Neo-Riemannian Excursions

Even with numerous scholars extending chromatic-voice leading models to tonal music (a.o. Rings 2011; Tymoczko 2011; Cohn 2012), the origins and formal functions of sequential chromatic third relations in the 18^th century remain opaque. Early instances of hexatonic cycles and Weitzmann regions cited by Cohn exhibit “double syntax”: Dominant-tonic syntax at the local level with middle-ground tonics related by parsimonious voice-leading. In this paper, I will triangulate a pluralist phenomenology of the chromatic major-third sequence that begins the first-movement development section of Mozart’s Divertimento for string trio in E♭, K. 563.  How, after the exposition’s B♭ major dominant area, does the sequence cast its terminal B♭ minor in a light that is equally expected and surprising? I argue that attention to register, voicing, and outer-voice counterpoint (salient to performers of this music) are key agents shaping the telos of these remarkable measures.

A neo-Riemannian allows us to model some of the terminal B♭ minor’s startling effect, revealing the end of the sequence as a break in a “zig-zag” path through incomplete N/R and L/P chains that flesh out successively more complete key areas. A Schenkerian approach captures the linear derivation of the implied F♯ minor from B♭ via chromatic 5-6 motion. Yet the prioritization of linearity risks flattening the intervallic diversity of Mozart’s outer-voice counterpoint. Far from mere surface phenomenon, this counterpoint serves as a cue to prima vista players as to where their parts may be headed, revealing this “purple patch” as chromatic structure with a social dimension.

The parallel tonal and narrative trajectories of Amy Beach’s “Autumn Song” (1904) culminate in a single moment: a wordless melisma that happens at the midpoint of an octatonic-triadic cycle. So named in order to distinguish it from the octatonic system of seventh chords, the octatonic-triadic cycle alternates R and P transformations, producing a cycle of eight triads drawn from one of the three forms of the octatonic scale. R/P cycles bear many similarities to the more prominent hexatonic (P/L) cycle, although because of their length they are often left incomplete. The greater voice-leading “work” required by R (which moves a single voice by whole step rather than half step) creates a voice-leading asymmetry by which clockwise and counter-clockwise movements through an odd number of stations involve different semitonal displacements. Each chord in the cycle thus has both an octatonic pole and two octatonic near poles, with which it shares no pitches. These near poles are separated by a different number of half-steps, and even different root intervals.

This paper argues that “Autumn Song” hinges on this textless moment of tonal instability: the pivotal moment’s enharmonic reinterpretation positions the song’s ordinarily proximate tonics (the initial minor and closing parallel major) on opposite sides of a tonally distant octatonic near pole. Beach problematizes the relationship between parsimonious voice leading and diatonic tonality, depicting musically—between verses of the poem—the psychological interiority of the protagonist, who emerges from the song’s rotational formal and harmonic processes with a changed outlook.

Since the publication of the influential and pioneering work of music theorist David Lewin (1987), group theory has informed many rigorous studies in music. However, mathematically-inclined music theorists need not be bound by group-theoretical thinking alone.  This paper examines how groups and, in particular, groupoids handle symmetry in music with groupoids exemplifying a "rogue" symmetry.  Music theorists have recently considered groupoids in a categorical setting applied to poly-Klumpenhouwer Networks (PK-Nets) [Popoff et al., 2019].  An equivalent algebraic approach considers a groupoid of order 12 within the Riemann group of UTTs, a normal subgroup of the group formalized by Julian Hook in his seminal paper “Uniform Triadic Transformations” [Hook, 2002].  This groupoid includes the important PLR transformations and the N transformation [Cohn, 2000] of Neo-Riemannian theory.  An examination of the algebraic- and music-theoretic structure of this groupoid follows in detail, and this paper demonstrates how tonic and dominant functions arise as epiphenomena of the groupoid structure.  All of the appealing musical properties of this groupoid are a result of the transformations having a distribution that can be described as maximally even which expands the analytic scope of this concept from diatonic scale theory [Clough and Douthett, 1991] to transformational theory.  Beethoven’s “An die ferne Geliebte” (1816) and Bartók’s “Music for Strings, Percussion, and Celesta” (1936) provide basic and advanced examples respectively of this analytic technique.

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