Music as Flow: A Formal Representation of Hierarchical Processes in Music

Modeling the temporal unfolding of musical events and its interpretation in terms of hierarchical relations is a common theme in music theory, cognition, and composition. To faithfully encode such relations, we need an elegant way to represent both the semantics of prolongation, where a single event is elaborated into multiple events, and process, where the connection from one event to another is elaborated into multiple connections. In existing works, trees are used to capture the former and graphs for the latter. Each such model has the potential to either encode relations between events (e.g., an event being a repetition of another), or relations between processes (e.g., two consecutive steps making up a larger skip), but not both together explicitly. To model meaningful relations between musical events and processes and combine the semantic expressiveness of trees and graphs, we propose a structured representation using algebraic datatype (ADT) with dependent type. We demonstrate its applications towards encoding functional interpretations of harmonic progressions, and large scale organizations of key regions. This paper offers two contributions. First, we provide a novel unifying hierarchical framework for musical processes and events. Second, we provide a structured data type encoding such interpretations, which could facilitate computational approaches in music theory and generation.