# Symbolic Music Generation with Non-Differentiable Rule Guided Diffusion

arXiv: https://arxiv.org/abs/2402.14285 · HTML: https://mlsanity.com/paper/2402.14285/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- Placed in Audio, speech and music, one of 14 directions in Diffusion models.
- Builds on 32 of its 52 references that are in the map.
- First posted on arXiv 22 Feb 2024.

## Placed in

- Diffusion models / [Audio, speech and music](https://mlsanity.com/diffusion/audio-speech-and-music/)

## Builds on (most cited first, 30 of 32)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2108.02938/) (2021)
- [Enabling Factorized Piano Music Modeling and Generation with the MAESTRO Dataset](https://mlsanity.com/paper/1810.12247/) (2018)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Noise2Music: Text-conditioned Music Generation with Diffusion Models](https://mlsanity.com/paper/2302.03917/) (2023)
- [MuseGAN: Multi-track Sequential Generative Adversarial Networks for Symbolic Music Generation and Accompaniment](https://mlsanity.com/paper/1709.06298/) (2017)
- [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [A Hierarchical Latent Vector Model for Learning Long-Term Structure in Music](https://mlsanity.com/paper/1803.05428/) (2018)
- [Compound Word Transformer: Learning to Compose Full-Song Music over Dynamic Directed Hypergraphs](https://mlsanity.com/paper/2101.02402/) (2021)
- [DiffCollage: Parallel Generation of Large Content with Diffusion Models](https://mlsanity.com/paper/2303.17076/) (2023)
- [POP909: A Pop-song Dataset for Music Arrangement Generation](https://mlsanity.com/paper/2008.07142/) (2020)
- [MidiNet: A Convolutional Generative Adversarial Network for Symbolic-domain Music Generation](https://mlsanity.com/paper/1703.10847/) (2017)
- [PopMAG: Pop Music Accompaniment Generation](https://mlsanity.com/paper/2008.07703/) (2020)
- [MuseMorphose: Full-Song and Fine-Grained Piano Music Style Transfer with One Transformer VAE](https://mlsanity.com/paper/2105.04090/) (2021)
- [Encoding Musical Style with Transformer Autoencoders](https://mlsanity.com/paper/1912.05537/) (2019)
- [Polyffusion: A Diffusion Model for Polyphonic Score Generation with Internal and External Controls](https://mlsanity.com/paper/2307.10304/) (2023)
- [MIDI-VAE: Modeling Dynamics and Instrumentation of Music with Applications to Style Transfer](https://mlsanity.com/paper/1809.07600/) (2018)
- [SDMuse: Stochastic Differential Music Editing and Generation via Hybrid Representation](https://mlsanity.com/paper/2211.00222/) (2022)

## Built on by (most cited first, 30 of 54)

- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [InverseBench: Benchmarking Plug-and-Play Diffusion Priors for Inverse Problems in Physical Sciences](https://mlsanity.com/paper/2503.11043/) (2025)
- [Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexity](https://mlsanity.com/paper/2405.15986/) (2024)
- [Inference-Time Text-to-Video Alignment with Diffusion Latent Beam Search](https://mlsanity.com/paper/2501.19252/) (2025)
- [Reangle-A-Video: 4D Video Generation as Video-to-Video Translation](https://mlsanity.com/paper/2503.09151/) (2025)
- [Diffusion Tree Sampling: Scalable inference-time alignment of diffusion models](https://mlsanity.com/paper/2506.20701/) (2025)
- [Training-Free Guidance Beyond Differentiability: Scalable Path Steering with Tree Search in Diffusion and Flow Models](https://mlsanity.com/paper/2502.11420/) (2025)
- [Variational Control for Guidance in Diffusion Models](https://mlsanity.com/paper/2502.03686/) (2025)
- [Ensemble Kalman Diffusion Guidance: A Derivative-free Method for Inverse Problems](https://mlsanity.com/paper/2409.20175/) (2024)
- [Split Gibbs Discrete Diffusion Posterior Sampling](https://mlsanity.com/paper/2503.01161/) (2025)
- [Neurosymbolic Diffusion Models](https://mlsanity.com/paper/2505.13138/) (2025)
- [Plug-and-Play Controllable Generation for Discrete Masked Models](https://mlsanity.com/paper/2410.02143/) (2024)
- [Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generation](https://mlsanity.com/paper/2505.22391/) (2025)
- [HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimization](https://mlsanity.com/paper/2511.08425/) (2025)
- [Neural Approximate Mirror Maps for Constrained Diffusion Models](https://mlsanity.com/paper/2406.12816/) (2024)
- [Free2Guide: Training-Free Text-to-Video Alignment using Image LVLM](https://mlsanity.com/paper/2411.17041/) (2024)
- [Diffusion-Sharpening: Fine-tuning Diffusion Models with Denoising Trajectory Sharpening](https://mlsanity.com/paper/2502.12146/) (2025)
- [Chemistry-Inspired Diffusion with Non-Differentiable Guidance](https://mlsanity.com/paper/2410.06502/) (2024)
- [Graph Guided Diffusion: Unified Guidance for Conditional Graph Generation](https://mlsanity.com/paper/2505.19685/) (2025)
- [Blade: A Derivative-free Bayesian Inversion Method using Diffusion Priors](https://mlsanity.com/paper/2510.10968/) (2025)
- [Calibrated Test-Time Guidance for Bayesian Inference](https://mlsanity.com/paper/2602.22428/) (2026)
- [Mamba-Diffusion Model with Learnable Wavelet for Controllable Symbolic Music Generation](https://mlsanity.com/paper/2505.03314/) (2025)
- [Tree-Guided Diffusion Planner](https://mlsanity.com/paper/2508.21800/) (2025)
- [SYMPLEX: Controllable Symbolic Music Generation using Simplex Diffusion with Vocabulary Priors](https://mlsanity.com/paper/2405.12666/) (2024)
- [BandCondiNet: Parallel Transformers-based Conditional Popular Music Generation with Multi-View Features](https://mlsanity.com/paper/2407.10462/) (2024)
- [System 0/1/2/3: Quad-process theory for multi-timescale embodied collective cognitive systems](https://mlsanity.com/paper/2503.06138/) (2025)
- [Diffusion-based Symbolic Music Generation with Structured State Space Models](https://mlsanity.com/paper/2507.20128/) (2025)
- [Prevailing Research Areas for Music AI in the Era of Foundation Models](https://mlsanity.com/paper/2409.09378/) (2024)
- [Efficient Fine-Grained Guidance for Diffusion Model Based Symbolic Music Generation](https://mlsanity.com/paper/2410.08435/) (2024)
- [Not that Groove: Zero-Shot Symbolic Music Editing](https://mlsanity.com/paper/2505.08203/) (2025)

## Abstract (arXiv)

We study the problem of symbolic music generation (e.g., generating piano rolls), with a technical focus on non-differentiable rule guidance. Musical rules are often expressed in symbolic form on note characteristics, such as note density or chord progression, many of which are non-differentiable which pose a challenge when using them for guided diffusion. We propose Stochastic Control Guidance (SCG), a novel guidance method that only requires forward evaluation of rule functions that can work with pre-trained diffusion models in a plug-and-play way, thus achieving training-free guidance for non-differentiable rules for the first time. Additionally, we introduce a latent diffusion architecture for symbolic music generation with high time resolution, which can be composed with SCG in a plug-and-play fashion. Compared to standard strong baselines in symbolic music generation, this framework demonstrates marked advancements in music quality and rule-based controllability, outperforming current state-of-the-art generators in a variety of settings. For detailed demonstrations, code and model checkpoints, please visit our project website: https://scg-rule-guided-music.github.io/.