It is twenty years since A Generative Theory of Tonal Music by Fred Lerdahl and Ray Jackendoff first appeared, published by MIT Press (Cambridge, MA, USA). It is a work that has never ceased to arouse the interest of musical thought, as is testified by the impressive number of references to this book within the domain of Sciences of Music, and by the amount of original research that has been conceived on the basis of its theoretical axioms.
           A recent work of Fred Lerdahl, Tonal Pitch Space, published by Oxford University Press (2001), introduces new perspectives and theoretical concepts. Some preliminary essays, published in various journals, had preceded this publication; these have been considerably revised and developed in the chapters of the recent book. They have also recently been the subject of work that tests the arguments advanced.
            The editorial committee of MUSICÆ SCIENTIÆ is happy to gather today, in the Spring 2003 issue, a selection of experimental contributions and theoretical reflections that show that Fred Lerdahl's new propositions could well arouse an interest equivalent to that sparked by A Generative Theory of Tonal Music. In the interest of familiarizing our readership with notions developed in the new work, it seemed opportune to conclude with an account written by two researchers particularly familiar with the concepts of Fred Lerdahl, the first from a psychological perspective, the second from a theoretical angle. It is hoped that their contributions and the set of paths opened by the authors of this issue will suggest ideas that lead to new explorations of the theoretical potential of Tonal Pitch Space.

This study conducted a perceptual analysis of an excerpt from Beethoven's Waldstein piano sonata in four experiments. The experiments were followed by an application of Lerdahl's (2001) Tonal Pitch Space theory, or TPS, to perceptual judgments of musical tension over the course of the excerpt. In the experiments, data were obtained for each of 15 time points corresponding to the 15 successive sonorities in the excerpt. Listeners in the first three experiments were piano students who learned and performed the excerpt before participating in the experimental tasks. In Experiment 1, judgments of phrase structure permitted examination of the perceptual segmentation of the excerpt. In Experiment 2, probe-tone ratings (Krumhansl and Kessler, 1982) were used to derive both a measure of perceived distance traveled in key space as well as a measure of temporal orientation – the degree to which listening at each time point was retrospective or prospective. In Experiment 3, a tension contour over the 15 time points was derived from listeners' continuous responding to perceived tension throughout the excerpt. In Experiment 4, judgments of consonance/dissonance for each sonority, isolated from context and presented in random order in a randomly selected transposition, were obtained from piano students unfamiliar with the excerpt. Strong correspondences were found between the perceptual measures and related music-theoretic predictors generated from TPS. The tension contour was regressed on TPS predictors. Both sensitivity to hierarchical structure and expectancies arising from voice leading, as well as the overall melodic contour, significantly contributed to the prediction of perceived tension.

The experiment reported here provides a comparison between listeners' judgments and theoretical predictions on harmonic tension and melodic attraction using Lerdahl's (2001) Tonal Pitch Space theory. A harmonic reduction of measures 1 to 8 from the first movement of Mozart's Piano Sonata, K282 was used for the experiment. The listeners heard the 24 sequential chord pairs from the harmonic reduction and were asked, first, to judge how strongly were attached the two chords to each other and, second, if there was an increase, decrease or no-change in tension in the progression from one chord to the next. The results from the experiment showed a significant correlation between theoretical predictions and listeners' tension judgments. However, a low correlation between Lerdahl's model and the attraction judgments demonstrated that distance in semitones is a more influential factor than anchoring strength in the perception of melodic attraction.  This conclusion followed from the high attraction ratings given to repeated chords or adjacent chords in which the bass is repeated or progresses by step. The most interesting result of this experiment, consistent with the theory, is the demonstration that listeners perceive an inverse relationship between tension and attraction. In other words, listeners give high-tension ratings to points of low melodic attraction and vice versa, even when they judge each factor separately.  Although we obtained a convergence of results between predictions and listeners' tension judgments and demonstrated a perceptual inverse relationship between tension and attraction, further tests are needed to determine how distance in semitones and anchoring strength combine to give a measure of melodic attraction.

This paper presents an approach to the pitch space of the seven diatonic modes. The proposed theory is an expansion of Fred Lerdahl's tonal pitch space model; its purpose is a more accurate description of the situations involved in the analysis of diatonic modal music. The original methodology and set of algebraic calculating formulae is retained, but it is applied on modal instead of tonal space, on the basis that the latter is a subset of the former. In connection to Fred Lerdahl's theory, melodic motion, chord attraction and various cadence types are described within the modal context. Apart from the calculation of the pitch and the chordal and regional space algebraic representations, geometrical representations of all three levels of the modal pitch space are also included. Finally, the stability conditions arising from the new model are used as criteria to build the time span reduction and prolongational reduction parts of the Generative Theory of Tonal Music (GTTM) analysis of modal music. Two short GTTM analyses of 20th century modal music are being presented to illustrate the new model's analytical use.

The theory of Tonal Pitch Space, unlike other spatial theories of tonality, is a theory of the tonal system rather than of the tonal discourse. The described space and the distances that can be measured in it are precompositional ; the functional constraints that may rule displacements in the space are not really considered because they do not belong to the level of the immanent system. Other theories on the contrary, among others the transformational theories, are more concerned with the description of movements and transformations and make use of the tonal space as a visualization of the constraints to which they are subjected. These two types of descriptions must be considered complementary.

Pitch space and Riemannian space offer divergent perspectives on tonal space in general. This article suggests that cognitive metaphor theory offers a means of understanding Riemannian space as a metaphorical mapping from pitch space; as an extension from a "deontic" to an "epistemic" musical category. My approach differs from other theorists of musical metaphor in considering mappings not between musical structure and extra-musical "real" space, but rather mappings between musical categories themselves. I illustrate this claim by reviewing recent writings both on pitch-space and Neo-Riemannian theory, and an analytical example by Chopin.