ANDREW BYRNE

completely systematic way. I felt that the generative properties of automata - that they evolve in completely logical yet ever-different and changing ways – could be harnessed to create melodies in a ‘Rzewskian’ style.

An intriguing iPhone app by Jeff Holtzkener provided a way forward. The app is based on Langton's Ant, a cellular automaton invented by scientist Christopher Langton in 1986. In it, an 'ant' travels across a grid, its journey dictated by simple rules. As with all cellular automata, the interest in Langton’s Ant comes from the evolution of the ant’s journey as complex and unexpected progressions result from simple rules. In the app, users are able to easily tweak the rules of the cellular automaton, thereby creating different outputs or ant journeys. I quickly found a great variety of possible pathways the ant can take across the grid. Some journeys periodic, some symmetrical, and some seemingly random.

I now had the tools to generate the music for the ANTS collection.

Mapping Langton’s Ant into Music
The app provides a visual representation of the ant on the grid as well as some simple midi sounds as accompaniment. At the earliest stage, I decided to replace these midi sounds with my own musical setting. The aim was to create a collection of musical pieces that could be performed independently in a concert setting.

The process of translating the ant journeys into music focused exclusively on pitch - specifically, matching the direction the ant travels to certain pitches or sounds. So, for example, if an ant can move in four possible directions on a grid of squares, there will be four discrete sounds/pitches in the corresponding melody (as in No. 1, for example). The order in which the four pitches appear (i.e., the character of the melody) is determined by twists and turns of the ant’s journey, which is shaped by the rules of the cellular automaton. The same process applies for journeys on hexagonal grids. There are six possible directions an ant can take, therefore, six sounds/pitches (nos. 2, 4, 5). And on octagonal grids, eight sounds/pitches (nos. 3, 6, 7).

From these limited materials, varied and complex melodies emerge.

Other musical details (rhythmic character, instrumentation, tempo, dynamics) are used to highlight characteristics of the pitch progressions. The aim is to make details of the ant journeys/melodic patterns as clear as possible. The logic of the cellular automaton is always respected.

For those interested in more, a detailed discussion of two ant melodies is included below. While I do provide some information on the inputs for the cellular automaton (the rules, cell shapes of the grid and the ant’s starting point) with the youtube videos and in musical scores, I want to focus to be on the musical realisations of the automata: (1) the distinctive features of each of ant journeys (now distilled into strings of numbers), and (2) how I translated the strings of numbers into simple and yet (I hope) engaging melodies in my homage to Rzewski’s works from the early 1970s .

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A Brief Analysis of Two Ant Melodies

ANT No. 5 – one line with six sounds, divided into two (ambling)
The rules and basic parameters for ANT No. 5 are the following: it is a hexagonal grid/six possible notes/sounds with the following rules L2, T, T, L1, L2, L1 (T = top, B = bottom, L1 = upper left, L2 = lower left, likewise for R). Ant starts facing left.

From these parameters, the automaton begins with this stream of notes
64426642 1153331553311153315 4442666 5531111533 226442266642264 3331555 4 311 6 553 26 5 426444266 553153 266444222 1111555333 222226442266642264 3331555 etc

I found it useful to conceive of the 6 ant directions as a circle (see below). Moving east translates as pitch 1, moving south-east as pitch 2, south-west pitch 3, and so on.

Hexagon (6 directions = 6 pitches)
5 6
4 1
3 2

This melody only ever moves in an anti-clockwise direction (refer hexagonal circle above) with lots of repeating notes. It also has a curious dual nature. The even notes only want to associate themselves with other even notes (6, 4, 2) and the same behaviour applies to the odd numbers (5, 3, 1). If I reorganise the original stream of notes into two streams or line (one even, one odd), this feature becomes clear. The melody jumps from groupings of even to odd notes, and vice versa, but even and odd never mingle.

part 1 1153331553311153315 5531111533 3331555 311 553 5 553153
part 2 64426642 4442666 226442266642264 4 6 26 426444266

part 1 (cont.) 1111555333 3331555 etc
part 2 (cont.) 266444222 222226442266642264 etc

The melody plays 6 even notes in the first phrase and then switcheds (i.e., moves one step rather than two on the hexagonal circle) into the odd world where it plays 19 odd numbers, then moves again into the even world with 7 notes, and then back to odd with 10 notes, and so on.

While this alternation between odd and even ‘phrases’ is a clear defining feature of this melody, it is interesting to note that the individual phrases are never identical and are always of different length. In this sense the melody is ever changing on the local level, while the alternation of the even and odd groupings is a constant.

In my arrangement for 3 percussionists, I highlight the duality of this automaton through instrumentation. The first part plays the three odd notes on the cowbell, cymbal, high drum; while the second plays the three even notes on low conga, floor tom, scratch pad. (The third plays all six notes on six instruments bass drum, clap,  floor tom, kickdrum, low tom, ride - always one instrument per note.) In addition, every time player 2 has a repeating note, it plays a dotted rhythm to create a lilting effect.

With a leisurely tempo and an assortment of lo-fi (and clearly artificial) percussion samples, ANT No. 5 has a whimsical, humorous air (at least to my ears) as it oscillates between the two 3-note groups.

ANT No. 2 – one line with six sounds (strolling)
This ant journey is also on a hexagonal grid (six sounds) and has the rules L1, L1, R1, R1 (T = top, B = bottom, L1 = upper left, L2 = lower left, likewise for R). The ant starts facing right.

Hexagon (6 directions = 6 pitches)
5 6
4 1
3 2

The cellular automaton generates with the following string of numbers
54321 65432121 6545 6543234321 6131 6545 6543234321 61 654321 65432 1232 1656 16543456 16543432 12345432 1232 1656 1232 1656 16543456 1654345432 12345432 1 654321 65432121 6545 6543234321 6121 6545 6543234321 654321 654321

This sequence of numbers falls into three different types of motives: material A that circles around note 6; material B that circles around note 1; and a refrain that runs directly in descending or ascending order from 6 to 1, and serves as a transition between material A and B. In addition the melody only ever moves around the circle by consecutive steps: note 1 can only ever be followed by 6 or 2; note 2 by 1 or 3, and so on. Lastly, there are no repeating notes in the melody.

(refrain)            54321 65432121
(material A)      6545 6543234321 6121 6545 6543234321 61
(refrain)            654321 65432
(material B)      1232 1656 16543456 16543432 12345432 1232 1656 1232 1656 16543456 1654345432 12345432 1
(refrain)            654321 65432121
(material A) 6545 6543234321 6121 6545 6543234321
(refrain)            654321 654321
spaces between the number sequences are there to more clearly show the groupings and don’t have any impact on the musical setting

In my arrangement, note 6 in material A, and note 1 in material B (marked in boldface above) are emphasised by having a crotchet duration while all other notes are quavers. The effect is to create a downbeat or arrival point at irregular intervals as the melodies twist around notes 1 or 6 in ever-changing patterns. In the first statement of material A, for example, there are six ‘bars’ beginning with note 6 as downbeat. The first is 5/8 (1 crotchet + 3 quavers) that moves anti-clockwise. The second is 11/8 (1 crotchet + 9 quavers) that generally moves anti-clockwise in a twisting manner travelling around the full circle and back to 6. The third is 5/8 and moves in clockwise direction. The fourth and fifth are the same as first and second. And so on. In contrast to Ant 5 melody , there is quite a lot of number melodic repetition in this piece. In the second statement of material A, for example, the first five bars are an exact repeat of the first five bars of first statement of material A.

The almost repeating but never exactly feature is suggestive of the irregular meters associated with Stravinsky’s music – although, in this case, it’s generated by a system.

The other 5 pieces that make up the ANTS collection span a spectrum of percussion sounds evoking deconstructed styles:
No. 1 is a bacchanalia with clapping and other sounds that perhaps recall a frenzied ‘flamenco-ish’ dance.
No. 3 uses a deconstructed drum kit to evoke a broken techno style.
No. 4 has an obstinate and obsessive character. A study of repeating notes.
No. 6 suggests on a shadowy world with wisps of snare, brushes, and cymbals sounds.
No. 7 introduces two ants, who march with a repeating 4-note pattern until they collide and veer off in different directions.

Information on the other five ANTS can be found in video-audio versions on youtube.

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Conclusion
From the discussion above, the reader could conclude that what I have described is not really composing. Haven't I just taken pre-existing material and arranged it for percussion instruments? Isn't the music generated by the cellular automaton with its rules (and with the help of an app) and not by me? The answer to these questions is, of course, 'yes'. But that is the point. In ANTS, I try and find a musical expression of cellular automata that stays true to the logic of the mathematical system while creating meaningful and compelling music for the performer and listener.

The pieces in ANTS constitute only a minuscule selection of possible Langton's Ant melodies. The decision to settle on seven melodies for the ANTS collection is entirely subjective on my part. My guiding principle was to create a balanced and varied collection of pieces. Likewise, Langton's Ant (like all cellular automata) never ends. Once an ant's journey begins, it continues indefinitely. The point where these pieces end is arbitrary. The melodies in the audio and score versions are less 4 minutes long, but they could be 40 minutes, 4 hours or 4 days long. The only limitations are the human performer and listener! 

ANTS exists in three versions: 
1. video + audio: combining video from the Langton's Ant app with the recording of my arrangements for percussion and drum machine samples - video (youtube) 
2. audio only: the recording of my arrangements for percussion and drum machine samples - audio (bandcamp) 
3. score for performers - score

The score only includes six melodies as ANTS (darting) is judged as unplayable but live performers. The scores are open-ended. Decisions are left up to the performer about the choice of instruments, dynamics, and tempo (and therefore the character of the pieces.