Pajara extensions

< 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