Neo-Riemannian theory's demonstration of the susceptibility to group-theoretic interpretation of elements of Hugo Riemann's theories has sparked a reappraisal of nineteenth-century harmonic theory, focusing on its nascent grouptheoretic content. This is evident in the resurgence of interest in Riemann's Schritt/Wechsel system (which can be interpreted as a formulation of a group isomorphic to the neo-Riemannian LPR group and familiar Tn/TnI group), and in Richard Cohn's explorations of connections between neo-Riemannian theory and Carl Friedrich Weitzmann's 1853 monograph on the augmented triad. In his essay “Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective,” Cohn presents a broader view, suggesting that “the term ‘neo-Riemannian’ is most pertinently viewed as synecdochally appropriating the name of Riemann to represent a tradition of German harmonic theory which his writings culminated and perpetuated,” and observing in particular that the neo- Riemannian approach “recuperates a number of concepts cultivated, often in isolation of each other, by individual nineteenth-century harmonic theorists.” Cohn identifies and locates historical precedents for six such concepts: triadic transformations, common-tone maximization, voice-leading parsimony, “mirror” or “dual” inversion, the Tonnetz, and enharmonic equivalence.

Neo-Riemannian theory has also fueled speculation about a group-theoretic orientation of theorists during the nineteenth century. This essay responds to Cohn's conjecture that the consonant triad might have contributed in the rise of a group-theoretic perspective. Specifically, Cohn suggests that the consonant triad's “over-determined” nature—that is, its status as an optimal structure from the perspectives of both acoustic generability and voice-leading parsimony—may have led nineteenth-century theorists to inadvertently cultivate the group-theoretic perspective later made explicit by the neo-Riemannian movement.

In fact, two implicitly group-theoretic models of triadic relations, reflecting the over-determined triad's twofold nature, are suggested in works by Moritz Hauptmann, Arthur von Oettingen, and Riemann: one privileges maximal common- tone retention and incremental voice leading, and one privileges acoustically proximate (fifth- and third-based) root-interval relations. Though both approaches support a fully chromatic perspective (the former gives rise to the neo-Riemannian LPR group and the latter to the isomorphic Schritt/Wechsel group), nineteenth-century authors associated the common-tone approach with diatonic tonal space and the root-interval approach with chromatic tonal space, underscoring the over-determined triad as a source of conflict between diatonic and chromatic conceptions of tonal organization. Though Hauptmann, Oettingen, and Riemann did not invoke the terminology of combinatorial group theory, their discussions of relationships among triads do appeal to a combinatorial perspective.