Sunday morning, November 8, 10-10:50 CST

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Transformational and Serial Techniques Poster Session

Richard Cohn (Yale University), Chair

Poster sessions begin with a short presentation from each of the poster presenters. A link to this Zoom webinar is just above. Fifteen minutes after the beginning of the session, every poster presenter will enter their own breakout room and entertain comments and questions.

N-dimensional Ski-hill Graphs and Complex Meters

Kája Lill  (University of Michigan)


Most rhythm and meter scholarship that addresses complex meters—meters that include quintuple and larger prime number pulse groupings—examines individual complex meters in isolation, like Justin London (2002) and Godfried Toussaint (2013). Some authors, like Fernando Benadon (2004) and Mark Gotham (2015), have begun to consider how different complex meters relate to one another. I contribute to this topic with a novel analytical method that synthesizes Gotham’s metric relationships and Benadon’s description of tempo- (metric-)modulation into a single graphical representation of complex meters. To do this, I build on the work of Richard Cohn (2001), Scott Murphy (2009), and Daphne Leong (2007) on ski-hill graphs—graphic representations of simple and compound meters that import analytical possibilities from the Tonnetz into the metrical realm—by expanding them from 2- to n-dimensions. N-dimensional ski-hill graphs are ideally suited for analyzing repertoire comprising changing complex meters and especially cases of tempo-modulation by providing succinct graphic representations of those meters and highlighting relationships that otherwise remain opaque. This poster first shows how I generalize Cohn’s 2-dimensional ski-hill graphs, which feature subdivisions by two or three, to account for subdivision by prime numbers larger than two or three. It then shows that by expanding ski-hill graphs into n-dimensions they can account for any number of prime number subdivisions. For demonstration, I provide analyses of complex meters utilizing 2-, 3-, and 4-dimensional ski-hill graphs respectively in Gustav Holst’s “Mars, The Bringer of War,” Elliot Carter’s Cello Sonata, and Leoš Janáček’s “Fanfare” from his Sinfonietta.

Reconsidering Negative Harmony: Melodic Dualism in Bárdos’ Scalar Schemata

M.A. Coury-Hall (New York City)


In this poster, we consider the unique mathematical transformation of pitches that preserves stability in the 7-tone scale systems identified by Hungarian music theorist Lajos Bárdos (1963). This transformation is referred to by several names in the literature including the popular term Negative Harmony and has its modern roots in Ernst Levy’s flawed work A Theory of Harmony. Although Levy’s work has dubious status among music theorists, his concept of Negative Harmony has continued to capture the imagination of composers and improvisers (most recently in jazz with Herbie Hancock and Jacob Collier as popular promoters of the idea). We show that this concept, a modern extension of Oettingen and Riemann’s concept of harmonic duality, which Riemann later notoriously reworked, can be used to construct the melodic Major/minor duality in the Bárdos collection of 7-tone scales, which includes the diatonic collection among other scales. Although a trailblazing theorist on this topic, Bárdos did not recognize the full mathematical and musical properties of this group and its structure-preserving properties. This poster seeks to fill a gap in the literature in the theory of these structures. We will show that the resulting melodic dualism (Cohn 2012) derived from a mathematical treatment of the transformation, in contrast with the harmonic dualism, and the technique’s efficient voice-leading (Tymoczko 2011) with respect to verticalization and horizontalization (Yust 2018) warrants further investigation by music theorists, especially in the context of the Bárdos Collection, and deserves a rigorous treatment in mathematical music theory.

Schubert, Schoenberg, and Some Extensions to Cohn’s Sum-Class System

David Orvek (Indiana University Bloomington)

David Orvek is a Ph.D. student at Indiana University’s Jacobs School of Music where he also works as an associate instructor for the undergraduate music theory sequence. He holds degrees in music theory from Southern Adventist University and The Ohio State University. His research to date has focused on transformational approaches to voice leading, particularly as it applies to music of the late-nineteenth and early-twentieth centuries. His presentation today is based on his 2019 master’s thesis entitled, “Generalized Transformational Voice-Leading Systems.” In addition to his interest in transformational theory, David has also done work with computer-assisted music analysis and corpus studies and is an active classical guitarist.


In this poster, I propose a generalized transformational system for studying voice leading within a single set class. Such a system not only provides ready-made transformational machinery for a variety of repertoires and analytical contexts, but also reveals similarities in the voice-leading possibilities within different set classes. While there exist transformations designed to act on large families of set classes (Straus 2011; Fiore and Satyendra 2005, among others), no studies to date have considered the relationship of these transformations to voice-leading efficiency or integrated them into a larger, generalizable transformational system. As I show in this poster, Richard Cohn’s (1998) “sum-class” system—originally designed for consonant triads—provides a framework for just such a system.

I begin by reviewing the basics of Cohn’s sum-class system through an analysis of a passage from Schubert’s G-minor violin sonata and then show how this system can be modified to accommodate other set classes. I then use this generalized sum-class system to observe the ways in which voice leading in passages from Schoenberg’s Pierrot lunaire and Das Buch der hängenden Gärten is similar to that seen in the Schubert excerpt. Finally, I conclude with a brief survey of my expanded sum-class system as manifest in other pitch structures. This not only suggests that the possibilities of within-set-class voice leading are quite similar for sets of the same cardinality, but also reveals that sum-class systems for some set classes are strongly related to familiar conceptual structures like the circle of fifths.

Mapping Schnittke’s Sequences in Bonded Uniform Triadic Transformation Spaces

Lauren Hartburg (Florida State University)


Uniform Triadic Transformations (UTTs [Hook 2002]) may be used to generate three-dimensional voice-leading spaces that accommodate Alfred Schnittke’s triadic language when tonal and twelve-tone systems fall short. Recent explorations of Schnittke’s triadic music include Segall’s PSM voice-leading spaces (2017) and Honarmand’s aggregate and quasi-aggregate completion (2019). I propose that the compound transformation PRP used in Honarmand’s analysis of Schnittke’s Piano Sonata no. 1 is not necessary since each of these motions can be understood as a single application of the UTT 〈−,3,2〉. In this study, I consider triadic passages from the first and third movements of Schnittke’s Piano Sonata no. 1 that employ the UTT 〈−,3,2〉. While the UTT 〈−,3,2〉 is the primary measure of voice-leading proximity in these passages, I propose secondary measures of distance based on common tone retention and parsimonious voice leading. For the sake of this study, harmonies adhering to the secondary parameters of voice-leading are limited to P, L, R, S, L', and R' (Morris, 1998). Through its union with Neo-Riemannian operations, UTT 〈−,3,2〉 space gains three-dimensional shape as every-other triad in the sequence “bonds” to a partner that is related by a secondary voice-leading measure. This bonding reveals a secondary UTT that can also be observed in a tile of the bonded-UTT space, constructed through the intersection of primary and secondary UTTs at 90o angles. These spaces are ideal for mapping Schnittke’s harmonic progressions as they allow for “slippage” within a primary sequential pattern to closely related substitute harmonies belonging to the secondary UTT.

George Theophilus Walker: A Unique, African-American, Voice in Twelve-Tone Music

Jack Boss (University of Oregon)

Tim S. Pack (University of Oregon)


The plenary session at the 2019 SMT, among other things, implored music theorists to expand our repertoire beyond the handful of white, male composers on whom we have focused for most of our history. It is possible to couple new repertoire with new ways of analyzing, but it is also fruitful to apply traditional analytic techniques to the music of underrepresented composers. In this way, we can comprehend how their music is similar to or different from composers in the canon by using the same “measuring stick” for both. Our poster applies traditional twelve-tone approaches, supported by observations about registral and rhythmic patterns, to the piano piece Spatials by African-American composer George Walker. Row counting the piece reveals that Walker reiterates the same row form within each of its six variations, only changing forms between variations. Nevertheless, he finds his own ways to create balance between unity and diversity in Spatials. As the presentation will show in more detail, he creates a large arch form by progressing from statements of octatonic subsets of the row as contiguous row segments in the early variations, to presentations of the same small group of set classes as “secondary harmonies” (according to Hyde 1980) in the middle, and back to statements of the contiguous versions at the end. The poster will illustrate numerous ways in which Walker creates diverse intervallic, rhythmic, and registral structures from a unified set-class repertoire following this large arch, and shows himself to be a unique voice within twelve-tone music.