The dictionary defines stochastic as (from the Greek stochastikos – skillful in aiming, from stochazesthai – to aim at, guess at, from stochos – target, aim, guess.) as a process that involving chance or probability. In music stochastic elements are randomly generated elements created by strict mathematical processes. Stochastic processes can be used in music either to compose a fixed piece, or produced in performance. Iannis Xenakis, an architect and composer who used probability, game theory, group theory, set theory, Boolean algebra, and frequently computers, to produce his scores, pioneered stochastic music. Earlier, John Cage and others had composed aleatoric music, which is created by chance processes but does not have the strict mathematical basis (Cage’s Music of Changes, for example, uses a system of charts based on the I-Ching).
Xenakis is particularly remembered for his pioneering electronic and computer music, and for the use of stochastic mathematical techniques in his compositions, including probability (Maxwell-Boltzmann kinetic theory of gases in Pithoprakta, aleatory distribution of points on a plane in Diamorphoses, minimal constraints in Achorripsis, Gaussian distribution in ST/10 and Atrees, Markovian chains in Analogiques), game theory (in Duel and Strategie), group theory (Nomos Alpha), and Boolean algebra (in Herma and Eonta). In keeping with his use of probabilistic theories, many of Xenakis’ pieces are, in his own words, “a form of composition which is not the object in itself, but an idea in itself, that is to say, the beginnings of a family of compositions”. The heavy reliance of Xenakis’ music upon mathematics, and probability theory in particular, led to criticism and a lack of appreciation by both the music community and the general public. In 1962 he published Musique Formelles-later revised, expanded and translated into Formalized Music: Thought and Mathematics in Composition in 1971-a collection of essays on his musical ideas and composition techniques, regarded as one of the most important theoretical works of 20th century music.