Lattice (music)

The lattice can be two-, three-, or n-dimensional, with each dimension corresponding to a different prime-number partial [pitch class].

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.

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 "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.

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.