Journal of Mathematics and Music Vol. 4, No. 3, November 2010, 1–11 Lerdahl’s tonal pitch space model and associated metric spaces Richard R. Randall a * and Bilal Khan b a School of Music, Carnegie Mellon University, Pittsburgh, PA, USA; b Department of Math & Computer Science, John Jay College of Criminal Justice, City University of NewYork, NewYork, NY, USA (Received 3 May 2007; final version received 7 July 2010 ) 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. Keywords: Lerdahl; tonal pitch space; music cognition; perception; metric space 1. Introduction The search for compelling representations of tonal hierarchy and its constituent harmonic rela- tions has a long-standing history. Lerdahl [1] describes geometric approaches to this problem that involve the collection and modelling of data from experiments in music cognition. The multidimensional-scaling models of Bharucha and Krumhansl [2], and Deutsch [3], for exam- ple, seek to encode cognitive relationships between chords within a single key area (region) as Euclidean distances. The work of Heinichen [4], Kellner [5], and Weber [6], is more speculative and when formalized, develops geometric representations of relationships between different key areas (regions) through their placement within a multidimensional space. Fred Lerdahl’s tonal pitch space (TPS) model [1] approximates the cognitive perceptual relation between chords by providing a combinatorial procedure for computing a distance value between two triads. The procedure employed by the TPS model is informed by experimental data and plausible hypotheses about how we perceive tonal relations. The model is extraordinarily powerful *Corresponding author. Email: randall@cmu.edu ISSN 1745-9737 print/ISSN 1745-9745 online © 2010 Taylor & Francis DOI: 10.1080/17459737.2010.529654 http://www.informaworld.com