Sunday, 22 January 2012

Iannis Xenakis and Cellular Automata

The architect/composer Iannis Xenakis is probably the most famous artist to have used CA in music composition practice. The following summary on Xenakis' use of cellular automata was originally presented in the review chapter of my PhD thesis (Burraston 2006), but there a few additions in this page. The Figures are from the thesis, and have been kept the same. 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 2004104200410 and this is said to be derived from rule 4200410 in Wolfram’s Scientific American article (Wolfram 1984). It should be noted that these two rules are not exactly the same, the differences in their behaviour can be seen below.


 Rule 4200410 after Wolfram’s Scientific American article

 Rule 2004104200410

The other two rules are given by Solomos are 2241410 and 2040410 and were used in a mixed form, examples of spacetime behaviour is shown below.

Rule 2241410

Rule 2040410

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”. This is in contrast to Wolfram’s (1983) earlier remarks, in regard to elementary rule 90, that “changes in boundary conditions apparently have no significant qualitative effect”. In a later paper Wolfram (1986) states : “The cycle structures of finite cellular automata depend in detail on the boundary conditions chosen” and provides tables highlighting these differences for two particular CA.

The 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.




Close up of Xenakis’ text/symbol pocket computer printout for Horos (left) presented in Solomos (2005). The same rule, 2004104200410 with 23 cells and periodic boundary, viewed as graphic blocks and stretched to line up with printout which shows the difference in the two evolutions (right). 

A close up of Xenakis’ pocket computer text/symbol printout presented in Solomos (2005) is shown above (left). The printout shows 22 printed columns of cells and hand annotations creating an extra column on the left. The same rule, 2004104200410 with 23 cells and periodic boundary, I have computed and is shown as graphic blocks alongside (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.  

I have made a short Mathematica program, which was used to generate the images on this page, a copy can be downloaded here.

References

Burraston, D. (2006) Generative Music and Cellular Automata. PhD Thesis, Univ. Technology Sydney, Australia. (Available on noyzelab research page)
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. (1983). Statistical Mechanics of Cellular Automata. Reviews of Modern
Physics, 55(3): 601-644 

Wolfram, S. (1984) Computer Software in Science and Mathematics. Scientific American. 251(3): 188-203
Wolfram, S. (1986) Random Sequence Generation by Cellular Automata. Advances in
Applied Mathematics, (7): 123-169 

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.


Note: Xenakis' pocket printout in figure above (left) is from the Xenakis archives and courtesy of Makis Solomos.