# Annotated bibliography for mathematical music visualization

Research date: 8 October 2026; implementation supplements added on 9 October.

This catalogue contains 63 references: 39 research publications or research manuscripts, 21 educational or official technical resources, and 3 existing projects. It is a scoped literature review, not an exhaustive systematic review. Access labels distinguish inspected full text sections from abstracts, metadata, and product descriptions. An available PDF is not a claim that every page has been read. DOI, repository, and publication dates are kept distinct where they differ.

Primary papers and official documentation support technical claims. Existing project pages establish prior products and inspiration; their claims of physical fidelity have not been independently tested. Recent arXiv versions are identified without assuming peer review. The proposals in the other research files are our synthesis, rather than findings reported by these sources.

## Mechanics and pattern formation

### P01 Vibration of plates

Leissa 1969. NASA SP 160; monograph. [Source](https://ntrs.nasa.gov/citations/19700009156).

**Access:** Metadata inspected; full PDF exceeded browser size limit.

Reference for plate modes and boundary conditions. Use its tables later to validate the chosen geometry; no numerical table values were extracted in this review.

### P02 Numerical Chladni figures

Müller 2013. European Journal of Physics 34, 1067–1074. [Source](https://arxiv.org/abs/1308.5523).

**Access:** Full text sections inspected.

Finite element membrane eigenmodes and an educational interface. Useful starting point for arbitrary domains; the membrane operator must not be substituted for plate bending without changing the model.

### P03 Port Hamiltonian formulation and symplectic discretization of plate models Part II Kirchhoff model for thin plates

Brugnoli et al. 2019. Applied Mathematical Modelling 75, 961–981; preprint 2018. [Source](https://arxiv.org/abs/1809.11136).

**Access:** Full text model sections inspected.

Thin plate equations, bending rigidity, and energy structure. The advanced port Hamiltonian discretization is an optional research extension, not a prerequisite for the first browser prototype.

### P04 Vibration driven random walk in a Chladni experiment

Grabec 2017. Physics Letters A 381, 59–64. [Source](https://doi.org/10.1016/j.physleta.2016.10.059).

**Access:** Publisher abstract and section excerpts inspected.

Experimental motivation for stochastic grain motion on a vibrating membrane. Supports a random walk approach; it does not calibrate grains on every material or mounting.

### P05 Chladni patterns explained by the space dependent diffusion of bouncing grains

Abramian, Protière, Lazarus and Devauchelle 2025. Physical Review Research 7, L032001. [Source](https://doi.org/10.1103/PhysRevResearch.7.L032001).

**Access:** Full text model and results sections inspected.

Key modern source for spatial diffusivity and grain accumulation. Its experimentally supported regime is dilute bouncing grains on a membrane. Primary PDF: https://journals.aps.org/prresearch/pdf/10.1103/PhysRevResearch.7.L032001

### P06 Stochastic models for Chladni figures

Arango and Reyes 2016. Proceedings of the Edinburgh Mathematical Society 59, 287–300. [Source](https://doi.org/10.1017/S0013091515000139).

**Access:** Full text stochastic model sections inspected.

Connects plate nodal sets with stochastic stability under stated assumptions. A contrasting particle model for the proposed comparison; theoretical conditions matter.

### P07 Diffusion Equation Generalized for Modeling of Chladni Patterns

Grabec and Sok 2023. Journal of Mechanical Engineering 69, 284–286. [Source](https://www.sv-jme.eu/article/diffusion-equation-generalized-for-modeling-of-chladni-patterns/).

**Access:** Indexed abstract and metadata inspected; PDF retrieval failed.

Continuum treatment of vibration driven random walks. Relevant to the density solver; inspect the full derivation before implementing this specific formulation.

### P08 Formation of inverse Chladni patterns in liquids at microscale roles of acoustic radiation and streaming induced drag forces

Lei 2017. Microfluidics and Nanofluidics 21, 50. [Source](https://link.springer.com/article/10.1007/s10404-017-1888-5).

**Access:** Full text available; abstract and model discussion inspected.

Shows why wet microscale particle patterns need radiation and streaming forces. An extension source rather than validation of a dry sand model.

### P09 Sorting particles in the air using direct and inverse Chladni patterns of a vibrating plate

Bourrud et al. 2025. Powder Technology 457, 120867. [Source](https://doi.org/10.1016/j.powtec.2025.120867).

**Access:** Indexed publisher abstract inspected; direct retrieval failed.

Recent experimental sorting with direct and inverse patterns. Useful evidence that particle properties affect the result; detailed sorting criteria require full text.

### P10 Numerical simulation of Faraday waves

Périnet, Juric and Tuckerman 2009. Journal of Fluid Mechanics; author manuscript. [Source](https://arxiv.org/abs/0901.0464).

**Access:** Full text introduction and method context inspected.

Three dimensional two fluid Navier–Stokes simulation compared with experiments. Demonstrates the complexity of genuine Faraday physics compared with a simple ripple shader.

### P11 Polygonal patterns of Faraday water waves analogous to collective excitations in Bose Einstein condensates

Liu and Wang 2024. Nature Physics 20, 287–293; first online 2023. [Source](https://www.nature.com/articles/s41567-023-02294-y).

**Access:** Publisher abstract and article context inspected.

Polygonal surface patterns in concave basins. Visually promising long term extension, with geometry and forcing that must be specified.

### P12 Musical Interfaces Visualization and Reconstruction of Music with a Microfluidic Two Phase Flow

Sze Yi Mak, Zida Li, Arnaud Frere, Tat Chuen Chan and Ho Cheung Shum 2014. Scientific Reports 4, 6675. [Source](https://www.nature.com/articles/srep06675).

**Access:** Full text experimental results sections inspected.

Physical music visualization using an unusually low interfacial tension fluid system. Inspires a fluid interface mode; it is not an ordinary bowl of water experiment.

### P13 Stable Fluids

Stam 1999. SIGGRAPH 1999, 121–128. [Source](https://www.dgp.toronto.edu/people/stam/reality/Research/pdf/ns.pdf).

**Access:** Full text numerical method sections inspected.

Semi Lagrangian advection and projection enable interactive flow. Numerical stability does not guarantee accurate small scale dynamics or negligible numerical dissipation.

### P14 Real Time Fluid Dynamics for Games

Stam 2003. Game Developers Conference tutorial. [Source](https://www.dgp.toronto.edu/people/stam/reality/Research/pdf/GDC03.pdf).

**Access:** Full text available; solver overview inspected.

Practical companion to Stable Fluids. Useful implementation reading; distinguish a graphics tutorial from a new experimental physics validation.

### P15 Visual Simulation of Smoke

Fedkiw, Stam and Jensen 2001. SIGGRAPH 2001, 15–22. [Source](https://www.dgp.toronto.edu/people/stam/reality/Research/pdf/smoke.pdf).

**Access:** Full text implementation sections inspected.

Vorticity confinement and smoke transport. Helps discuss detail lost on coarse grids and the extra forces introduced to recover visual richness.

### P16 Complex Patterns in a Simple System

Pearson 1993. Science 261, 189–192. [Source](https://arxiv.org/abs/patt-sol/9304003).

**Access:** Full text equations and pattern discussion inspected.

Gray–Scott reaction–diffusion patterns from finite perturbations. The paper explicitly distinguishes its selected regime from stable classical Turing patterns.

### P17 The chemical basis of morphogenesis

Turing 1952. Philosophical Transactions of the Royal Society B 237, 37–72. [Source](https://doi.org/10.1098/rstb.1952.0012).

**Access:** Bibliographic record located; publisher and university PDF retrieval failed.

Historical foundation of reaction–diffusion pattern formation. Retained as foundational reading, not used as a source for uninspected numerical or experimental details.

### P18 Deterministic Nonperiodic Flow

Lorenz 1963. Journal of the Atmospheric Sciences 20, 130–141. [Source](https://doi.org/10.1175/1520-0469(1963)020%3C0130:DNF%3E2.0.CO;2).

**Access:** Publisher abstract inspected.

Nonlinear phase space trajectories and sensitivity to initial conditions. A music driven Lorenz scene would be a designed extension, not a physical acoustic prediction.

### P19 From Kuramoto to Crawford exploring the onset of synchronization in populations of coupled oscillators

Strogatz 2000. Physica D 143, 1–20. [Source](https://doi.org/10.1016/S0167-2789(00)00094-4).

**Access:** Full text governing equations inspected.

Review of synchronization and the order parameter. Mathematical basis for oscillator rings; no claim that arbitrary music naturally produces Kuramoto dynamics.

### P20 Oscillators that sync and swarm

O’Keeffe, Hong and Strogatz 2017. Nature Communications 8, 1504. [Source](https://www.nature.com/articles/s41467-017-01190-3).

**Access:** Full text collective state sections inspected.

Combines spatial attraction and phase dynamics. Offers rings and clusters as an unusual visualization; audio forcing and monochrome phase encoding are proposed adaptations.

### P21 Can One Hear the Shape of a Drum

Kac 1966. American Mathematical Monthly 73, 1–23. [Source](https://www.math.ucdavis.edu/~hunter/m207b/kac.pdf).

**Access:** Original paper PDF available; introductory material inspected.

Spectral geometry as an educational theme. This paper poses the famous question; it is not the later negative solution.

### P22 One cannot hear the shape of a drum

Gordon, Webb and Wolpert 1992. Bulletin of the American Mathematical Society 27, 134–138. [Source](https://arxiv.org/abs/math/9207215).

**Access:** Author manuscript abstract inspected.

Nonisometric isospectral planar domains. A precise counterexample to the idea that frequency information always determines a unique shape.

### P23 Flocks Herds and Schools A Distributed Behavioral Model

Reynolds 1987. SIGGRAPH Computer Graphics 21, 25–34. [Source](https://www.red3d.com/cwr/papers/1987/boids.html).

**Access:** Author publication page and abstract inspected.

Local behavioral rules produce flocking. An optional generative art mode; its behavior is not a simulation of acoustic sand.

## Music visualization and perception

### P24 CymaSense a real time 3D cymatics based sound visualisation tool

McGowan, Leplâtre and McGregor 2017. DIS Companion 2017. [Source](https://doi.org/10.1145/3064857.3079159).

**Access:** Indexed institutional abstract inspected; repository returned 403.

Relevant prior cymatics visualization work. Prevents an unsupported first ever novelty claim; detailed algorithms need direct paper access.

### P25 An Audio Driven System For Real Time Music Visualisation

Graf, Opara and Barthet 2021. AES 150 Convention, Paper 10498. [Source](https://arxiv.org/abs/2106.10134).

**Access:** Full text architecture and evaluation sections inspected.

Modular feature mapping and user evaluation. Reports useful creative possibilities alongside usability problems; motivates separating simple controls from technical inspection.

### P26 An Artistic Visualization of Music Modeling a Synesthetic Experience

Adiletta and Thomas 2020. arXiv project report; publication status not independently established. [Source](https://arxiv.org/abs/2012.08034).

**Access:** Full text design and interview sections inspected.

FFT features drive particle art informed by interviews. Individual descriptions and key emotion mappings should not be treated as universal psychological laws.

### P27 Development and evaluation of a mixed reality music visualization for a live performance based on music information retrieval

Erdmann, von Berg and Steffens 2025. Frontiers in Virtual Reality 6. [Source](https://www.frontiersin.org/journals/virtual-reality/articles/10.3389/frvir.2025.1552321/full).

**Access:** Full text methods and results inspected.

Audio linked and randomized visuals compared in a within participant study. Relevant to synchronization and aesthetic evaluation; MR and live performance limit transfer to this website.

### P28 Audiovisual correspondence between musical timbre and visual shapes

Adeli, Rouat and Molotchnikoff 2014. Frontiers in Human Neuroscience 8, 352. [Source](https://www.frontiersin.org/journals/human-neuroscience/articles/10.3389/fnhum.2014.00352/full).

**Access:** Full text methods and results inspected.

Timbre associations were observed with colored and grayscale shapes. Useful support for shape based design; spectral centroid alone is not a complete timbre model.

### P29 Dynamic complexity in audiovisual aesthetics

Clemente, Board, Pearce and Orgs 2024. Psychology of Aesthetics Creativity and the Arts; accepted manuscript record. [Source](https://discovery.ucl.ac.uk/id/eprint/10198080/).

**Access:** Institutional abstract inspected.

Complexity and audiovisual congruence influence liking in controlled abstract stimuli. Motivates measuring visual complexity rather than assuming more particles always improve art.

### P30 musicolors Bridging Sound and Visuals For Synesthetic Creative Musical Experience

ChungHa Lee and Jin Hyuk Hong 2025. arXiv version; publication status not independently established. [Source](https://arxiv.org/abs/2503.14220).

**Access:** Abstract and metadata inspected.

Web visualization library and qualitative creative use study. Relevant to preset sharing and creative control, not evidence of a physical sound to color law.

### P31 Visual and Auditory Aesthetic Preferences Across Cultures

Harin Lee et al. 2025. arXiv v2; record indicates CogSci 2025 presentation. [Source](https://arxiv.org/abs/2502.14439).

**Access:** Abstract and metadata inspected.

Reports shared patterns and cultural variation in aesthetic judgments. Useful guard against extrapolating one small convenience sample to all listeners.

### P32 Abstraction Beats Realism Physiological Visualizations Enhance Arousal Synchrony in VR Concert Recreations

Meng et al. 2026. Recent arXiv version; publication status not independently established. [Source](https://arxiv.org/abs/2603.19730).

**Access:** Abstract and metadata inspected.

A frontier example of evaluating abstract VR visuals using physiology. It does not establish that less physical accuracy is better for this browser project.

## Audio analysis

### P33 A Tutorial on Onset Detection in Music Signals

Bello et al. 2005. IEEE Transactions on Speech and Audio Processing 13, 1035–1047. [Source](https://doi.org/10.1109/TSA.2005.851998).

**Access:** Indexed original paper abstract inspected; located PDF retrieval failed.

Foundational comparison of onset methods. Supports treating transient detection as a distinct task from tempo and beat tracking.

### P34 Harmonic Percussive Separation Using Median Filtering

FitzGerald 2010. DAFx 2010. [Source](https://dafx.de/paper-archive/2010/DAFx10/DerryFitzGerald_DAFx10_P15.pdf).

**Access:** Full text method sections inspected.

Median filtering separates harmonic and percussive spectrogram structure approximately. Useful future preprocessing; sustained and transient components are not perfectly isolated instruments.

### P35 librosa Audio and Music Signal Analysis in Python

McFee et al. 2015. SciPy 2015, 18–24. [Source](https://brianmcfee.net/papers/scipy2015_librosa.pdf).

**Access:** Original paper available; overview inspected.

Research reference implementation for offline feature comparison. Paper describes an old library version; implementation should pin and document actual installed versions.

### P36 FMP Notebooks Educational Material for Teaching and Learning Fundamentals of Music Processing

Müller and Zalkow 2019. ISMIR 2019. [Source](https://www.audiolabs-erlangen.de/content/05_fau/professor/00_mueller/05_notebookFMP/2019_MuellerZalkow_FMP_ISMIR.pdf).

**Access:** Full text available; educational overview inspected.

Reproducible explanations of signal processing with code. Useful model for accompanying the demo with readable mathematical experiments.

### P37 madmom a new Python Audio and Music Signal Processing Library

Böck et al. 2016. Author paper version. [Source](https://arxiv.org/abs/1605.07008).

**Access:** Abstract and metadata inspected.

Reference for high level rhythm analysis. A research baseline option rather than a required browser dependency.

## Browser engineering and publication

### D01 Discrete Short Time Fourier Transform

AudioLabs. [Source](https://www.audiolabs-erlangen.de/resources/MIR/FMP/C2/C2_STFT-Basic.html).

**Access:** Equations and examples inspected.

Defines the window, hop, complex coefficients and spectrogram. Basis for the time and frequency resolution discussion.

### D02 Spectral Based Novelty

AudioLabs. [Source](https://www.audiolabs-erlangen.de/resources/MIR/FMP/C6/C6S1_NoveltySpectral.html).

**Access:** Equations and code inspected.

Log compression, positive spectral differences and novelty normalization. Useful for detecting attacks; an attack is not necessarily a beat.

### D03 Log Frequency Spectrogram and Chromagram

AudioLabs. [Source](https://www.audiolabs-erlangen.de/resources/MIR/FMP/C3/C3S1_SpecLogFreq-Chromagram.html).

**Access:** Resource inspected.

Pitch class aggregation provides an optional harmony representation. It does not directly recover clean notes from every polyphonic mix.

### D04 AnalyserNode

MDN. [Source](https://developer.mozilla.org/en-US/docs/Web/API/AnalyserNode).

**Access:** API methods inspected.

Quick waveform and spectrum analysis. Frequency output does not supply complex phase coefficients.

### D05 AudioWorklet

MDN. [Source](https://developer.mozilla.org/en-US/docs/Web/API/AudioWorklet).

**Access:** API documentation inspected.

Dedicated audio processing context for custom analysis and reduced models. Requires careful scheduling and secure deployment.

### D06 Web Audio API

W3C. [Source](https://www.w3.org/TR/webaudio/).

**Access:** Cross origin security and analyser sections inspected.

Normative details of audio analysis and cross origin silence. Anchor for distinguishing playable media from analyzable media.

### D07 WebGPU API

MDN. [Source](https://developer.mozilla.org/en-US/docs/Web/API/WebGPU_API).

**Access:** Capabilities and compatibility status inspected.

Compute shaders are attractive for dense simulation. Availability is not universal; feature detection and a fallback remain necessary.

### D08 WebGL2RenderingContext

MDN. [Source](https://developer.mozilla.org/en-US/docs/Web/API/WebGL2RenderingContext).

**Access:** API documentation inspected.

Candidate graphics backend for the broadly accessible implementation. Specific floating point rendering capabilities need runtime checks.

### D09 MediaDevices getUserMedia

MDN. [Source](https://developer.mozilla.org/en-US/docs/Web/API/MediaDevices/getUserMedia).

**Access:** API documentation inspected.

Microphone input depends on permissions and secure contexts. Useful later input mode.

### D10 MediaRecorder

MDN. [Source](https://developer.mozilla.org/en-US/docs/Web/API/MediaRecorder).

**Access:** API documentation inspected.

Candidate interactive recording mechanism. Codec support and determinism must be checked; a recording is not automatically a reproducible offline render.

### D11 YouTube IFrame Player API

Google. [Source](https://developers.google.com/youtube/iframe_api_reference).

**Access:** Documented player methods inspected.

Provides player controls and state, without a documented raw audio sample interface for our analysis pipeline.

### D12 YouTube API Services Developer Policies

Google. [Source](https://developers.google.com/youtube/terms/developer-policies).

**Access:** Content access and component separation restrictions inspected.

Constrains downloading and separating audiovisual content through API clients. A YouTube extraction backend is outside the proposed first version.

### D13 What is GitHub Pages

GitHub. [Source](https://docs.github.com/en/pages/getting-started-with-github-pages/what-is-github-pages).

**Access:** Official documentation inspected.

Static hosting for code and documentation; computation can stay on the visitor's device.

### D14 GitHub Pages limits

GitHub. [Source](https://docs.github.com/en/pages/getting-started-with-github-pages/github-pages-limits).

**Access:** Free eligibility and usage limits inspected.

Free public repository hosting is suitable for a portfolio demo, within published site and traffic limits.

### D15 Pages limits

Cloudflare. [Source](https://developers.cloudflare.com/pages/platform/limits/).

**Access:** Free plan and asset limits inspected.

Alternative static host. Bundled sample audio must fit asset limits; functions have separate quotas.

### D16 GitHub and Software

Zenodo. [Source](https://help.zenodo.org/docs/github/).

**Access:** Official documentation inspected.

Archive a software release as a persistent research artifact.

### D17 DOI versioning

Zenodo. [Source](https://zenodo.org/help/versioning).

**Access:** Official documentation inspected.

Distinguishes a particular version DOI from a concept DOI covering versions. Use exact versions for reproducible research.

### D18 Endorsement

arXiv. [Source](https://info.arxiv.org/help/endorsement.html).

**Access:** Official requirements inspected.

Endorsement may be required depending on account and category. Submission access is not guaranteed.

### D19 Content Moderation

arXiv. [Source](https://info.arxiv.org/help/moderation/index.html).

**Access:** Official guidance inspected.

Moderation checks suitability. Posting a preprint is distinct from peer reviewed acceptance.

### D20 Paper Proceedings

NIME. [Source](https://nime.org/papers/).

**Access:** Official proceedings inspected.

A relevant community for expressive musical interfaces. Future submission tracks and dates need checking when the project is ready.

### D21 2026 Call For Late Breaking Demos

ISMIR. [Source](https://ismir2026.ismir.net/call-for-late-breaking-demo).

**Access:** Current call inspected.

As of this review, submissions are closed. The call says LBD abstracts are not official proceedings papers; this is guidance for a future cycle.

## Existing projects

### C01 Cymatic Wizard

[Project](https://cymaticwizard.com/). **Access:** Product page inspected.

Browser upload and microphone visualization already exist. Physics assertions on the page have not been independently validated here.

### C02 Cymatica

[Project](https://www.cymatica.fm/). **Access:** Product page inspected.

Commercial mobile cymatics is an established product direction. Platform and GPU performance claims are vendor statements.

### C03 Sentinel Cymatics Lab

[Project](https://github.com/jkh2/Sentinel-Cymatics-Lab). **Access:** README inspected; implementation not audited.

Open browser precedent with audio reactivity and several pattern modes. Treat its model fidelity claims as unverified until code and behavior are examined.

## Reading order

Begin with P05 and P06 for grain transport, P03 for plate mechanics, P02 for the membrane distinction, D01 and D02 for signal processing, and P25 and P27 for visualization evaluation. Read P13 and P16 before adding fluid and reaction–diffusion modes. P19 and P20 are the strongest mathematical extensions beyond waves.

## Search coverage and remaining gaps

The search followed six branches: Chladni mechanics and stochastic grains; waves and fluid interfaces; nonlinear pattern formation and synchronization; music information retrieval; audiovisual perception; browser implementation and research dissemination. Searches used exact titles, topic combinations, author publication pages, publisher pages, institutional repositories, and citations from relevant papers.

Unresolved access gaps include the full Leissa monograph, CymaSense proceedings paper, the located Bello PDF, the original Turing paper, and the 2023 generalized diffusion PDF. Abstracts or metadata for these are retained with limits, rather than silently promoted to full paper readings. A later formal review should record databases, exact search strings, deduplication, inclusion criteria, and citation chaining in a reproducible search log. This catalogue is a substantial starting map for that work.



## Implementation supplement — 9 October 2026

### P38 Curl-Noise for Procedural Fluid Flow

Bridson, Hourihan and Nordenstam 2007. Author-hosted three-page manuscript. [Full text](https://www.cs.ubc.ca/~rbridson/docs/bridson-siggraph2007-curlnoise.pdf).

**Access:** Full text inspected.

The paper constructs procedural velocities through curl to avoid sources and sinks and discusses modulation and boundaries. The Flow orb uses this construction principle with a custom tangent trigonometric potential. It does not reproduce the paper’s Perlin-noise method or supply a pressure/viscosity solver. This is an implementation supplement to the original 61-reference review.

### P39 Numerical Modelling of Liquid Droplet Dynamics in Microgravity

Easter, Bojarevics and Pericleous (2011). Journal of Physics: Conference Series 327, 012027. DOI 10.1088/1742-6596/327/1/012027. [Institutional full text](https://gala.gre.ac.uk/id/eprint/7058/1/Numerical_Modelling_Liquid_Droplet_dynamics.pdf).

**Access:** Full PDF inspected; section 2 checked for version 0.8.

The spherical-harmonic decomposition and small-amplitude frequency ordering inform the liquid envelope. Musical forcing, visible time, damping and flow trails are project choices. The electromagnetic fluid solver in the paper is not implemented. This supplement brings the catalogue to 63 references.