Laka: Difference between revisions

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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]]. The canonical [[extension]] to the [[13-limit]] tempers out [[352/351]], [[640/637]], [[729/728]] and [[847/845]], and to the no-17 [[19-limit]], [[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.  
 
[[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.
 
<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>


Revision as of 14:42, 19 April 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 tempers out 352/351, 640/637, 729/728 and 847/845, and to the no-17 19-limit, 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

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

Notes

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