A Computational Composer's Assistant for Atonal Counterpoint.(Brief Article)

Every pitch in rigorous common-practice tonal counterpoint (for instance, a Bach fugue) is subject to definable constraints in a matrix of at least two dimensions: the melodic and the harmonic. In a strict canon, for example, each pitch forms a part of a thematic pattern that recurs elsewhere in the texture, while that same pitch functions to define or to embellish a related series of harmonies. Sixteenth-century composers such as Palestrina and Lassus and 20th century composers such as Bartok, Stravinsky, and Schoenberg have constructed counterpoints according to very different--but equally rigorous--melodic and harmonic constraints. Owing to the dual harmonic/melodic constraints on each pitch in any closely-controlled modal, tonal, or atonal style, independent contrapuntal lines which align to form coherent harmonies often appear to be felicitous solutions to a complex set of simultaneous equations: discoveries as much as creations.

Counterpoint Assistant (CPA), a composer's assistant program written in Macintosh Common LISP (MCL), is a mathematical tool for discovering such solutions--albeit by an indirect empirical method (Jones 1993). CPA is designed to reveal transpositions and temporal offsets of user-created melodic lines that can be superimposed to meet user-specified constraints on harmony, texture, and counterpoint. While some music theorists have elaborated the complex relational and generative structures that underlie what appear to be relatively simple contrapuntal systems (Morris 1995), the simpler empirical approach implemented in CPA is an attempt to facilitate the exploration of a much wider range of contrapuntal relationships specifically for use in composition. CPA differs from programs which automatically generate counterpoint in 16th-century style (Schottstaedt 1984; Polito, Daida, and Bersano-Begey 1997) or in specific contemporary styles (Hunt 1974; Bell 1995) in that CPA itself provides no specific harmonic or melodic information about the counterpoint to be generated. While CPA does facilitate a chromatic rather than a diatonic approach to melodic transposition and to harmonic control, every individual melodic and harmonic element in the counterpoints produced by CPA is specifically composed or closely constrained by the user. Rather than replacing the function of the composer, CPA is designed to assist the composer in developing the composer's own coherent matrix of melodic and harmonic elements.

Typically, the user composes and enters a contrapuntal sequence of notes consisting of one to six given lines. The user then defines, by means of explicit harmonic, melodic, contrapuntal, and textural constraints, the remaining lines needed to complete the desired counterpoint. CPA begins by creating and testing all possible combinations of these given lines and computed lines--lines whose constraints the user has specified. These combinations often number in the tens of thousands. Combinations of lines (i.e., counterpoints) that do not meet all of the linear, harmonic, contrapuntal, and textural constraints provided by the user are eliminated; only those that meet all of the user's constraints are returned as output. The output may consist of zero to hundreds of counterpoints, depending upon the constraints provided by the user.

While CPA can be adapted for use in diatonic or quasi-tonal counterpoint, the program is designed using a fully chromatic harmonic and melodic matrix that suggests freely atonal, serial, or post-serial composition. While the program allows computations of long sections, CPA is designed for intensive and detailed manipulation of shorter segments of material--segments the length of a phrase or a short subsection that can serve as "building blocks" of larger compositional units.

Using the Program