Sieve Analysis and Construction: Theory and Implementation

Sieve Theory was used in order to construct symmetries at a desired degree of complexity. This was achieved by the combination of two or more modules, where each module is notated as an ordered pair (M, I) that indicates a modulus (period) and a residue (an integer between zero and M-1) within that modulus. The abstract image of a sieve is that of a selection of points on a straight line; according to Xenakis “Every well-ordered set can be represented as points on a line, if it is given a reference point for the origin and a length u for the unit distance, and this is a sieve” (Xenakis 1992, 268). Modules are combined by the set-theoretical operations of union, intersection and complementation. Given the possibility of multiple notations of the same sieve, the use of each logical operation depends on both the type of formula we choose and on the type of sieve in question. This paper is based on typology of both the different types of sieves as well as the available formulae types for a given sieve. We categorise sieves according to their symmetry and periodicity. Xenakis refers to these two notions as two distinct levels of identity: in the opening of his article on sieves he talks about “spatial identities” and “identities in time”, correspondingly; he then refers to these levels as being internal and external to the sieve (Xenakis 1992, 268). Symmetry is evident in the sieve's intervallic structure and periodicity in its periodic nature.