Pajara

Pajara, 22 & 32, is a regular temperament wherein the octave is split into two tritone periods, and the generator is a fifth (3/2). A fifth minus a tritone is 16/15 (Diaschismic tempering), and therefore the 5/4 major third is found two generators below the tritone. Pajara makes the further equivalence of 5/4 plus a period to 7/4 (Jubilismic tempering) and therefore twice 4/3 is 7/4 (Archy tempering). The result is a 10-form system generated by a fifth tuned somewhere around 710 cents. There are five patent tunings of Pajara: 12, 22, 54, 32, and 10 (which is also the 20edo val for the 7-limit).

Extensions

There are two main extensions of Pajara to the 11-limit: pajarous (10 & 22) and a temperament called "undecimal pajara" (12 & 22, which is supported by only those two patent vals) which is pending a rename. Undecimal pajara is best flat of 22edo; pajarous is best sharp of 22edo.

Tuning considerations

Most optimization methods place the optimal tuning of Pajara's perfect fifth at around 707 cents (hence the focus on 12 & 22 as an extension). However, Pajara is usually not the best interpretation of those structures. EDOs with tunings of fifths flat of that of 22edo do not support Pajara in the patent val (except for 12edo), and 12edo alongside rank-2 Pajara structures with that tuning are generally extremely inaccurate in the 7-limit due to the fact that Archy temperament forces 9/8 and 8/7 together.

Conversely, systems with fifths sharp of 32edo narrow the distinction between 5/4 and 6/5 considerably, detuning 6/5 to a neutral sound. Therefore, the tuning range of Pajara can be considered to lie between about 709 to 712.5 cents.

Generator chain

The following chart assumes pajarous.

Period 1					Period 2
Up			Down		Up			Down
#	Cents	JI	Cents	JI	#	Cents	JI	Cents	JI
0	0	1/1	600	7/5	0	600	12/7	1200	2/1
1	110	18/17, 17/16, 16/15, 15/14	490	4/3	1	710	3/2	1090	28/15, 15/8, 32/17, 17/9
2	220	8/7, 9/8	380	5/4	2	820	8/5	980	7/4, 16/9
3	330	6/5, 11/9	270	7/6	3	930	12/7	870	5/3, 18/11
4	440	9/7	160	12/11, 10/9	4	1040	9/5, 11/6	760	14/9
5	550	11/8	50	25/24, 49/48	5	1150	96/49, 48/25	650	16/11

List of patent vals

Due to being a weak extension of Archy, the patent vals of Pajara are a subset of the Archy edos. Specifically, they must be even (and therefore, some edos are doubled compared to the Archy table).

EDO	Extension	Generator tuning	7/4 tuning	25/24 tuning
10	Pajarous	480c	960c	0c
32	Pajarous	487.5c	975c	37.5c
54	Pajarous	488.9c	977.8c	44.4c
22	Pajarous, Undecimal pajara	490.9c	981.8c	54.5c
12	Undecimal pajara	500c	1000c	100c