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'''Laka''' is the [[Rank-3 temperament|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.  
{{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♯).  


[[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.  
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.  


<blockquote>
See [[Aberschismic family #Laka]] for technical data.  
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.
</blockquote>
—[[Graham Breed]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101682.html#101776 Yahoo! Tuning Group | ''Laka 17-limit minimax planar temperament'']</ref>
 
It corresponds to the fact that [[41edo|41et]] and [[53edo|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]] [[17/1|17]], as [[19/1|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 ==
== Interval lattice ==
<gallery>
<gallery>
File:Lattice Laka.png|13-limit laka
File:Lattice Laka.png|13-limit laka
File:Lattice Laka19.png|2.3.5.7.11.13.19 subgroup laka
File:Lattice Laka19.png|2.3.5.7.11.13.19-subgroup laka
</gallery>
</gallery>


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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* [[Dekany laka]] – a transversal scale
* [[Dekany laka]] – a transversal scale


== Notes ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.5133{{c}}, ~5/4 = 385.5563{{c}}
| CWE: ~3/2 = 702.6175{{c}}, ~5/4 = 386.4170{{c}}
| POTE: ~3/2 = 702.6640{{c}}, ~5/4 = 386.8005{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.4078{{c}}, ~5/4 = 385.5405{{c}}
| CWE: ~3/2 = 702.5780{{c}}, ~5/4 = 386.7718{{c}}
| POTE: ~3/2 = 702.6464{{c}}, ~5/4 = 387.2662{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | No-17 19-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.4062{{c}}, ~5/4 = 385.5254{{c}}
| CWE: ~3/2 = 702.5613{{c}}, ~5/4 = 386.6230{{c}}
| POTE: ~3/2 = 702.6221{{c}}, ~5/4 = 387.0532{{c}}
|}


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