Three Conceptions of Musical Distance

This paper considers three conceptions of musical distance (or inverse “similarity”) that produce three different musico-geometrical spaces: the first, based on voice leading, yields a collection of continuous quotient spaces or orbifolds; the second, based on acoustics, gives rise to the Tonnetz and related “tuning lattices”; while the third, based on the total interval content of a group of notes, generates a six-dimensional “quality space” first described by Ian Quinn. I will show that although these three measures are in principle quite distinct, they are in practice surprisingly interrelated. This produces the challenge of determining which model is appropriate to a given music-theoretical circumstance. Since the different models can yield comparable results, unwary theorists could potentially find themselves using one type of structure (such as a tuning lattice) to investigate properties more perspicuously represented by another (for instance, voice-leading relationships).