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" or chroma.
Examples of musical lattices include the Tonnetz of Euler (1739) 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. 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.)". 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:
- Gilmore, Bob (2006). "Introduction", p.xviii, "Maximum Clarity" and Other Writings on Music, edited by Bob Gilmore. Urbana: University of Illinois Press. ISBN 0-252-03098-2.
- Johnston, Ben (2006). "Rational Structure in Music", "Maximum Clarity" and Other Writings on Music, edited by Bob Gilmore. Urbana: University of Illinois Press. ISBN 0-252-03098-2.
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