Musical composition and 2D cellular automata based on music intervals

Igor Lugo; Martha G. Alatriste-Contreras

REVIEW 4 major objections 5 minor 1 cited by

Musical composition and 2D cellular automata based on music intervals

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A 2D cellular automaton whose rules use melodic and harmonic intervals can organize a random array of notes into a chosen key, and the authors argue this captures part of the musical creativity process.

desk verdict A reproducible CA-music rule set whose key-tone convergence is largely built into the rules; the creativity claim overreaches. read the letter →

arxiv 2411.19844 v1 pith:GMX5NZCT submitted 2024-11-29 cs.SD cs.CYeess.ASstat.AP

Igor Lugo , Martha G. Alatriste-Contreras This is my paper · ORCID

classification cs.SDcs.CYeess.ASstat.AP MSC 68Q8000A65

keywords cellularautomatamusicalcompositionmusicintervalskeytoneEminorgammadistributionMooreneighborhoodcreativity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that a two-dimensional cellular automaton whose update rules encode melodic and harmonic intervals can take a random array of notes and organize it into a coherent pattern within a chosen key, and that this offers a computational window into musical creativity. The authors define four interval-based rules and show that, with E minor as the key tone, the proportion of cells belonging to the key rises to near totality within about five iterations, unlike random or deterministic control rules. If correct, the result would mean that a small set of music-theoretic rules can replicate a core aspect of composition, and that statistical analysis of the resulting arrays can distinguish rule-based 'music' from noise.

What carries the argument

The update mechanism is a 2D cellular automaton on a 50-by-50 grid with a Moore neighborhood. Each cell's state is a note in scientific pitch notation; the paper partitions notes into two melodic-interval sets relative to the key tone—(root, 3rd, 5th, 7th) and (2nd, 4th, 6th)—and applies four local rules (STAY FLAT, LEAP EMOTION, EXTEND/SUSPEND/ADD, SEARCH FOR SCALE TONE) that decide the next note based on how many neighbors lie in those sets. This machinery carries the argument by converting music-theoretic proximity into cellular-neighborhood counts and by making the key-tone sets absorbing under the dynamics.

What would settle it

Run a listening test or music-theoretic analysis comparing the output of the musical rule with the output of the deterministic rule; if listeners cannot reliably distinguish the musical-rule output as more musical, the claim that interval-based rules replicate musical creativity fails. Alternatively, implement a control rule that simply forces all cells into the key-tone set without interval structure and show that it produces the same key-tone proportion dynamics, demonstrating that the interval rules are not the cause.

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Extended reading notes

Core claim

The paper's central claim is that a 2D cellular automaton whose transition rules distinguish chord-tone intervals (root, 3rd, 5th, 7th) from extension and suspension tones (2nd, 4th, 6th), and which updates each cell based on how many Moore-neighbors fall in those sets, drives a random uniform array of notes to near-total membership in a target key (E minor) within about five iterations. The authors argue that this interval-based coordination is a form of harmonization that resembles the traditional compositional process, and that the resulting statistical signature (a gamma distribution with low variance and high skewness and kurtosis) is characteristic of musical audio signals, whereas the random and deterministic controls yield noise-like outputs.

Load-bearing premise

The load-bearing premise is that the fraction of cells whose notes belong to the E-minor key tone is an adequate proxy for musical organization and 'pleasing' output; the paper offers no perceptual or music-theoretic validation of this equivalence, yet its central conclusion depends on it.

Editorial extensions

If this is right

Random arrays of notes can be harmonized into a chosen key by local interval rules, offering a new instrument for composers to generate key-consistent material.
The musical rule produces a distinct statistical profile (low variance, high skewness and kurtosis) that could serve as a quantitative marker distinguishing rule-based music from noise.
If the claim holds, musical creativity in composition can be partially encoded in a few lines of code, supporting the idea that simple local rules generate complex organized structures.
The approach is generic in key: changing the key tone changes the interval sets, so the same framework should work for any major or minor tonality without altering the rule logic.
The conceptual validation framework—comparing musical, random, and deterministic rules—provides a template for testing other musically grounded transition rules in cellular automata.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The harmonization may be an artifact of the rule design: cells outside the key sets are forced to copy a neighbor, and key-set cells with few key-neighbors jump to another key-set note, so the key-tone set is an absorbing state; perceptual or music-theoretic validation would be needed to support the 'creativity' claim.
A stronger control would be a rule that also drives cells into the key but without the interval structure (e.g., a majority rule on key membership), isolating whether the interval sets themselves matter beyond simply constraining states to the key.
The gamma-distribution finding likely reflects the bounded, skewed nature of the proportion data rather than musicality; applying the same analysis to non-musical binary or categorical patterns could test whether the statistical signature is specific to music.
The model could be extended to generate actual audio and subjected to listening tests, or to other tonal systems (e.g., non-Western scales), to test whether the claimed resemblance to 'pleasing music' holds beyond the authors' visual inspection.

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Signed reviews

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a cleanly written, reproducible CA-music paper whose central claim is not supported because the key-tone convergence is built into the rules. I agree with the stress-test note; it lands.

What's actually new: a 2D CA whose update rules are defined by melodic/harmonic intervals (first set root/3rd/5th/7th, second set 2nd/4th/6th) applied to random note arrays. That specific combination is a genuine niche extension of the CA-music line from Xenakis, Wolfram, and Miranda. The authors also give complete pseudocode and OSF code/data, and the simulations run as described.

The soft spot is load-bearing, not cosmetic. In Algorithm 1, cells in the first or second interval sets either keep their state (STAY FLAT) or randomly draw from those key-tone sets (LEAP EMOTION, EXTEND/SUSPEND). The remaining cells—those not in the key—copy a random neighbor's state (SEARCH FOR SCALE TONE). Since the initial uniform random array has about 58% key-tone cells, a non-key cell has a high probability of copying a key-tone neighbor, and that fraction rises as the CA runs. So the jump in key-tone proportion by iteration 5 is a direct consequence of the rule definitions. The paper calls this 'harmonization' and 'pleasing music,' but the only measure is proportion of key-tone cells. No spatial analysis, no melodic/harmonic structure, no null model that only biases toward the key, no listening test or music-theoretic evaluation. The comparison cases don't isolate intervals either: the random rule has no key target, and the deterministic rule targets a single note under a different threshold.

Statistical details are also shaky: one seed, no error bars, and in Table 2 the random case's KS p-value is 0.049, below 0.05, yet still labeled best fit. Minor, but it adds to the impression that the quantitative story is post hoc.

Bottom line: the paper is a coherent, reproducible exploration of a niche rule set, but the creativity/harmonization claim overreaches. The useful core is the rule design and the observed dynamics; the interpretation needs a null model and a real evaluation of musical structure. This is worth a referee's time—it is not incoherent and the code/data are there—but the right outcome is major revision at best, and likely reject unless the authors reframe the claim.

Who it's for: people working on CA-based algorithmic composition who want a concrete example of interval rules; also useful as a teaching case of circularity in generative-music claims. I would not cite it in my own work.

Referee Report

4 major / 5 minor

Summary. The paper proposes a two-dimensional cellular automaton whose cell states are musical notes in scientific pitch notation and whose update rules are defined in terms of melodic and harmonic intervals relative to a chosen key tone (E minor). Starting from a uniformly random 50x50 array, the musical rule set (Algorithm 1) is compared with a fully random rule (Algorithm 2) and a deterministic rule targeting A4 (Algorithm 3). The main reported findings are that the musical rule drives the proportion of key-tone cells to nearly 100% within five iterations, that pairwise Spearman correlations between the three final arrays are near zero, and that the final arrays are best described by gamma distributions with skewed shapes. The paper interprets these results as evidence that interval-based cellular automata can coordinate random note arrays into organized, music-like patterns and can 'replicate some aspects of the creativity used in musical composition' (Section 5.1).

Significance. If the central claim were supportable, a cellular-automaton model that uses music-theoretic intervals to produce structured musical output would be a relevant contribution to computational creativity and algorithmic composition. The manuscript has genuine strengths: the algorithm is stated explicitly, the implementation details are reproducible, and the OSF repository makes code and data available. However, the significance of the paper as a claim about musical creativity is currently low. The only quantitative measure of 'organization' is the proportion of cells belonging to the key tone, and that measure is largely built into the update rules. There is no perceptual validation, no music-theoretic analysis of the generated note arrays, and no null model that isolates the role of interval structure from the role of target-set membership. The statistical analysis is also fragile, resting on a single seed and marginal goodness-of-fit results. The paper's broad conclusion therefore goes beyond what the evidence demonstrates.

major comments (4)

[Section 3.2, Algorithm 1; Section 4, Figure 3] The rise in key-tone proportion shown in Figure 3(a) is nearly tautological. For cells in the first or second interval sets, every update rule (STAY FLAT, LEAP EMOTION, EXTEND/SUSPEND) either keeps the current note or draws uniformly from one of those two sets, both of which are subsets of the E-minor key. For cells outside those sets, SEARCH FOR SCALE TONE copies a random neighbor's state, and because the initial uniform array already contains a substantial fraction of key-tone cells, non-key cells are likely to be replaced by key-tone notes. No control rule with the same key-tone target bias but without interval-based neighborhoods is run, so the comparison in Algorithms 2 and 3 does not establish that intervals, rather than target-set membership, drive the observed convergence.
[Section 4, Figure 3; Section 5] The paper's only quantitative evidence for 'organization' or 'pleasing music' is the proportion of cells in the key tone across iterations. There is no spatial autocorrelation or clustering analysis of the final note array, no examination of melodic or harmonic structure in the generated note sequences, no comparison with any musical corpus, and no listening test or music-theoretic evaluation. The statement that the outputs 'closely resemble the pleasing music' (Section 4) and that the musical rule produces 'organized sounds closely related to music' (Section 5) is therefore not supported by the data presented.
[Section 3.3; Table 2; Table 1] The statistical analysis is weaker than reported. In Table 2, the KS p-value for the random case is 0.049, which is below 0.05, so the gamma distribution is actually rejected at the 5% level for that case; the musical and deterministic p-values (0.066 and 0.056) are marginal. In addition, the entire study uses one random seed and one final realization per case, so there is no uncertainty quantification or error bars for Figure 3 or for the Spearman correlations in Table 1. The correlations are computed on single final arrays and do not characterize the convergence dynamics.
[Section 3.3, Algorithms 2 and 3] The control cases are not matched to the musical rule, so they cannot isolate the role of interval structure. The random rule has no key-tone target at all, while the deterministic rule targets the single pitch A4 with a different neighbor threshold (at least five neighbors) than the musical rule uses (thresholds of at least three, one or two, and zero). The observed differences among the three cases could therefore be due to the size of the target set, the threshold values, or the amount of stochasticity, rather than to the use of melodic and harmonic intervals.

minor comments (5)

[Section 3.2] The Moore neighborhood is described as 'defined by the fourth orthogonal and diagonal adjacent cells,' which is unclear; it should say the eight surrounding cells (four orthogonal and four diagonal). Figure 2 also contains typographical errors: 'Neighborhods' and 'Von Neuman.'
[Section 1] The phrase 'caring out the musical composition' should be 'carrying out the musical composition'; similar typographical issues appear elsewhere (for example, 'existed knowledge').
[Section 4] The sentence 'Subfigure (c) displays a gradual rise ... We can see that reached the 30 iteration ... closed to 80%' should be reworded for grammatical correctness, and 'Emin' should be written as 'E minor.'
[Section 4, Table 1] The text says 'the null hypothesis that two cases have no ordinal correlation is accepted,' but a non-significant p-value only means failure to reject, not acceptance of the null.
[Section 4, Tables 2 and 3] The KS test is reported without the sample size, and the gamma distribution is listed with an estimated location parameter; the text should clarify how many data points were used and what the location parameter represents.

Circularity Check

2 steps flagged · score 8.0 of 10

The E-minor harmonization is built into Algorithm 1's target sets, and the gamma-as-musicality interpretation rests on the authors' own prior arXiv result.

self definitional [Section 3.2, items 3-4 and Algorithm 1; Section 4, Figure 3(a)]
"The first set of melodic intervals is defined by the tuple of (root, 3rd, 5th, 7th) and the second set is defined by (2nd, 4th, 6th). ... (d) SEARCH FOR SCALE TONE. ... Each cell outside the first or second sets of melodic intervals updates its state by randomly selecting one value related to its neighboring cells. ... Subfigure (a) shows a sharp rise in the number of cells in the tonal key. We can see that after 5 iterations the number of cells in the key tone is almost complete, i.e., the array has been harmonized."

The claimed result—that the musical interval rules harmonize random notes and suggest that such rules matter for generating large-scale patterns of organized notes—is the direct output of the rule's own definition. The first and second melodic interval sets are exactly the seven pitch classes of the E-minor key (E, F#, G, A, B, C, D). Algorithm 1 keeps every cell in those sets inside them (STAY FLAT, LEAP EMOTION, EXTEND/ADD) and forces every cell outside them to copy a random neighbor. Because the initial uniform SPN array already has about 58% key-tone cells (7 of 12 pitch classes), the copy step drives the minority toward key-tone cells; the sharp rise in Figure 3(a) and saturation by iteration 5 is the rule saturating on its built-in target set.
self citation load bearing [Section 2 (Literature review) and Section 5 (Discussion)]
"In the same way, the work of Lugo and Alatriste-Contreras [20] studied the spectrum of audio signals ... They found that the best-fit statistical distributions related to those signals were skewed statistical distributions, and they suggested that these type of distributions are the most common statistical attributes in the nature of audio signals related to music. ... the gamma distribution associated with the musical rule shows the natural characteristic of audio signals related to music."

The only external evidence offered that a gamma fit indicates musicality is the authors' own prior work, cited as [20] (Lugo and Alatriste-Contreras, arXiv:2404.18355). That work is not independently established in the present paper and is not machine-checked or reproduced against an external benchmark. Section 5 then uses this self-cited finding to interpret the fitted gamma for the musical rule as the natural characteristic of audio signals related to music and to conclude that the output is closely related to musical sounds rather than noise. The interpretive conclusion therefore reduces to a self-citation chain.

full rationale

Score is high because the paper's central empirical claim—that interval-based CA rules coordinate random note arrays into a key and resemble pleasing music—is not discovered from the dynamics; it is embedded in Algorithm 1's state sets and copying rule. The only measured outcome (key-tone proportion) is the same set the rules are restricted to, so the rise in Figure 3(a) is a saturation effect of the rule definition. The comparison to random and deterministic rules does not isolate interval structure, because the random rule has no key-tone bias and the deterministic rule uses a different threshold and a single target note A4; this weakens the causal claim but is a validation-design issue rather than circularity. Separately, the step from gamma distribution to musical sound rests on the authors' own prior arXiv finding, which is a load-bearing self-citation. Prior CA/music literature (Xenakis, Wolfram, Miranda) is cited only for background, not for the key-tone result, so no other circularity is found.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The model depends on several hand-chosen parameters (grid size, iteration count, thresholds in the rules) and domain assumptions about Western tonality and the relation between key-tone proportion and musical quality. No new physical or conceptual entities are introduced. The most consequential input is the implicit assumption that being in the key tone equals being musical, which is never tested.

free parameters (6)

Array dimension = 50x50
Chosen 'designed for simplicity' with no sensitivity analysis or justification of why this size matters.
Number of iterations = 30
Chosen because the model 'converges into a fixed pattern' after this number, a post hoc stopping rule with no formal convergence criterion.
Key tone = E minor
Selected ex-ante without justification and without testing whether the results generalize to other keys.
STAY FLAT neighbor threshold = >= 3
Hand-set threshold in Algorithm 1; no justification is given for the value 3.
LEAP EMOTION neighbor count = 1 or 2
Hand-set threshold in Algorithm 1; no justification is given.
Deterministic rule threshold = neighbors >= 5
Hand-set in Algorithm 3; no justification is given.

assumptions (6)

domain assumption The Western twelve-tone equal temperament with A4=440 Hz is the relevant musical system.
Section 3.1: all notes and intervals are drawn from SPN in twelve-tone equal temperament; the results are specific to this musical system.
domain assumption Music intervals defined by root, 3rd, 5th, 7th (first set) and 2nd, 4th, 6th (second set) are meaningful tonal sets for harmonization.
Section 3.2: the entire rule set assumes these intervals capture the essence of tonal music, without empirical justification in the paper.
domain assumption A 2D CA with Moore neighborhood is an appropriate model for musical composition.
Section 3.2: the authors follow the Game of Life formulation, but do not compare with other CA topologies or other generative models.
domain assumption The proportion of cells in the key tone is a valid measure of musical organization or pleasingness.
Section 4, Figure 3: the paper equates harmonization with the number of cells whose notes belong to the key, but no perceptual or music-theoretic link is established.
domain assumption The final array values can be treated as a continuous sample suitable for gamma distribution fitting via KS test.
Section 3.3 and Table 2: the paper fits gamma distributions to the final array data, but the nature of the data (note indices, frequencies, or counts) is never clearly specified.
ad hoc to paper The authors' prior result that music audio signals follow skewed/gamma distributions (Ref [20]) is a valid benchmark.
Section 5: the paper uses its own earlier finding as the standard for what music-like data should look like, making the validation self-referential.

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Cite this review

Pith. "Pith review of Musical composition and 2D cellular automata based on music intervals." pith.science (2026). https://pith.science/paper/GMX5NZCT

@misc{pith2026241119844,
  author       = {Pith},
  title        = {Pith review of: Musical composition and 2D cellular automata based on music intervals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GMX5NZCT}},
  note         = {Machine review of arXiv:2411.19844}
}

read the original abstract

This study is a theoretical approach for exploring the applicability of a 2D cellular automaton based on melodic and harmonic intervals in random arrays of musical notes. The aim of this study was to explore alternatives uses for a cellular automaton in the musical context for better understanding the musical creativity. We used the complex systems and humanities approaches as a framework for capturing the essence of creating music based on rules of music theory. Findings suggested that such rules matter for generating large-scale patterns of organized notes. Therefore, our formulation provides a novel approach for understanding and replicating aspects of the musical creativity.

Figures

Figures reproduced from arXiv: 2411.19844 by the authors.

**Figure 1.** Scientific pitch notation and frequencies based on the the twelvetone equal temperament in which A4 = 440hz tion, we used the Musicpy library (https://musicpy.readthedocs.io/en/latest/) because it shows a concise syntax for writing code and generating music. Therefore, based on those sources, we can generate our CA for exploring the effect of rules based on musical intervals in an array of random notes. Furthermo… view at source ↗

**Figure 2.** Neighborhods. (a) Von Neuman and (b) Moore. 3. First and Second melodic intervals. These are two variables related to the key tone. The first set of melodic intervals is defined by the tuple of (root, 3rd, 5th, 7th) and the second set is defined by (2nd, 4th, 6th). The first interval is the common three notes in consecutive thirds (triads), plus the seventh note. The second set is related to those notes associated w… view at source ↗

**Figure 3.** Number of cells and iterations in each case. (a) Musical, (b) Random, and (c) Deterministic associated with notes in the key tone. Subfigure (a) shows a sharp rise in the number of cells in the tonal key. We can see that after 5 iterations the number of cells in the key tone is almost complete, i.e., the array has been harmonized. Subfigure (b) displays the common behavior of random numbers generated by a uniform d… view at source ↗

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Reference graph

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