Probability. Is based on the fact that the ratio s/n between a certain outcome of an event s and the total number of possible outcomes n is stable (approaches a limit). See History and Mathematical treatment in http://en.wikipedia.org/wiki/Probability the Relation to randomness section presents a more philosophical view. For a little more math: http://en.wikipedia.org/wiki/Probability_theory Stochastic distributions. From stochos, Greek word for goal, aim, or guess. See definition and references to music: http://en.wikipedia.org/wiki/Stochastic The goal here is represented by a defined direction of a process or by well defined average states of a system (music) without precisely controlling the details of the process. Xenakis was the first composer to use stochastic distributions as a tool for creating a music (as he writes in his book Formalized Music) similar to the sound made by cicadas or of the rain falling on a tin roof. Pithoprakta, is the first work using this technique (1956) and contains what Xenakis calls “clouds of sounds”: no particular sound can be heard separately although the overall effect is obvious in the same way clouds are made of droplets of water undistinguishable from afar. Later, composers such as Ligeti (Atmospheres) and Penderecki (Threnody) have used similar textures (produced with non-mathematical means) nowadays known as sound mass. A first step in Xenakis' work is to distribute attack points on a timeline in an asymmetrical way. In order to do this he defines a series of segments (durations between attacks) in increasing order of magnitude and calculates their probabilities using a formula for continuous probability: P(x) = δe-δxdx where x is the size of the segment, δ is the linear density (average attacks per time unit) and dx a very small quantity. Other formulae are used to calculate pitch intervals, each sound's duration, etc. Glissandi are also used and Xenakis uses the Gaussian or Normal distribution to calculate their “speed” (how fast a glissando covers a certain pitch interval). http://en.wikipedia.org/wiki/Normal_distribution Since the composer controls both the average or mean value and the spread (standard deviation), it is customary to talk about controlled randomness in such cases. In the early 1960s, Xenakis used a FORTRAN computer program running on an IBM mainframe computer to produce a series of pieces, ST, for various numbers of instruments. They were molded after his piece for a small chamber ensemble Achorripsis. For more details, consult Iannis Xenakis – Formalized Music, on reserve, ML3800X46 F61992