# Optimization of brass instruments based on physical modeling sound simulations

Robin Tournemenne, Jean François Petiot, Joël Gilbert

► To cite this version:

Robin Tournemenne, Jean François Petiot, Joël Gilbert. Optimization of brass instruments based on physical modeling sound simulations. ISMA 2017, 2017, Montréal, Canada. hal-01716553

HAL Id: hal-01716553

https://hal.science/hal-01716553v1

Submitted on 3 Dec 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

# OPTIMIZATION OF BRASS INSTRUMENTS BASED ON PHYSICAL
MODELING SOUND SIMULATIONS

Jean-François PETIOT¹, Robin TOURNEMENNE¹, Joël GILBERT²

(1): Laboratoire des Sciences du Numérique de Nantes (UMR CNRS 6004), Ecole Centrale de Nantes, France – Jean-François.Petiot@ec-nantes.fr

(2): Laboratoire d'Acoustique de l'Université du Maine (UMR CNRS 6613), Le Mans, France

## ABSTRACT

* physical modeling based simulations are integrated in the optimization process of the bore of a brass instrument, a trumpet

* the novelty of the approach lies in the fact that the objective function concerns the sound of the instrument, as simulated when in interaction with a virtual musician

* given the computationally expensive function evaluation and the unavailability of gradients, a surrogate-assisted optimization framework is implemented using the mesh adaptive direct search algorithm (MADS) [1].

### 1. Physical modelling of brass

<figure><img src="image_6.png" /></figure>

Coupling of a non-linear **excitator** (characterized by mechanical parameters) with a linear **resonator** (characterized by the input impedance $Z_{in}$)

* 3 time-varying variables: opening height $h(t)$ of the lips, volume flow $u(t)$ and pressure in the mouthpiece $p(t)$

$$ 
\left\{ 
\begin{array}{l}
P(j\omega) = Z_{in}(j\omega) \cdot U(j\omega) \\
\\
\frac{d^2h(t)}{dt^2} + \frac{2\pi f_l}{Q_l} \frac{dh(t)}{dt} + (2\pi f_l)^2 (h(t) - h_0) = \frac{P_m - p(t)}{\mu_l} \\
\\
u(t) = b \cdot h(t) \sqrt{\frac{2(P_m - p(t))}{\rho}}
\end{array}
\right. 
$$

### 2. Simulations with the harmonic balance technique

In permanent regime (frequency domain)

$$ p(t) = \sum_{n=1}^{N} A_n \cos(2\pi nFt + \phi_n) $$

<figure><img src="image_7.png" /></figure>

Amplitudes and phases of the
harmonics: $A_l, \phi_l, ..., A_N, \phi_N$
Playing frequency: $F$

Control parameters of the simulations
(virtual embouchure) [2]

<table><thead><tr><td>Definition</td><td>Notation</td><td>Value</td></tr></thead><tbody><tr><td>Resonance frequency of the lips</td><td>f<sub>l</sub> (Hz)</td><td>130 to 480</td></tr><tr><td>Mass per area of the lips</td><td>μ<sub>l</sub> (kg/m<sup>2</sup>)</td><td>1 to 6</td></tr><tr><td>Pressure in the mouth</td><td>P<sub>m</sub> (kPa)</td><td>6 to 9</td></tr><tr><td>Width of the lips</td><td>b (mm)</td><td>10</td></tr><tr><td>Rest value of the opening height</td><td>H<sub>0</sub> (mm)</td><td>0.1</td></tr><tr><td>Quality factor of the resonance</td><td>Q<sub>l</sub></td><td>3</td></tr></tbody></table>

### 3. Problem formulation: minimization of an objective function $J$

$$ \min_{x} J(x, \varphi) $$

x: design variables (bore of the resonator)

φ: set of embouchures (virtual musician)

Optimization framework

Set φ of embouchures
(Pm, μl, fl)

<figure><img src="image_8.png" /></figure>

Example of objective function $J$ (representative of the intonation)

$$ J(x, \varphi) = \frac{\sum_{i \in \text{notes}} |ETD(x, i, \varphi)|}{P-1} $$

ETD: Equal Tempered Deviation (cent)

$$ ETD(x, i, \varphi) = \alpha_{ref \to i} - 1200 \cdot \log_2 \left( \frac{\bar{F}_i(x, \varphi)}{\bar{F}_{ref}(x, \varphi)} \right) $$

### 4. Optimization method

The objective function $J(x,φ)$:

* Potentially heavy calculation time

* Derivative not available ($J(x,φ)$ is stochastic due to random sampling of the set of embouchures $φ$)

<figure><img src="image_9.png" /></figure>

Surrogate-assisted
derivative-free optimization
(NOMAD software)

<figure><img src="image_10.png" /></figure>

Mesh Adaptive Direct
Search (MADS)

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

Different Surrogate models:

* Polynomial response surfaces

* Kernel Smoothing

* Radial Basis Functions

## RESULTS Optimization of the leadpipe of a trumpet

Objective function: intonation of the regimes 2, 3, 4, 5 (notes)

<figure><img src="image_13.png" /></figure>

Two cases:

* 2-d problem (2 diameters in the leadpipe)

* 5-d problem (4 diameters in the leadpipe+ mouthpiece depth)

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>Results of J and ETD for
the 2-d problem</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Results of J and ETD for
the 5-d problem</figcaption></figure>

## CONCLUSIONS

* Alternative to an optimization based on the input impedance [2]

* Improvements of the intonation of trumpets based on simulations [3], with a reasonable computation time

* Possibility to define objective functions based on the spectrum of the sounds

* Possibility to optimize the spectrum of sounds subjected to constraints on the intonation

<figure><img src="image_17.png" /></figure>

**Acknowledgments** to B. Talgorn and Pr. M. Kokkolaras from McGill University (Montreal) for their participation to this work

## REFERENCES

[1] Tournemenne R., Petiot J-F., Talgorn B., Kokkolaras M., Gilbert J. (2016). Brass instruments design using physics-based sound simulation models and surrogate-assisted derivative-free optimization. *Journal of Mechanical Design*, april 2017, Vol. 139, 0141401-1-011401-9.

[2] Wilfried Kausel (2001): Optimization of Brasswind Instruments and its Application in Bore Reconstruction, *Journal of New Music Research*, 30:1, 69-82.

[3] Petiot J-F., Gilbert J. Comparison of trumpets' sounds played by a musician or simulated by physical modeling. *Acta Acustica united with Acustica*. Vol. 99, (2013), 629-641. DOI 10.3813/AAA.918642.