Economou A and Grasl T
Xenakis’ sieves are a powerful formal tool to create integer-sequence generators that can be used for the generation of various numerical patterns to represent pitch scales, rhythm sequences, as well as patterns of loudness, density, timber and so forth. The key idea in the design of the sieves is the notion of a vocabulary of elementary modules that can be combined one with another under the operations of union and intersection in various ways to produce a series of patterns that rise from the simplest possible ones to highly expressive structures simulating almost random-like linear distributions of points on a line. Significantly Xenakis asserted that his algebra of sieves revealed insightful views to fundamental aspects of composition by “giving when it exists, a more hidden symmetry derived from the decomposition of a modulus”. It is this analytical power of the formalism to reveal hidden symmetries, emergent structures, and ordering systems that is revisited and reworked here.
The key idea for the work outlined here is that the construction of the sieves when it is produced by division rather that by aggregation and concatenation provides readily available nested patterns that clearly foreground various emergent numerical and spatial structures. The formal tool for this inquiry is taken by Polya’s theorem of counting non-equivalent configurations with respect to a given permutation group. Here the theorem is applied within an automated computational framework to produce all possible non-equivalent configurations that can be extracted from a regular division of a linear module into an n‑number of aliquot parts. The software generates automatically the cyclic index of any symmetry group of any finite linear shape, allows for the computation of a figure inventory of variables upon the cycle index and visualizes the result of all non-equivalent configurations with corresponding isomorphic two-dimensional subshapes of regular n‑gons. Significantly the software illustrates all such possible configurations in partial order lattices too to show the nested orders and symmetries of the spatial sieves.
Xenakis Past, Present, and Future
28-30 January 2010
New York (USA)