What Is Transformational Theory?

Meg Higa

The transformational theory of music is a mathematical attempt to explain its nature, structure and effect on human experience. Students of music theory, even the ancient Greeks, have known that music can be explained by science and math, as well as by aesthetic pleasure. The advent of sophisticated electronics and powerful computers of the late 20th century finally enabled attempts at modeling music numerically. Transformational theory was first proposed by a mathematician and musician at Harvard University in the US. Professor David Lewin’s 1987 book was titled, “Generalized Musical Intervals and Transformations.”

Transformational theory was first proposed by a musician and mathematician at Harvard University.
Transformational theory was first proposed by a musician and mathematician at Harvard University.

The diatonic scale used in tonal music — just a piano’s white keys, for example — is a very small set of seven elements with a starting point {C,D,E,F,G,A, & B}. This is its conventional designation. There’s no reason not to designate them numerically {1,2,3,4,5,6,7}. The full chromatic scale of atonal music with no starting point — the inclusion of a piano’s black keys — is still a small set of just twelve elements. Nearly all of the world’s music is contained within this small set.

Musical set theory borrows from the mathematics of sets and sequences to this limitation of twelve elements. Their infinitely variable sequences explain the world’s nearly infinite catalog of songs. A pianist instructed to play three ascending notes in succession — do-re-mi, for example, using the Latin convention — would be represented by the sequence {C,D,E}. Transformational theory dispenses with the set altogether, arguing that individual musical elements need not be specified if the rules and relationships of changing sounds can be defined.

In the three-note example of the above paragraph, the sequence can be represented {n, n+1, n+2}. The numbers represent the musical interval, or pitch space, already well-defined by, not only a piano’s spacing of keys, but also the science of sound waves. A vocalists who requests accompanying music in a “different key” to better suit her range is representing the variable “n” in the sequence. Transformational theory would describe that the element “n” undergoes a sequential transformation equivalent to the three ascending notes.

Further pared down to its essence, transformational theory defines a musical composition as a “sonic space,” designated “S,” which contains just a single element “n.” All of the many musical notes in the composition can be mapped onto this space according to their transformational operation “T,” in relation to “n.” For example, the dramatic piano technique of striking all the white keys from left to right in one quick sweep might be spatially represented as a spiraling helix in the shape of a metal spring. Music is expressed as a network, rather than a collection of symbols.

David Lewin passed away in 2003 without publishing much of his theoretical papers. Advanced mathematicians, computer programmers and theorists of music have since advanced and refined his original framework. One group of researchers fed the entirety of several 18th century orchestral symphonies, including one of composer Ludwig Beethoven’s, to a computer programmed with the mathematics of transformational theory. Each piece of music resulted in a graphic of the geometric shape called a torus, more commonly known as a doughnut with a hole.

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