John Roeder

Harmonic Implications of Schoenberg's "Observations of Atonal Voice Leading"

Modern music that uses chords of six or more parts seems to be at a stage corresponding to the first epoch of polyphonic music. Accordingly, one might reach conclusions concerning the constitution of chords through a procedure similar to figured bass more easily than one could clarify their function by the methods of reference to degrees. For it is apparent, and will probably become increasingly clear, that we are turning to a new epoch of polyphonic style, and as in the earlier epochs, harmonies will be a product of the voice leading: justified solely by the melodic lines! (Schoenberg [1922] 1978, 389).

A Declarative Model of Atonal Analysis

Most computational models of musical understanding have focused on procedural aspects of analysis, suggesting techniques for parsing, comparing, and transforming various representations of a piece, or adapting discovery procedures of artificially intelligent (AI) inference systems, which plan and follow agendas and goals. Much contemporary AI research, however, also focuses on declarative aspects of knowledge, attempting to define data representations and relations that are commensurate with human cognition. Naturally, musical analysis has both procedural and declarative aspects: the declarative determines what the form of the analysis is, and the procedural determines how the analysis is obtained. However, a predominantly procedural analysis risks sacrificing the form of musical understanding to obtain efficiency or compatibility with a particular computer language. In this article I argue that, for a significant body of twentieth-century music, a declarative system models the structure of analytical understanding better than do existing procedural programs, and I present a functioning declarative system that infers complex musical structures from the elementary musical relations that it identifies.

A Transformational Space for Elliott Carter’s Recent Complement-Union Music

Elliott Carter’s recent music exploits a special combinatorial property of the all-trichord hexachord. I show how this property can be reconceived in terms of interesting and analytically significant musical transformations: three involutions on the pitch-class aggregate which constitute a Klein four-group, and which have a natural interpretation as the symmetry group on a particular 12-vertex geometrical structure. Accordingly the opening of Carter’s Figment II for solo cello can be analyzed transformationally as a complete traversal of this structure by just a few, striking, characteristic gestures.