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AN INFORMAL INTRODUCTION TO SOME FORMAL CONCEPTS FROM LEWIN'S TRANSFORMATIONAL THEORY

AN INFORMAL INTRODUCTION TO SOME FORMAL CONCEPTS FROM LEWIN'S TRANSFORMATIONAL THEORY into GIS theory to represent a transformational perspective. Parts two and three introduce three types of mappings including the interval function of a GIS. A knowledge of these is a prerequisite to the study of GISs and neo-Riemannian groups. Part four explores the Cartesian perspective offered by the definition of a GIS and demonstrates the kinship between Cartesian and transformational perspectives by constructing a GIS from a simply-transitive-group system. Part five outlines relationships between atonal theory’s system of transpositions and inversions, neo-Riemannian triadic systems, and Lewin’s simply-transitive-group systems.4 The last part of the article considers “non-commutative” GISs. Two of Lewin’s innovative findings will be illustrated. First, given any non-commutative GIS, one can always discover another system, its dual, which exists in a sort of “parallel universe” of transformations. Second, in any noncommutative GIS, transposing a pair of notes by the same amount can actually change the interval between the notes! In this circumstance, there always exist operations (which are not transpositions) that preserve intervals. Non-commutative GISs, dual GISs, and non-transposing, intervalpreserving operations are especially original topics in GMIT not treated elsewhere. I. A Transformational Perspective. In GMIT, Lewin presents the idea of “interval” first in terms of a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Music Theory Duke University Press

AN INFORMAL INTRODUCTION TO SOME FORMAL CONCEPTS FROM LEWIN'S TRANSFORMATIONAL THEORY

Satyendra, Ramon
Journal of Music Theory , Volume 48 (1) – Jan 1, 2004
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Publisher
Duke University Press
Copyright
Copyright 2004 by Yale University
ISSN
0022-2909
eISSN
1941-7497
DOI
10.1215/00222909-48-1-99
Publisher site
See Article on Publisher Site

Abstract

into GIS theory to represent a transformational perspective. Parts two and three introduce three types of mappings including the interval function of a GIS. A knowledge of these is a prerequisite to the study of GISs and neo-Riemannian groups. Part four explores the Cartesian perspective offered by the definition of a GIS and demonstrates the kinship between Cartesian and transformational perspectives by constructing a GIS from a simply-transitive-group system. Part five outlines relationships between atonal theory’s system of transpositions and inversions, neo-Riemannian triadic systems, and Lewin’s simply-transitive-group systems.4 The last part of the article considers “non-commutative” GISs. Two of Lewin’s innovative findings will be illustrated. First, given any non-commutative GIS, one can always discover another system, its dual, which exists in a sort of “parallel universe” of transformations. Second, in any noncommutative GIS, transposing a pair of notes by the same amount can actually change the interval between the notes! In this circumstance, there always exist operations (which are not transpositions) that preserve intervals. Non-commutative GISs, dual GISs, and non-transposing, intervalpreserving operations are especially original topics in GMIT not treated elsewhere. I. A Transformational Perspective. In GMIT, Lewin presents the idea of “interval” first in terms of a

Journal

Journal of Music Theory – Duke University Press

Published: Jan 1, 2004

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