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Musical composition and 2D cellular automata based on music intervals

This study proposes a theoretical framework utilizing 2D cellular automata governed by musical intervals to model and replicate the essence of musical creativity through the generation of organized large-scale patterns from random note arrays.

Original authors: Igor Lugo, Martha G. Alatriste-Contreras

Published 2026-07-01

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CC BY 4.0

Original authors: Igor Lugo, Martha G. Alatriste-Contreras

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant checkerboard, but instead of black and white squares, every square holds a random musical note. Some are high, some are low, some are sharp, and some are flat. At the start, this board looks like a chaotic mess of noise, much like a room full of people shouting different words at once.

This paper asks a simple question: Can we teach this chaotic board to "sing" a beautiful song just by giving it a few simple rules?

The authors, Igor Lugo and Martha G. Alatriste-Contreras, say yes. They built a digital "musical garden" using a concept called a 2D Cellular Automaton. Think of this not as a complex computer program, but as a set of very simple, neighborly instructions that every note on the board follows simultaneously.

The Rules of the Game

In their model, every note looks at its immediate neighbors (the squares touching it) and decides what to do next based on music theory—specifically, the "distance" between notes, known as intervals.

They created four main "personality types" for the notes to follow:

The Stay-Flat Rule: If a note is surrounded by many "friendly" neighbors (notes that fit well together in a chord), it stays exactly the same. It's like a person at a party who feels comfortable and decides to keep telling the same story.
The Leap Emotion Rule: If a note has a few friendly neighbors, it might jump up or down to a different note that still fits the group. It's like someone at the party deciding to change the subject to something related but new.
The Extension Rule: If a note is lonely (has no friendly neighbors), it reaches out to find a note that adds a little "spice" or "suspense" to the mix, like adding a 9th or 11th note to a chord.
The Search Rule: If a note is completely out of place, it looks at its neighbors and tries to pick a note that fits the group better.

The Experiment

The researchers ran three different scenarios on their 50x50 grid of random notes:

The Musical Rule: The notes followed the specific music theory rules described above.
The Random Rule: Notes just picked new notes completely by chance, ignoring their neighbors.
The Deterministic Rule: Notes followed a rigid, non-random rule to force them into a specific pattern.

What Happened?

The results were like watching a messy room clean itself up.

The Musical Rule: Within just a few steps (iterations), the chaotic noise transformed into a harmonious, organized structure. The notes "self-organized" into a specific key (they used E minor). It was as if the room instantly found a rhythm and everyone started singing in harmony.
The Random Rule: The notes remained chaotic, jumping up and down with no pattern, like static on a radio.
The Deterministic Rule: The notes organized, but in a very rigid, predictable way that lacked the "feel" of the musical rule.

The "Fingerprint" of Music

The authors didn't just listen; they measured the data. They found that the "Musical Rule" created a specific statistical pattern (called a Gamma distribution) that is highly skewed.

To use an analogy: If you were to measure the height of people in a random crowd, you'd get a bell curve. But if you measured the "musicality" of the notes in their simulation, the results looked like a mountain with a long tail. The authors suggest this specific "shape" of data is a fingerprint of what we humans perceive as music. The random noise didn't have this shape; only the rule-based music did.

The Big Takeaway

The paper concludes that musical creativity might be simpler than we think. It suggests that the complex, beautiful process of composing music can be broken down into simple, local rules—like a set of instructions for a cellular automaton.

Just as a single cell in a biological organism follows simple rules to create a complex body, a single note following simple interval rules can create a complex, pleasing piece of music. The authors argue that their model captures the essence of how humans turn random ideas into organized art, proving that you don't need a genius to compose; you just need the right set of rules to guide the chaos into harmony.

Technical Summary: Musical Composition and 2D Cellular Automata Based on Music Intervals

Problem Statement
The paper addresses the challenge of modeling musical creativity and composition using computational frameworks. While previous studies by Xenakis and Wolfram demonstrated that simple computational rules could generate complex structures, the exploration of Cellular Automata (CA) specifically grounded in music theory—particularly the use of melodic and harmonic intervals to harmonize random note arrays—remains under-analyzed. The authors seek to determine if CA rules based on music theory can coordinate large-scale sets of random notes into a specific key and if such transitional rules can resemble traditional compositional approaches.

Methodology
The study employs a theoretical approach combining complex systems theory and humanities perspectives. The core methodology involves the implementation of a 2D Cellular Automaton (CA) using Python (libraries: NumPy, SciPy, Matplotlib, Musicpy).

System Configuration:
- Grid: A 50x50 array (2,500 cells) representing a finite plane with closed boundaries.
- State: Each cell holds a note defined by Scientific Pitch Notation (SPN) within the Western twelve-tone equal temperament system (A4 = 440Hz).
- Initial Conditions: The array is initialized with notes randomly selected from a uniform distribution across the full SPN range (C0 to B8).
- Neighborhood: The Moore neighborhood (8 adjacent cells: 4 orthogonal, 4 diagonal) is used.
- Variables: The system utilizes "First Melodic Intervals" (root, 3rd, 5th, 7th) and "Second Melodic Intervals" (2nd, 4th, 6th, and extended notes like 9th, 11th, 13th).
Transition Rules (Algorithm 1):
The study proposes four specific musical rules to update cell states simultaneously based on the number of neighbors falling within the defined interval sets:
1. Stay Flat: If a cell is within an interval set and has $\ge$ 3 neighbors in the same set, it retains its current state.
2. Leap Emotion: If a cell is within an interval set and has 1 or 2 neighbors in the same set, it updates to a random value from that interval set.
3. Extend, Suspend, or Add: If a cell is within an interval set and has 0 neighbors in the same set, it updates to a random value from the other interval set (simulating extended or suspended chords).
4. Search for Scale Tone: If a cell is outside the interval sets, it updates to a random value selected from its neighbors.
Comparative Cases:
To validate the model, the authors compare the "Musical Rule" (Algorithm 1) against two baselines:
- Random Rule: Transitions are based on random SPN selection, ignoring current state and neighbors.
- Deterministic Rule: Transitions depend strictly on neighbor counts relative to a key tone, updating to a fixed note (A4) if a threshold is met.
Data Analysis:
The simulation runs for 30 iterations. The authors analyze the proportion of cells converging to the key tone (E minor), calculate Spearman correlation coefficients between cases, and perform goodness-of-fit tests (Kolmogorov–Smirnov) to identify the best-fitting statistical distributions for the resulting data.

Key Results

Convergence Dynamics: The "Musical Rule" case demonstrated a sharp rise in the proportion of cells matching the key tone, reaching near-complete harmonization after approximately 5 iterations. In contrast, the "Random Rule" showed volatile fluctuations, and the "Deterministic Rule" showed a gradual rise, reaching only ~80% harmonization by iteration 30.
Statistical Independence: Spearman correlation analysis revealed negligible correlation coefficients between the three cases (all $p$ -values > 0.05), confirming that the musical rule produces a distinct dynamic pattern compared to random or purely deterministic approaches.
Distribution Analysis: All three cases were best described by a Gamma distribution. However, the parameters differed significantly:
- The Musical Rule exhibited the smallest variance but the highest skewness (3.09) and kurtosis (14.39), indicating a highly asymmetric distribution with fat tails.
- The Random Rule showed the highest variance and the lowest skewness/kurtosis.
- The Deterministic Rule fell between the two.
  The authors note that the high skewness and kurtosis in the musical case align with statistical patterns previously identified in natural audio signals of music.

Key Contributions

Novel CA Formulation: The paper introduces a specific 2D CA configuration where state updates are governed by music theory concepts (intervals, triads, extensions) rather than abstract mathematical rules.
Bridging Disciplines: It provides a framework connecting complex systems modeling with the humanities, suggesting that the "mental activity" of musical creativity can be captured through simple code based on theoretical rules.
Empirical Validation: The study offers quantitative evidence (via statistical moments and distribution fitting) that rules based on music theory generate organized, "musical" patterns distinct from noise or purely deterministic outputs.

Significance and Claims
The authors claim that their formulation offers a novel approach to understanding and replicating aspects of musical creativity. They posit that:

Replication of Creativity: The CA model replicates a common strategy used by musicians: determining sequences of actions (notes) based on rules (intervals) to generate coherent structures from disorder.
Nature of Musical Signals: The findings suggest that sounds perceived as "music" are associated with highly skewed statistical distributions (specifically Gamma distributions with high kurtosis), which the musical rule successfully reproduces.
Complementary Tool: The study does not claim to replace human composers but suggests that CAs based on music theory can serve as instruments to assist musicians and complement scientific and humanities research into the composition process.

The paper concludes modestly, stating that while the model shows potential, future work is required to evaluate this approach using other computational models, such as complex networks.