Tiling Six-Part Double Canons on Trichords

THE STRUCTURE OF MORPHOLOGICAL SPACE DAVID KANT AND LARRY POLANSKY 1. INTRODUCTION HE STUDY OF CONTOUR—important in music theory, cognition, and ethnomusicology—is motivated by an interest in melodic similarity and classification. Contour theory attempts to categorize, clarify, analyze, and define basic melodic principles, as well as, more generally, morphology—musical phenomena quantifiable in some parameter(s) as change over time. Contour is fundamental to perception. As such, an understanding of contour relationships, such as distance functions (metrics), in contour space is essential. A contour is an ordered set of directional relationships between (quantifiable) elements, prioritizing “up/down/equal” over “how much up or down.” The study of contour has often consisted of categorical classification and a search for archetypes. The up/down motion of things changing in time is one way to understand morphology, or to paraphrase Henry Cowell, the “nature of melody” in terms of a restrictive yet perceptually primal feature. The study of contour relationships (particularly similarity) and categorizations is fundamentally an effort to describe contour space. T 442 Perspectives of New Music This paper presents new mathematical and computational tools to visualize and understand the structure of contour space. By integrating degrees of magnitude into contour representations—ranking (Marvin and Laprade 1987) and n-ary contour (Polansky and Bassein 1992)— we propose a formal unification of contour space with morphological space. Since contour is an equivalence relation, we first examine contour equivalence relations on the space of morphologies. Next, using basis coordinate space (or simply basis space), we formally describe how contour archetypes generate, and organize, all possible morphologies, creating a single, highly-structured mathematical space, suggesting a reconsideration of the usual distinctions between contour and morphology . In this paper we offer a formal description of the continuum of direction and magnitude, via consideration of problems and ideas raised by contour theory regarding the structure of combinatorial contour (CC-) space. 2 THE STUDY OF MUSICAL CONTOUR 2.1 CONTOUR THEORIES Some formulations of contour are reductive, typological, and categorical (Adams 1976; Huron 1995; Seeger 1960; Kolinski 1965a, 1965), while others focus on measurable distances and relationships between contours (Polansky 1981, 1987, 1996; Marvin and Laprade 1987; Marvin 1994; Morris 1987, 1993, 2001; Johnson 2001a, 2001b; Quinn 1999). Theorists such as Morris, Marvin and Laprade, and Friedmann (1985, 1987) utilize techniques, terminology, concepts and standard transformational operations from post-tonal music theory (inversion, retrograde, embedding), whereas Polansky and others have focused on generalized mathematical tools. 2.2 LINEAR AND COMBINATORIAL CONTOUR The distinction between linear contour—adjacent directional relationships —and combinatorial contour—the network of such relationships —is fundamental. Linear contour (LC ) is defined as a vector of adjacent ternary directional relationships in a morph (M),1 a finite ordered list of values.2,3 Combinatorial contour (CC ) is defined by the The Structure of Morphological Space 443 half-matrix of pairwise relationships in a morph (Morris 1987, 2001; Polansky 1996, 1987, 1981; Quinn 1999; Marvin and Laprade 1987). By convention, we use −1, 0, and 1 to represent directional relationships “is less than,” “is equal to,” and “is greater than.” The CC half-matrix is often expressed in the literature as a vector, by convention: relationships to the first element (top row), relationships to the second element (second row), etc. The morph, LC, and CC all have different lengths. For a morph of length L, the length of the LC vector Llc is L − 1, and the length of the CC vector Lcc is (L2 − L)/2. Example 1 shows LC and CC vectors for the morph [3, 4, 5, 1]m. Contour—both LC and CC—is an equivalence relation on all possible morphologies. Many different morphs have the same LC or CC representation, but CC further distinguishes morphs that are equivalent as LCs. CC restricts the range of magnitude variation of represented values more than LC does, and thus is a more “accurate” representation. Since CCs are equivalent to ranked morph elements, greater contour lengths L allow for greater resolution of rank and greater approximation of magnitude.4 Below, we describe a geometric representation which includes linear and combinatorial contours in the same space, and consider some ramifications of this unified space. 2...