Geometry and the quest for theoretical generality

This paper reconsiders scale-theoretical ideas from the standpoint of voice leading. I begin by showing that Clough and Douthett's definition of maximal evenness makes covert reference to voice-leading distances, albeit disguised by the form of their equations. I then argue that theirs is a special case of the broader problem of quantizing continuous chords to a scale, deriving some new results in this more general setting - including an analogue of the cardinality equals variety property. In the second part of the paper, I show how Clough and Douthett's J-function began as a device for generating maximally even collections, only later evolving into a tool for studying the voice leadings between them. Particularly important here are Julian Hook's signature transformations, which I generalize to a wider range of collections. I conclude with a few remarks about history and methodology in music theory.