Cellular Automata in Xenakis' "Horos"

Apr 15, 2023

Iannis Xenakis' Horos[1] is widely regarded as an undisputed masterpiece of late 20th-century orchestral literature. Perhaps the most striking aspect of the work is its remarkable orchestration, which is partially the consequence of mapping Cellular Automata (abbreviated as CA)[2] #496553230, and to a lesser extent, the mixture of CA#701602730 and CA#203549480.3 Readers interested in a detailed analysis of the work should consult the following three sources: Peter Hoffman's "Music Out of Nothing?" and "Towards an ‘automated Art," and Makis Solomos' "Cellular automata in Xenakis' music." While both authors display great insight into the inner workings and mapping strategies Xenakis used, future analyses would benefit from in-depth visualizations. I have set out to re-render some of the CA systems Xenakis used with the hope of clarifying the impact these systems have over specific passages in the work. I have also included the programming code necessary to reproduce them in the Wolfram Mathematica software.

The instances in which Xenakis mentions CA systems are restricted to only a few primary sources — namely his book “Formalized Music,” as well as two interviews: Varga's “Conversations with Iannis Xenakis,” and Restagno's “Un'autobiografia dell'autore raccontata da Enzo Restagno.” All three sources provide tremendous insight into the reasons for Xenakis’ application of CA. The following passage is taken from the 1992 preface to his revised 2nd edition of “Formalized Music” book:

Another approach to the mystery of sounds is the use of cellular automata which I have employed in several compositions these past few years. This can be explained by an observation which I made: scales of pitch (sieves) automatically establish a kind of global musical style, a sort of macroscopic “synthesis’ of musical works, much like a ‘spectrum of frequencies, or iterations’, of the physics of particles. Internal symmetries or their dissymmetries are the reason behind this. Therefore, through a discerning logico-aesthetic choice of ‘non-octave’ scales, we can obtain very rich simultaneities (chords) or linear succession which revive and generalize tonal, modal or serial aspects. It is on this basis of sieves that cellular automata can be useful in harmonic progressions which create new and rich timbric fusions with orchestral instruments. Examples of this can be found in works of mine such as Ata, Horos, etc. (Xenakis XII)

I find the passage salient, since, essentially, he has written about his experience composing Horos. Hoffman puts Xenakis' praxis succinctly as: “[combining] scales of durations and scales of pitch ('sieves') in order to create complex temporal evolutions of orchestra clusters” (171). In the following interview with Varga, Xenakis goes into the technical details regarding his use of CA:

In all these years I've been working on the theoretical construction of sieves - that is, of scales, with the help also of the computer. Apart from that the only new procedure I've used is the so-called cellular automata. Horos was the first piece where I put them to use. The method helps in deciding how to go from the notes of one chord to those of another within a rational, perceptible structure. […] Let’s say you have a grid on your screen, with vertical and horizontal lines forming small squares, that is, cells. These are empty. It’s for the composer (whether working with pictures or with sounds) to fill them. How? One way is through probabilities, for instance by using the Poisson distribution, as I did 30 years ago in Achorripsis. There’s also another way, with the help of a rule that you work out for yourself. Let’s suppose the vertical lines represent a chromatic scale, or semitones, quarter-tones and so on. Any kind. You start at a given moment, that is, at the given vertical line, at a given pitch – in other words, a cell – and you say: here’s a note played by an assigned instrument. What’s the next moment going to be? What notes? In accordance with your rule, the cell which has been filled gives birth to say, one or two adjacent cells. In the next step each cell will create one or two notes. Your rule helps to fill the entire grid. These are the cellular automata. It's related to the nature of fluids, for instance. For me the sound is a kind of fluid in time - that's what gave me the idea to transfer one area to the other. I was also attracted by the simplicity of it: it's a repetitious, a dynamic procedure which can create a very rich output. (199-200).

Examining Xenakis’ sketches, there is convincing evidence that his first encounter with cellular automata was through Stephen Wolfram’s article “Computer Software in Science and Mathematics,” published in 1984 in Scientific American (Solomos [3]).[4] The article presents a coherent foundation for integrating computation in order to explore new ways of thinking in science and simulating certain aspects of natural phenomena. According to Wolfram, CA systems are analogous to natural systems in terms of exhibiting complex and unpredictable behavior that cannot be easily simplified or reduced (computational irreducibility). Wolfram’s categorization of CA in four classes of behavior is demonstrated in Figure 1 below. This is precisely the page where we encounter CA#496553230 with totalistic rule 4200410 (boxed in red in the top-right image).

Fig.1 Wolfram’s pg.199 from his 1984 article on computation. The top right image is a simulation of CA#496553230 with rule #4200410.

Examples 1-3 below display my visualizations of all three types of cellular automata used by Xenakis. Note that Example 1 is the same CA Xenakis encountered in Wolfram's article and subsequently used in Horos. The instrumental distribution is represented by the following color rules:

Strings = Red;

Woodwinds = Green;

Brass = Yellow.

Ex.1 Evolution of CA#496553230.

Ex.2 Evolution of CA#701602730.

Ex.3 Evolution of CA#203549480.

Figure 2 below displays the evolution of CA#496553230 as rendered from Xenakis’ pocket computer. Circled in red, "we can read the following handwritten annotations: ‘Japon 86’, ‘(4200410)’: [which] is the automaton [that] begins in m.10." (Solomos 10)

Fig. 2 Printed sheets of paper from Xenakis’ pocket computer showing Horos’ cellular automata evolutions. Archives Xenakis, Bibliothèque nationale de France.

To illustrate how pitch, temporal unfolding, and instrumentation come together in certain passages of Horos, Example 4 generates a 33-step evolution of CA#496553230 in a left-to-right horizontal grid form. This is the same CA evolution displayed in Figure 2 above. Example 5 further visualizes the interaction between the pitch scale represented on the vertical (Y) axis, the 33-step time evolution represented in the horizontal (X) axis, and the instrumental distribution represented by the following color rules:

4 = Strings = Red;

2 = Woodwinds = Green;

1 = Brass = Yellow;

0 and 3 = Silence = White.

CODE

Transpose[CellularAutomaton[{496553230, {5, 1}, 1}, {{1}, 0}, {33, {-11, 11}}]] // Grid

Ex.4 Left-to-right 33-step evolution of CA#496553230, starting with a single cell (1) against a (0) background.

CODE

Rotate[ArrayPlot[CellularAutomaton[{496553230, {5, 1}, 1}, {{1}, 0}, {33, {-11, 11}}], ColorRules → {4 → Red, 3 → White, 2 → Green, 1 → Yellow, 0 → White}], 90 Degree]

Ex.5 Left-to-right 33-step evolution of CA#496553230, starting with a single cell (1→Brass) against a (0→Silence) background. (Y) axis displays the scale used in the work.

Figure 3 shows the first instance of CA#496553230 in the work in m.10 and mm.14-15. Except for the few deviations as a result of Xenakis' creative intervention, the first 31 steps have been transcribed accurately. Compare it with Example 5 above and follow along with the audio here.

Fig.3 mm.10-15 from Horos’ published score. CA#496553230 is applied at mm.10, 14, and 15. Editions Salabert. 

Hoffmann puts forth the idea that “Xenakis’ occupation with cellular automata [...] reveals more than just the using of a mechanism to create complex textures. Cellular automata allow to study both the universal features and the fundamental limitations of algorithmic action of automata as such, including the computer. Using cellular automata for music composition therefore means, at least in [the] case of Iannis Xenakis, to profoundly reflect about the nature of a machine music, i.e. about Automated Art." (171). While Hoffmann is correct in pointing out Xenakis’ interest in automated art, in my opinion, Xenakis is always ultimately guided by his musical intuition. From this perspective, mapping CA systems to musical parameters provides several concrete advantages, particularly in the temporal domain.

In the context of Xenakis’ later output, the rhythmic scheme of Figure 3 is fairly common. His later works mark a strange adoption of regularly pulsing rhythms. While his characteristic layering of complex polyrhythms (e.g. Herma etc.) is interjected in smaller doses, Xenakis’ larger structural outlines consist of rather repetitive and relatively simple, straight, ‘grid-like’ rhythms. This stage of development was reached before his first use of cellular automata systems in 1986. However, it is entirely plausible that an algorithmic system like CA further attracted Xenakis since it resonated with the rhythmic strategies used in his later works. From a compositional perspective, the adoption of grids where each cell represents the smallest rhythmical value is intriguing since creative solutions must be used in order to render the music less pedantic and stiff. Examples 6-8 below highlight the various instrumental groups, thus presenting an interesting visual (and perceptual) counterpoint among these different families as they unfold in time.

CODE

Rotate[ArrayPlot[CellularAutomaton[{496553230, {5, 1}, 1}, {{1}, 0}, {33, {-11, 11}}], ColorRules → {4 → Red, 3 → White, 2 → Green, 1 → Yellow, 0 → White}], 90 Degree]

Ex.5 33-step evolution of CA#496553230, all instruments.

Ex.6 33-step evolution of CA#496553230, only brass.

Ex.7 33-step evolution of CA#496553230, only woodwinds.

Ex.8 33-step evolution of CA#496553230, only strings.

Although simple in principle, these visualizations clarify that even an exact mapping of Example 5 into pulsing 16th-note rhythms, once orchestrated among different instrumental groups, result in large-scale interdependent behaviors of considerable complexity. The counterpoint and repetitions mark interesting compositional potential. Xenakis further improves on the system by manually adjusting the formal panels (in a cut-and-paste fashion) and introducing smaller (probabilistic) rhythmic values within the regular grid. 

For the sake of completeness, I have included the other two CA used in the work below. It is noteworthy to observe the changes in behavior of particular instrumental groups given a change in the behavior of the CA as a whole. The application of these two CA is not as straightforward as the first one. In mm. 67-71 Xenakis interweaves the two automata, going back and forth, and also introduces chords from the first CA, which was originally implemented in m.10, and mm.14-15 (see Solomos pg.11-15 for a detailed analysis). The reader is strongly encouraged to run these simulations themselves, change the number of steps, and freely "zoom in" on various sectors of the system as they pertain to Xenakis' work.

CODE

Rotate[ArrayPlot[CellularAutomaton[{701602730, {5, 1}, 1}, {{1}, 0}, {33, {-11, 11}}], ColorRules → {4 → Red, 3 → White, 2 → Green, 1 → Yellow, 0 → White}], 90 Degree]

Ex.9 33-step evolution of CA#701602730, all instruments.

Ex.10 33-step evolution of CA#701602730, only brass.

Ex.11 33-step evolution of CA#701602730, only woodwinds.

Ex.12 33-step evolution of CA#701602730, only strings.

CODE

Rotate[ArrayPlot[CellularAutomaton[{203549480, {5, 1}, 1}, {{1}, 0}, {33, {-11, 11}}], ColorRules → {4 → Red, 3 → White, 2 → Green, 1 → Yellow, 0 → White}], 90 Degree]

Ex.13 33-step evolution of CA#203549480, all instruments.

Ex.14 33-step evolution of CA#203549480, only brass.

Ex.15 33-step evolution of CA#203549480, only woodwinds.

Ex.16 33-step evolution of CA#203549480, only strings.

The exploration of cellular automata in Xenakis' composition of Horos provides valuable insights into his approach to integrating complex systems that can be perceived through musical mapping. The visualizations presented in this writing help demonstrate the intricate interplay between pitch, temporal evolution, and instrumental distribution in Horos. They reveal the complex and interdependent behaviors that emerge when the CA systems are mapped among different instrumental groups, highlighting the potential of CA systems as a compositional tool for musical expression.

FOOTNOTES

[1] Performance by the Luxembourg Philharmonic Orchestra

[2] For a quick overview into the world of Cellular Automata, see this introduction from Wolfram MathWorld site: https://mathworld.wolfram.com/CellularAutomaton.html 

[3] In Stephen Wolfram’s shorthand, these codes correspond to 5-color totalistic rules #4200410, #2241410, and #2040410. The full rules are: #2004104200410, #2414102241410, and #0404102040410. "A totalistic cellular automaton is a cellular automata in which the rules depend only on the total (or equivalently, the average) of the values of the cells in a neighborhood. These automata were introduced by Wolfram in 1983. ... The evolution of a one-dimensional totalistic cellular automaton can completely be described by a table specifying the state a given cell will have in the next generation based on the average value of the three cells consisting of the cell to its left, the value the cell itself, and the value of the cell to its right." See here for more information: https://mathworld.wolfram.com/TotalisticCellularAutomaton.html 

E.g. if a 3-cell "neighborhood" sum equals either:

0,1,2,3,4,5,6,7,8,9,10,11, or 12,

then, the next step central cell value correspondingly becomes either:

0,1,4,0,0,2,4,0,1,4,0,0, or 2; thus forming rule #2004104200410

[4] Solomos credits Hoffmann as the first scholar to put forth such hypothesis. However, I am referring the reader to consult Solomos' paper since he has provided a copy of the aforementioned sketches from the Xenakis archives, also reproduced in this writing.

BIBLIOGRAPHY

Hoffmann, P. (2002). “Towards an ‘automated Art’: Algorithmic Processes in Xenakis’ Compositions”, in Xenakis studies: in memoriam, edited by J. Harley, Contemporary Music Review 21, 2-3, Oxfordshire, Routledge : 121-131.

Hoffmann, P. (2009). Music Out of Nothing? A Rigorous Approach to Algorithmic Composition by Iannis Xenakis. phd diss. Technischen Universität Berlin

Restagno, E. (1988). “Un'autobiografia dell'autore raccontata da Enzo Restagno”, in E. Restagno (ed.), Xenakis, Torino, EDT/Musica

Solomos, M. (2005). “Cellular automata in Xenakis’ music. Theory and Practice.”, in Makis Solomos, Anastasia Georgaki, Giorgos Zervos (eds.), Definitive Proceedings of the International Symposium Iannis Xenakis.

Varga, B. A. (1996). Conversations with Iannis Xenakis, London, Faber and Faber

Wolfram, S. (1984). “Computer Software in Science and Mathematics”, Scientific American 251, 3: 188-203.

Weisstein, E. W. (n.d.). Cellular Automaton. MathWorld. Retrieved from https://mathworld.wolfram.com/CellularAutomaton.html

Weisstein, E. W. (n.d.). Totalistic Cellular Automaton. MathWorld. Retrieved from https://mathworld.wolfram.com/TotalisticCellularAutomaton.html 

Xenakis, I. (1992). Formalized Music, English translation C. Butchers, G. H. Hopkins, J. Challifour, second edition with additional material compiled, translated and edited by S. Kanach, Stuyvesant (New York), Pendragon Press.

Xenakis, I. (2007). Horos [recorded by Luxembourg Philharmonic Orchestra cond. by A. Tamayo]. On Iannis

Xenakis Orchestral Works Vol.3 [CD]. Timpani Label.

Xenakis, I. (1986). Horos [Score]. Editions Salabert EAS 18414