Lattice

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Lattice (music)

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

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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.

Higher dimensions

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.

Isomorphic keyboards

See also: Alternative keyboards

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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 Jankó keyboard, the Wicki–Hayden note layout and digital controllers like the Lumatone and Linnstrument.

Gallery

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.