Geometry and the quest for theoretical generality

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Abstract

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.

Original languageEnglish (US)
Pages (from-to)127-144
Number of pages18
JournalJournal of Mathematics and Music
Volume7
Issue number2
DOIs
StatePublished - Jul 1 2013

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Music
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • geometry
  • maximal evenness
  • signature transformations
  • transformational theory
  • voice leading

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