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