Geometric pattern matching framework for music analysis and discovery
A General Theoretical Framework for Transformed Geometric Pattern Matching in Music
SoundComputational Geometry
Summary
Matching and discovering patterns in music can be complicated because musical elements can be transformed in various ways like transposing or stretching notes. The authors created a formal framework to describe these transformations and how they affect pattern matching. They also defined key ideas like when patterns match exactly or in a maximal way and explored ways to group similar transformations. This helps make musical pattern analysis more rigorous and clear.
What this means in practice
- •For music software developers: Design algorithms that detect recurring musical motifs despite changes like transposition or inversion using a formal transformation framework.
- •For digital audio toolmakers: Implement pattern discovery tools that group equivalent musical patterns under transformation groups to aid in music editing and analysis.
A theory result. No direct application yet.
Authors
David Meredith
Abstract
In this paper, we present a framework of concepts, terminology and notation for geometric pattern matching, centred around the idea of a {\em transformation class}. After formally defining the concept of a transformation class, we provide detailed definitions of a number of specific transformation classes that are particularly relevant to pattern matching and pattern discovery in music, which is the primary application domain within which the framework has been developed. We then provide formal definitions of the notions of {\em match}, {\em occurrence}, {\em maximal match\/} and {\em maximal occurrence\/} and prove some results relating to these concepts. We then define and explore in detail the idea of an {\em occurrence set\/} and the more general notion of a {\em heterogeneous occurrence set}. Finally, we explore in detail the concept of {\em transformational equivalence}, including a consideration of {\em transformation groups}, along with the group-theoretic and transformational equivalence properties of selected transformation classes.