Counterpoint

My talk will connect recent work on voice-leading geometry to the hierarchical ideas of Schenkerian theory.  Its basic thesis is that musical objects such as chords, sets, motives, and scales define two scalar contexts: the extrinsic or containing scale and the intrinsic scale consisting of the object’s own notes.  Attending to the intrinsic scale reveals a quadruple hierarchy in which voices move along possibly scale-like chords that move within scales that are themselves moving through a chromatic aggregate (fixed or continuous, equal-tempered or unequal).  Reconceiving chords as abstract and scale-like objects leads to methodologically unproblematic reformulations of many characteristic Schenkerian claims.

I begin by introducing several practical applications for the notion of the intrinsic scale, including motivic development, generalizing the notion of chordal inversion, harmonizing melodies, modeling efficient voice leading, and extending neo-Riemannian theory to arbitrary chord-and-scale environments.  I then introduce a remarkable set of two-dimensional spiral diagrams that represent each level of the quadruple hierarchy: voice in chord, chord in scale, and scale in aggregate.  These diagrams subsume virtually every existing model of voice leading, including Heinichen’s circle of major and minor keys, Euler’s Tonnetz as reinterpreted by Richard Cohn, Douthett and Steinbach’s “Cube Dance” and “Power Towers,” the circle of fifths, and many others.  By combining multiple diagrams we can connect nonhierarchical voice-leading geometry to the hierarchical structures of Schenkerian analysis, using standard scalar terminology to reinterpret such notions as “verticalization,” “the imaginary continuo,” and “register transfer.”  

Amplifying recent scholarship, our paper dismantles Stravinsky’s iconoclastic self-image as a radical modernist via an examination of his neoclassical style through a tonal lens. Stravinsky frequently adapts clear tonal models to create music that not only “gestures” towards common-practice tonality, but inhabits it (Concerto for Piano and Winds, Largo; Violin Concerto, Aria II).

Both Lynne Rogers’s “dissociation” and Donald Traut’s expanded notion of “displacement” have informed analyses of Stravinsky’s counterpoint. However, we argue that (1) “dissociated” passages often preserve underlying tonal schemas; (2) Traut’s pervasive normalization of perceived displacements is too reductive. 

In our expanded analytical purview, we identify seven contrapuntal techniques Stravinsky employed (including displacement): (1) Harmonic ambiguity, where two possible harmonic interpretations coexist. (2) Displacement of upper-voice notes in relation to the bass; conversely, added bass notes can intrude into the timespan of a well-paced upper voice. (3) Dissonances above a pedal can repeat persistently, regardless of chord changes, and are not obligated to resolve. (4) Common Dissonance types (e.g., suspensions) may be “incorrect” in their melodic or metrical treatment.

Three further techniques are more particular to Stravinsky: (1) The Elision of the end of a harmonic progression into the beginning of another, causing two different harmonic functions to overlap. (2) Heterophony, where two different elaborations of the same schema sound simultaneously—giving rise to forbidden dissonances. (3) Finally, in the case of a Persistent bass, the bass note holds on to the previous harmony for too long. Thus, the upper voices should be taken as structural.

• Example 10
• Examples Packet

In an effort to elucidate aspects of harmonic practice that originated in contrapuntal patterns, this paper traces a specific set of voice-leading configurations in which the generative roles of dissonance, counterpoint, and harmonic progression are blurred. These configurations, originating in early eighteenth-century practice, feature upper-voice “suspension-braids,” which are notable for their interlocking qualities and for their compatibility with an extended range of thoroughbass patterns and bass motions. Segments of such patterns were partially assimilated into later compositional practice and also appeared in treatises on fundamental bass and scale-step systems of harmony, but these configurations revealed a degree of friction between counterpoint-based idioms and the emerging harmonic theories attempting to explain them according to singular principles. The continued appearance of these configurations in subsequent compositional practice therefore raises the question of what aspects of dissonance and counterpoint have been both embedded in harmonic practice and increasingly subordinated to chord structure in the harmonic theories that stretch from Rameau to Schenker. This dialectical engagement between historical theory and compositional practice offers a critique of our inherited harmonic theories, exposing competing origin myths for dissonance and discrepancies over its autonomy relative to harmonic principles. Tracing this history also reveals that the essential connection between rhythm/meter, dissonance treatment, and harmonic progression clearly present in practice (the “rhythmic harmony” acknowledged by Kirnberger) becomes increasingly subordinated to harmonic principles in nineteenth-century theory. The historical path of embedded dissonance appears to exemplify Adorno’s notion of sedimentation, which offers several interesting disciplinary and aesthetic conclusions about harmonic theory.