Lerdahl's tonal pitch space model and associated metric spaces

Abstract

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

Acknowledgements

The authors wish to thank the reviewers for their comments and suggestions, many of which were incorporated into the final submission.

Notes

Noll and Garbers 7 provide an excellent critique of Lerdahl's TPS model in the context of theoretical problems associated with his attempt to combine a principle of hierarchy with a principle of shortest path. Their discussion is comprehensive and addresses issues outside the scope of what we present here.

An earlier form of this methodology was introduced in 8 in a broader context and in 9.

We identify the basic model as L to distinguish it from the larger TPS framework.

The data in Table 2 is shown in Table 10, 8.2. However, 2 is given as the source.