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

Topics in Geometrical Music Theory

Rachel Hall (Saint Joseph’s University), Chair

Inés Thiebaut (University of Utah) and Nicholas Nelson (Stony Brook University)

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

Leah Frederick (Indiana University) 

Generic (Mod-7) Voice-Leading Spaces

Jason Yust (Boston University)

Generalized Trichordal and Tetrachordal Tonnetze: Geometry and Analytical Applications

Julian Hook (Indiana University) 

Generalized Normal Forms

Abstracts

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

Inés Thiebaut (University of Utah) and Nicholas Nelson (Stony Brook University)

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

Leah Frederick (Indiana University)

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

Jason Yust (Boston University)

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

Julian Hook (Indiana University)

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.