Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited


lattice diagram


    a visual representation of the mathematical relationships of musical ratios in 2-, 3-, or multi-dimensional space, consisting of points which represent the ratios as positions calculated according to the Fundamental Theorem of Arithmetic.

    Lattices may be based upon two types of factoring: either odd or prime - similar to the two types of limit. In either case, a vector is drawn or imagined to represent each factor, with exponents represented as a series of points at regular equal distances along that vector.

    Angles between and lengths of the vectors are not standardized; simple triangular or rectilinear lattices are popular for ASCII text use in emails and internet discussion groups. I use a more complex formula of the rectilinear type which provides a unique angle and vector-segment-length for each prime axis.

    In my theoretical usage, lattices are a graphical expression of my own theory of sonance.

    The precursors to musical lattice diagrams are the Lambdoma, Ellis's Duodenarium and Riemann's Tonnetz matrix charts. Harry Partch's Tonality Diamond is related but slightly different.

    Adrian Fokker apparently designed the first 3-dimensional lattices with factors of 3, 5, and 7 represented.

    John Chalmers has made very complex diagrams of triangles representing tetrachords, following similar principles (see his book Divisions of the Tetrachord and this webpage.)

    Probably the most complex diagrams have been designed by Erv Wilson, many of whose lattices form beautiful mandala-like designs (see articles in several issues of Xenharmonikon, reproduced at The Wilson Archives).

    Other theorists known for their lattice diagrams are Ben Johnston, David Canright, Graham Breed, Paul Erlich, Dave Keenan, and myself.

    [from Joe Monzo, JustMusic: A New Harmony]

    . . . . . . . . . . . . . . . . . .

    I think we should call the various lattice diagrams "tonal lattices" to distinguish them from the partially ordered sets subject to certain requirements of greatest lower and least upper bounds on pairs of elements that other mathematicians call lattices.

    [from John Chalmers, personal communication]


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