Laka
Laka is the rank-3 temperament tempering out 540/539 and 5120/5103, with the canonical extension to the 13-limit tempering out 352/351, 640/637, 729/728 and 847/845. 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.
Gene Ward Smith considered laka to be a 17-limit temperament, assigning †442/441 (41g & 53 & 58) as the main extension. It should be noted that 41 & 53g & 58 also makes for a possible extension.
It's the way the numbers fall. The Laka geometry happens to work reasonably well in the 13-limit but not so well in the 17-limit. There isn't one obvious 17-limit extension and none of them are competitive with other 17-limit temperaments.
It corresponds to the fact that 41et and 53et tune the 13-limit quite well but fail at the 17-limit. As such, laka makes the most sense as a 2.3.5.7.11.13.19 subgroup temperament, omitting harmonic 17, as 19 is easily available in a 24-tone scale, shown in the lattice below. This again is related to the fact that 41et and 53et are good in the said subgroup.
See Hemifamity family #Laka for technical details.
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
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