Local Compositions

Summary

Local compositions are introduced as elementary objects of music. They derive from powerset denotators. It is shown that all denotators may be transformed into local compositions. A special type of local compositions can be defined from a fixed set of denotators of a given “ambient space”. With these so-called objective local compositions the problem of universal construction of new concepts from given ones cannot be solved. This imposes a deeper theory of non-objective functorial local compositions. The basic vocabulary as well as an introductory list of common local compositions—such as scales, ordinary and fractal chords, meters, rhythms, and motives—are presented. The chapter concludes with a discussion of tangent objects in local theory, a concept framework which leads to alterations and related results by Mason and Mazzola.

Ich kann ein lokales Objekt, eine lokale Struktur nicht abstrakt schaffen; so elementar ich sie auch immer denke: ich bin bereits gezwungen, sie zu kodieren, um sie niederschreiben zu können – im Grunde fällt der Lokalstruktur auch eine alphabetische Rolle zu (. . . )

Pierre Boulez [138, II, p.66]