Iannis Xenakis and Cellular Automata - D Burraston

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The following summary was originally presented in (Burraston 2006), and has a few additions.The architect/composer Iannis Xenakis, one of Olivier Messiaen's students, is probably the most famous artist to have used CA in composition practice. His interest was based on the simplicity of the CA process to produce a rich output (Varga 1996). Xenakis used CA calculated with his "pocket computer" to determine the succession of chords within a rational, perceptible structure for sections of his orchestral composition Horos in 1986. Xenakis may have used CA in several other compositions before his death in 2001, although the complete extent of his application is still unknown.  Xenakis (1992) states in the preface of his book "Formalized Music" that the use of CA "can be found in works of mine such as Ata, Horos, etc."

         Greater detail on the CA used in Horos and the mapping process can be found in (Hoffmann 2002, Solomos 2005), where it is shown that Xenakis applied the output to bars 10, 14-15, 16-18 and 67-71. Mapping to musical output was based on pre-determined pitches and the state of the cell chose the instrumentation. The presence or absence of cells controlled the musical events. Xenakis was also quite happy to interfere with the output as he saw fit. The approach taken utilised a 5 state totalistic nearest neighbour (v5k3) automaton of one dimension and three different rules. The first rule used was 20041042004105 and this is said to be derived from rule 42004105 in Wolfram's Scientific American article (Wolfram 1984b). It should be noted that these two rules are not exactly the same, the differences in their behaviour can be seen in Fig 2.16 (top left and right). The other two rules are given by Solomos are 22414105 and 20404105 and were used in a mixed form, examples of spacetime behaviour is shown in Fig 2.16 (bottom left and right).

                      

                      

Fig 2.16 Rule 42004105 from Wolfram's Scientific American article (top left). Rule 20041042004105 (top right), 22414105 (bottom left) and 20404105 (bottom right) used by Xenakis in Horos.

         Xenakis may have used fixed rather than periodic boundary conditions. It is difficult to ascertain precisely from these papers how the CA used by Xenakis implemented the boundary. The importance of specifying the type of boundary conditions was stated by Wuensche and Lesser (1992), if they are fixed the system will become more disordered, because the wiring is atypical at edge locations. he type of boundary conditions one chooses will depend on the application or investigation requirements.  Solomos (2005) states : 1) the far right column of cells "does not intervene in next line calculations" and 2) the handwritten annotations on the far left column are "manually added by Xenakis". This seems to imply that the right column boundary is fixed at null (0). The other options are choosing a fixed state value between 1 and 4. It is not clear whether the handwritten left column is a manually added part of the CA calculation, or added arbitrarily and not part of this calculation.

 

 

Fig 2.17 Close up of Xenakis’ text/symbol pocket computer printout for Horos (left) presented in Solomos (2005). The same rule, 20041042004105 with 23 cells and periodic boundary, viewed as graphic blocks and stretched to line up with printout  (right).

         A close up of Xenakis pocket computer text/symbol printout presented in Solomos (2005) is shown in Fig 2.17 (left). The printout shows 22 printed columns of cells and hand annotations creating an extra column on the left. The same rule, 20041042004105 with 23 cells and periodic boundary, is shown as graphic blocks in Fig 2.17 (right). Comparing the pocket computer printout with the data it can be seen that differences occur at timestep 12 as the CA reaches its boundaries. Note that this rule will evolve to all 0's (single cycle attractor) with periodic boundary conditions for a system size of 22 or 23 cells from a single seed. A discrepancy arises in the spacetime evolution depicted in Fig 1 of Hoffmann's '2002) paper, as that shows a 21 cell system. Solomos (2005) examined more of Xenakis scores between 1986 and 1990, suggesting that a further 8 pieces may have used CA in some way.

 

Xenakis pocket printout image courtesy of Makis Solomos

References

Burraston, D. (2006) Generative Music and Cellular Automata. PhD Thesis, Univ. Technology Sydney, Australia.

Hoffmann, P. (2002) Towards an “Automated Art”: Algorithmic Processes in Xenakis’ Compositions. Contemporary Music Review 21 Nos 2/3: 121-131

Solomos, M. (2005) Cellular Automata in Xenakis’ Music. Theory and Practice. In Georgaki, A. & Solomos, M. Eds. Proceedings of the International Symposium Iannis Xenakis. pp120-137

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

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

Wuensche, A. & Lesser, M. (1992) The Global Dynamics of Cellular Automata : An Atlas of Basin of Attraction Fields of One-Dimensional Cellular Automata. Addison-Wesley. (Available as PDF from www.ddlab.com)

Xenakis, I. (1992) Formalised Music. (Revised Edition). Pendragon Press.

Appendix

A Mathematica notebook to generate the spacetime diagrams is here.