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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, "(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) ==Formalism== 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. There are, however, any number of situations in which "transformations" can extend to an entire space. Here, transformational theory provides a degree of abstraction that could be a significant music-theoretical asset. One transformational network can describe the relationships among musical events in more than one musical excerpt, thus offering an elegant way of relating them. For example, figure 7.9 in Lewin's GMIT can describe the first phrases of both the first and third movements of Beethoven's Symphony No. 1 in C Major, Op. 21. In this case, the transformation graph's objects are the same in both excerpts from the Beethoven Symphony, but this graph could apply to many more musical examples when the object labels are removed. Further, such a transformational network that gives only the intervals between pitch classes in an excerpt may also describe the differences in the relative durations of another excerpt in a piece, thus succinctly relating two different domains of music analysis. Lewin's observation that only the transformations, and not the objects on which they act, are necessary to specify a transformational network is the main benefit of transformational analysis over traditional object-oriented analysis. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Transformational theory」の詳細全文を読む スポンサード リンク
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