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Studio E • Friday afternoon, 2:00–5:00

# Topics in Geometrical Music Theory

Rachel Hall (Saint Joseph’s University), Chair

Near-Symmetry: A Theory of Chord Quality with Implications for Voice Leading

Generic (Mod-7) Voice-Leading Spaces

Generalized Trichordal and Tetrachordal Tonnetze: Geometry and Analytical Applications

Generalized Normal Forms

## Abstracts

Near-Symmetry: A Theory of Chord Quality with Implications for Voice Leading

Traditional methods of musical analysis tend to treat chord symmetry as a binary property: chords are either symmetrical or they are not. Symmetry is thus understood as a static property that inheres in a chord. This paper proposes instead a dynamic, transformational approach to chord asymmetry by addressing two questions: 1) is it possible to render an asymmetrical chord symmetrical by adjusting one of its tones by some distance (cardinality stays the same)? And 2) is it possible to render it symmetrical by adjoining one or more tones (cardinality increases)? The authors refer to these distinct symmetrical potentialities as ‘degrees of near–symmetry.’

To measure the symmetrical potentiality of a collection, the authors draw and extend upon the literature on parsimonious voice-leading distances as elucidated in Callender, Tymoczko and Quinn (2008) and Douthett and Steinbach (1998), as well as on the atonal voice-leading literature by Straus (1997, 2003, 2005, 2011) and transformational theory by Lewin (1982, 1987, 1992). Having calculated the near–symmetry degrees of all pc-sets from cardinalities 3 through 6, the authors explore relationships between pc-sets not generally considered to be closely related but that share the same (or similar) symmetrical potentiality and examine how these otherwise-disparate sets are deployed in similar manners across the work of various composers. The authors also explore the various ways in which pc-sets with special degrees of near–symmetry behave in certain parsimonious and SUM-class voice-leading spaces.

Generic (Mod-7) Voice-Leading Spaces

In the burgeoning field of geometric music theory, scholars have
explored ways of spatially representing voice leadings between chords. The
*OPTIC* spaces provide a way to examine all “classes” of *n*-note
chords formed under various types of equivalence: octave, permutational,
transpositional, inversional, and cardinality. Although it is possible to map
diatonic progressions in these spaces, they often appear irregular since the
spaces are constructed with the fundamental unit of a mod-12 semitone, rather
than a mod-7 diatonic step. Outside of geometric music theory, the properties
of diatonic structure have been studied more broadly: Clough has established
framework for describing diatonic structure analogous to that of Forte’s set
theory; Hook provides a more generalized, “generic,” version of this work to
describe any seven-note scale. This paper employs these theories in order to
explore the fundamental difference between mod-12 and mod-7 spaces: that is,
whether the spaces are fundamentally discrete or continuous.

After reviewing the construction of these voice-leading spaces, this
paper will present the mod-7 *OPTIC-*,* OPTI*-, *OPT*-, and
*OP*-spaces of 2- and 3-note chords. Although these spaces are
fundamentally discrete, they can be imagined as lattice points within a
continuous space. This construction reveals that the chromatic (mod-12)
and generic (mod-7) voice-leading lattices both derive from the same
topological space. In fact, although the discrete versions of these
lattices appear to be quite different, the topological space underlying
each of these graphs depends solely on the number of notes in the chords
and the particular *OPTIC* relations applied.

Generalized Trichordal and Tetrachordal Tonnetze: Geometry and Analytical Applications

Some recent work on generalized *Tonnetze *has examined the
topologies resulting from Richard Cohn’s common-tone based formulation,
while other work has reformulated the *Tonnetz* as a network of
voice-leading relationships and investigated the resulting geometries.
This paper considers the original common-tone based formulation and takes
a geometrical approach, showing that *Tonnetze* can always be
realized in toroidal spaces, and that the resulting spaces always
correspond to one of the possible Fourier phase spaces. We can optimize
the given *Tonnetz* to the space (or vice-versa) using the DFT. Short
analytical examples from Stravinsky’s “Owl and the Pussycat” and
Shostakovich’s String Quartet no. 12 demonstrate how the embedding in
phase spaces broadens the potential application on non-triadic trichordal
*Tonnetze*.

Two-dimensional *Tonnetze* may be understood as simplicial
decompositions of the 2-torus into regions associated with the
representatives of a single Forte set class, making simplicial
decompositions of the 3-torus a natural generalization to tetrachords.
This means that a three-dimensional *Tonnetz* is a network of
*three* tetrachord-types related by shared trichordal subsets.
Essential to constructing the three-dimensional *Tonnetze* is the
duplication of interval classes with distinguishable intervallic axes. I
illustrate one possible three-dimensional *Tonnetz*, whose duplicated
ic3s can be enharmonically distinguished as minor thirds or augmented
seconds, in an analysis of Brahms’ Sarabande WoO 5/1 and its reuse in the
Op. 88 Quintet. Duplicated intervals in other three-dimensional
*Tonnetze* may be understood through Hauptmannian or tuning-theory
based distinctions or distinctions between chordal and non-chordal
intervals.

Generalized Normal Forms

This talk strengthens the connections between pitch-class set theory
(Forte et al.) and geometric music theory (Callender, Quinn, Tymoczko) by
showing that generalized versions of “normal forms” or “prime forms” may
be derived under any combination of the *OPTIC* equivalence
relations. In this conception, the usual “normal order” of a collection of
notes is its *OPC* normal form, inasmuch as all collections sharing
the same normal order are related by some combination of octave,
permutational, and cardinality equivalence. The familiar “prime form” is
the *OPTIC* normal form, which relies on transpositional and
inversional equivalence as well. Calculation of normal forms corresponding
to other subsets of the *OPTIC* relations helps to clarify ways in
which different sets or strings of notes may be related; as more relations
are added, more things become equivalent, and normal forms become simpler.
Normal forms provide a systematic means, previously lacking, for labeling
maps of *OPTIC* spaces, and may be used to define *normal
regions*, useful aids to visualizing the smaller spaces that arise
through the addition of new relations to those already present in some
larger space. The talk will review the *OPTIC* relations, present a
detailed algorithm for the calculation of all normal forms, and offer
examples of normal forms, normal regions, and ways in which they may be
used.