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Generalized Musical Intervals and Transformations$
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David Lewin

Print publication date: 2007

Print ISBN-13: 9780195317138

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780195317138.001.0001

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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

Chapter:
(p.88) 5 Generalized Set Theory (1):Interval Functions; Canonical Groups and Canonical Equivalence; Embedding Functions
Source:
Generalized Musical Intervals and Transformations
Author(s):

David Lewin

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780195317138.003.0005

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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