Maximally Even Sets

A geometry of music: harmony and counterpoint in the extended common practice, by Dmitri Tymoczko, New York, Oxford University Press, 2011, xviii + 450 pp., US$39.95, ISBN 978-0-19-533667-2 This is...

A geometry of music: harmony and counterpoint in the extended common practice

Four experiments are reported in which the materials are derived from two 12-tone serial compositions (Schoenberg9s Wind Quintet and String Quartet, No. 4). Two experiments use the probe tone method (Krumhansl & Shepard, 1979) to assess factors contributing to tone prominence in serial music. The contexts in Experiment 1 are musically neutral statements of the complete or incomplete tone rows; the contexts in Experiment 4 are excerpts from the two pieces. Two experiments use a classification task to evaluate whether the prime form of the row is perceived as similar to its mirror forms (inversion, retrograde, and retrograde inversion). The materials are neutral presentations of the forms (Experiment 2) or excerpts from the pieces (Experiment 3). Large individual differences are found. A subgroup of listeners, with more music training on average, show the following effects in the probe tone experiments: low ratings for tones sounded more recently in the contexts and high ratings for tones not yet sounded; low ratings for tones fitting with local tonal implications; similar patterns for the neutral contexts and the musical excerpts. The remaining listeners show the opposite effects. Classification accuracy of mirror forms is above chance and is higher for the neutral sequences than the musical excerpts; performance is correlated with music training. The experiments show that some, but not all, listeners can perceive invariant structures in serial music despite mirror transformations, octave transpositions of tones, and variations of rhythm and phrasing.

The Perception of Tone Hierarchies and Mirror Forms in Twelve-Tone Serial Music

The General Computational Theory of Musical Structure (GCTMS) is a theory that may be employed to obtain a structural description (or set of descriptions) of a musical surface. This theory is based on general cognitive and logical principles, is independent of any specific musical style or idiom, and can be applied to any musical surface. The musical work is presented to GCTMS as a sequence of discrete symbolically represented events (e.g. notes) without higher-level structural elements (e.g. articulation marks, time-signature etc.)-although such information may be used to guide the analytic process. The aim of the application of the theory is to reach a structural description of the musical work that may be considered as 'plausible' or 'permissible' by a human music analyst. As style-dependent knowledge is not embodied in the general theory, highly sophisticated analyses (similar to those an expert analyst may provide) are not expected. The theory gives, however, higher rating to descriptions that may be considered more reasonable or acceptable by human analysts and lower to descriptions that are less plausible. The analytic descriptions given by GCTMS may be said to relate to and may be compared with the intuitive 'understanding' a listener has when repeatedly exposed to a specific musical work. Although the theory does not make any claim of simulating cognitive processes as these are realised in the mind, it does give insights into the intrinsic requirements of musical analytic tasks and its results may be examined with respect to cognitive validity. The proposed theory comprises two distinct but closely related stages of development: a) the development of a number of individual components that focus on specialised musical analytic tasks, and b) the development of an elaborate account of how these components relate to and interact with each other so that plausible structural descriptions of a given musical surface may be arrived at. A prototype computer system based on the GCTMS has been implemented. As a test case, the theory and prototype system have been applied on various melodic surfaces from the 12-tone equal-temperament system. iii Acknowledgements I would like to express my gratitude to: My supervisors, Mr Peter Nelson and Dr Alan Smaill, for their guidance and support throughout this research study, and for all their invaluable comments and advice that helped me crystallise the views presented in this thesis. My colleagues from the Faculty of Music and the Department of Artificial Intelligence at the …

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Towards a General Computational Theory of Musical Structure

The early atonal music of Schoenberg and his school is emblematic of the difficulties in comprehending twentieth-century music. Music theorists have tried to analyze this music either through a modified Schenkerian approach or through pitch-set theory. After discussing limitations in these approaches, this paper develops at length a different theory for atonal music, one that adapts Jackendoff's and my A Generative Theory of Tonal Music in appropriate ways. Analyses of three Schoenberg pieces (from op. 11 and op. 19) illustrate the theory. The paper closes with an informal discussion of atonal pitch space.

Atonal prolongational structure

Basic atonal theory

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Basic atonal theory

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Maximally Even Sets

A geometry of music: harmony and counterpoint in the extended common practice

The Perception of Tone Hierarchies and Mirror Forms in Twelve-Tone Serial Music

Towards a General Computational Theory of Musical Structure

Atonal prolongational structure

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