- Transformational theory
Transformational theory is a branch of music theory developed by
David Lewinin the 1980s, and formally introduced in his most influential work, "Generalized Musical Intervals and Transformations" (1987). Musical transformations, which are operations defined on a mathematical group that represents potential musical events, can be used to analyze both tonal and atonal music. The resulting analyses are compelling visually and metaphorically because they show through the use of arrows how one musical event is transformed into another as an audible process in a piece of music.
The idea of a musical transformation revolves around Lewin's observation that the mathematical groups defined by
musical set theory(as developed by Milton Babbitt, Allen Forte, et al) have identity elements that are arbitrarily defined. Both pitch space and pitch-class space are better defined as torsors, which are a type of mathematical set that Lewin calls a Generalized Interval System (GIS). While traditional musical set theory focuses on the makeup of musical objects and describes operations that can be performed on them, transformational theory focuses on the intervals or types of musical motion that can be described between the musical events in real music. 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?'" (GMIT, p. 159)
The greater abstraction that is achieved by removing definitive labels from musical objects and only defining the operations that generate one from another is transformational theory's greatest 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 seen in the illustration given here can describe the first phrases of both the first and third movements of Beethoven's Symphony No. 1 in C Major, Op. 21. Further, a transformational network that gives the intervals between pitch classes in an excerpt may also describe the differences in the relative durations (measured in eighth notes, for example) of another excerpt in a piece, thus succinctly relating two different domains of music analysis.
Although transformation theory is 20 years old, it did not become a widespread theoretical or analytical pursuit until the late 1990s. Following Lewin's revival (in GMIT) of
Hugo Riemann's three contextual inversion operations on triads (parallel, relative, and Leittonwechsel) as formal transformations, the branch of transformation theory called Neo-Riemannian theorywas popularized by Brian Hyer (1995), Michael Kevin Mooney (1996), Richard Cohn (1997), and an entire issue of the Journal of Music Theory (42/2, 1998). Transformation theory has received further treatment by Fred Lerdahl (2001), Julian Hook (2002), David Kopp (2002), and many others.
*Lewin, David. "Generalized Musical Intervals and Transformations" (Yale University Press: New Haven, CT, 1987)
*Lewin, David. "Transformational Techniques in Atonal and Other Music Theories", "Perspectives of New Music", xxi (1982–3), 312–71
*Lewin, David. "Musical Form and Transformation: Four Analytic Essays" (Yale University Press: New Haven, CT, 1993)
*Lerdahl, Fred. "Tonal Pitch Space" (Oxford University Press: New York, 2001)
*Hook, Julian. "Uniform Triadic Transformations" (Ph.D. dissertation, Indiana University, 2002)
*Kopp, David. "Chromatic Transformations in Nineteenth-century Music" (Cambridge University Press, 2002)
*Hyer, Brian. "Reimag(in)ing Riemann", "Journal of Music Theory", 39/1 (1995), 101–138
*Mooney, Michael Kevin. "The `Table of Relations' and Music Psychology in Hugo Riemann's Chromatic Theory" (Ph.D. dissertation, Columbia University, 1996)
*Cohn, Richard. "Neo-Riemannian Operations, Parsimonious Trichords, and their "Tonnetz" Representations", "Journal of Music Theory", 41/1 (1997), 1–66
* [http://www.math.ucr.edu/home/baez/week234.html This Week's Finds in Mathematical Physics (Week 234)] by John Baez
Musical set theory
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