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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.  
'''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]].  


[[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]] 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.  
 
<blockquote>
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.  
See [[Hemifamity family #Laka]] for technical details.  
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<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]]
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