Laka: Difference between revisions

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'''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.

'''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]] 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.

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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

== ~~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| ]]

[[Category:Rank-3 temperaments]]

[[Category:Hemifamity family]]

[[Category:Swetismic temperaments]]

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-'''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.
+'''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]] 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.
@@ Line 11: / Line 13: @@
 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 17: / Line 19: @@
 * [[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:Rank-3 temperaments]]
 [[Category:Hemifamity family]]
 [[Category:Swetismic temperaments]]