Mathematical Tools in Music I

Stochastic distributions. Distributions used by Xenakis:

The range of probability values; "e", the base of natural (Napierians) logarithms used to describe natural growth processes.

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

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 a has a 50-50 chance to be followed by either a or b and b has a 70% chance to be followed by a and only 30% chance to be followed by b itself. The numbers on each column have to add up to 1 (something will happen for sure). A system like the one above, will end up spending more time on a because of the bias in the second column. This tendency leads to a stationary or steady or ergodic state. Xenakis has used this property in his works by using a (much more complex) matrix then arbitrarily intervening in the process once it reaches that stationary state. After a while, the system is allowed again to follow its course. The title of his piece Eonta or Beings alludes to the fact that, in such cases, the Markov chain exhibits a mind of its own, like a being.

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, 31 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

Since they are a general way of dealing with such procedures, sieves can be applied not only to the pitch domain but also to rhythms, dynamics, timbres, articulations and any other sound parameters. Moreover, sieves can also be weighted giving the elements which pass through them different probabilies of occurrence - another useful musical tool.

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