Stochastic Methods

Another way of creating a similar soundscape was devised by Iannis Xenakis using his mathematician and architect background.

To create similarly dense structures, Xenakis used stochastic processes and formulæ^3.3.

Stochastic functions are basically conditional random transition tables. The basic rule for them is that an element 12#12 may be followed by another element 13#13 with a probability factor 14#14. This is commonly expressed as: 15#15

Those tables of probabilities are better known as Markov Chain Processes.

Figure: Example of a simple Markov chain transitions table. The first line read as: ``there is a 0.25 chance (25%) that the element a changes to (or will be followed by the) element b''

16#16

Those elements (or states) a, b...can be notes, chords, rhythms or any musical data.

In Figure:3.3, a fragment of the drawing for Bars 52-57[3] of Pithoprakta is shown.

Glissandi lines for the strings instruments, previously obtained according to transition tables similar to the one shown above are drawn on millimetric squared paper. The starting and ending times and pitches are set. Those are then transcribed to the music paper.

Figure: I. Xenakis Drawings for the strings glissandi in Pithoprakta

17#17

Figure: I. Xenakis Score (Pithoprakta) form the drawing above.

18#18

Those random-based processes as used by Xenakis but also by John Cage^3.4are radically different from Ligeti's composition methods.