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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