A Minimal Mathematical Model for Conducting Patterns

Tom Verhoeff

math.HOApr 11, 2026

A Minimal Mathematical Model for Conducting Patterns

Authors: Tom Verhoeff

Organizations: Dept. Math. & CS, Eindhoven University of Technology, The Netherlands

Abstract

We present a minimal mathematical model for conducting patterns that separates geometric trajectory from temporal parametrization. The model is based on a cyclic sequence of preparation and ictus points connected by cubic Hermite segments with constrained horizontal tangents, combined with a quintic timing law controlling acceleration and deceleration. A single parameter governs the balance between uniform motion and expressive emphasis. The model provides a compact yet expressive representation of conducting gestures. It is implemented as the interactive Wolfram Demonstration "Conducting Patterns" and is used in the Crusis web app.

Figures & tables

Figure 1: Baton with trail tracing a conducting pattern for a 4-beat time signature.

Figure 2: Pattern curve with anchor points at local extremes for 4-beat time signature.

Figure 3: Speed plot for 4-beat pattern with $\beta=0.7$ .

Figure 4: Screenshot of our Wolfram Demonstration Conducting Patterns .

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 5 — Figure 5: Common pattern curves for 2-, 3-, and 6-beat time signatures.

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Generating expressive conducting gestures from music is a challenging cross-modal motion synthesis problem: the output must follow long-range musical structure, preserve beat-level synchronization, and remain plausible as a fine-grained 3D human performance. Existing conducting-motion studies are often limited by sparse pose representations, small-scale data, or evaluation protocols that do not directly measure whether music and gesture are mutually aligned. This paper presents TransConductor, a Transformer-based framework for music-driven conducting gesture generation. We introduce ConductorMotion, a SMPL-parameter data construction pipeline that recovers detailed body motion from conducting videos and forms a dataset targeted at professional conducting gestures. Given acoustic descriptors extracted from audio and an initial pose, TransConductor uses a Trans-Temporal Music Encoder and a Trans-Temporal Conducting Gesture Decoder to autoregressively predict SMPL pose parameters. To better assess artistic correspondence, we further build a retrieval-based evaluation model that embeds music and gestures into a shared space and yields FID, modality distance, multi-modality distance, and diversity metrics. Experiments show that TransConductor outperforms dance-generation and conducting-generation baselines, while ablations verify the benefits of the Transformer backbone and the proposed alignment loss.

Abstract

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