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Athanassios Economou and Thomas Grasl
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Xenakis Matters: Contexts, Processes, Applications
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Communications in Computer and Information Science (CCIS)
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Xenakis, Sieves, Symmetry, Polya’s Theorem of Counting, n-chords
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The sieve is a powerful formal tool developed by Xenakis 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 sieves is the notion of a vocabulary of elementary modules that can be combined one with another under operations of union and intersection in various ways. Sieves produce a series of patterns that rise from the simplest ones possible to highly expressive structures, simulating almost random-like linear distributions of points on a line. Significantly, Xenakis asserted that his algebra of sieves revealed insights into fundamental aspects of composition by “giv[ing], when it exists, a more hidden symmetry derived from the decomposition of a modulus”. It is this implicit power of this formalism to reveal hidden symmetries and emergent structures that is revisited and reworked here. The formal tools for the inquiry on sieves are taken from La-grange’s theorem of subgroup numerical relation and Pólya’s theorem of counting non-equivalent configurations with respect to a given permutation group. Here the theorems are 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 automatically generates 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 correspond-ing isomorphic two-dimensional subshapes of regular n-gons. Significantly, the software illustrates all such possible configurations in partial order lattices as well, in order to show the nested orders and symmetries of the spatial sieves.
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