In musical tuning, a lattice
When listed in a spreadsheet a lattice may be called a tuning table.
The points in a lattice represent pitch classes (or pitches if octaves are represented), and the connectors in a lattice represent the intervals between them. The connecting lines in a lattice display intervals as vectors, so that a line of the same length and angle always has the same intervalic relationship between the points it connects, no matter where it occurs in the lattice.
Repeatedly adding the same vector (repeatedly stacking the same interval) moves you further in the same direction. Lattices in just intonation (limited to intervals comprising primes, their powers, and their products) are theoretically infinite (because no power of any prime equals any power of another prime). However, lattices are sometimes also used to notate limited subsets that are particularly interesting (such as an Eikosany illustrated further below or the various ways to extract particular scale shapes from a larger lattice).
Examples of musical lattices include the tonnetz of Euler (1739) and Hugo Riemann and the tuning systems of composer-theorists Ben Johnston and James Tenney. Musical intervals in just intonation are related to those in equal tuning by Adriaan Fokker's Fokker periodicity blocks. Many multi-dimensional higher-limit tunings have been mapped by Erv Wilson. The limit is the highest prime number used in the ratios that define the intervals used by a tuning.
Thus Pythagorean tuning, which uses only the perfect fifth (3:2) and octave (2:1) and their multiples (powers of 2 and 3), is represented through a two-dimensional lattice (or, given octave equivalence, a single dimension), while standard (5-limit) just intonation, which adds the use of the just major third (5:4), may be represented through a three-dimensional lattice though
In other words, the circle of fifths on one dimension and a series of major thirds on those fifths in the second (horizontal and vertical), with the option of imagining depth to model octaves:
---A ---E ---B ---F- --5:3--5:4-15:8-45:32- \ / \ / \ / \ / \ / \ / \ / \ / --F----C----G----D--- = --4:3--1:1--3:2--9:8- / \ / \ / \ / \ / \ / \ / \ / \ -D--A-âÂÂ-EâÂÂ--B--- -16:15-8:5--6:5--9:5--
Erv Wilson has made significant headway with developing lattices than can represent higher limit harmonics, meaning more than 2 dimensions, while displaying them in 2 dimensions.
To the right are templates Wilson used to generate what he called an Euler lattice after the German mathematician who introduced the tonnetz it is modeled after. Each prime harmonic (each vector representing a ratio of or where is a prime) has a unique spacing, avoiding clashes even when generating lattices of multidimensional, harmonically based structure.