Distances
Julian Hook · 2023
Abstract Concepts of distance capture important musical intuitions in many spaces, intuitions not adequately captured by interval functions. This chapter describes distance functions and distance spaces, known to mathematicians as metrics and metric spaces. Some distance functions, especially those in real and modular distance spaces, are closely related to interval functions, but others are not. Among the concepts studied are isometries (distance-preserving mappings), distances derived from graphs or groups, relationships between distances and path lengths, and products and quotients of distance spaces. Several kinds of distance functions may be defined in OPTIC spaces, including the max distance, the Euclidean distance, and the voice-leading distance. As is often the case, spaces involving cardinality equivalence introduce complexities that do not arise in other spaces.