Cellular automata (CA) provide a means for the generation of complex emergent structures from the local interaction of simple, usually orthogonally-interconnected units. They have become a popular paradigm for exploring the analogies between mathematical models and biological phenomena. The motivation for the use of CA in computer-aided composition is cited by Miranda as being an expert system’s hard- wired inability to compose new musical styles [Miranda 2003]. Some types of CA bear a strong relationship to chaotic dynamic systems because they exhibit unpredictable behaviour at the macroscopic level despite being deterministic. This was formally identified by Wolfram, who devised a widely-referenced taxonomy for describing CA types [Wolfram 2002]. CA can also been described mathematically in terms of finite state automata [Neumann and Burks 1966], and ‘static’ L-Systems [DuBois 2003].
A CA consists of grid of cells which begin in an arbitrary initial configuration and update their states at every time-step during execution. At a given time-step, t, the new state of a cell is determined by the state of its orthogonal neighbours at time t−1 using a set of evolution rules specified before run-time. Cell states are usually binary or ternary, and cell types are often classified using a ‘KxRy’ notation, where x refers to the number of immediate neighbours and y refers to the radius of influence. CA are also classified according to their number of possible evolution rules, which is a function of the number of possible cell states, the radius of cell influence and
§2.2 Formal Computational Approaches 25
the number of immediate neighbours. Wolfram’s taxonomy identified four different classes of CA behaviour [Wolfram 2002]:
Type 1 ‘convergent’, in which a static uniform grid state is quickly reached;
Type 2 ‘steady cycle’, in which stable repeating patterns quickly emerge;
Type 3 ‘chaotic’, in which no stable patterns emerge and any apparent structures are transient;
Type 4 ‘complex’, in which interesting patterns are perceivable but no stability occurs until after a large number of time steps.
The mapping of a CA to a musical parameter space is non-trivial, and as important to the act of composition as choosing the rule set. Frequently the resulting patterns are mapped to pitches restricted to a certain scale, such as chromatic, pentatonic or diatonic [Millen 2004]. Miranda distinguished between simplistic mappings of grid cells to MIDI note numbers and the more sophisticated method of mapping structural changes in groups of cells to higher-level musical structures [Miranda 2003]. Bilotta et al. identified analogous mapping categories of ‘local’ and ‘global’ [Bilotta et al. 2001]. They also use ‘indirect’ methods of manipulating the structure of the information con- tained in the CA before translating it into music [Bilotta and Pantano 2002]. Resultant structures characterised by researchers as ’gliders’, ’beetles’, ’solitons’, ’spiders’ and ’beehives’ contain varying degrees of recognisable musical harmonies when mapped directly from cell states [Bilotta and Pantano 2002].
Miranda presented a CA system for algorithmic composition called CAMUS for mapping Conway’s Game of Life to a harmonic musical output using each cell’s coor- dinates [Miranda 2003]. Bilotta et al. described a series of musical works produced using a genetic algorithm to further evolve the musical information resulting from a mapping of a binary CA’s output to musical parameters [Bilotta et al. 2000]. They con- cluded that type 1 CA are good for rhythmic generation, types 2 and 4 are good for harmonic generation, and type 3 are less useful except with very simple initial con- ditions. CA also feature in several interactive compositional or improvisational tools. Millen presented such a system wherein the musical parameters that the cells map to can be altered by the user during performance in reaction to visual observation of the grid state [Millen 2004].
Dorin used boolean networks (BNs) instead of CA to produce complex polyrhythms [Dorin 2000]. BNs are one-dimensional configurations of binary state machines — that is, each unit performs a boolean operation using the inputs from its two neighbours. An autonomous, synchronous boolean network is a special case of a CA [Dorin 2000]. Dorin observed that it is rare for a BN’s stable pattern to be bro- ken even when significantly perturbed in real-time, and that this makes them ideal for generating rhythmic material for live applications. Dorin also produced a CA mounted on the faces of a virtual cube called LIQUIPRISM, distinguishing it from the more common form of CA environment which models the surface of a torus [Dorin 2002]. A stochastic element is introduced by occasionally activating cells which have
been in ‘off’ states after substantial periods of inactivity. The mapping from the CA to music in any given time-step is done through a process of eliminating cells which are not moving from off to on and then selecting a maximum of two cells from each face. Each face maps to a MIDI channel being fed into a synthesiser.
Miranda believes that CAs are appropriate tools for generating new material, but concedes that they seem better suited to synthesis than composition. In his estimation the musical results “tend to lack the cultural references that we normally rely on when appreciating music” [Miranda 2003]. Bilotta et al. noted that as a general rule, only a very small subset of the available rule sets give ‘appreciable’ musical results, but that certain configurations can generate ‘pleasant’ harmony [Bilotta et al. 2001]. Dorin has demonstrated that the combination of musical and visual output of CAs can manifest as effective multimedia art [Dorin 2002].