Class notes are reminders of some of the main points discussed in class. They are not always presented in a rigorous format.
Directional processes.
Markov chains. A random process with a finite number of possible outcomes (state space) where an outcome or state depends only on the immediately preceding outcome is a Markov chain.
See also: http://en.wikipedia.org/wiki/Markov_chain.
A transitional matrix describes such a process:
↓ | a b
__|______
a | 0.2 0.8
b | 0.8 0.2
In this example, a has a 20% chance to remain an a and 80% chance to become a b while b has a 80% chance to become an a and 20% chance to remain a b. If we start with 100 as and 0 bs we can see that after a while
stage a b
0 100 0
1 20 80
2 68 32
3 39 61
4 57 43
5 46 54
6 52 48
7 49 51
8 50 50
9 50 50
… ------
the system reaches a stationary state (example taken from Formalized Music). Xenakis experimented with Markov chains in the late 1950s and 1960s in works like Analogiques A et B and Eonta (Beings). The last title illustrates his metaphor according to which a Markov chain behaves like a “being” with a will of its own when arbitrarily interrupted by the subjective intervention of the composer.
While Xenakis employes this technique in organizing “frames” of sounds with particular distributions of pitches in different registers, durations, dynamics or densities (see Formalized Music), many other composers have used and are using Markov chains to create sequences of pitches, durations, textures, etc. Along with stochastic distributions, it is one of most popular techniques especially among composers using computers.
Later in his career Xenakis started using random walks (http://en.wikipedia.org/wiki/Random_walk), related in a way to Markov chains, which he sometimes uses to create arborescences or directed acyclic graphs.
http://en.wikipedia.org/wiki/Arborescence_%28graph_theory%29
Through composed music is another example of process advancing in time without returning to the original state. An interesting case, Ockeghem's Missa prolationum (late 15th. century), contains canons at gradually increasing intervals with the four voices sing in four different mensurations (time signatures) and thus each advancing at its own pace.
http://en.wikipedia.org/wiki/Missa_prolationum
The Illiac Suite for string quartet (1956) by Lejaren Hiller was the first major composition realized with the assistance of a computer, the Illiac I computer (http://en.wikipedia.org/wiki/ILLIAC). Using the Monte Carlo method, a stochastic procedure, sounds were selected randomly, one at a time out of the fourteen white keys of two octaves. Rules of species counterpoint determined if the next choice was a valid one. The process would start from the beginning in case of failure. Since the process consisted of choosing sequentially the sounds of the composition, it can be considered directional.