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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:Rank-3 temperaments]]
[[Category:Hemifamity family]]
[[Category:Hemifamity family]]
[[Category:Swetismic temperaments]]
[[Category:Swetismic temperaments]]

Latest revision as of 07:13, 22 October 2025

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
    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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