Mathematics and the Twelve-Tone System: Past, Present, and Future

The first major encounter of non-trivial mathematics and non-trivial music was in the conception and development of the twelve-tone system from the 1920s to the present. Although the twelve-tone system was formulated by Arnold Schoenberg, it was Milton Babbitt whose ample but non-professional background in mathematics made it possible for him to identify the links between the music of the second-Viennese school and a formal treatment of the system. This paper sketches a rational reconstruction of the twelve-tone system as composers and researchers applied mathematical terms, concepts, and tools to the composition and analysis of serial music. It also identifies some of the major trends in twelve-tone topics that have led up to the present, provides a brief account of our present mathematical knowledge of the system, and suggests some future directions for research, including open questions and unproven conjectures.