Set-Class Similarity, Voice Leading, and the Fourier Transform

Tymoczko, Dmitri

doi:10.1215/00222909-2009-017

Cited by 20 publications

(15 citation statements)

References 11 publications

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“…All in all, a good correlation between phase distance and voice-leadings should be seen as more than a coincidence, but less than mandatory. However, when taken together with the correlation established by [19] between magnitude of coefficients and neighbourhood to special pc-sets, it strengthens the case of Fourier coefficients in the study of voice-leading. In [14] Guerino Mazzola introduced 13 a topological formalism for gestures, i.e.…”

Section: 2supporting

confidence: 65%

The Torii of Phases

Amiot¹

2013

Mathematics and Computation in Music

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Section: 2supporting

confidence: 65%

The Torii of Phases

Amiot¹

2013

Mathematics and Computation in Music

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“…That is, when we say that the C major triad is close to an augmented triad, "close" means specifically that we only need to move the notes a short distance along the pccircle to get to the augmented triad. This explains, on a very general level, Tymoczko's (2008b) result relating Fourier magnitudes to voice-leading distance from the nearest subset of a perfectly-even chord (see also Callender 2007). Voice-leading distance also plays a fundamental role in Cohn's (2012) theories of chromatic harmony, specifically the small voice-leading distance from major and minor triads to perfectly even three-note chords, which is evident in the large | !…”

Section: (2) Fourier Phase Spacementioning

confidence: 86%

Schubert’s Harmonic Language and Fourier Phase Space

Yust¹

2015

Journal of Music Theory

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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.

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“…The la)er topic is potentially confusing, insofar as it applies the DFT to pitch collections differently from any other part of the book (and also unfortunately reuses the term "diatonicity" in a different sense), though Amiot does give ample warning of the switch. Section 5.4 improves upon a result of Tymoczko's (2008) that relates the DFT to voice leading. Tymoczko showed that the size of a DFT coefficient correlates with the voice-leading distance from the nearest subset of a perfectly even collection (with possible duplications).…”

Section: Overviewmentioning

confidence: 99%