Types and Applications of P3,0 Seventh-Chord Transformations in Late Nineteenth-Century Music

I. P3,0 in the Context of Neo-Riemannian Voice-Leading Transformations[1.1] P3,0 refers to one class of parsimonious voice-leading connections between seventh chords introduced in an article by Jack Douthett and Peter Steinbach (1998) as Pm,n. In addition to common tones that may be held between the two chords, the subscripts indicate the number of voices that move by half step (m) or whole step (n) in progressing from one chord to another. Thus, the smaller the sum of m and n, the more parsimonious the connection, and the more closely related the two chords are from a neo-Riemannian perspective. P3,0 indicates a relationship between two seventh chords in which holding one tone in common and moving each of the other three by half step will transform one chord into the other.[1.2] Numerous other studies of voice-leading transformations in nineteenth-century chromatic harmony appeared around the time Pm,n was identified and in the years since. A comprehensive survey of these investigations is not practical here, but it is useful to note that studies of transformations based on such common-tone-plus-stepwise voice leading tend to fall into three general categories, as listed below:1. Triadic transformations, usually among major and minor triads, or set class 3-11 (037); see Cohn 1996, Cohn 1997, and Kopp 2002.(1)2. Seventh-chord transformations among chord types of the same set class, especially dominant and half-diminished, or set class 4-27 (0258); see Lewin 1996, 205ff., Childs 1998, and Callender 1998.3. "Cross-type" transformations among:a. triads and seventh chords; see Capuzzo 2004, Hunt 2007, and Hook 2007;(2) orb. triads or seventh chords of non-equivalent set class; see Douthett and Steinbach 1998 and Tymoczko 2011, 284ff.(3)[1.3] These and related studies typically construct Tonnetze or similar networks to illustrate how specific kinds of voice-leading transformations create discernible, logical pathways through harmonic spaces.(4) Frequently, the networks presented have been limited to transformations among consonant triads, and only a small percentage of these studies have sought to explore through analysis how transformational voice-leading interacts with voice-leading associated with traditional harmonic practice. Steven Rings, for instance, has observed that a neo-Riemannian analysis often "[begins] with the most chromatically exceptional moment in the piece and [moves] outward from there to construct a broader interpretation" (2011, 492). In these cases, considerations of key and harmonic syntax are undervalued in favor of tonally neutral chord networks.(5)[1.4] One of the advantages of revisiting Pm,n is that it is inherently generic. That is, while it has potential for the construction of networks and associated graphs (as Douthett and Steinbach so effectively demonstrate), it also possesses the flexibility to operate independently of such networks, and in ways that facilitate the analysis of music that blends tonally unencumbered voice-leading transformations with tonal harmonic syntax.[1.5] Regarding transformations in the second and third categories above, Pm,n is particularly attractive for its applicability to a large number of seventh-chord connections involving parsimonious voice leading and chords of either the same or different set classes. To date, however, analytical studies that examine seventh-chord connections from this perspective have tended to focus on transformations with specific subscript values. Douthett and Steinbach, for example, explore only P1,0 and P0,1 among chords of different set classes (dominant/half-diminished sevenths and minor sevenths) and P2,0 among chords of a single set class (dominant/half-diminished sevenths). The P3,0 relationship has not been discussed in any detail as a specific transformation class, yet it holds considerable interest in its ability to account for a variety of seventh-chord successions in late nineteenth-century music. …