The papers listed with their abstracts below present in turn the theory behind the algorithm, and the application of that algorithm in Coord.
A self-similar map of rhythmic components in Journal of Mathematics and Music, Volume 10, Issue 1, 2016
This paper defines a collection of rhythmic building blocks produced by generative operations that fuse metrical anticipation and parallelism. It connects aspects of musical expectation with Arthur Komar's constraints on the generation of rhythmic derivations. A correspondence between those constraints and the divisibility of binomial coefficients is used to map derived rhythmic elaborations and syncopations separately onto Pascal's triangle and jointly onto the Sierpinski gasket. This mapping provides concise means to directly enumerate and compare rhythmic configurations. An Online Supplement provides musical examples and discusses potential applications. [A limited number of free accesses are available upon request.]
Navigating Outcomes of Rhythmic Anticipation [opens pdf] in Proceedings of MUME 2018 - The Sixth International Workshop on Musical Metacreation, held at the Ninth International Conference on Computational Creativity, ICCC 2018, Salamanca
An approach to computer-aided improvisation that leverages aspects of low-level rhythmic coherence is demon- strated. Nested anticipations at distinct metrical levels determine rhythmic patterns formed by the anticipations’ collective outcomes. A connection to number theory provides a self-similar map of rhythmic building blocks, affording control over relative degrees of syncopation and elaboration. The result is real-time navigation and manipulation of rhythmic patterns by means of operations that reflect subjective musical goals.
Building Blocks of Rhythmic Expectation [opens pdf] in Proceedings of MUME 2016 - The Fourth International Workshop on Musical Metacreation, held at the Seventh International Conference on Computational Creativity, ICCC 2016, Paris
Rhythmic expectation is a key aspect of musical experience, but it has traditionally lacked a specific, low-level representation analogous to that provided by pitch and meter. A representation that encodes intersections of anticipation and repetition will be demonstrated to provide a degree of real-tme control over variation and hybridization of rhythms. The encoding connects low-level musical structure to pattern formation of binomial coefficients on Pascal’s triangle and self-similarity in the Sierpinski gasket. A generative music application demonstrates use of these encodings as building blocks upon a computational musical landscape.
Short, repetitive melodic figures are a common feature of various musical genres, including electronic dance music (EDM). The rhythms of such figures are localized to the extent that surface patterns and perceived structure can be algorithmically connected at a low level in order to create intuitively related variations without explicitly simulating compositional styles or strategies.
This paper describes creation of localized rhythmic variations using an approach that is generative in two senses of the word. It applies a finite set of rules to generate all rhythmic patterns that are well-formed according to strictly defined rhythmic relationships. Then it uses the resulting patterns as building blocks with which to dissect, compare and generate actual rhythms.
The set of all potential building blocks forms a hierarchy that encodes intersections of basic rhythmic expectation. That hierarchy has a self-similar structure that crystallizes parallelism within and between those structural intersections into an enumerable set of surface subpatterns, facilitating reuse of the overall generative analysis to create specific rhythmic variations.
Rhythms that have been dissected into these building blocks can be manipulated at a high level of abstraction, enabling organic rhythmic variation via real-time improvisation.