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+[The following note originally appeared on the emusic-l mailing list.  It is

+reprinted here with the author's permission]

+

+From xrjdm@FARSIDE.GSFC.NASA.GOV Wed Nov 23 11:26:39 1994

+Date: Tue, 4 Oct 1994 15:09:23 -0500

+From: Joe McMahon <xrjdm@FARSIDE.GSFC.NASA.GOV>

+Reply to: Electronic Music Discussion List <EMUSIC-L@AMERICAN.EDU>

+To: Multiple recipients of list EMUSIC-L <EMUSIC-L@AMERICAN.EDU>

+Subject: Automata: the long-awaited summary

+

+Back in August, I think, I promised to post a quick intro to cellular

+automata and how they can be used as a sound-generation tool. Since I'm

+going to take a couple of different sources and sum them up with little or

+no direct attribution, combined with my own opinions, I'll give everybody

+my references *first* so they can delete the article and draw their own

+conclusions if they so prefer.

+

+The primary reference that got me started on all this is one in the CMJ:

+Vol 14, No. 4, Winter 1990: "Digital Synthesis of Self-modifying Waveforms

+by Means of Cellular Automata" (Jacques Chareyon). Those who are already

+familiar with automata may just skip to that article and forget about the

+rest of this one.

+Note: the article gives a mail address for M. Chareyon, but he did not

+answer an inquiry about any available recordings using this technique in

+1990.

+

+So. Anyone still here? Good.

+

+Cellular automata are a mathematical concept first introduced in the late

+1940's. Generally speaking, a cellular automaton consists of a grid of

+cells. Each cell may take on any of a number of values - binary automata

+(cell on or cell off) are the most commonly studied. Each cell has a

+neighborhood, defined more simply as other cells which influence its state.

+The exact nature of this influence is defined by what are called transition

+rules. The cellular automaton starts off with some cells in any of the

+allowable states. for each "step" in the automaton's history, the

+neighborhood of every cell is checked, and the state of the cell is

+updated. All updates occur simultaneously.

+

+The transition rule must describe the resulting state of a cell for every

+possible configuration of other cells in the neighborhood. For large

+numbers of states, the amount of memory required to hold the transition

+rule becomes increasingly large, Therefore, some automata use what is known

+as a "totalistic" rule. These rules simply sum the values of the cells in

+the neighborhood and then assign a result on this basis. The resulting

+tables are far smaller.

+

+Many readers may already be familiar with John Horton Conway's game of

+"Life". This is a two-dimensional binary automaton with a totalistic rule.

+This makes for a very small rule set:

+

+  i) If fewer than two filled cells (cells with value 1) surround a cell,

+     it becomes empty next generation.

+ ii) If more than three filled cells surround a cell, it becomes empty

+     next generation.

+iii) If exactly three cells filled cells surround a cell, it becomes

+     filled on the next generation.

+

+This corresponds to a totalistic rule set with a total of 8(2-1)+1 or 9

+rules (one each for the sum values of 0 (no cells with a value) through 9

+(all cells with a value) ).If the transition rule were represented as a

+non-totalistic one, the rule set would need 2**8 or 256 entries. There are

+many interesting totalistic automata, so giving up detailed description of

+every nuance of the transitions to save memory space isn't a big sacrifice.

+

+Interesting as two dimensional automata are, they really aren't terribly

+useful for music making. There have been some experiments which have

+attempted to use a two-dimensional automaton to generate MIDI events -

+synthesis at the note level, using :

+

+Battista, T. and M. Giri, 1988. "Composizione Tramite Automi Cellulari."

+Atti del VII Cooloquio di Informatica Musicale. Rome, Italy: Edizione Arti

+Grafiche Ambrosini, pp. 181-182.

+

+Edgar, R. and J. Ryan, 1986. "LINA" Exhibition of the 1986 International

+Computer Music Conference, San Francisco: Computer Music Association.

+

+I have not heard any of the music from these efforts, so I certainly can't

+pass any judgement on them. For the purposes of this summary, we'll just

+look at one-dimensional automata. These use a linear array of cells, with

+the neighborhood generally being one or two cells on either side of each

+cell.

+(This is the type of automaton dealt with in M. Chareyon's article, which I

+will be paraphrasing broadly hereafter).

+

+M. Chareyon's automata are wavetables. A digitized signal is stored as a

+linear array of numbers in memory. A totalistic rule is used to determine a

+lookup value which indexes into an array containing the resulting value;

+this is saved into a second array. After the first array is completely

+processed, the roles of the two are swapped and the process is repeated.

+

+The limiting factor in this process is the number of bits of resolution

+being used to generate the sound. For a totalistic rule using a two-cell

+neighborhood and 12-bit individual samples, we have 3*(2*12) = 12288

+entries in the rule table. At 2 bytes each, this is  24K of storage. If we

+go to 16-bit sample resolution, we have 196608 entries at 2 bytes each for

+a total of 393216 bytes, or 384K.

+

+The key point of M. Charyeon's method is the use of small neighborhoods

+with large numbers of cellular states. Since the computation of the new

+wavetable is all table lookup, very complex transition rules can be

+precomputed and loaded into the tables, allowing the synthesis to

+essentially be a fast sum-and-lookup loop to calculate each new wavesample.

+>From the article, it appears that M. Chareyon was able to produce 2 or 3

+voices in realtime on a Mac II with a Digidesign Sound Accelerator board.

+It seems that it would probably be possible to use an AV Mac to do it

+without the board.

+

+This LASy (Linear Automaton Synthesis) method is closely related to the

+Karplus-Strong plucked-string algorithm, in that a wavesample is run

+through an algorithm which recirculates the samples to "self-modify" the

+wave. In fact, a judicious choice of table entries allows one to very

+simply simulate the K-S algoritm directly.

+

+So what are the sounds like? Some automata produce waveforms which quickly

+"ramp-up" to complex spectra and then drop off quickly. Others move to a

+steady state and then remain there. Yet others produce never-ending and

+unpredictable waveforms, whose harmonic content is constantly changing.

+

+Obviously enough, the original wavesample can be obtained mathematically,

+or by actual sampling and using LASy as a waveshaper. As M. Chareyon notes,

+a quick estimate of the number of possible automata for a 2-neighbor

+totalistic rule using a 256-entry wavetable with 12-bit entries is

+(2**12)**256 * (2**12)**(3*2**12) or about 10**4500 possible automata. Of

+course, many, many of these would not be suitable for music (e.g., the 4096

+automata in which all values go to one vlaue in one step, etc.); however,

+the number of musically useful automata is still likely to be an immense

+number.

+

+M. Chareyon provides a number of examples of ways to fill out the rule

+tables and a number of hints on creating wave tables - generally speaking,

+one can create a function which is used to compute the values to be placed

+into the table and then fill it so it can simply be loaded and used by the

+basic algorithm. His experience in using LASy is that he manages

+approximately 50% of the time to produce sounds with the desired

+characteristics, and that about 10% of the remaining time he gets

+unexpected but useful results which can be used as starting points for

+further exploration.

+

+Again, the important point is that the basic automaton uses wavesamples at

+full resolution, calculating a new wavesample for each step of the

+automaton; the next wavesample can be played while the new one is being

+calculated. Because of the large number of states, mathematical tools for

+the analysis of automata and the construction of automata with specifically

+desired qualities require too much storage and compute time to make them

+useful for LASy purposes.

+

+Again, much of this article is paraphrased from M. Chareyon's article; I

+take no credit for any of the work in this note. I'm just summarizing.

+

+The following other articles were referenced by M. Chareyon's article:

+

+Burks, A., ed. 1970. Essays on Cellular Automata. Champaign/Urbana, IL:

+University of Illinois Press.

+

+Chareyon, J. 1988a. "Sound Synthesis and Processing by Means of Linear

+Cellular Automata." Proceedings of the 1988 Internation Computer Music

+Conference. San Francisco: Computer Music Association.

+

+Chareyon, J. 1988b. "Wavetable come Automa Cellulare: una Nuova Tecnica di

+Sintesi." Atti del VII Colloquio di Informatica Musicale, Rome, Italy:

+Edizioni Arti Grafiche Ambrosini, pp. 174-177.

+

+Farmer, D., T. Toffoli, and S. Wolfram, eds. 1984. Cellular Automata.

+North-Holland Physics Publishing. [One of the definitive works on cellular

+automata - fairly heavy math, not a popular presentation - JM]

+

+Gardner, M. 1970. "The Fantastic Combinations of John Conway's New Solitare

+Game 'Life'". Scientific American 223(4) 120-123. [A good introduction to

+cellular automata, focusing on 'life' in specific. Useful intro if my

+1-paragraph summary of automata was confusing :) - JM]

+

+ --- Joe M.

+

+--

+"At the end of the hour, we'll have information on the sedatives used by

+the artists,,," (MST3K)

+

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