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.