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

# Transformations

Ed Gollin (Williams College), Chair

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

Modal Spelled Pitch Classes

## Abstracts

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

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

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 _{7} × ℤ_{7},*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.