Music & Mathematics (03)

YouTube also offers other videos that allow you to listen to a score along with the music.

Here's one of Metastaseis (Metastasis), an orchestral piece composed in 1954 by Xenakis, in which the sound events were defined almost entirely statistically.

In fact, Xenakis's compositional process is closely linked to mathematics. To solve problems such as the distribution of sounds and figures, density, duration, and the notes themselves, Xenakis often uses statistical distributions, combinatorics, but also physical laws and symbolic logic.
His approach is consistent with his critique of integral serialism expressed in his work "The Crisis of Series Music," which can be summarized as follows:

The application of the series to all compositional parameters in order to achieve total control over the composition produces an uncontrollable complexity that "prevents the listener from following the interweaving of lines and has the macroscopic effect of an uncalculated and unpredictable dispersion of sounds across the entire spectrum."
This results in a clear contradiction between the intent compositional method that aims to be totally deterministic and the sound effect.
Since serialism allows the series to be transposed onto each of the 12 notes and also allows the use of inversion, retrograde, and retrograde inversion with related transpositions, then it is nothing more than a particular case of combinatorics [i.e., the branch of mathematics that studies ways to group and/or order the elements of a finite set of objects according to given rules - from Wikipedia].
Consequently, combinatorics constitutes a generalization of the serial method and therefore a transcendence of it.

From these two considerations and from Xenakis's physical and mathematical, as well as musical, background, a totally new compositional approach is born. If the contradiction of serialism is due to its complexity, which generates an uncalculated dispersion of sounds, it is therefore a question of finding new criteria for control. Xenakis's deductive line can be summarized in the following steps:

Integral serialism, adopting the mechanisms of transposition, inversion, and retrograde, almost spontaneously led to combinatorics.
Statistical sciences, of which combinatorics is a part, have developed mathematical and conceptual tools capable of studying and defining complex phenomena.
Probability calculus, conceptually and historically linked to combinatorics and its powerful generalization, presents itself as the most suitable tool for determining the macroscopic effect generated by highly complex linear polyphony.
The series is therefore replaced with the concept of a sound ensemble organized through statistical distributions that allow for the control of various sonic characteristics such as pitch, density, duration, timbre, etc.
The level of control [that on which the composer works] rises from the single sound. to the sound ensemble. The microscopic aspect of the composition may be disordered, but at the macroscopic level, one has a clear perception of a sound figure.

Let's clarify these postulates (especially postulate 5).
If I have 40 strings play a long note chosen at random but with a uniform distribution (all notes have the same probability) between C3 and C4, the perception will be that of a chromatic cluster of 8va.
Another example: if n instruments play random notes with pitches uniformly distributed between C3 and F3 and durations chosen equally between sixteenth and eighth notes, the perception will be that of a band with internal movement whose speed depends on the average duration and whose timbre depends on the instrumental distribution.
The composer's task is no longer to choose individual notes, a task left to statistical distribution, but to determine the shape of the ensemble by establishing which statistical distributions govern the various sound parameters and the movements of the ensemble through controlling the parameters of these distributions.

Note that Xenakis's conceptual approach is not dissimilar from that of Stockhausen, when he created the concept of the group and shifted his focus from the single sound to the group, and is analogous to that of Ligeti when he composed in groups, introducing the idea of micro-polyphony.

Xenakis's position, also determined by his mathematical knowledge, is, however, extreme. He completely abandons the level of the single sound event, to place himself at a higher level, that of sets of sounds. If we consider that even what the instrumental composer defines as a "single sound" is, in reality, a conglomeration of simple sounds (harmonics and/or sinusoidal partials), this position is amply justified. The possible levels of control over the sound event are many: from "composing the sound" to "composing the sound." from electronics to additive synthesis, to the organization of the event without a precise determination of the sonic result (Fluxus). It's up to the composer to decide where to place himself.

Finally, note that Xenakis' music is statistical, not random. Chance is there, but it is cancelled out by the multiplicity of events.

Here's the video. If you like, you can also watch it on YouTube, which also allows you to zoom in to full screen.

And to give you a better understanding, here's an isolation of the famous glissandi, whose textures reproduce the surface structure of the Philips Pavilion, designed by Le Corbusier with the assistance of Xenakis in 1958 for the performance of Edgar Varèse's Poème Électronique. (images here).

Back