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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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Transformation Graphs and Networks (1): Intervals and Transpositions

Transformation Graphs and Networks (1): Intervals and Transpositions

(p.157) 7 Transformation Graphs and Networks (1): Intervals and Transpositions
Generalized Musical Intervals and Transformations

David Lewin

Oxford University Press

Instead of starting with a Generalized Interval System (GIS) and deriving certain characteristic transformations therefrom, it is possible to start with a family of characteristic transformations on a musical space and derive a GIS structure therefrom. That is, instead of regarding the i-arrow as a measurement of extension between points s and t observed passively “out there” in a Cartesian res extensa, one can regard the situation actively, like a singer, player, or composer, thinking: “I am at s; what characteristic transformation do I perform in order to arrive at t?” This chapter explores this conceptual interrelation between interval-as-extension and transposition-as-characteristic-motion-through-space. After developing the mathematics that shows a logical equivalence between GIS structures and certain structures of transformations on spaces, the work proceeds by example. Passages from Schoenberg, Wagner, Brahms, and Beethoven indicate how suggestive it can be to consider networks of “intervals” and networks of “transpositions” (modulations and so forth) as various aspects of the same basic phenomenon.

Keywords:   Generalized Interval System, music theory, musical spaces, interval, transposition

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