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'''Laka''' is a [[rank-3 temperament|rank-3]] [[regular temperament|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]].  
{{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 = ?
}}
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]] 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.  
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 [[Hemifamity family #Laka]] for technical details.  
See [[Aberschismic family #Laka]] for technical data.  


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


== Chords ==
== Chords and harmony ==
Laka enables [[essentially tempered chord]]s of [[swetismic chords|swetismic]] in the [[11-odd-limit]], in addition to [[major minthmic chords|major minthmic]], [[huntmic chords|huntmic]], [[squbemic chords|squbemic]] and [[cuthbert chords|cuthbert]] in the [[13-odd-limit]].
Laka enables [[essentially tempered chord]]s of [[swetismic chords|swetismic]] in the [[11-odd-limit]], in addition to [[major minthmic chords|major minthmic]], [[huntmic chords|huntmic]], [[squbemic chords|squbemic]] and [[cuthbert chords|cuthbert]] in the [[13-odd-limit]].


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[[Category:Laka| ]] <!-- main article -->
[[Category:Laka| ]] <!-- main article -->
[[Category:Rank-3 temperaments]]
[[Category:Rank-3 temperaments]]
[[Category:Hemifamity family]]
[[Category:Aberschismic family]]
[[Category:Swetismic temperaments]]
[[Category:Swetismic temperaments]]

Latest revision as of 07:00, 26 July 2026

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
    13-limit laka
  • 2.3.5.7.11.13.19-subgroup 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

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 ¢
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