Diaschismic

Diaschismic, [10 & 12], is the temperament (usually 2.3.5.17) generated by a half-octave representing 45/32~17/12~24/17~64/45 and a semitone representing 135/128~18/17~17/16~16/15, which is tuned to around 105 cents. 3/2 is a period plus a generator; 5/4 is a period minus two generators. (Note that 3/2 can be taken as the generator instead of 17/16.) In other words, diaschismic equates a stack of two 5/4's and two 9/8's to the octave; equivalently, it equates a stack of one 5/4 and two 16/15's to the half-octave.

12edo and 22edo are notable Diaschismic tunings. Some tunings of Diaschismic that are more accurate in the 5-limit are 34edo and 46edo.

Extensions

Archytas (10 & 12) (Pajara)

Main article: Pajara

The edo join [10 & 12] given above represents the extension pajara, which tempers out 64/63 such that a sharpened generator of about 110 cents is also 15/14 and stacks twice to a flatly tuned 8/7. The semioctave is interpreted as 7/5 and 10/7. This is a particularly common practical 7-limit interpretation of diaschismic supported by most of the small diaschismic edos, and especially 22edo.

Further extensions of Pajara are discussed on its own page.

Aberschismic (46 & 58)

A more accurate but complex option is to interpret diaschismic as a hemifamity temperament. Here, the comma 81/80 represented by six generators is also 64/63 and half of 50/49, so that 8/7 is a comma above 9/8 (being itself a comma above 10/9)..

In the 11-limit, an obvious mapping is to treat the neogothic major third 81/64 as 14/11, as in hemifamity; this sets 11/8 to be offset from the tritone by two commas. As a result, 11/9 is a comma sharp of 6/5. 16/13 is a further comma sharp.

Alternatively, another mapping offsets 11 from the fourth by two commas instead, finding 13 at 8 generators up and setting the tritone to 91/64.

Only the mappings for the 2.3.5.7.17 group will be given here.

Generator chain

Period 1					Period 2
Up			Down		Up			Down
#	Cents	JI	Cents	JI	#	Cents	JI	Cents	JI
0	0	1/1	600	24/17, 64/45	0	600	17/12, 45/32	1200	2/1
1	104	18/17, 17/16, 16/15	496	4/3	1	704	3/2	1096	15/8, 32/17, 17/9
2	208	9/8	392	5/4	2	808	8/5	992	16/9
3	312	6/5	288	32/27	3	912	27/16	888	5/3
4	416	81/64	184	10/9	4	1016	9/5	784	128/81
5	520	27/20	80	21/20, 25/24	5	1120	40/21, 48/25	680	40/27
6	24	81/80, 64/63	576	7/5	6	624	10/7	1176	63/32, 160/81
7	128	27/25, 15/14	472	21/16	7	728	32/21	1072	50/27, 28/15
8	232	8/7	368	21/17	8	832	34/21	968	7/4
9	336	17/14	264	7/6	9	936	12/7	864	28/17
10	440	9/7	160	35/32	10	1040	51/28	760	14/9

List of patent vals

The following patent vals support 2.3.5 Diaschismic, not including vals contorted in 2.3.5.

Edo	Extension	Generator (3/2)
2		600.000
14		685.714
12	Pajara, Septimal diaschismic	700.000
70	Septimal diaschismic	702.857
58	Septimal diaschismic	703.448
46	Septimal diaschismic (alternate 11 mapping)	704.348
126	(alternate 11 mapping)	704.762
80	(alternate 11 mapping)	705.000
114		705.263
148		705.405
34	Septimal diaschismic (alternate 11 mapping)	705.882
124		706.452
90		706.667
56		707.143
78		707.692
22	Pajara	709.091
54	Pajara	711.111
32	Pajara	712.500
10	Pajara	720.000