# Generalized Set Theory (1):Interval Functions; Canonical Groups and Canonical Equivalence; Embedding Functions

# Generalized Set Theory (1):Interval Functions; Canonical Groups and Canonical Equivalence; Embedding Functions

This chapter begins a study of generalized set theory, that is, the interrelationships among finite sets of objects in musical spaces. The first construction studied is the Interval Function between sets *X* and *Y*; this function assigns to each interval *i* in a Generalized Interval System the number of ways *i* can be spanned between a member of *X* and a member of *Y*. Then the Embedding Number of *X* in *Y* is studied; this is the number of distinct forms of *X* that are subsets of *Y*. To study that number, the meaning of a “form” of the set *X* must be established—a notion that involves stipulating a Canonical Group of operations. Both the Interval Function and the Embedding Number generalize Forte’s Interval Vector. Passages from Webern, Chopin, and Brahms illustrate applications of the constructs.

*Keywords:*
music theory, musical spaces, Interval Function, Embedding Number, Forte, Interval Vector, Generalized Interval System

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