On the Concept of the Scale

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.