arxiv: 2604.10356 · v1 · submitted 2026-04-11 · 🧮 math.HO · cs.GR· cs.RO

Recognition: unknown

A Minimal Mathematical Model for Conducting Patterns

Tom Verhoeff

Pith reviewed 2026-05-10 15:33 UTC · model grok-4.3

classification 🧮 math.HO cs.GRcs.RO

keywords conducting patternsmathematical modelcubic Hermite segmentsquintic timing lawgesture representationpreparation and ictus points

0 comments

The pith

A minimal model represents conducting gestures as cycles of preparation and ictus points joined by cubic segments with a quintic timing law.

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

The paper sets out to show that conducting patterns can be captured with a small set of mathematical elements rather than complex motion data. It separates the shape of the gesture from how fast it unfolds, using a repeating chain of preparation and beat points linked by cubic curves that stay level at the ends, then driving progress along those curves with a quintic speed profile. One adjustable value lets the motion range from steady to more accented. A sympathetic reader would care because this keeps the description short enough to use directly in calculations or programs while still allowing the gestures to feel musical and varied.

Core claim

The central claim is that conducting patterns admit a minimal representation consisting of a cyclic sequence of preparation and ictus points connected by cubic Hermite segments whose tangents are constrained to be horizontal, together with a quintic timing law that governs acceleration and deceleration along the path, where one parameter sets the degree of expressive variation from uniform motion.

What carries the argument

The key mechanism is the separation of the geometric trajectory, defined by preparation and ictus points with cubic Hermite interpolation and horizontal tangent constraints, from the temporal parametrization given by a quintic timing law with a single expressiveness parameter.

Load-bearing premise

The assumption that real conducting gestures are adequately captured by cyclic sequences of preparation and ictus points joined by cubic Hermite segments with horizontal tangents and timed by a quintic law with one parameter.

What would settle it

High-speed video of conductors performing standard patterns could be compared directly to the model's generated paths and velocity curves to check for systematic mismatches in shape or acceleration profile.

Figures

Figures reproduced from arXiv: 2604.10356 by Tom Verhoeff.

**Figure 2.** Figure 2: Pattern curve with anchor points at local extremes for 4-beat time signature. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Speed plot for 4-beat pattern with β = 0.7. To give us some way to vary the speed along the curve, we introduce the speed balance β ∈ [0, 1]. Defining amin = 1 − β, amax = 1 + β, we then have • prep → ictus: a = amin, b = amax, • ictus → prep: a = amax, b = amin. with interpretation • β = 0: uniform motion, • β = 1: maximal contrast (zero at preparation, fastest at ictus). 5 [PITH_FULL_IMAGE:figures/full_… view at source ↗

**Figure 4.** Figure 4: Screenshot of our Wolfram Demonstration Conducting Patterns. The model also features in the web app Crusis, a conducting companion [12]. It offers a pattern editor based on this model. Crusis incorporates some enhancements beyond a dot traversing the pattern curve, such as • drawing a fat dot at the baton’s tip location on the pattern curve, • changing the color of that fat dot momentarily at the downbeat,… view at source ↗

**Figure 5.** Figure 5: Common pattern curves for 2-, 3-, and 6-beat time signatures. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A clean, explicit parametric model for conducting gestures using constrained cubic Hermite segments and a quintic timing law with one free expressiveness parameter.

read the letter

The paper's main contribution is a straightforward construction that separates the geometric path of a conducting gesture from its timing. It defines a cycle of preparation and ictus points joined by cubic Hermite splines with horizontal tangent constraints at the ictus, then drives the parametrization with a quintic polynomial whose single parameter shifts between uniform speed and accented motion. This is implemented in a Wolfram Demonstration and used in the Crusis web app, which makes the whole thing immediately reproducible and testable by anyone with the software.

Referee Report

0 major / 2 minor

Summary. The paper introduces a minimal mathematical model for conducting patterns that separates the geometric trajectory from the temporal parametrization. It uses a cyclic sequence of preparation and ictus points connected by cubic Hermite segments with horizontal tangent constraints, combined with a quintic timing law controlled by a single expressiveness parameter that balances uniform and accented motion. The model is presented as compact and expressive, with an implementation in the Wolfram Demonstration 'Conducting Patterns' and application in the Crusis web app.

Significance. Should the construction prove as described, it offers a simple, reproducible parametric representation of conducting gestures that could be valuable for computational musicology, animation, and educational tools. The explicit use of standard primitives (cubic Hermite splines and quintic polynomials) and the provision of working code enhance the model's accessibility and verifiability, directly supporting the claim of compactness without requiring data fitting.

minor comments (2)

[Abstract] The abstract could more precisely state the range of the single free parameter and its effect on the timing law to better convey the model's expressiveness.
[Implementation] While the Wolfram implementation is mentioned, including a brief description of how the model parameters map to the interactive controls would aid readers in reproducing the gestures.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. We are pleased that the separation of geometry and timing, the use of standard primitives, and the accessibility via code and demonstrations were viewed favorably.

Circularity Check

0 steps flagged

No significant circularity in constructive model definition

full rationale

The paper presents an explicit parametric construction for conducting gestures: a cyclic sequence of preparation and ictus points joined by cubic Hermite segments (with horizontal tangent constraints) driven by a quintic timing law with one free parameter. The central claim—that this yields a compact, expressive representation—is satisfied directly by the existence and reproducibility of the construction itself (including the Wolfram implementation). No predictions are derived from data fits, no quantities are defined in terms of themselves, and no self-citations or uniqueness theorems are invoked to justify the model. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The model introduces one tunable parameter and rests on domain assumptions about the structure of conducting gestures without postulating new physical entities or fitting to empirical datasets.

free parameters (1)

expressiveness parameter
Single parameter that governs the balance between uniform motion and expressive emphasis.

axioms (1)

domain assumption Conducting gestures consist of cyclic sequences of preparation and ictus points connected by cubic Hermite segments with constrained horizontal tangents.
This structure is taken as the basis for the geometric trajectory component of the model.

pith-pipeline@v0.9.0 · 5369 in / 1259 out tokens · 67834 ms · 2026-05-10T15:33:25.449607+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 1 canonical work pages

[1]

Atherton,Vertical Plane Focal Point Conducting, Ball State University, Muncie, IN, 1989

L. Atherton,Vertical Plane Focal Point Conducting, Ball State University, Muncie, IN, 1989

1989
[2]

Baumeister,Conducting Patterns 2/4, 4/4, 3/4, 6/8 Time Signature Basics [video], YouTube,https://youtu.be/cfPoxZeOXjg, 2024

A. Baumeister,Conducting Patterns 2/4, 4/4, 3/4, 6/8 Time Signature Basics [video], YouTube,https://youtu.be/cfPoxZeOXjg, 2024

2024
[3]

Time-Optimal Control of Robotic Manipula- tors Along Specified Paths,

J. Bobrow, S. Dubowsky, J. Gibson, “Time-Optimal Control of Robotic Manipula- tors Along Specified Paths,”International Journal of Robotics Research,4(3):3–17, 1985

1985
[4]

A Class of Local Interpolating Splines,

E. Catmull, R. Rom, “A Class of Local Interpolating Splines,” inComputer Aided Geometric Design, 1974

1974
[5]

Farin,Curves and Surfaces for Computer-Aided Geometric Design, Academic Press, 2002

G. Farin,Curves and Surfaces for Computer-Aided Geometric Design, Academic Press, 2002

2002
[6]

Ensemble musicians’ synchronization with conductors,

R. Luck, P. Toiviainen, “Ensemble musicians’ synchronization with conductors,” Music Perception, 24(2), 2006

2006
[7]

Marrin Nakra,Inside the Conductor’s Jacket, PhD thesis, MIT, 2000

T. Marrin Nakra,Inside the Conductor’s Jacket, PhD thesis, MIT, 2000

2000
[8]

Force & Motion: Conducting to the Click,

R. Polfreman, B. Oliver, C. Metcalf, D. Halford, “Force & Motion: Conducting to the Click,”Proc. MOCO 2019, ACM, DOI: 10.1145/3347122.3347139. Also see https://generic.wordpress.soton.ac.uk/conductormocap/. 10

work page doi:10.1145/3347122.3347139 2019
[9]

Prautzsch, W

H. Prautzsch, W. Boehm, M. Paluszny,Bezier and B-Spline Techniques, Springer, 2002

2002
[10]

Rudolf,The Grammar of Conducting, 3rd ed., Schirmer, 1994

M. Rudolf,The Grammar of Conducting, 3rd ed., Schirmer, 1994

1994
[11]

An Interactive Conducting System using Kinect,

L.-W. Toh, W. Chao, Y.-S. Chen, “An Interactive Conducting System using Kinect,”IEEE International Conference on Multimedia and Expo (ICME), San Jose, CA, pp. 1–6, 2013

2013
[12]

Verhoeff,Crusis: Automated Conductor Display,https:// t-verhoeff-software.pages.tue.nl/crusis/, 2026

T. Verhoeff,Crusis: Automated Conductor Display,https:// t-verhoeff-software.pages.tue.nl/crusis/, 2026

2026
[13]

Conducting Patterns,

T. Verhoeff, “Conducting Patterns,”Wolfram Demonstrations Project, 2026 (Sub- mitted, under review)

2026
[14]

Gesture and movement in music performance,

G. W¨ ollner, “Gesture and movement in music performance,”Frontiers in Psychol- ogy, 2012. A Example 2-, 3-, and 6-beat conducting patterns To illustrate the use of our model, we show, in addition to the 4-beat pattern in Figure 2, parameterizations of the common 2-, 3-, and 6-beat time signatures. The choice of anchor point positions and roundness values...

2012