Kleismic

Kleismic

Subgroups	2.3.5, 2.3.5.13
Reduced mapping	⟨1; 6 5 14]
ET join	15 & 19
Generators (CWE)	~6/5 = 317.1¢
MOS scales	3L 1s, 4L 3s, 4L 7s, 4L 11s, 15L 4s
Ploidacot	alpha-hexacot
Comma basis	15625/15552 (2.3.5); 325/324, 625/624 (2.3.5.13)
Minimax error	5-odd-limit: 1.35¢; 2.3.5.13 15-odd-limit: 2.35¢
Target scale size	5-odd-limit: 7 notes; 2.3.5.13 15-odd-limit: 15 notes

Kleismic, [15 & 19], is a high-accuracy temperament (usually seen in its basic form as a 2.3.5 temperament) that equates a stack of six 6/5 minor thirds to one 3/1. Via a structurally induced extension to 2.3.5.13, it equates three 6/5's to one semitwelfth 26/15 and equates 25/24 to 26/25 and 27/26.

Kleismic harmony is naturally based on splitting 5/3 into 4/3 and 5/4 thus making 3:4:5 or 12:15:18 triads. Indeed, Kleismic[15] (4L11s) can be constructed from the generator sequence GS(3:4:5)[19] by tempering out four kleismas.

Kleismic, despite generating a heptatonic scale, is not particularly usefully a 7-form temperament; this is because 3/2 is an imperfect sixth rather than a perfect fifth. In the 11-form, however, it is much better as 5/4 and 4/3 are mapped to the same degree, resulting in a dichotomy of [0 4 8]/11 triads similar to the [0 2 4]/7 ones found in fifth-centric temperaments.

Interval chain

In the following table, odd harmonics 1–15 are labeled in bold.

#	Cents*	Approximate ratios
0	0.0	1/1
1	317.1	6/5
2	634.2	13/9, 36/25
3	951.3	26/15
4	68.4	25/24, 26/25, 27/26
5	385.5	5/4
6	702.6	3/2
7	1019.6	9/5
8	136.7	13/12, 27/25
9	453.8	13/10
10	770.9	25/16, 39/25
11	1088.0	15/8
12	205.1	9/8
13	522.2	27/20
14	839.3	13/8
15	1156.4	39/20
16	273.5	75/64
17	590.6	45/32
18	907.7	27/16
19	24.7	65/64, 81/80

* In 2.3.5.13-subgroup CWE tuning, octave reduced

List of patent vals

The following patent vals support 2.3.5.13 Kleismic. Contorted vals are not included.

Edo	Generator tuning	Fifth tuning
15	320.000	720.000
34	317.647	705.882
155	317.419	704.516
121	317.355	704.132
208	317.308	703.846
295	317.288	703.729
87	317.241	703.448
401	317.207	703.242
314	317.197	703.185
227	317.181	703.084
367	317.166	702.997
507	317.160	702.959
140	317.143	702.857
613	317.129	702.773
473	317.125	702.748
333	317.117	702.703
526	317.110	702.662
193	317.098	702.591
439	317.084	702.506
246	317.073	702.439
299	317.057	702.341
53	316.981	701.887
72	316.667	700.000
19	315.789	694.737