Laka: Difference between revisions
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{{Infobox regtemp | |||
| Title = Laka | |||
| Subgroups = 2.3.5.7.11; 2.3.5.7.11.13 | |||
| Comma basis = [[540/539]], [[5120/5103]]; <br>[[352/351]], [[540/539]], [[729/728]] | |||
| Edo join 1 = 41 | Edo join 2 = 53 | Edo join 3 = 58 | |||
| Mapping = 1; 1 0 -6 15 12; 0 1 1 -1 -1 | |||
| Generators = 3/2; 5/4 | Generators tuning = 702.6; 386.8 | |||
| Optimization method = CWE | |||
| Odd limit 1 = 11 | Mistuning 1 = 1.74 | Complexity 1 = ? | |||
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 3.20 | Complexity 2 = ? | |||
}} | |||
'''Laka''' is a [[rank-3 temperament]] generated by a perfect fifth of [[~]][[3/2]] and a step for the [[81/80|syntonic]]~[[64/63|septimal comma]] to reach the interval classes of [[5/1|5]], [[7/1|7]], and higher [[prime harmonic|primes]]. Using an arrow to represent this comma step, we have [[5/4]] at the down major third (C–vE), [[7/4]] at the down minor seventh (C–vBb), and [[11/8]] at the up augmented third (C–^E#), [[tempering out]] [[540/539]], which makes it a member of [[swetismic temperaments]]. | '''Laka''' is a [[rank-3 temperament]] generated by a perfect fifth of [[~]][[3/2]] and a step for the [[81/80|syntonic]]~[[64/63|septimal comma]] to reach the interval classes of [[5/1|5]], [[7/1|7]], and higher [[prime harmonic|primes]]. Using an arrow to represent this comma step, we have [[5/4]] at the down major third (C–vE), [[7/4]] at the down minor seventh (C–vBb), and [[11/8]] at the up augmented third (C–^E#), [[tempering out]] [[540/539]], which makes it a member of [[swetismic temperaments]]. | ||
Revision as of 15:51, 24 July 2026
| Laka |
352/351, 540/539, 729/728
13-limit 21-odd-limit: 3.20 ¢
13-limit 21-odd-limit: ? notes
Laka is a rank-3 temperament generated by a perfect fifth of ~3/2 and a step for the syntonic~septimal comma to reach the interval classes of 5, 7, and higher primes. Using an arrow to represent this comma step, we have 5/4 at the down major third (C–vE), 7/4 at the down minor seventh (C–vBb), and 11/8 at the up augmented third (C–^E#), tempering out 540/539, which makes it a member of swetismic temperaments.
The canonical extension to the 13-limit finds 13/8 at the up augmented fifth (C–^G#), tempering out 352/351, 640/637, 729/728 and 847/845, and a no-17 19-limit extension is available by recognizing 19/16 at the 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 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 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
| 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 ¢ |
| 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 ¢ |
| 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 ¢ |