Lerdahl's tonal pitch space model and associated metric spaces

Richard R. Randall, Bilal Khan

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


This paper explores the boundary separating a theoretically derived model of chord relations from an empirically derived model. Fred Lerdahl's tonal pitch space (TPS) model approximates cognitive perceptual relations by providing a combinatorial procedure for computing the distance value between any two chords in a key. If TPS posits a hypothesized model of perception, then we would like to know if, and the extent to which, it differs from experimental data it claims to approximate. To achieve such a comparison, we develop three conceptual tools. First, we develop normalized canonical representations of each model, thereby avoiding comparisons affected by design choices. Second, we develop a distance measure that allows us to accurately compare the TPS model with another model derived from perceived chord relations described by Bharucha and Krumhansl. Finally, we use the distance measure to inform the design of a third model. These three models are shown to create a metric space of metric tonal models. The proposed distance measure and the method of normalization are applicable to any model with formal properties described herein and have the potential to focus experimental design and strengthen the relationship between experimental data and analytic systems.

Original languageEnglish (US)
Pages (from-to)121-131
Number of pages11
JournalJournal of Mathematics and Music
Issue number3
StatePublished - Nov 2010
Externally publishedYes


  • Lerdahl
  • metric space
  • music cognition
  • perception
  • tonal pitch space

ASJC Scopus subject areas

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


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