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~~.

{{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 ==

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>

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]].

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* [[Dekany laka]] – a transversal scale

~~[[Category~~:~~Temperaments]]~~

== 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:Laka| ]]

[[Category:~~Hemifamity~~ family]]

[[Category:Rank-3 temperaments]]

[[Category:Aberschismic family]]

[[Category:Swetismic temperaments]]

@@ Line 1: / Line 1: @@
-'''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 ==
 <gallery>
 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>
 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]].
@@ Line 26: / Line 30: @@
 * [[Dekany laka]] – a transversal scale
-[[Category:Temperaments]]
+== 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:Laka| ]] <!-- main article -->
-[[Category:Hemifamity family]]
+[[Category:Rank-3 temperaments]]
+[[Category:Aberschismic family]]
 [[Category:Swetismic temperaments]]