Laka: Difference between revisions
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| Odd limit 2 = 13-limit 21 | Mistuning 2 = 3.20 | Complexity 2 = ? | | Odd limit 2 = 13-limit 21 | Mistuning 2 = 3.20 | Complexity 2 = ? | ||
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''' | The '''laka''' [[rank-3 temperament]] is one of the [[11-limit]] [[extension]]s of [[aberschismic]], inheriting the [[chain of fifths|chain-of-fifths-based]] structure and the generic comma step that results from equating the [[81/80|syntonic]] and [[64/63|septimal commas]]. It [[tempering out|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]] | 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 [[ | See [[Aberschismic family #Laka]] for technical data. | ||
== Interval lattice == | == Interval lattice == | ||
Latest revision as of 07:00, 26 July 2026
| Laka |
352/351, 540/539, 729/728
13-limit 21-odd-limit: 3.20 ¢
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
| 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 ¢ |