A HYBRID COMPOSITIONAL SYSTEM: PITCH-CLASS COMPOSITION WITH TONAL SYNTAX.(music)

I. THEORIES OF COMPOSITION AND THE HYBRID SYSTEM [1]

THEORIES OF COMPOSITION (i.e., theories for the generation of musical structures) form a spectrum whose range illustrates to what degree a theory determines [2] the organization of a composition's musical surface. At one end of the range, lie aleatoric theories, such as those developed by John Cage and employed in compositions such as Music for Piano (1952--56). [3] Since the materials and/or the procedures for organizing the materials are randomly produced with indeterminate composing theories, the specific nature of the musical surface cannot be predicted from the theory. [4] At the other end of the range, lie deterministic or algorithmic composing theories, such as those employed by, Lejaren Hiller, Larry Austin, and others. Since the materials and/or the procedures for organizing the materials are algorithmic, "once certain variables specified by the theory have been defined, a piece of music will emerge." [5] Tonal composing theories, such as the rules of counterpoint or the syntax associated with harmon ic progressions, lie somewhere in between the middle and the deterministic end of the range, because these theories determine many details of the compositional surface but they do not produce pieces. [6] Twelve-tone theories of composition are scattered throughout this spectrum of choice. A simple or minimal twelve-tone theory that only specifies an ordering for the twelve pitch classes along with a set of operations, such as transposition, inversion, and their retrogrades, does not specify how one row relates to any other. Since this theory determines very little about the organization of these musical structures (sequentially, polyphonically, or by any other musical parameter), very few details of a composition's musical surface (i.e., its score) can be predicted from the theory. Therefore, its position on the spectrum would be closer to aleatoric theories. A twelve-tone system incorporating a theory of hexachordal combinatoriality would move it closer to the middle of the spectrum, since the constraints im posed by hexachordal combinatoriality would limit the sequential and polyphonic organization of the twelve-tone rows on a composition's musical surface.

Theories of composition could also form a spectrum whose range illustrates to what degree a theory determines the organization of structural levels relating the musical surface to some background structure. At one end of the range lie completely bottom up approaches to composition, such as motivic composition. [7] Since motivic theories only generate surface structures, they say nothing about background structures or the relation of the surface to a background. At the other end of the range would be theories, such as a rational reconstruction of Schenker's analytical methodology, that would specify at every level the transformational relationships connecting the musical surface to a background structure. [8] Once again, a simple or minimal twelve-tone theory simply producing an ordering of the twelve pitch classes along with a set of basic operations determines very little about the organization of structural levels relating the musical surface to a background structure, presumably a single row form. Consequ ently, its position on the spectrum would be closer to the bottom up approaches to composition. A twelve-tone theory incorporating a theory of array structures, such as Morris, Composition with Pitch Classes, would move it closer to the middle of the spectrum, since it would specify more precisely the organization of structural levels relating the musical surface (the faster unfolding columnar aggregates of the array) to the background structure (the slower unfolding twelve-tone rows forming the linear aggregates of the array).

The hybrid compositional system, which will be outlined in this paper, lies somewhere in between the middle and the algorithmic end of the range of theories that specify the organization of a composition's musical surface, because the theory determines many details but it does not specify every detail of a composition's structure. The hybrid compositional system also lies somewhere in between the middle and the end of the range where theories specify at every level the transformational relationships connecting the musical surface to a background structure, since the system contains a set of transformational relationships connecting most of the musical surface to a background structure. The system is a composing language that combines structural elements from nontonal theories, such as twelve-tone arrays and self-deriving rows, and aspects of tonal theories, such as structural levels and prolongation. The system facilitates constructing complex and coherent musical structures, some of which function in a mann er similar or analogous to the musical structures in tonal systems.

Although the concept of incorporating structures usually associated with tonal music into nontonal systems appears contradictory, the contradiction is nominal, rather than structural. [9] If the diatonic set generates the tonal system's structural elements and relations, and if it is one member of a system of sets classified by their [T.sub.n]/[T.sub.n]I types, and if some sets in the system do not generate tonal structures, then in one sense, the labels tonal and nontonal simply denote sets possessing different structural properties. However, the labels do not imply or lead to the conclusion that differently labeled collections cannot generate similar or equivalent structures, and the labels do not imply or lead to the conclusion that every member of the system does not or cannot generate similar or equivalent structures. Discovering a syntactic connection between the tonal interpretation of the [T.sub.n]/[T.sub.n]I set [0,1,3,5,6,8,10] and other members of the [T.sub.n]/[T.sub.n]I system, and constructing a system based on those commonalties, would be a method of incorporating tonal structures into systems based on set classes other than the diatonic. David Lewin suggests this method of incorporating structures usually associated with tonal music into nontonal systems in his article "A Formal Theory of Generalized Tonal Functions." [10] Lewin transforms the arrangement of tonic, dominant, and subdominant triads linked by common tones (see Example 1) into an expression for constructing systems of tonal functions given a tonic pitch-class T, a dominant interval d, and a mediant interval m.

Each system consists of tonic, dominant, subdominant, mediant, and submediant triads. Lewin's generalization is not limited to any particular dominant or mediant interval, such as perfect fifth and major and minor thirds. Example 2 illustrates the triadic system determined by the ordered triple (C, 3,1) (d = 3 and m = 1). In the (C, 3,1) system, all the triads are [T.sub.n]/[T.sub.n]I [0,1,3] type trichords. Thus, Lewin shows a way to generalize tonal functions to sonorities not usually considered tonal.

Even if the tonal interpretation of the [T.sub.n]/[T.sub.n]I set [0,1,3,5,6,8,10] shared no syntactic features with sonorities not usually considered tonal, the diatonic set may not determine every syntactic property of the tonal system. [11] Since some structures, such as those associated with Schenker's conception of tonality, may be token representations of more general syntactic properties, another method of incorporating tonal structures into systems using [T.sub.n]/[T.sub.n]I sets might be creating a functionally analogous system based on generalizations of Schenkerian tonal syntactic features and interpreting those features within the context of a particular [T.sub.n]/[T.sub.n]I set type. For, example, concepts such as neighbor-note, passing note, prolongation, structural level, and Ursatz, may not be limited to the triads generated by the diatonic set. On the other hand, musical structures traditionally thought of as nontonal, such as tone rows and arrays, may be adapted to simulate tonal functions, or their structure could be the basis of determining tonally analogous functions.