In musical tuning, a lattice "is a way of modeling the tuning relationships in a just intonation system. It is an array of points in a periodic multidimensional pattern. Each point on the lattice corresponds to a ratio (i.e., a pitch, or an interval with respect to some other point on the lattice). The lattice can be two-, three-, or n-dimensional, with each dimension corresponding to a different prime-number partial"[1] or chroma.
For example Hugo Riemann's Tonnetz (1739), and Ben Johnston's tuning systems. Adriaan Fokker's Fokker periodicity blocks are used to mathematically relate musical intervals in just intonation to those in equal tuning. The limit is the highest prime-number partial used in 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, 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 "a twelve-note 'chromatic' scale may be represented as a two-dimensional (3,5) projection plane within the three-dimensional (2,3,5) space needed to map the scale. (Octave equivalents would appear on an axis at right angles to the other two, but this arrangement is not really necessary graphically.)"[1]. 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 using depth to model octaves:
A----E----B----F#+ | | | | F----C----G----D | | | | Db---Ab---Eb---Bb
Equals the ratios:
5/3--5/4-15/8--45/32 | | | | 4/3--1/1--3/2---9/8 | | | | 16/15-8/5-6/5---9/5