Lattice: Difference between revisions

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== 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 a prominent example of a lattice representing [[5-limit]] intervals. In its conventional hexagonal arrangement, the [[3/2|perfect fifth]] (3/2) and [[5/4|major third]] (5/4) intervals are positioned 60 degrees apart, with the [[5/3|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.

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== Higher dimensions ==

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~~.

Lattices naturally extend into higher dimensions as more prime factors are incorporated into the tuning system. After applying octave reduction, a 5-limit system requires two dimensions, a 7-limit system requires three dimensions, and each additional prime factor adds another dimension to the full representation.

{{w|Vogel's Tonnetz}} extends Euler's 5-limit Tonnetz to the 7-limit, resulting in a three-dimensional diagram.

This additional dimension allows for the representation of relationships involving the prime 7, adding intervals like the [[harmonic seventh]] (7/4) and the [[septimal minor third]] (7/6) alongside the 5-limit intervals.

Higher-dimensional lattices have been used by composers like [[Ben Johnston]], and [[Kyle Gann]] to organize their harmonic materials.

The lattice structure can guide everything from chord progressions to large-scale form, with paths through the lattice corresponding to harmonic progressions.

[[File:Lattice_7lim.png|300px|thumb|none|A 3-dimensional lattice for the [[7-limit]]. Made using [[Scale Workshop]].]]

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Isomorphic keyboards are a practical application of lattices in instrument design.

An isomorphic keyboard is an instrument with a two-dimensional grid of buttons or keys, arranged so that any given sequence or combination of musical intervals maintains the "same shape" regardless of where it occurs.

An isomorphic keyboard is an instrument with a two-dimensional grid of buttons or keys, arranged so that any given sequence or combination of musical intervals maintains the same shape, regardless of where it occurs.

This corresponds directly to a lattice: the keyboard layout is essentially a lattice mapped to physical space.

This mapping creates consistent fingering patterns for musicians, where a particular chord shape or scale pattern can be transposed by simply moving the same finger configuration to a different position on the keyboard.

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Examples of isomorphic keyboard designs include the {{w|Jankó keyboard}}, the {{w|Wicki–Hayden note layout}} and digital controllers like the [[Lumatone]] and [[Linnstrument]].

[[File:Lattice_bosanquet.png|300px|thumb|none|An isomorphic meantone layout (Bosanquet).]]

[[File:Lattice_bosanquet.png|300px|thumb|none|An isomorphic [[meantone]] layout (Bosanquet).]]

== In mathematics ==

The lattices used in music theory correspond to mathematical structures of the same name.<ref group="note">Not to be confused with a different structure in order theory, which is also called a {{w|Lattice (order)|lattice}}.</ref>

In mathematical terms, a lattice is defined as a free abelian group embedded into {{w|Euclidean space}}.

As expected, each point in the lattice represents a pitch class (or pitch), and the vectors between points represent musical intervals.

The lattice is generated by a set of basis vectors corresponding to prime harmonics, or other fundamental intervals.

The embedding into Euclidean space then also induces a norm onto intervals, which serves as a way of measuring [[complexity]].

An example is the construction of [[Tenney-Euclidean]] interval space, which provides a metric for measuring harmonic distances between pitches.

We map a ''p''-limit JI space into R^n by representing each ratio as a [[monzo|vector]], according to its prime decomposition.

The axes are then scaled according to the logarithms of their respective primes, so prime 2 has length log2(2) = 1, prime 3 has length log2(3), and prime 5 has length log2(5), etc.

For the 5 limit, we get the embedding:

:<math>

2^x \cdot 3^y \cdot 5^z \to \left[x, y \log_2(3), z \log_2(5) \right] \in \mathbb{R}^3

</math>

When distances are measured in this space, we get the [[Tenney–Euclidean_metrics#TE_norm|Tenney-Euclidean distance]], a useful measure which is often used for finding optimal temperaments.

== Gallery ==

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* [http://www.huygens-fokker.org/docs/fokkerpb.html "Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7-) Harmonic Lattice of Notes"] by [[Adriaan Fokker]] (1969)

* [http://tonalsoft.com/monzo/lattices/lattices.htm "Harmonic Lattice Diagrams"] by [[Joseph Monzo|Joseph L. Monzo]] (1998)

== Notes ==

[[Category:Lattice| ]]

@@ Line 12: / Line 12: @@
 == 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 Tonnetz is a prominent example of a lattice representing [[5-limit]] intervals. In its conventional hexagonal arrangement, the [[3/2|perfect fifth]] (3/2) and [[5/4|major third]] (5/4) intervals are positioned 60 degrees apart, with the [[5/3|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.
@@ Line 18: / Line 18: @@
 == Higher dimensions ==
-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.
+Lattices naturally extend into higher dimensions as more prime factors are incorporated into the tuning system. After applying octave reduction, a 5-limit system requires two dimensions, a 7-limit system requires three dimensions, and each additional prime factor adds another dimension to the full representation.
+{{w|Vogel's Tonnetz}} extends Euler's 5-limit Tonnetz to the 7-limit, resulting in a three-dimensional diagram.
+This additional dimension allows for the representation of relationships involving the prime 7, adding intervals like the [[harmonic seventh]] (7/4) and the [[septimal minor third]] (7/6) alongside the 5-limit intervals.
+Higher-dimensional lattices have been used by composers like [[Ben Johnston]], and [[Kyle Gann]] to organize their harmonic materials.
+The lattice structure can guide everything from chord progressions to large-scale form, with paths through the lattice corresponding to harmonic progressions.
 [[File:Lattice_7lim.png|300px|thumb|none|A 3-dimensional lattice for the [[7-limit]]. Made using [[Scale Workshop]].]]
@@ Line 26: / Line 33: @@
 {{Wikipedia|Isomorphic keyboard}}
 Isomorphic keyboards are a practical application of lattices in instrument design.
-An isomorphic keyboard is an instrument with a two-dimensional grid of buttons or keys, arranged so that any given sequence or combination of musical intervals maintains the "same shape" regardless of where it occurs.
+An isomorphic keyboard is an instrument with a two-dimensional grid of buttons or keys, arranged so that any given sequence or combination of musical intervals maintains the same shape, regardless of where it occurs.
 This corresponds directly to a lattice: the keyboard layout is essentially a lattice mapped to physical space.
 This mapping creates consistent fingering patterns for musicians, where a particular chord shape or scale pattern can be transposed by simply moving the same finger configuration to a different position on the keyboard.
@@ Line 37: / Line 44: @@
 Examples of isomorphic keyboard designs include the {{w|Jankó keyboard}}, the {{w|Wicki–Hayden note layout}} and digital controllers like the [[Lumatone]] and [[Linnstrument]].
-[[File:Lattice_bosanquet.png|300px|thumb|none|An isomorphic meantone layout (Bosanquet).]]
+[[File:Lattice_bosanquet.png|300px|thumb|none|An isomorphic [[meantone]] layout (Bosanquet).]]
+== In mathematics ==
+{{Wikipedia|Lattice (group)}}
+The lattices used in music theory correspond to mathematical structures of the same name.<ref group="note">Not to be confused with a different structure in order theory, which is also called a {{w|Lattice (order)|lattice}}.</ref>
+In mathematical terms, a lattice is defined as a free abelian group embedded into {{w|Euclidean space}}.
+As expected, each point in the lattice represents a pitch class (or pitch), and the vectors between points represent musical intervals.
+The lattice is generated by a set of basis vectors corresponding to prime harmonics, or other fundamental intervals.
+The embedding into Euclidean space then also induces a norm onto intervals, which serves as a way of measuring [[complexity]].
+An example is the construction of [[Tenney-Euclidean]] interval space, which provides a metric for measuring harmonic distances between pitches.
+We map a ''p''-limit JI space into R^n by representing each ratio as a [[monzo|vector]], according to its prime decomposition.
+The axes are then scaled according to the logarithms of their respective primes, so prime 2 has length log<sub>2</sub>(2) = 1, prime 3 has length log<sub>2</sub>(3), and prime 5 has length log<sub>2</sub>(5), etc.
+For the 5 limit, we get the embedding:
+:<math>
+^x \cdot 3^y \cdot 5^z \to \left[x, y \log_2(3), z \log_2(5) \right] \in \mathbb{R}^3
+</math>
+When distances are measured in this space, we get the [[Tenney–Euclidean_metrics#TE_norm|Tenney-Euclidean distance]], a useful measure which is often used for finding optimal temperaments.
 == Gallery ==
@@ Line 57: / Line 83: @@
 * [http://www.huygens-fokker.org/docs/fokkerpb.html "Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7-) Harmonic Lattice of Notes"] by [[Adriaan Fokker]] (1969)
 * [http://tonalsoft.com/monzo/lattices/lattices.htm "Harmonic Lattice Diagrams"] by [[Joseph Monzo|Joseph L. Monzo]] (1998)
+== Notes ==
+<references group="note" />
 [[Category:Lattice| ]] <!-- main article -->