A New Use of the Basic Mathematical Idea of Twelve-Tone Music
Ward Douglas Maurer · 2006
We here briefly describe a collection of pieces which we have written, and which have been performed for large audiences, in which mathematics is used. Specifically, every one of the 12 major chords, and every one of the 12 minor chords, appears in each of these pieces. We argue that this is a more pleasing use of the number 12 in music than the twelve-tone system of Schonberg. The Twelve-Tone System One of the most obvious connections between mathematics and music is that of twelve-tone music, pioneered by Schonberg and also used by Anton Webern, Alban Berg, and Hanns Eisler (see [1, 2, 3]). A piece of music is made up of tone rows, each of which consists of the 12 notes of the chromatic scale in some order. Mathematical transformations of a tone row may be used; for example, it may be transposed into a different key, or it may be played backwards. Also, it may be inverted, meaning that downward and upward intervals are interchanged. Any combination of these three transformations may be used. Also, adjacent notes in a tone row may be played together, as a chord. The number of possible variations of a tone row depends on that tone row; as a simple example, a chromatic scale (which is a tone row), inverted and then played backwards, gives the original scale, so it is not a different variation. In academic music departments in the US, twelve-tone music was popular for a long time. Today, however, it seems to have fallen on hard times. Much of the blame for this has been placed on Schonberg’s own initial insistence that the twelve-tone method, rather than the classical ideas of musical harmony, should be the basic principle behind composition: “As originally designed by Schonberg, the [twelve-tone] method was intended to preclude tonality” [4]. This was fascinating for a long time but has generally lost its appeal. Contrary to popular belief, however, not all twelve-tone music is atonal; both Berg and Schonberg wrote twelve-tone music in which some semblance of tonal harmony was preserved (also see [4]). Our Alternate Use Of Twelve Tones Our own interest in the mathematical ideas of twelve-tone music was an outgrowth of our interest in the composition of satirical songs. Not that we had any idea of satirizing the twelve-tone system itself (although Leonard Bernstein has done this). Rather, we were interested in expanding the range of tonal music, which normally uses only a tiny fraction of the available chord changes. Our interest was drawn, in this connection, to Bach’s Well-Tempered Clavier, containing 24 preludes and fugues, one in each of the 12 major and 12 minor keys. However, we wanted to write, not 24 pieces, but a single piece. We were also fascinated by the Symphony in D Minor by Cesar Franck, an early exponent of chromaticism (later greatly expanded by Schonberg). Franck, although his music was unabashedly tonal, used sequences of chord changes which had never been heard before.