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Voice Leading Spaces and Transformation

Robert L. Wells (University of Mary Washington), Chair

Collection Space: Systematizing Parsimonious Transformations in French Scalar Tonality

Matthew Kiple (Temple University)


The “scalar tonality” associated with the music of early twentieth-century Russian and French composers represents one path at a tonal crossroad, located at what Cohn (2012) posits as a third stage of tonal evolution. Analytical methods used to demystify this music focus on context rather than systemic pitch hierarchies, emphasizing harmony’s additive rather than reductive properties. Most scholars demonstrate how collections such as whole-tone, octatonic, and acoustic, etc., supplant the diatonic collection as primary referential objects, but a comprehensive system of parsimonious transformations among referential collections has yet to be excavated. In lieu of such a system, I offer what I call “collection space”—a parsimonious intercardinal voice-leading space for maximally and nearly even sets—as a transformational space representative of Cohn’s third tonal-evolutionary stage.

This paper is divided into three parts: (1) a rebuttal of criticisms toward graphical spaces of voice-leading relations among intercardinal multisets, (2) the construction of collection space as an exclusive network of parsimonious connections among maximally and nearly even collections of cardinalities four through eight, analogized with Cohn’s pan-triadic and Tristan-genus systems, and (3) a demonstration with animated analyses of music by Fauré, Debussy, and Lili Boulanger, coordinated with live piano demonstrations. I also illustrate how collection space connects systemically to pan-triadic and Tristan-genus systems via what I call “pivotal subsets.” My intention is to elucidate the abundance of possible voice-leading connections and harmonic juxtapositions characteristic of scalar-tonal French music, and to demonstrate how collection space reflects unequivocally this music’s tonal iridescence.

Generic (Mod-7) Approaches to Chromatic Voice Leading

Leah Frederick (Oberlin College & Conservatory)

Leah Frederick is Visiting Assistant Professor of Music Theory and Aural Skills at Oberlin College & Conservatory. Her recent research uses mathematical techniques to examine the relationship between diatonic and chromatic conceptions of musical space. Her work on voice leading in mod-7 space was awarded the Society for Music Theory’s 2020 SMT-40 Dissertation Fellowship and Music Theory Midwest’s 2018 Arthur J. Komar Award. Her article, “Generic (Mod-7) Voice-Leading Spaces,” was published in the Journal of Music Theory in 2019.

Frederick holds a Ph.D. in Music Theory from Indiana University and Bachelor’s degrees in both Mathematics and Viola Performance from Penn State University. While at Indiana University, she served as editor of the Indiana Theory Review and received the Wennerstrom AI Fellowship for outstanding teaching. Frederick is also a violist, and her writing on the viola repertoire has appeared in the Journal of the American Viola Society.


Recent extensions to Clough’s (1979) diatonic set theory have adapted mathematical approaches to capture voice leading in mod-7 space. Although often used to describe diatonic progressions, these transformational and geometric systems are constructed from generic pitch space, meaning that each element in these spaces represents an entire equivalence class containing a letter name with any accidental attached. Any generic voice-leading structure can be interpreted as a chromatic progression by inflecting each generic chord with a different scalar collection; thus, chromatic voice leading is understood as two concurrent levels of voice leadings: one at the level of the generic structure and another at the level of the underlying scale.

This paper combines existing transformational and geometric tools for describing voice leadings between generic chords (Frederick 2018, 2019) with analogous approaches to scales (Hook 2008, 2011; Tymoczko 2004, 2011) to provide a new perspective on chromatic voice leading informed by diatonic set theory. Unlike the mod-12 neo-Riemannian approach, this mod-7 conception of chromatic voice leading can efficiently describe both functional and non-functional chromatic relationships, as well as differentiate between enharmonically equivalent spellings of chromatic chords. This paper introduces the chromatic voice-leading transformation group, which acts on the infinite set of closed-position triads belonging to any diatonic collection. This system acts only on complete closed-position triads and diatonic collections; however, it is possible to capture similar information about the voice leading in progressions with non-triadic chords and non-diatonic scales using geometric techniques for both chords and scales.

Parsimony in Microtonal Music

Greg Hartmann (The Graduate Center, CUNY)

Pianist Greg Hartmann is currently pursuing his doctoral studies at the Graduate Center, CUNY, as a student of Julian Martin. Greg recently won first prize in the 2018 Memphis International Piano Competition, second prize in the top division of the 2019 Schubert Club Scholarship Competition, third prize in the 2019 Thousand Islands International Piano Competition, third prize in the 2018 High Point University Piano Competition, and received the Jung-Springberg Award for Outstanding Musicianship in the 2018 Kuleshov International Piano Competition. He was also the first prizewinner in the 2016 Walter A. and Dorothy J. Oestreich Concerto Competition, 2016 Rochester Symphony Young Artist Competition, and the 2016 Lakeshore Wind Ensemble Young Artist Competition. Also an accomplished composer, Greg won the 2018 Paula Nelson-Guenther Young Composer Competition with his orchestral work Requiem for a Memory: Nocturne for Orchestra. He has performed concerti with the New Albany Symphony Orchestra, Rochester Symphony, Lakeshore Wind Ensemble, Concord Chamber Orchestra, and Waukesha Area Chamber Orchestra. He also maintains an interest in Music Theory, in which he also holds a Master’s degree. Greg has performed in masterclasses for many renowned artists including Robert McDonald, James Tocco, Daniel Shapiro, Eugene Pridonoff, Roland Krueger, James Giles, and Douglas Humpherys, and has participated in music festivals including the Aspen Music Festival and School, Pianofest in the Hamptons, the Bowdoin International Music Festival, the Sejong International Music Festival, Euro Music Festival and Academy, and the Gijón International Piano Festival. In his free time, Greg enjoys tennis, running, and ping pong.


Parsimonious voice leading has been well studied for tonal music, but the existing literature defines parsimony in a way that severely limits its application to pitch-class spaces with a cardinality larger than 12. Instead of conceiving of parsimony as a minimal number of the smallest step in a given cardinality, we can consider allowing the equivalent of one standard whole step of voice leading motion (up to 1/6 of an octave or c/6 steps in any cardinality c.) This prevents the motion from becoming imperceptibly small at higher cardinalities and allows for parsimonious trichords analogous to (037) in cardinalities that are not a multiple of three.

Following Straus’s (2005) similarity measure for set-classes, I also construct a concept of fuzzy set-classes. This allows the comparison of set-classes in higher cardinalities with familiar set-classes (where c = 12) and allows the analyst to emphasize the similarity of two sets that may not belong to the same set-class.

The expanded application of these existing tools to this new repertoire not only allows us to more fully understand the voice leading structures in microtonal pieces, but also reveals parallels between techniques of microtonal composition and the more well-studied techniques of tonal composition. For instance, I demonstrate that it uncovers operations in microtonal music analogous to the familiar Neo-Riemannian P, L, and R operations. Overall, I show that an extension in the definition of parsimony reveals structures in microtonal music analogous to structures in tonal music.