8 1 5|, &lt;0 1 -2 -2 5 2|]
 
{| class="wikitable"
|-
|-
! | Eigenmonzo
! Edo<br>generator
! | Fifth
! Eigenmonzo<br>(unchanged interval)
! Generator (¢)
! Comments
|-
|-
| | 4/3
|  
| | 701.955
| 4/3
| 701.955
|
|-
|-
| | 13/11
|  
| | 703.597
| 13/11
| 703.597
|
|-
|-
| | 10/9
|  
| | 704.399
| 10/9
| 704.399
|
|-
|-
| | 6/5
|  
| | 705.214 (5 limit minimax)
| 6/5
| 705.214
| 5 limit minimax
|-
|-
| | 3125/2592
|  
| | 705.447 (5 limit least squares)
| 3125/2592
| 705.447
| 5 limit least squares
|-
|-
| | 20\34
| 20\34
| | 705.882
|  
| 705.882
|
|-
|-
| | 5/4
|  
| | 706.843
| 5/4
| 706.843
|
|-
|-
| | 33\56
| 33\56
| | 707.143
|  
| 707.143
|
|-
|-
| | 35/27
|  
| | 707.246 (9 limit least squares)
| 35/27
| 707.246
| 9 limit least squares
|-
|-
| | 9/7
|  
| | 708.771 (9, 13 and 15 limit minimax)
| 9/7
| 708.771
| 9, 13 and 15 limit minimax
|-
|-
| | 71663616/52521875
|  
| | 708.814 (7 limit least squares)
| 71663616/52521875
| 708.814
| 7 limit least squares
|-
|-
| | 13\22
| 13\22
| | 709.091
|  
| 709.091
|
|-
|-
| | 11/10
|  
| | 709.286 (11 limit minimax)
| 11/10
| 709.286
| 11 limit minimax
|-
|-
| | |0 17 -16 -16 26&gt;
|  
| | 709.883 (11 limit least squares)
| {{vector|0 17 -16 -16 26}}
| 709.883
| 11 limit least squares
|-
|-
| | 11/8
|  
| | 710.264
| 11/8
| 710.264
|
|-
|-
| | 15/11
|  
| | 710.508
| 15/11
| 710.508
|
|-
|-
| | |0 16 -20 -20 29 8&gt;
|  
| | 710.665 (13 limit least squares)
| {{vector|0 16 -20 -20 29 8}}
| 710.665
| 13 limit least squares
|-
|-
| | |0 7 -29 -21 35 11&gt;
|  
| | 710.879 (15 limit least squares)
| {{vector|0 7 -29 -21 35 11}}
| 710.879
| 15 limit least squares
|-
|-
| | 7/6
|  
| | 711.043 (7 limit minimax)
| 7/6
| 711.043
| 7 limit minimax
|-
|-
| | 32\54
| 32\54
| | 711.111
|  
| 711.111
|
|-
|-
| | 16/15
|  
| | 711.731
| 16/15
| 711.731
|
|-
|-
| | 14/11
|  
| | 711.785
| 14/11
| 711.785
|
|-
|-
| | 12/11
|  
| | 712.341
| 12/11
| 712.341
|
|-
|-
| | 19\32
| 19\32
| | 712.500
|  
| 712.500
|
|-
|-
| | 13/10
|  
| | 713.553
| 13/10
| 713.553
|
|-
|-
| | 8/7
|  
| | 715.587
| 8/7
| 715.587
|
|-
|-
| | 11/9
|  
| | 715.803
| 11/9
| 715.803
|
|-
|-
| | 15/13
|  
| | 717.420
| 15/13
| 717.420
|
|-
|-
| | 14/13
|  
| | 717.925
| 14/13
| 717.925
|
|-
|-
| | 15/14
|  
| | 719.443
| 15/14
| 719.443
|
|-
|-
| | 16/13
|  
| | 720.264
| 16/13
| 720.264
|
|-
|-
| | 13/12
|  
| | 738.573
| 13/12
| 738.573
|
|}
|}


==Spectrum of Pajaro Tunings by Eigenmonzos==
=== Pajaro ===
 
{| class="wikitable center-all left-4"
[[Gencom|Gencom]]: [7/5 3/2; 40/39 50/49 55/54 64/63]
 
[[Gencom_map|Gencom map]]: [&lt;2 2 7 8 1 11|, &lt;0 1 -2 -2 5 -3|]
 
{| class="wikitable"
|-
|-
! | Eigenmonzo
! Edo<br>generator
! | Fifth
! Eigenmonzo<br>(unchanged interval)
! Generator (¢)
! Comments
|-
|-
| | 4/3
|  
| | 701.955
| 4/3
| 701.955
|
|-
|-
| | 10/9
|  
| | 704.399
| 10/9
| 704.399
|
|-
|-
| | 6/5
|  
| | 705.214 (5 limit minimax)
| 6/5
| 705.214
| 5 limit minimax
|-
|-
| | 3125/2592
|  
| | 705.447 (5 limit least squares)
| 3125/2592
| 705.447
| 5 limit least squares
|-
|-
| | 20\34
| 20\34
| | 705.882
|  
| 705.882
|
|-
|-
| | 5/4
|  
| | 706.843
| 5/4
| 706.843
|
|-
|-
| | 33\56
| 33\56
| | 707.143
|  
| 707.143
|
|-
|-
| | 35/27
|  
| | 707.246 (9 limit least squares)
| 35/27
| 707.246
| 9 limit least squares
|-
|-
| | 9/7
|  
| | 708.771 (9 limit minimax)
| 9/7
| 708.771
| 9 limit minimax
|-
|-
| | 71663616/52521875
|  
| | 708.814 (7 limit least squares)
| 71663616/52521875
| 708.814
| 7 limit least squares
|-
|-
| | 13\22
| 13\22
| | 709.091
|  
| 709.091
|
|-
|-
| | 11/10
|  
| | 709.286 (11 limit minimax)
| 11/10
| 709.286
| 11 limit minimax
|-
|-
| | |0 17 -16 -16 26&gt;
|  
| | 709.883 (11 limit least squares)
| {{vector|0 17 -16 -16 26}}
| 709.883
| 11 limit least squares
|-
|-
| | 11/8
|  
| | 710.264
| 11/8
| 710.264
|
|-
|-
| | 15/11
|  
| | 710.508
| 15/11
| 710.508
|
|-
|-
| | 7/6
|  
| | 711.043 (7 limit minimax)
| 7/6
| 711.043
| 7 limit minimax
|-
|-
| | 32\54
| 32\54
| | 711.111
|  
| 711.111
|
|-
|-
| | |0 31 -15 -15 34 -22&gt;
|  
| | 711.580 (13 limit least squares)
| {{vector|0 31 -15 -15 34 -22}}
| 711.580
| 13 limit least squares
|-
|-
| | |0 27 -19 -16 40 -24&gt;
|  
| | 711.631 (15 limit least squares)
| {{vector|0 27 -19 -16 40 -24}}
| 711.631
| 15 limit least squares
|-
|-
| | 16/15
|  
| | 711.731
| 16/15
| 711.731
|
|-
|-
| | 14/11
|  
| | 711.785
| 14/11
| 711.785
|
|-
|-
| | 12/11
|  
| | 712.341
| 12/11
| 712.341
|
|-
|-
| | 19\32
| 19\32
| | 712.500
|  
| 712.500
|
|-
|-
| | 18/13
|  
| | 712.676 (13 and 15 limit minimax)
| 18/13
| 712.676
| 13 and 15 limit minimax
|-
|-
| | 13/11
|  
| | 713.849
| 13/11
| 713.849
|
|-
|-
| | 13/12
|  
| | 715.357
| 13/12
| 715.357
|
|-
|-
| | 8/7
|  
| | 715.587
| 8/7
| 715.587
|
|-
|-
| | 11/9
|  
| | 715.803
| 11/9
| 715.803
|
|-
|-
| | 15/14
|  
| | 719.443
| 15/14
| 719.443
|
|-
|-
| | 16/13
|  
| | 719.824
| 16/13
| 719.824
|
|-
|-
| | 15/13
|  
| | 723.871
| 15/13
| 723.871
|
|-
|-
| | 14/13
|  
| | 728.298
| 14/13
| 728.298
|
|-
|-
| | 13/10
|  
| | 745.786
| 13/10
| 745.786
|
|}
|}


==Spectrum of Pajaric Tunings by Eigenmonzos==
=== Pajaric ===
 
{| class="wikitable center-all left-4"
[[Gencom|Gencom]]: [7/5 3/2; 40/39 45/44 50/49 56/55]
 
[[Gencom_map|Gencom map]]: [&lt;2 2 7 8 7 11|, &lt;0 1 -2 -2 0 -3|]
 
{| class="wikitable"
|-
|-
! | Eigenmonzo
! Edo<br>generator
! | Fifth
! Eigenmonzo<br>(unchanged interval)
! Generator (¢)
! Comments
|-
|-
| | 15/11
|  
| | 663.049
| 15/11
| 663.049
|
|-
|-
| | 11/10
|  
| | 682.502
| 11/10
| 682.502
|
|-
|-
| | 14/11
|  
| | 691.246
| 14/11
| 691.246
|
|-
|-
| | 4/3
|  
| | 701.955
| 4/3
| 701.955
|
|-
|-
| | 13/11
|  
| | 703.597
| 13/11
| 703.597
|
|-
|-
| | 10/9
|  
| | 704.399
| 10/9
| 704.399
|
|-
|-
| | 6/5
|  
| | 705.214 (5 limit minimax)
| 6/5
| 705.214
| 5 limit minimax
|-
|-
| | 3125/2592
|  
| | 705.447 (5 limit least squares)
| 3125/2592
| 705.447
| 5 limit least squares
|-
|-
| | 20\34
| 20\34
| | 705.882
|  
| 705.882
|
|-
|-
| | |0 -32 11 11 -1&gt;
|  
| | 706.676 (11 limit least squares)
| {{vector|0 -32 11 11 -1}}
| 706.676
| 11 limit least squares
|-
|-
| | 5/4
|  
| | 706.843 (11, 13 and 15 limit minimax)
| 5/4
| 706.843
| 11, 13 and 15 limit minimax
|-
|-
| | 33\56
| 33\56
| | 707.143
|  
| 707.143
|
|-
|-
| | 35/27
|  
| | 707.246 (9 limit least squares)
| 35/27
| 707.246
| 9 limit least squares
|-
|-
| | 9/7
|  
| | 708.771 (9 limit minimax)
| 9/7
| 708.771
| 9 limit minimax
|-
|-
| | 71663616/52521875
|  
| | 708.814 (7 limit least squares)
| 71663616/52521875
| 708.814
| 7 limit least squares
|-
|-
| | 13\22
| 13\22
| | 709.091
|  
| 709.091
|
|-
|-
| | |0 -46 10 10 -4 17&gt;
|  
| | 709.889 (13 limit least squares)
| {{vector|0 -46 10 10 -4 17}}
| 709.889
| 13 limit least squares
|-
|-
| | |0 -47 9 11 -5 19&gt;
|  
| | 710.032 (15 limit least squares)
| {{vector|0 -47 9 11 -5 19}}
| 710.032
| 15 limit least squares
|-
|-
| | 7/6
|  
| | 711.043 (7 limit minimax)
| 7/6
| 711.043
| 7 limit minimax
|-
|-
| | 32\54
| 32\54
| | 711.111
|  
| 711.111
|
|-
|-
| | 16/15
|  
| | 711.731
| 16/15
| 711.731
|
|-
|-
| | 19\32
| 19\32
| | 712.500
|  
| 712.500
|
|-
|-
| | 18/13
|  
| | 712.676
| 18/13
| 712.676
|
|-
|-
| | 13/12
|  
| | 715.357
| 13/12
| 715.357
|
|-
|-
| | 8/7
|  
| | 715.587
| 8/7
| 715.587
|
|-
|-
| | 15/14
|  
| | 719.443
| 15/14
| 719.443
|
|-
|-
| | 16/13
|  
| | 719.824
| 16/13
| 719.824
|
|-
|-
| | 15/13
|  
| | 723.871
| 15/13
| 723.871
|
|-
|-
| | 11/9
|  
| | 726.296
| 11/9
| 726.296
|
|-
|-
| | 14/13
|  
| | 728.298
| 14/13
| 728.298
|
|-
|-
| | 13/10
|  
| | 745.786
| 13/10
| 745.786
|
|-
|-
| | 12/11
|  
| | 750.637
| 12/11
| 750.637
|
|}
|}
[[Category:Pajara]]
[[Category:Temperament extensions]]
{{Todo| cleanup | expand }}

Latest revision as of 09:31, 25 December 2025

< Pajara

The 11-limit pajara temperament has various extensions to the 13-limit. The following temperaments are discussed in this article:

  • Tridecimal pajara (12 & 22) – tempering out 50/49, 64/63, 65/63, 99/98;
  • Pajarina (12f & 22) – tempering out 50/49, 64/63, 78/77, 99/98;
  • Pajarita (12f & 22f) – tempering out 40/39, 50/49, 64/63, 66/65;

In addition, we also consider the 13-limit extensions of important alternative 11-limit mappings. These are:

  • Tridecimal pajarous (10 & 22) – tempering out 50/49, 55/54, 64/63, 65/63;
  • Pajaro (10 & 22f) – tempering out 40/39, 50/49, 55/54, 64/63.
  • Pajaric (10 & 12f) – tempering out 40/39, 45/44, 50/49, 56/55.

Tuning spectra

Pajara

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
4/3 701.955
14/11 704.377
10/9 704.399
6/5 705.214 5 limit minimax
3125/2592 705.447 5 limit least squares
20\34 705.882
11/9 706.574
5/4 706.843
[0 -50 5 5 29⟩ 707.106 11 limit least squares
33\56 707.143
12/11 707.234
35/27 707.246 9 limit least squares
15/11 707.390
11/8 708.114
11/10 708.749 11 limit minimax
9/7 708.771 9 limit minimax
71663616/52521875 708.814 7 limit least squares
[0 -49 9 9 37 -19⟩ 708.901 13 limit least squares
[0 -51 7 10 42 -22⟩ 709.038 15 limit least squares
13\22 709.091
7/6 711.043 7 limit minimax
32\54 711.111
13/11 711.151 13 and 15 limit minimax
16/15 711.731
19\32 712.500
13/10 713.553
8/7 715.587
15/13 717.420
14/13 717.925
15/14 719.443
16/13 720.264
13/12 738.573

Pajarous

Edo
generator 		Eigenmonzo
(unchanged interval) 		Generator (¢) Comments
4/3 701.955
13/11 703.597
10/9 704.399
6/5 705.214 5 limit minimax
3125/2592 705.447 5 limit least squares
20\34 705.882
5/4 706.843
33\56 707.143
35/27 707.246 9 limit least squares
9/7 708.771 9, 13 and 15 limit minimax
71663616/52521875 708.814 7 limit least squares
13\22 709.091
11/10 709.286 11 limit minimax
[0 17 -16 -16 26⟩ 709.883 11 limit least squares
11/8 710.264
15/11 710.508
[0 16 -20 -20 29 8⟩ 710.665 13 limit least squares
[0 7 -29 -21 35 11⟩ 710.879 15 limit least squares
7/6 711.043 7 limit minimax
32\54 711.111
16/15 711.731
14/11 711.785
12/11 712.341
19\32 712.500
13/10 713.553
8/7 715.587
11/9 715.803
15/13 717.420
14/13 717.925
15/14 719.443
16/13 720.264
13/12 738.573

Pajaro

Edo
generator 		Eigenmonzo
(unchanged interval) 		Generator (¢) Comments
4/3 701.955
10/9 704.399
6/5 705.214 5 limit minimax
3125/2592 705.447 5 limit least squares
20\34 705.882
5/4 706.843
33\56 707.143
35/27 707.246 9 limit least squares
9/7 708.771 9 limit minimax
71663616/52521875 708.814 7 limit least squares
13\22 709.091
11/10 709.286 11 limit minimax
[0 17 -16 -16 26⟩ 709.883 11 limit least squares
11/8 710.264
15/11 710.508
7/6 711.043 7 limit minimax
32\54 711.111
[0 31 -15 -15 34 -22⟩ 711.580 13 limit least squares
[0 27 -19 -16 40 -24⟩ 711.631 15 limit least squares
16/15 711.731
14/11 711.785
12/11 712.341
19\32 712.500
18/13 712.676 13 and 15 limit minimax
13/11 713.849
13/12 715.357
8/7 715.587
11/9 715.803
15/14 719.443
16/13 719.824
15/13 723.871
14/13 728.298
13/10 745.786

Pajaric

Edo
generator 		Eigenmonzo
(unchanged interval) 		Generator (¢) Comments
15/11 663.049
11/10 682.502
14/11 691.246
4/3 701.955
13/11 703.597
10/9 704.399
6/5 705.214 5 limit minimax
3125/2592 705.447 5 limit least squares
20\34 705.882
[0 -32 11 11 -1⟩ 706.676 11 limit least squares
5/4 706.843 11, 13 and 15 limit minimax
33\56 707.143
35/27 707.246 9 limit least squares
9/7 708.771 9 limit minimax
71663616/52521875 708.814 7 limit least squares
13\22 709.091
[0 -46 10 10 -4 17⟩ 709.889 13 limit least squares
[0 -47 9 11 -5 19⟩ 710.032 15 limit least squares
7/6 711.043 7 limit minimax
32\54 711.111
16/15 711.731
19\32 712.500
18/13 712.676
13/12 715.357
8/7 715.587
15/14 719.443
16/13 719.824
15/13 723.871
11/9 726.296
14/13 728.298
13/10 745.786
12/11 750.637
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