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 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 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.

[[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.

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 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).]]

== ~~Examples~~ ==

== Gallery ==

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

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== See also ==

* [[Fokker block]]

* [[Keyboard]]

== External links ==

@@ 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  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 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.
+[[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.
+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 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).]]
-== Examples ==
+== Gallery ==
 <gallery>
 File:Lattice Marvel.png|A square lattice representing [[marvel]] temperament.
@@ Line 27: / Line 49: @@
 == See also ==
 * [[Fokker block]]
+* [[Keyboard]]
 == External links ==