Lattice: Difference between revisions

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A '''~~harmonic~~ lattice ~~diagram~~''' (~~'''lattice''') is an instrument to visualize relations of tones, which projects the multidimensional~~ [[~~just intonation~~|JI]] ~~interval space onto~~ a ~~plane in such a way that intervals~~ that differ by ~~one particular JI relation will always~~ appear at fixed relative positions to each other. ~~Most commonly, the dimensionality of JI [[subgroup]]s is reduced~~ by ~~one by means of [[octave equivalence]]~~, ~~so that intervals an octave apart are mapped to the same point~~.

A '''lattice''' is a geometric construction that organizes pitches (or [[pitch class|pitch classes]]) according to their intervallic relationships.

In a lattice, pitches are represented by points, and tones that differ by a specific interval consistently appear in fixed relative positions to each other.

These points can be connected by lines to highligh specific consonances.

This creates a structure where pitch relationships can be analyzed through their geometric arrangement, which can be visualized in a lattice diagram.

Most lattice diagrams reduce the dimensionality by one through [[octave equivalence]], meaning intervals separated by octaves are treated as equivalent points.

This reduction makes a lattice diagram more comprehensible when projected onto a two-dimensional plane.

While lattices are often used to represent [[just intonation]], they can also be applied to tempered spaces, in particular [[rank-3]] systems generated by two distinct intervals (aside from the [[period]]) and which thus function equivalently to three-prime [[JI subgroup]]s.

== Tonnetz ==

~~{{Wikipedia|Vogel's Tonnetz}}~~

The Tonnetz is a prominent example of a lattice representing [[5-limit]] intervals. In its conventional hexagonal arrangement, the perfect fifth (3/2) and major third (5/4) intervals are positioned 60 degrees apart, with the major sixth ([[5/3]]) placed 60 degrees from the major third and 120 degrees from the perfect fifth. This arrangement results in [[5-limit]] major triads ([[4:5:6]]) and minor triads ([[10:12:15]]) forming triangular patterns within the lattice structure.

The ~~'''~~Tonnetz~~'''~~ is ~~the lattice that represents~~ a ~~tuning system's mapping~~ of [[5-limit]] intervals, ~~conventionally arranged in a hexagonal fashion such that [[~~5/4~~]] and [[3/2]]~~ are 60 degrees apart, with [[5/3]] ~~being~~ 60 degrees from ~~5/4~~ and 120 degrees from ~~3/2, so that~~ the 5-limit [[4:5:6~~|major (4:5:6)~~]] and [[10:12:15~~|minor (10:12:15)~~]] ~~chords form triangles~~. ~~Such lattices~~ can ~~also~~ be ~~extended~~ to other subgroups with 3 primes ~~in them~~, where a respective fundamental chord such as ~~[[4~~:5:~~7]] or [[~~8~~:11:14]]~~ can be assigned to a triangle~~. '''Vogel's Tonnetz''' is a 7-limit extension of Euler's 5-limit Tonnetz~~.

The Tonnetz can be generalized to other subgroups with 3 primes, such as [[2.3.7]], where a respective fundamental chord such as 6:7:8 can be assigned to a triangle.

~~It is also possible~~ to ~~create lattices for [[regular temperament~~|~~tempered]] systems, in particular~~ [[~~rank-3 temperament|rank~~-3]] ~~systems generated by two distinct intervals (aside from~~ the [[~~equave~~]] ~~or fraction thereof) and which thus function equivalently to three~~-~~prime JI~~ subgroups.

Lattices can be extended to incorporate more dimensions. For instance, {{w|Vogel's Tonnetz}} extends Euler's [[5-limit]] Tonnetz to the [[7-limit]], resulting in a 3-dimensional diagram. Similarly, other lattices can be constructed for different subgroups or temperaments.

== Examples ==

File:Lattice Marvel.png|A square lattice representing [[marvel]] ~~temperament~~.

File:Lattice Marvel.png|A square lattice representing [[marvel]] temperament.

File:41equal lattice 5-limit.png|A hexagonal lattice representing [[41edo]]'s mapping of [[5-limit]] intervals, using [[ups and downs notation]].

File:Lattice32.png|A hexagonal lattice representing a subset of [[7-limit]] intervals, using [[color notation]].

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 {{Wikipedia|Lattice (music)}}
-A '''harmonic lattice diagram''' ('''lattice''') is an instrument to visualize relations of tones, which projects the multidimensional [[just intonation|JI]] interval space onto a plane in such a way that intervals that differ by one particular JI relation will always appear at fixed relative positions to each other. Most commonly, the dimensionality of JI [[subgroup]]s is reduced by one by means of [[octave equivalence]], so that intervals an octave apart are mapped to the same point.
+A '''lattice''' is a geometric construction that organizes pitches (or [[pitch class|pitch classes]]) according to their intervallic relationships.
+In a lattice, pitches are represented by points, and tones that differ by a specific interval consistently appear in fixed relative positions to each other.
+These points can be connected by lines to highligh specific consonances.
+This creates a structure where pitch relationships can be analyzed through their geometric arrangement, which can be visualized in a lattice diagram.
+Most lattice diagrams reduce the dimensionality by one through [[octave equivalence]], meaning intervals separated by octaves are treated as equivalent points.
+This reduction makes a lattice diagram more comprehensible when projected onto a two-dimensional plane.
+While lattices are often used to represent [[just intonation]], they can also be applied to tempered spaces, in particular [[rank-3]] systems generated by two distinct intervals (aside from the [[period]]) and which thus function equivalently to three-prime [[JI subgroup]]s.
+== Tonnetz ==
 {{Wikipedia|Tonnetz}}
-{{Wikipedia|Vogel's Tonnetz}}
+The Tonnetz is a prominent example of a lattice representing [[5-limit]] intervals. In its conventional hexagonal arrangement, the perfect fifth (3/2) and major third (5/4) intervals are positioned 60 degrees apart, with the major sixth ([[5/3]]) placed 60 degrees from the major third and 120 degrees from the perfect fifth. This arrangement results in [[5-limit]] major triads ([[4:5:6]]) and minor triads ([[10:12:15]]) forming triangular patterns within the lattice structure.
-The '''Tonnetz''' is the lattice that represents a tuning system's mapping of [[5-limit]] intervals, conventionally arranged in a hexagonal fashion such that [[5/4]] and [[3/2]] are 60 degrees apart, with [[5/3]] being 60 degrees from 5/4 and 120 degrees from 3/2, so that the 5-limit [[4:5:6|major (4:5:6)]] and [[10:12:15|minor (10:12:15)]] chords form triangles. Such lattices can also be extended to other subgroups with 3 primes in them, where a respective fundamental chord such as [[4:5:7]] or [[8:11:14]] can be assigned to a triangle. '''Vogel's Tonnetz''' is a 7-limit extension of Euler's 5-limit Tonnetz.
+The Tonnetz  can be generalized to other subgroups with 3 primes, such as [[2.3.7]], where a respective fundamental chord such as 6:7:8 can be assigned to a triangle.
-It is also possible to create lattices for [[regular temperament|tempered]] systems, in particular [[rank-3 temperament|rank-3]] systems generated by two distinct intervals (aside from the [[equave]] or fraction thereof) and which thus function equivalently to three-prime JI subgroups.
+Lattices can be extended to incorporate more dimensions. For instance, {{w|Vogel's Tonnetz}} extends Euler's [[5-limit]] Tonnetz to the [[7-limit]], resulting in a 3-dimensional diagram. Similarly, other lattices can be constructed for different subgroups or temperaments.
 == Examples ==
 <gallery>
-File:Lattice Marvel.png|A square lattice representing [[marvel]] temperament.
+File:Lattice Marvel.png|A square lattice representing [[marvel]] temperament.
 File:41equal lattice 5-limit.png|A hexagonal lattice representing [[41edo]]'s mapping of [[5-limit]] intervals, using [[ups and downs notation]].
 File:Lattice32.png|A hexagonal lattice representing a subset of [[7-limit]] intervals, using [[color notation]].