Laka

Laka
Subgroups	2.3.5.7.11; 2.3.5.7.11.13
Comma basis	540/539, 5120/5103; ; 352/351, 540/539, 729/728
Reduced mapping	⟨1; 1 0 -6 15 12; 0 1 1 -1 -1]
ET join	41 & 53 & 58
Generators (CWE)	~3/2 = 702.6 ¢, ~5/4 = 386.8 ¢
MOS scales	n/a
Ploidacot	n/a
Minimax error	11-odd-limit: 1.74 ¢; ; 13-limit 21-odd-limit: 3.20 ¢
Target scale size	11-odd-limit: ? notes; ; 13-limit 21-odd-limit: ? notes

The laka rank-3 temperament is one of the 11-limit extensions of aberschismic, inheriting the chain-of-fifths-based structure and the generic comma step that results from equating the syntonic and septimal commas. It tempers out 540/539, which makes it a member of swetismic temperaments, and as a consequence, 11/8 is mapped to the comma-up augmented third (C–^E♯).

The canonical extension to the 13-limit maps 13/11 to the diatonic minor third, which is the exact mean of 6/5 and 7/6, and implies 13/8 is the comma-up augmented fifth (C–^G♯) and that 352/351, 640/637, 729/728 and 847/845 are tempered out. Additionally, a no-17 19-limit extension is available by recognizing 19/16 at the comma-down augmented second (C–vD♯), tempering out 400/399, 456/455 and 495/494. The lattice structure is very comparable to that of pele, but it is more complex as many of the simple divisive ratios are further away from the origin.

See Aberschismic family #Laka for technical data.

Interval lattice

13-limit laka
2.3.5.7.11.13.19-subgroup laka

These lattices show laka as generated by ~2, ~3/2, and ~7/4 for a direct comparison with pele.

Chords and harmony

Laka enables essentially tempered chords of swetismic in the 11-odd-limit, in addition to major minthmic, huntmic, squbemic and cuthbert in the 13-odd-limit.

Scales

Dekany laka – a transversal scale

Tunings

11-limit norm-based tunings
	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 702.5133 ¢, ~5/4 = 385.5563 ¢	CWE: ~3/2 = 702.6175 ¢, ~5/4 = 386.4170 ¢	POTE: ~3/2 = 702.6640 ¢, ~5/4 = 386.8005 ¢

13-limit norm-based tunings
	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 702.4078 ¢, ~5/4 = 385.5405 ¢	CWE: ~3/2 = 702.5780 ¢, ~5/4 = 386.7718 ¢	POTE: ~3/2 = 702.6464 ¢, ~5/4 = 387.2662 ¢

No-17 19-limit norm-based tunings
	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 702.4062 ¢, ~5/4 = 385.5254 ¢	CWE: ~3/2 = 702.5613 ¢, ~5/4 = 386.6230 ¢	POTE: ~3/2 = 702.6221 ¢, ~5/4 = 387.0532 ¢