In musical tuning, a lattice "is a way of modeling the tuning relationships of a just intonation system. It is an array of points in a periodic multidimensional pattern. Each point on the lattice corresponds to a ratio. The lattice can be two-, three-, or n-dimensional, with each dimension corresponding to a different prime-number partial." When listed in a spreadsheet a lattice may be referred to as a tuning table. The points in a lattice represent pitch classes, 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 moves you further in the same direction. Lattices in just intonation are theoretically infinite. However, lattices are sometimes also used to notate limited subsets that are particularly interesting. Examples of musical lattices include the Tonnetz of Euler and Hugo Riemann and the tuning systems of Ben Johnston. 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 and octave and their multiples, is represented through a two-dimensional lattice, while standard just intonation, which adds the use of the just major third, may be represented through a three-dimensional lattice though "a twelve-note 'chromatic' scale may be represented as a two-dimensional projection plane within the three-dimensional space needed to map the scale. ". In other words, the circle of fifths on one dimension and a series of major thirds on those fifths in the second, with the option of imagining depth to model octaves: 5-limit A----E----B----F#+ 5/3--5/4-15/8-45/32 | | | | | | | | F----C----G----D = 4/3--1/1--3/2--9/8 | | | | | | | | -Ab-—-Eb—--Bb 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. Here is a template he used to generate what he called an "Euler" lattice after where he drew his inspiration. Each prime harmonic has a unique spacing, avoiding clashes even when generating lattices of multidimensional, harmonically based structure. Wilson would commonly use 10-squares-to-the-inch graph paper. That way, he had room to notate both ratios and often the scale degree, which explains why he didn't use a template where all the numbers where divided by 2. The scale degree always followed a period or dot to separate it from the ratios. Examples:
