Jason Yust

Tonal prisms: iterated quantization in chromatic tonality and Ravel's ‘Ondine’

The mathematics of second-order maximal evenness has far-reaching potential for application in music analysis. One of its assets is its foundation in an inherently continuous conception of pitch, a feature it shares with voice-leading geometries. This paper reformulates second-order maximal evenness as iterated quantization in voice-leading spaces, discusses the implications of viewing diatonic triads as second-order maximally even sets for the understanding of nineteenth-century modulatory schemes, and applies a second-order maximally even derivation of acoustic collections in an in-depth analysis of Ravels ‘Ondine’. In the interaction between these two very different applications, the paper generalizes the concepts and analytical methods associated with iterated quantization and also pursues a broader argument about the mutual dependence of mathematical music theory and music analysis.

Applications of DFT to the Theory of Twentieth-Century Harmony

Music theorists have only recently, following groundbreaking work by Quinn, recognized the potential for the DFT on pcsets, initially proposed by Lewin, to serve as the foundation of a theory of harmony for the twentieth century. This paper investigates pcset “arithmetic” – subset structure, transpositional combination, and interval content – through the lens of the DFT. It discusses relationships between interval classes and DFT magnitudes, considers special properties of dyads, pcset products, and generated collections, and suggest methods of using the DFT in analysis, including interpreting DFT magnitudes, using phase spaces to understand subset structure, and interpreting the DFT of Lewin’s interval function. Webern’s op. 5/4 and Bartok’s String Quartet 4, iv, are discussed.

A space for inflections: following up on JMM’s special issue on mathematical theories of voice leading

Journal of Mathematics and Musics recent special issue 7(2) reveals substantial common ground between mathematical theories of harmony advanced by Tymoczko, Hook, Plotkin, and Douthett. This paper develops a theory of scalar inflection as a kind of voice-leading distance using quantization in voice-leading geometries, which combines the best features of different approaches represented in the special issue: it is grounded in the concrete sense of voice-leading distance promoted by Tymoczko, invokes scalar contexts in a similar way as filtered point-symmetry, and abstracts the circle of fifths like Hooks signature transformations. The paper expands upon Tymoczkos ‘generalized signature transform’ showing the deep significance of generalized circles of fifths to voice-leading properties of all collections. Analysis of Schuberts Notturno for Piano Trio and ‘Nacht und Träume’ demonstrate the musical significance of inflection as a kind of voice leading, and the value of a robust geometrical understanding of it.

Restoring the structural status of keys through DFT phase space

One of the reasons for the widely felt influence of Schenker’s theory is his idea of long-range voice-leading structure. However, an implicit premise, that voice leading is necessarily a relationship between chords, leads Schenker to a reductive method that undermines the structural status of keys. This leads to analytical mistakes as demonstrated by Schenker’s analysis of Brahms’s Second Cello Sonata. Using a spatial concept of harmony based on DFT phase space, this paper shows that Schenker’s implicit premise is in fact incorrect: it is possible to model long-range voice-leading relationships between objects other than chords. The concept of voice leading derived from DFT phases is explained by means of triadic orbits. Triadic orbits are then applied in an analysis of Beethoven’s Heiliger Dankgesang, giving a way to understand the ostensibly “Lydian” tonality and the tonal relationship between the chorale sections and “Neue Kraft” sections.

Analysis of analysis: Using machine learning to evaluate the importance of music parameters for Schenkerian analysis

While criteria for Schenkerian analysis have been much discussed, such discussions have generally not been informed by data. Kirlin [Kirlin, Phillip B., 2014 “A Probabilistic Model of Hierarchical Music Analysis.” Ph.D. thesis, University of Massachusetts Amherst] has begun to fill this vacuum with a corpus of textbook Schenkerian analyses encoded using data structures suggested byYust [Yust, Jason, 2006 “Formal Models of Prolongation.” Ph.D. thesis, University of Washington] and a machine learning algorithm based on this dataset that can produce analyses with a reasonable degree of accuracy. In this work, we examine what musical features (scale degree, harmony, metrical weight) are most significant in the performance of Kirlins algorithm.

Mathematics and Computation in Music

The present paper is concerned with the existence, meaning and use of the phases of the (complex) Fourier coefficients of pc-sets, viewed as maps from ZZc to C. It explores a particular cross-section of the most general torus of phases, representing pc-sets by the phases of the third and fifth coefficients. On this 2D torus, triads take on the wellknown configuration of the Tonnetz. Some other (sequences of) chords are viewed in this space as examples of its musical relevance. The end of the paper uses the model as a convenient universe for drawing gestures – continuous paths between pc-sets.

Probing Questions about Keys: Tonal Distributions through the DFT

Pitch-class distributions are central to much of the computational and psychological research on musical keys. This paper looks at pitch-class distributions through the DFT on pitch-class sets, drawing upon recent theory that has exploited this technique. Corpus-derived distributions consistently exhibit a prominence of three DFT components, \(f_5\), \(f_3\), and \(f_2\), so that we might simplify tonal relationships by viewing them within two- or three-dimensional phase space utilizing just these components. More generally, this simplification, or filtering, of distributional information may be an essential feature of tonal hearing. The DFTs of probe-tone distributions reveal a subdominant bias imposed by the temporal aspect of the behavioral paradigm (as compared to corpus data). The phases of \(f_5\), \(f_3\), and \(f_2\) also exhibit a special linear dependency in tonal music giving rise to the idea of a tonal index.

Harmonic qualities in Debussy's “Les sons et les parfums tournent dans l'air du soir”

This analysis of the fourth piece from Debussys Préludes Book I illustrates typical harmonic techniques of Debussy as manipulations of harmonic qualities. We quantify harmonic qualities via the magnitudes and squared-magnitudes of the coefficients of the discrete Fourier transform (DFT) of pitch class sets, following Ian Quinn. The principal activity of the piece occurs in the fourth and fifth coefficients, the octatonic and diatonic qualities, respectively. The development of harmonic ideas can therefore be mapped out in a two-dimensional octatonic/diatonic phase space. Whole-tone material, representative of the sixth coefficient of the DFT, also plays an important role. I discuss Debussys motivic work, how features of tonality – diatonicity and harmonic function – relate to his musical language, and the significance of perfectly balanced set classes, which are a special case of nil DFT coefficients.

Introduction to the special issue on pedagogies of mathematical music theory

A discipline defines itself by its pedagogy. In established fields, standard curricula provide a basis of shared knowledge, organize, classify, and categorize that knowledge, and tell the story of the discipline, its classic results, and its canonical figures. Mathematical music theory, as the present volume attests, is a field of study that is relatively new, even while being at the same time ancient. Many of the contributions testify to how mathematical music theory traces its origins back to the musica speculativa tradition of the middle ages and its roots in Classical thought. After Boethius, however, our shared intellectual forebears seem to skip ahead some generations to the trailblazers of the later twentieth century, with Milton Babbitt, David Lewin, and especially John Clough, standing out as those who shaped an emerging area of study. Institutionally, mathematical music theory is quite young, with the present journal now in its eighth year and the Society for Mathematics and Computation in Music between its fourth and fifth biennial meetings. This youth is revealed in this special issue also, with many of the authors asking the essential questions of an emerging field: Where is our place in the academic institutions of the twenty-first century? How do we teach our subject, and in what contexts? How do we advertise what we do to the general population? What are our canonical results? This special issue coalesced out of two recent events. The first was the panel discussion “Mathematical Music Theory in Academia: Its Presence, Role, and Objectives in Departments of Mathematics, Music, and Computer Science” organized by Mariana Montiel at the Fourth International Conference for Mathematics and Computation in Music, June 2013, in Montreal. Many of the practical and existential issues confronting this intrinsically multidisciplinary field were addressed in the comments of panelists Guerino Mazzola, David Clampitt, Thomas Noll, Thomas Fiore, Emmanuel Amiot, and Anja Volk, a group representative of the field’s diversity of disciplinary backgrounds and professional affiliations. The second, more direct catalyst was a panel discussion hosted by the Mathematics of Music Analysis interest group at the October 2013 meeting of the Society for Music Theory (SMT). The panelists Jonathan Kochavi, Timothy Johnson, and Mariana Montiel discussed mathematical music theory in the pedagogy of music and mathematics. Jonathan Kochavi’s paper here is adapted from his talk at the SMT meeting, while Mariana Montiel’s and Francisco Gómez’s contribution is partially based on her presentation. The idea for the session originally came from Robert Peck, whose work also appears in this issue. In addition to the invited papers of Kochavi, Montiel–Gómez, and Peck, we also solicited reflections, reports, and pedagogical essays from music theorist Thomas Noll and mathematicians Rachel Hall and James Hughes.