Stochastic Processes

• Various musical practices over the years have made use of random or stochastic processes.

• A stochastic process is defined as one in which a sequence of values is drawn from a corresponding sequence of jointly distributed random variables.

• For example, Mozart's Musikalisches Wurfelspiel (“Dice Music”) prescribes the random ordering and assembly of various musical fragments to create a new version of the piece every time it is performed.

• A similar, though more recent example, includes John Cage's Reunion, which is performed by playing chess on a specially designed chess board, such that the players' moves trigger four compositions that are distributed to eight speakers surrounding the audience.

• Computers are good at generating random numbers and distributions, which can subsequently be used to implement a stochastic approach.

• Iannis Xenakis was a pioneer in using a computer to calculate various probability theories to aid in the creation of his compositions Atrées (1962) and Morsima-Amorsima (1962).

• Another approach involves the use of conditional probabilities, which can be represented in the form of a matrix called a transition table. The table below specifies transition probabilities from the current notes C, D, E and G along the top to the next notes C, D, E and G along the side. For example, if the current note is a G, there is a 50$\%$ likelihood the next note will be either an E or a G.

  	C	D	E	G
  C	0.0	0.3	0.0	0.0
  D	1.0	0.3	0.3	0.0
  E	0.0	0.4	0.6	0.5
  G	0.0	0.0	0.1	0.5

• A transition table can be used to find the probability of note sequences. Using the table above, the probability of generating the sequence D, E, G if the current note is a D is 0.04 (D to E is 0.4 and then E to G is 0.1 ... the probability of both is the product of the individual probabilities). The probability of generating the sequence C, E, G is 0.

• An example in Common Music recreates music in a style reminiscent of 19th-century American composer Stephen Foster using a transition table derived from Foster's music by Harry Olson in the 1950s.