The transformational theory of music is an attempt to mathematically explain the nature, structure, and impact of music on human experience. Music theory students, dating back to the ancient Greeks, have long known that science and math, as well as aesthetic pleasure, can be used to explain music. Late-twentieth-century advances in sophisticated electronics and powerful computers finally made numerical modeling of music possible. A mathematician and musician at Harvard University in the United States was the first to propose transformational theory. “Generalized Musical Intervals and Transformations,” by Professor David Lewin, was published in 1987.
The diatonic scale, which is used in tonal music and can be found on a piano’s white keys, is a very small set of seven elements with the following starting points: C,D,E,F,G,A, & B. This is how it’s usually referred to. There’s no reason why they shouldn’t be numbered 1,2,3,4,5,6,7. With no starting point — the inclusion of a piano’s black keys — the full chromatic scale of atonal music is still a small set of only twelve elements. This small collection contains nearly all of the world’s music.
To this limitation of twelve elements, musical set theory borrows from the mathematics of sets and sequences. Their nearly infinite catalog of songs is explained by their infinitely variable sequences. The sequence C,D,E would represent a pianist instructed to play three ascending notes in succession — do-re-mi, for example, using the Latin convention. Individual musical elements do not need to be specified if the rules and relationships of changing sounds can be defined, according to transformational theory.
The sequence can be represented as n, n+1, n+2 in the three-note example from the previous paragraph. The numbers represent the musical interval, or pitch space, which is well-defined not only by the spacing of keys on a piano, but also by sound wave science. The variable “n” in the sequence is represented by a vocalist who requests accompanying music in a “different key” to better suit her range. The element “n” undergoes a sequential transformation equivalent to the three ascending notes, according to transformational theory.
In its most basic form, transformational theory defines a musical composition as a “sonic space,” designated “S,” containing only one element “n.” The transformational operation “T” in relation to “n” allows all of the composition’s many musical notes to be mapped onto this space. The dramatic piano technique of striking all the white keys in one quick sweep from left to right, for example, could be represented spatially as a spiraling helix in the shape of a metal spring. Rather than a collection of symbols, music is expressed as a network.
In 2003, David Lewin died without having published many of his theoretical papers. Since then, advanced mathematicians, computer programmers, and music theorists have improved and refined his framework. One group of researchers fed the entire contents of several 18th-century orchestral symphonies, including one by Ludwig Beethoven, to a computer programmed with transformational theory mathematics. Each piece of music resulted in a graphic of a torus, which is a geometric shape that looks like a doughnut with a hole.