Transformational theory

Transformational theory is a branch of music theory David Lewin developed in the 1980s, and formally introduced in his 1987 work, Generalized Musical Intervals and Transformations. The theory—which models musical transformations as elements of a mathematical group—can be used to analyze both tonal and atonal music.

The goal of transformational theory is to change the focus from musical objects—such as the "C major chord" or "G major chord" -- to relations between objects. Thus, instead of saying that a C major chord is followed by G major, a transformational theorist might say that the first chord has been "transformed" into the second by the "Dominant operation." (Symbolically, one might write "Dominant(C major) = G major.") While traditional musical set theory focuses on the makeup of musical objects, transformational theory focuses on the intervals or types of musical motion that can occur. According to Lewin's description of this change in emphasis, "[The transformational] attitude does not ask for some observed measure of extension between reified 'points'; rather it asks: 'If I am at s and wish to get to t, what characteristic gesture should I perform in order to arrive there?'" (from "Generalized Musical Intervals and Transformations", hereafter GMIT, p. 159)

The formal setting for Lewin's theory is a set S (or "space") of musical objects, and a set T of transformations on that space. Transformations are modeled as functions acting on the entire space, meaning that every transformation must be applicable to every object.

Lewin points out that this requirement significantly constrains the spaces and transformations that can be considered. For example, if the space S is the space of diatonic triads (represented by the Roman numerals I, ii, iii, IV, V, vi, and vii°), the "Dominant transformation" must be defined so as to apply to each of these triads. This means, for example, that some diatonic triad must be selected as the "dominant" of the diminished triad on vii. Ordinary musical discourse, however, typically holds that the "dominant" relationship is only between the I and V chords. (Certainly, no diatonic triad is ordinarily considered the dominant of the diminished triad.) In other words, "dominant," as used informally, is not a function that applies to all chords, but rather describes a particular relationship between two of them.

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