Lattice: Difference between revisions
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{{Wikipedia|Lattice (music)}} | {{Wikipedia|Lattice (music)}} | ||
A ''' | 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|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 | 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. | ||
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 == | == Examples == | ||
<gallery> | <gallery> | ||
File:Lattice Marvel.png|A square lattice representing [[marvel]] | 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: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]]. | File:Lattice32.png|A hexagonal lattice representing a subset of [[7-limit]] intervals, using [[color notation]]. | ||
Revision as of 17:01, 25 April 2025
A lattice is a geometric construction that organizes pitches (or 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 subgroups.
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 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.
Lattices can be extended to incorporate more dimensions. For instance, 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
-
A square lattice representing marvel temperament. -
A hexagonal lattice representing 41edo's mapping of 5-limit intervals, using ups and downs notation. -
A hexagonal lattice representing a subset of 7-limit intervals, using color notation. -
A torus of notes in 15edo, notated by porcupine notation with LH-NiceIonian as the base scale.
External links
- Beginner's guide to lattices: See chapter 1.3 of Alternative Tunings: Theory, Notation and Practice by Kite Giedraitis (2016)
- "lattice/lattice diagram" on Tonalsoft Encyclopedia
- "Octave Equivalent Music Lattices" by Graham Breed (2008)
- "Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7-) Harmonic Lattice of Notes" by Adriaan Fokker (1969)
- "Harmonic Lattice Diagrams" by Joseph L. Monzo (1998)