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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.

[[File:Lattice_5lim.png|300px|thumb|none|The [[5-limit]] Tonnetz lattice.]]

== Higher dimensions ==

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]].]]

== Isomorphic keyboards ==

:''See also: [[Keyboard#Alternative keyboards|Alternative keyboards]]''

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.

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.

~~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~~.

Unlike lattice diagrams that often reduce dimensionality through octave equivalence, isomorphic keyboards must physically represent all pitches.

This effectively limits them to rank 2 systems, where the two-dimensional physical layout must accommodate the full range of playable notes.

Quite often these are still tuned to [[EDOs]], which are rank-1 systems.

In such cases, there is an additional interval on the keyboard that is a [[unison]], although it may be out of reach on the keyboard.

== ~~Examples~~ ==

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

Isomorphic layouts are not limited to keyboards, [[skip fretting]] systems are also examples.

[[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 ==

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

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File:Torus.png|A '''torus''' of notes in 15edo, notated by porcupine notation with LH-NiceIonian as the base scale.

</gallery>

== See also ==

* [[Fokker block]]

* [[Keyboard]]

== External links ==

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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.
+[[File:Lattice_5lim.png|300px|thumb|none|The [[5-limit]] Tonnetz lattice.]]
+== Higher dimensions ==
+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]].]]
+== Isomorphic keyboards ==
+:''See also: [[Keyboard#Alternative keyboards|Alternative keyboards]]''
+{{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.
+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.
-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.
+Unlike lattice diagrams that often reduce dimensionality through octave equivalence, isomorphic keyboards must physically represent all pitches.
+This effectively limits them to rank 2 systems, where the two-dimensional physical layout must accommodate the full range of playable notes.
+Quite often these are still tuned to [[EDOs]], which are rank-1 systems.
+In such cases, there is an additional interval on the keyboard that is a [[unison]], although it may be out of reach on the keyboard.
-== Examples ==
+Examples of isomorphic keyboard designs include the {{w|Jankó keyboard}}, the {{w|Wicki–Hayden note layout}} and digital controllers like the [[Lumatone]] and [[Linnstrument]].
+Isomorphic layouts are not limited to keyboards, [[skip fretting]] systems are also examples.
+[[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 ==
 <gallery>
 File:Lattice Marvel.png|A square lattice representing [[marvel]] temperament.
@@ Line 24: / Line 73: @@
 File:Torus.png|A '''torus''' of notes in 15edo, notated by porcupine notation with LH-NiceIonian as the base scale.
 </gallery>
+== See also ==
+* [[Fokker block]]
+* [[Keyboard]]
 == External links ==
@@ Line 31: / Line 84: @@
 * [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 -->