Mathematical Concepts and Techniques

Joseph Schillinger System of Music Composition (SSMC), 1946, based on mathematical processes, pieneered algorithmic composition "beyond style". Among students: Gershwin, Earl Brown, Henry Cowell, Benny Goodman, and Glenn Miller. Collaborated with Leon Theremin.

Stochastic distributions

From stochos, a Greek word for goal, aim, or guess. Th 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 indistinguishable from afar.

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. Here is an example page 26 (35) of how this is computed.

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). Other formulae are used to calculate pitch intervals, each sound's duration etc. Glissandi are also employed and Xenakis uses the Gaussian or Normal distributioan to calculate their 'speed' (how fast a glissando covers a certain pitch interval).

Xenakis has a slightly different expression: page 25 (33).

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 modeled after his piece for a small chamber ensemble Achorripsis where the matrix controlling the activity of various groups of instruments was created by using the Poisson distribution (p. 23/29), also used in Pithoprakta for calculating the density of 'clouds of sound'.

Other Mathematical devices

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.
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 such as Analogique 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.

Sieves

Part of Number Theory sieves are logical filters that use modulo and Boolean operations to select a collection of elements out of a continuum such as selecting the C major scale out of all possible Equal Temperament pitches.

Notation: 30 3 denotes the modulo and the index 0 the residual class. Using the chart at the top of the page,

30 = {0. 3. 6. 9, ...}, represents a fully diminished chord C, Eb, F#, A.
30 ∪ 31 = {0, 1, 3, 4, 6, 7, 9, 10, ...}, an octatonic scale (Messiaen mode).

First introduced by Xenakis, sieves are a powerful tool that can be used to create scales, rhythms, or any other selection of materials a composer might need.

Xenakis and other composers also employed

Xenakis also used Game Theory

Cellular automota, genetic algorithms have also been used as well as Artificial Intelligence techniques.

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