Computing Musical Sound

Summary The links between mathematics and music are ancient and profound. The numerology of musical intervals is an important part of the theory of music: it has also played a significant scientific role. Musical notation seems to have inspired the use of cartesian coordinates. But the intervention of numbers within the human senses should not be taken for granted. In the Antiquity, while the pythagorician conception viewed harmony as ruled by numbers, Aristoxenus objected that the justification of music was in the ear of the listener rather than in some mathematical reason. Indeed, the mathematical rendering of the score can yield a mechanical and unmusical performance. With the advent of the computer, it has become possible to produce sounds by calculating numbers. In 1957, Max Mathews could record sounds as strings of numbers, and also synthesize musical sounds with the help of a computer calculating numbers specifying sound waves. Beyond composing with sounds, synthesis permits to compose the sound itself, opening new resources for musicians. Digital sound has been popularized by compact discs, synthesizers, samplers, and also by the activity of institutions such as IRCAM. Mathematics is the pervasive tool of this new craft of musical sound, which permits to imitate acoustic instruments; to demonstrate auditory illusions and paradoxes; to create original textures and novel sound material; to set up new situations for real-time musical performance, thanks to the MIDI protocol of numerical description of musical events. However one must remember Aristoxenus’ lesson and take in account the specificities of perception.