The idea of harmonic space is a powerful one, not primarily because it makes visualization of harmonic objects possible, but more fundamentally because it gives us access to a range of metaphors commonly used to explain and interpret harmony: distance, direction, position, paths, boundaries, regions, shape, and so on. From a mathematical perspective, these metaphors are all inherently topological. The most prominent current theoretical approaches involving harmonic spaces are neo-Riemannian theory and voice-leading geometry. Recent neo-Riemannian theory has shown that the Tonnetz is useful for explaining a number of features of the chromatic tonality of the nineteenth century (e.g., Cohn 2012), especially the common-tone principles of Schubert's most harmonically adventurous progressions and tonal plans (Clark 2011a-b). One of the drawbacks of the Tonnetz, however, is its limited range of objects, which includes only the twenty-four members of one set class. The voice-leading geometries of Callender, Quinn, and Tymoczko 2008 and Tymoczko 2011 also take chords as objects. But because the range of chords is much wider-all chords of a given cardinality, including multisets and not restricted to equal temperament-the mathematical structure of voice-leading geometries is much richer, a continuous geometry as opposed to a discrete network. Yet voice-leading geometries also differ fundamentally from the Tonnetz in what it means for two chords to be close together: in the Tonnetz, nearness is about having a large number of common tones, not the size of the voice leading per se. In the harmonic space described in this paper, Fourier phase space, the conception of distance is similar to that of the Tonnetz. But it also has the richer mathematical structure and wider range of objects that one associates with voice-leading geometries. The first part of the paper describes the particular virtues of the Tonnetz's common-tone based conception of distance for analysis of Schubert, and also how, on the other hand, its highly circumscribed range of musical objects poses severe limitations on its application. The second section describes a Fourier phase space, based on the discrete Fourier transform (DFT) on pitchclass sets described in Quinn (2006), and shows how it retains the music-analytic virtues of the Tonnetz while expanding its range of objects and embedding it in more mathematically robust space.
The discrete Fourier transform on pitch-class sets, proposed by David Lewin and advanced by Ian Quinn, may provide a new lease on life for Allen Forte's idea of a general theory of harmony for the twentieth century based on the intervallic content of pitch-class collections. This article proposes the use of phase spaces and Quinn's harmonic qualities in analysis of a wide variety of twentieth-century styles. The main focus is on how these ideas relate to scale-theoretic concepts and the repertoires to which they are applied, such as the music of Debussy, Satie, Stravinsky, Ravel, and Shostakovich. Diatonicity, one of the harmonic qualities, is a basic concern for all of these composers. Phase spaces and harmonic qualities also help to explain the “scale-network wormhole” phenomenon in Debussy and Ravel and better pinpoint the role of octatonicism in Stravinsky's and Ravel's music.
This book presents a theory of temporal structure for music, making two main arguments. The first is that a single model of temporal structure, expressible in the form of a certain type of mathematical network, is common to all modalities, particularly rhythm, tonality, and form. As a result, we can develop tools to talk about the experience of musical time in abstraction from any particular modality, and make analogies from structural phenomena in one modality to another (e.g., formal counterpoint). The second argument is that each of these modalities is in principle independent: it has its own set of structuring criteria, and it may lead to structures that agree or disagree with each other. The resulting coordination or disjunction between modalities is of more direct aesthetic importance, typically, than anything that can be said about one isolated parameter alone. These claims have deep ramifications for theories of rhythm, tonality, and form: for instance, that it is possible to discuss formal structure without necessary reference to tonal features. Theories of harmony, key, formal function, hypermeter, and closure are developed in conjunction with analysis of a wide range of eighteenth- and nineteenth-century composers, surveys of classical repertoire, and observations about the history of musical styles. A number of mathematical tools for temporal structure are also proposed.
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