Sunday 11-12:30 ET

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Poster Session 5: Fourier Analysis / Linear Approaches

Richard Cohn (Yale University), Chair

Musical Maps and Chord Cartographies: Mapping Harmony in Fourier Space

Jennifer Harding (University of Massachusetts Amherst)

bio for Jennifer Harding

Jennifer Harding is a Lecturer of Music Theory at University of Massachusetts Amherst. She holds graduate degrees in music theory and violin performance from Florida State University and Northern Illinois University. Her research uses computational approaches to music analysis to answer questions about harmony in the music of a wide range of composers. Jennifer's recent dissertation, “Applications of the Discrete Fourier Transform to Music Analysis," offers several approaches for investigating harmony through the lens of the Fourier transform.


Musical maps are a visual representation of data from another domain, invoking spatial metaphors to render abstract information more concrete. Recent scholarship using the discrete Fourier transform (DFT) has yielded fruitful theoretical and analytical results by evaluating the “fuzzy membership” of both pitch classes and rhythms to cyclic patterns. Visualizations are helpful for conveying and exploring the complex numerical results of applying the DFT to these musical parameters. Justin Hoffman’s 2008 cartographies plot the location of pitch-class multisets (sets that allow for more than one instance of a pitch class) within spaces generated by the different Fourier components. I map the harmonies from three different musical excerpts onto these spaces, providing insight into both the music and the spaces themselves.

The sonorities in a J. S. Bach chorale exhibit clustering within the fifth Fourier component space (correlating with “diatonicity”). An excerpt from Thomas Adès’s The Four Quarters shows a very different pattern of clustering in the space of the fourth Fourier component, which corresponds to “octatonicity.” The visualization clarifies the relative saturation of the two fully-diminished-seventh chords that comprise an octatonic collection within the sonority. Finally, I map the opening of the “Chorale” from Charles Ives’s Three Quarter-Tone Pieces onto several Fourier component spaces, revealing striking similarities to the chorale by Bach. These similarities suggest that using Fourier space to describe pitch-class collections can be a compelling and methodical way to approach microtonal and other unusual collections.

Supplementary Material(s)

Analyzing Hemiolas with the Discrete Fourier Transform

Aditya Chander (Yale University)


Hemiolas are defined by the conflict of pulses in a 3:2 ratio in at least one level of a metric hierarchy. Existing analytical models of hemiolas capture this conflict but typically overlook other qualities that may distinguish one hemiola from another, particularly the phase difference of the conflicting pulses and their relative strengths. These omissions limit the analytical power of methods developed for hemiola analysis such as ski-hill graphs and semimeters.      

The Discrete Fourier Transform (DFT) can be leveraged to increase the richness of hemiola analysis, providing more nuanced characterisations of the pulses at play. The DFT operates on a sampled time-domain signal (rhythm), outputting the magnitude (strength) and phase of the frequency components (pulse periods) of that signal. The DFT has previously been used to analyse chord quality, large-scale metric form and rhythmic balance, but not hemiolas. 

The DFT captures local variations in pulse strength and phase more effectively than previously developed methods for hemiola analysis due to its note onset sensitivity. This property is demonstrated through its application to an excerpt from Jean Sibelius’s Violin Concerto. Its analytical power offers avenues for further research in music encoding, music perception, and performance timing analysis. While the DFT alone cannot provide a theory of musical meter, it illuminates how rhythms reinforce specific metric interpretations, a question of fundamental importance in hemiola analysis.

Contextualizing Triadic Post-Tonality in Three Preludes from Dmitri Shostakovich’s 24 Preludes and Fugues, Op. 87

Trevor Hofelich (Florida State University)

bio for Trevor Hofelich

Trevor Hofelich is a second-year doctoral student studying music theory at Florida State University, where he is the recipient of the Gail Hendrick Endowment Scholarship. His analytical interests lie within twentieth-century Russian music with an emphasis on tonal analogues and prolongational techniques in the music of Dmitri Shostakovich. He presented his paper, “Contextualizing Triadic Post-Tonality in Three Preludes from Dmitri Shostakovich’s 24 Preludes and Fugues, Op. 87,” at the MTSNYS virtual conference in 2021. In 2019, he earned his master’s degree in Music Theory from Indiana University and was an Adjunct Assistant Professor of Music at Hofstra University in 2019–2020. He has worked independently with professors Joseph Kraus, David Loeb, Lynne Rogers, and Carl Schachter. He holds a master’s degree in Music Composition from Mannes, where he also studied Violin Performance with Sally Thomas. In 2017, Mannes awarded Trevor the Bohuslav Martinu Composition Award for his orchestral piece, Stories of a Phantom, and in 2015 he received the Felix Salzer Techniques of Music Award. Additionally, he appears as a solo violinist on David Loeb’s albums Violin with an Asian Soul in 2018, and Travelogue in 2016.


The majority of Shostakovich studies has privileged analytical investigations of his symphonies and chamber works, but not works for solo piano. The Op. 87 preludes exhibit structural and harmonic tendencies similar to his larger works while demonstrating meta-compositional musing on Baroque preludes. This reflection often takes place through a triadic post-tonal lens that distorts structural direction in favor of expansion. I situate Preludes 4, 13, and 22 in linear frameworks and engage with motivic threads between them.

In Preludes 4 and 13, I illustrate how linear descents occur in multiple dimensions. Octave descents in each cycle of Prelude 4 resemble those in preludes from Bach’s WTC, Book I, reflecting pathways based on the rule of the octave. The composing-out of a descending fourth in triadic post-tonal contexts with the coupling of ^3 throughout, however, produces a static Urlinie. A melodic descending fourth generates harmonic content in Prelude 13. While parsimonious voice leading might be explored in mm. 44–47 of the coda, this passage is better considered the result of motivic nesting, where a diatonic tetrachord is chromatically embellished in an inner voice. 

Like Prelude 4, the structural immobilization of Prelude 22 results from large-scale melodic inactivity. An inert ^3 is prolonged throughout, overriding the direction generated by chromatic meandering in the foreground. Recurring neighboring motions and their chromatic intensifications, combined with the static middleground, contribute to a petrified musical exterior.

This paper explores how Shostakovich’s synthesis of orthodox compositional practice and triadic post-tonality engages with structural hearing. 

Supplementary Material(s)

Urlinie Play and Musical Narrative

Benjamin K. Wadsworth (Kennesaw State University)

Meghan O’Harra (University of Massachusetts Amherst)


Brent Yorgason’s (2020) model of Urlinie Play extends the Schenkerian analytical project to trace rivalries between structural lines that are not limited to the soprano, a situation typical of 19th-century music. Yorgason distinguishes between focal and non-focal (“shadow”) Urlinien, defining categories of Urlinie Play that vary in their degree of structural dependence between rival lines. This elegant typology, however, is not a useful heuristic for questions of musical narrative. To connect Urlinie Play with musical narrative, we examine whether aspects of the Urlinie Play model align with recent theories of musical narrative, particularly Hatten (2004) and Almén (2008). Only Yorgason’s strategy of Competition (a rivalry between two nearly equal lines) suggests the mapping of registral rivalries onto other expressive rivalries, for instance between major and minor modes. We next redefine Urlinie Play as a competition between registers that unfolds throughout a work. In this temporal process, a series of registral objects, each corresponding to a focal line, creates a narrative that can be traced on different levels of abstraction. At the most general level, we propose a new, four-fold set of narrative archetypes that vary mode (major or minor) and direction of focal register (up or down). After tracing these narratives in Chopin’s Preludes, we have found that the archetypes predict stable semantic meanings across different works. In conclusion, reorienting Yorgason’s model around registral competition results in cogent narrative analyses that will interest scholars and performers alike.