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{{Wikipedia|Lattice (music)}} | |||
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}} | |||
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. | |||
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]]. | |||
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> | |||
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 == | |||
<gallery> | |||
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]]. | |||
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 == | |||
* Beginner's guide to lattices: See chapter 1.3 of [http://www.tallkite.com/AlternativeTunings.html ''Alternative Tunings: Theory, Notation and Practice''] by [[Kite Giedraitis]] (2016) | |||
* [http://www.tonalsoft.com/enc/l/lattice.aspx "lattice/lattice diagram"] on [[Tonalsoft Encyclopedia]] | |||
* [http://x31eq.com/lattice.htm#7limit "Octave Equivalent Music Lattices"] by [[Graham Breed]] (2008) | |||
* [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 --> | |||