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Studio B • Thursday afternoon, 2:00–3:30

Transformations

Ed Gollin (Williams College), Chair

Robert Wells (University of Mary Washington)

David Lewin and the “GIS that Wasn’t”: Interactions Between Musical and Mathematical Thought in GMIT

Nathan Lam (Indiana University)

Modal Spelled Pitch Classes

Abstracts

David Lewin and the “GIS that Wasn’t”: Interactions Between Musical and Mathematical Thought in GMIT

Robert Wells (University of Mary Washington)

David Lewin’s generalized interval system (GIS) and transformational theories have profoundly impacted the music-theoretic discipline since the publication of Generalized Musical Intervals and Transformations (1987), inspiring wide-ranging analytical and theoretical studies. While the mathematical underpinnings of Lewin’s theories are well known, Hook (2007b) and Hall (2009) have noted Lewin’s frequent departures in GMIT from standard mathematical writing, often blurring the boundaries between music and mathematics. Tymoczko (2009), too, observes how Lewin’s musical assumptions often color his mathematics, while the mathematics sometimes limits musical applications.

To more precisely characterize Lewin’s approach to mathematical music-theoretic writing, the current paper will consider how Lewin’s framing of the GIS construct early in GMIT demonstrates an idiosyncratic bridging of musical and mathematical realms. Specifically, the first part of this paper will consider how Lewin’s int functions define diverse mappings between musical spaces (involving pitch, rhythmic, and harmonic objects) and mathematical ones (generally numerical spaces or mathematical groups). The second part of this paper will then consider, using Lewin’s “failed” duration GIS as a case study, how Lewin’s implicit restrictions on these spaces and their interrelations can be unnecessarily limiting. A temporal analysis of a Bach crab canon will exemplify how less restrictive boundaries on musical and mathematical domains can motivate new avenues of transformational research.

Modal Spelled Pitch Classes

Nathan Lam (Indiana University)

This paper proposes modal spelled pitch class (mspc) as an extension upon Julian Hook’s spelled heptachords (2011). Mspcs formally specify the tonic for the analysis of diatonic, centric music. The theory’s utility is similar to that of Steven Rings’s Tonal GIS (2011); however, while Rings’s theory is based on mod-12 pitch classes, Hook’s system is situated on the line of fifths, and it represents diatonic objects more efficiently and elegantly.

Mspcs take the form of an ordered triple that includes the key signature, the generic pitch class (gpc, letter name without accidentals) of the note in question, and the tonic’s gpc. The structure of the group is ℤ × ℤ7 × ℤ7, and the group forms an interval space (Lewinian GIS). Although the three components could be transposed independently, I will focus on the coordinated transpositions of these components. Coordinated transpositions correspond to familiar compositional devices such as diatonic transposition, parallel inflection, scale-degree reinterpretation, chromatic modulation, etc., which are all abstract transpositions within my framework.

I will demonstrate analytical usage that intersects with the triadic-transformational analyses of nineteenth-century chromatic mediants, and burgeoning research in diatonic modality. The paper concludes with a few notable mspc T-nets, including Schubert’s Piano Sonata in B-flat major and Holst’s First Suite for Military Band.