Transformational theory

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.

The "transformations" of transformational theory are typically modeled as functions that act over some musical space S, meaning that they are entirely defined by their inputs and outputs: for instance, the "ascending major third" might be modeled as a function that takes a particular pitch class as input and outputs the pitch class a major third above it.

[2] For example, a single pair of pitch classes (such as C and E) can stand in multiple relationships: E is both a major third above C and a minor sixth below it.

In the opening pages of GMIT, Lewin suggests that a subspecies of "transformations" (namely, musical intervals) can be used to model "directed measurements, distances, or motions".

Although transformation theory is more than thirty years old, it did not become a widespread theoretical or analytical pursuit until the late 1990s.

Transformation theory has received further treatment by Fred Lerdahl (2001), Julian Hook (2002), David Kopp (2002), and many others.

Some authors, such as Ed Gollin, Dmitri Tymoczko and Julian Hook, have argued that Lewin's transformational formalism is too restrictive, and have called for extending the system in various ways.

Others, such as Richard Cohn and Steven Rings, while acknowledging the validity of some of these criticisms, continue to use broadly Lewinnian techniques.