Frank Foulks

The theory of linear arrays provides a definition of linear order from the reflexive, symmetric, but non-transitive relation of matching. However, a distance function is not generally available for the elements of a linear array. Given the original intended interpretation of the matching predicate as holding between phenomenal qualia, this result presents an apparent contradiction to the existence of human practices, specifically the tradition of musical practice described by common-practice music theory, that involve precise judgments of phenomenal distance. This paper resolves that seeming contradiction by constructing a notion of the musical scale out of the theory of linear arrays. First, the theory of linear arrays is briefly reviewed. Then, criteria of adequacy for the construction are adduced from common-practice music theory, the theory of measure and the semiotic theory of notation. The construction is presented along with results indicating its fulfillment of the criteria of adequacy. Finally, the relations between several variables introduced in the course of the construction are investigated

In the early twentieth century Arnold Schoenberg introduced into music an atonal style of polyphonic composition. The artistic success of his and his colleagues' oeuvre insured the wide diffusion of his technique of the twelve tone series. In the period immediately following World War II, the serial technique exercised a dominant influence over advanced musical practice. However, in the decades that followed, composers and theorists alike began to entertain doubts about the relation between the derivation of a composition from a series and its audible musical structure. A reconsideration of the theory of atonal polyphony must rest on rigorous analyses both of the structure of the musical materials, and of the semiotic relation between score and performance, which is constitutive of musical works in the western common practice tradition. ;The aim of this dissertation is to advance the project of founding a reconsideration of the theory of atonal polyphony by constructing the concrete entities needed to provide a phenomenalistic semantic framework for the semiotics of contrapuntal theory. The starting point is Nelson Goodman's logical constructionalism, and, in particular, his theory of signs of Languages of Art, and his phenomenalist system of The Structure of Appearance. Chapter One is a contribution to phenomenalist theory. It introduces a systematic notion of the maximal phenomenal continuity that explicates the concept of the sense content, as it occurs in natural language philosophical discourse. Chapter Two begins with a discussion of the idealizations that are appropriate to a semantics of contrapuntal theory. This discussion results in a systematic notion of the primitive concrete entity of contrapuntal theory, the note, which is shown to be a species of sense content, as defined in Chapter One. By aggregating notes, the construction subsumes under the systematic notion of the sub-polyphony both whole compound polyphonic occurrences and their musically significant melodic and harmonic sub-divisions. The resulting explication of the concrete notions of contrapuntal theory both connects phenomenalist theory with a familiar realm of human experience, and provides a basis for a systematic examination and revision of contrapuntal theory