**Stochastic distributions**. Distributions used by Xenakis:

- continuous probability
- Poisson law (rare events)
- Normal/Gaussian distribution

The main elements that the composer can control when using such distributions are:

- a mean or average value around which everything is clustered
- how wide is the pool of values from which choices are made or standard deviation

How does Xenakis choose his values (sounds). His claim that this technique
allows him to follow a strict logic (constraints or rules), the stochastic
distribution(s) and, at the same time, allows him the freedom of choice of
deciding at each moment what value will be assigned, as long as it does
not contradict the rule (distribution). Examples from his book and scores.
Achorripsis, the *thesis of minimum constraints*, and the
ST computer program.

This music is similar to (or describes) phenomena characterized by a statistical outlook: the overall shape is important and the details are either not significant or impossible to control. Xenakis uses the metaphor cloud of sounds: individual sounds are like droplets of water in a cloud. We do not see individual droplets but we perceive easily the outline and volume of a cloud in the sky. The use of stochastic distributions to create such musical textures seems natural to Xenakis since physical phenomena of the same kind were already studied with their help. Music and Science use a common tool.

**Markov Chains** describe the chance that a system will go
from one state to another in one step. They are a mathematical tool
which may describe sequences of symbols or sounds such as natural
speech or musical melodies (lines, or strings of sounds). Edgar Alan Poe
describes a similar situation in his short story The Golden Bug.
An encoded document is deciphered by replacing the most frequent symbol
encountered with the letter "e" (the most frequent letter in English), then
replacing reoccurring strings of 5 symbols with the group _the_, a.s.o.
Markov chains, named after a Russian mathematician, are usually described
by transformation matrices of the form:

| | V | a b _____|______________ | a | 0.5 .70 | b | 0.5 .30 |which shows that

**Set Theory and Sieves**. Basic operations with sets of
pitches (union, intersection, complement, etc.) are used by Xenakis in
some pieces to define different pools of sound. The entire work becomes
then an aural rendition of such operations Herma, for piano
solo).

**Sieves** are logical filters which allow the extraction of selec
elements from a continuum, e.g. the C major scale out of all sounds of
the chromatic scale. They are based on residual modulo m
algebras. In the notation used by Xenakis, **3**1
denotes the class of all numbers which, when divided by **3**, have
a remnant of **1**: 1, 4, 7, 10, 13,...

The expressions defining scales used in tonal music, for example, are rather involved and represent a rather awkward way to derive these scales. However, the advantage of the proceedure becomes obvious when one tries either to

- describe more exotic scales based on non-Western tunnings (such as a Byzantine scale involving 72 discreete divisions of the octave)
- transpose such constructs on differents sounds (the major scale from
**C**to**F#**, for example.

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