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IDENTIFICATION OF LIP-PARAMETERS
ASSOCIATED TO DIFFERENT TRUMPETS USING
CONSTRAINED CONTINUATION

Vincent Fréour, Hideyuki Masuda, Bruno Cochelin, Christophe Vergez

► To cite this version:

Vincent Fréour, Hideyuki Masuda, Bruno Cochelin, Christophe Vergez. IDENTIFICATION OF LIP-PARAMETERS ASSOCIATED TO DIFFERENT TRUMPETS USING CONSTRAINED CONTINUATION. Forum Acusticum, Sep 2023, Turin, Italy. hal-04800014

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Distributed under a Creative Commons Attribution 4.0 International License
# IDENTIFICATION OF LIP-PARAMETERS ASSOCIATED TO DIFFERENT TRUMPETS USING CONSTRAINED CONTINUATION

Vincent Fréour<fnref n="1" />*

Hideyuki Masuda<fnref n="1" />

Bruno Cochelin<fnref n="2" />

Christophe Vergez<fnref n="2" />

## ABSTRACT

The relationship between the player's global perception and acoustical characteristics of a musical instrument is very complex but nevertheless an important topic of interest for musical instrument makers. One of our hypotheses is that some aspects of the biomechanical control required by the player to achieve a given task (embouchure adjustments for instance) can contribute to explain the player's perception of an instrument: for an exact same musical task, if two instruments require different lip adjustments, there are likely to be perceived as different in terms of "blowing feeling" by the player. In this study we follow this idea in order to compare numerically two trumpets. Our strategy is based on the application of constrained continuation where the physical model studied is enriched with target trajectories for some of the output variables, and where two lip parameters of the physical model are relaxed. These constraints are established from playing measurements on one player and two trumpets. The evolutions of the relaxed parameters along the solution branches are then compared between the two instruments, and confronted to the blowing feeling reported by the player.

**Keywords:** musical acoustics, numerical continuation, brass instruments, physical modelling

*Corresponding author: vincent.freour@music.yamaha.com.

Copyright: ©2023 Fréour et al. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 Unported License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

## 1. PLAYING MEASUREMENTS AND PHYSICAL MODEL

Playing measurements are conducted on two trumpets. The mouthpiece is equipped with a pressure sensor (EN-DEVO 8510B) that records the acoustic pressure in the mouthpiece cup *p*. A similar sensor is used to measure the quasi-static pressure in the oral cavity of the player *p*₀. The player is instructed to play a crescendo maneuver for a Bb4 (about 466 Hz) while trying to maintain the pitch as stable as possible.

The physical model considered is based on four equations: two coupled mechanical equations to represent the dynamics of a two d.o.f. lip model as presented in [1], a nonlinear flow equation and a modal representation of the air-column to represent the nonlinear characteristics and the linear air-column resonator respectively, as detailed in [2]. The choice of the two d.o.f lip model for the lips is motivated by some limitations observed in previous work with the simple one d.o.f. lip model, where constrained continuation could be successfully performed, but the exact value of the target fundamental frequency could not be reached [2]. Indeed a drawback of the one d.o.f. outward-striking lip model is a tendency to produce a sharp intonation. We try here to overcome this issue by considering a more sophisticated model that includes two mechanical modes.

In addition, two equations are considered in order to account for the constraints obtained from playing measurements. A first equation ("loudness equation") sets the relationship between the L2 norm of the mouthpiece pressure *p*, and the mouth pressure *p*₀, and a second equation ("intonation frequency") forces the fundamental fre-

<footnote n="1">¹Yamaha Corporation, Research and Development Division, 10-1 Nakazawa-cho, Naka-ku, 430-8650 Hamamatsu, Japan</footnote>

<footnote n="2">²Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR7031, 4 impasse Nikola Tesla, 13453 Marseille Cedex 13, France</footnote>
quency to a constant value $f_{0c}$ (since small variations of intonation are observed in the player's performance along the crescendo maneuver):

$$||p||_{L2} = a\sqrt{p_0^2 - p_{0s}^2} \quad \text{and} \quad f_0 = f_{0c}, \quad (1)$$

where $a$, $p_{0s}$ and $f_{0c}$ are set by an optimization routine in order to match the playing measurements.

Since two equations have been added to the original model without constraints, two parameters need to be relaxed to keep the system balanced. We choose here to relax two lips parameters: $\zeta$, a dimensionless lip parameter that depends on the lip opening width $b$, and $f_x = \omega_x/2\pi$, the natural frequency of the outward striking (x-direction) mechanical mode of the lips.

## 2. NUMERICAL RESULTS

This system of equations can then be recast into a quadratic system, and the continuation of periodic solutions can be performed using the Asymptotic Numerical Method (ANM) [3] implemented in the software ManLab<fnref n="1" />. The constrained bifurcation diagram obtained for the first trumpet is given in Fig. 1, along with the evolution of $\zeta$ and $f_x$ with respect to the continuation parameter $p_0$. Note that the natural frequency $f_y$ of the second lip mechanical mode – that behaves as an inward-striking reed – was kept constant and set above 466 Hz.

It can be seen that a periodic solution which successfully follows the constraints established from playing measurements can be found over the $p_0$ range covered by the player. Furthermore significant variations in the lip parameters are observed along the solution branch. After running similar calculations with trumpet 2, the variation of lip parameters for the two trumpets are gathered in Table 1 for two scenarios: 1- the variations $\Delta\zeta$ and $\Delta f_x$ along the solution branch up to a same reference amplitude of $p$, 2- the variations $\Delta\zeta'$ and $\Delta f_x'$ above 1.6 kPa up to a same reference amplitude of $p$. This second scenario consists in ignoring the piano range at the beginning of the maneuver.

These results illustrate the noticeable differences in lip parameter values between the two trumpets. Over the full $p_0$ range, larger variations of both $\zeta$ and $f_x$ are observed for trumpet 1. However, if the piano range is ignored, larger variations of $f_x$ are observed for trumpet 2.

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Results from constrained continuation applied to trumpet 1.</figcaption></figure>

<table><caption>**Table 1.** Variations of $\zeta$ and $f_x$ along the solution branch.</caption><thead><tr><th></th><th>Δζ</th><th>Δζ'</th><th>Δf<sub>x</sub></th><th>Δf'<sub>x</sub></th></tr></thead><tbody><tr><td>Trumpet 1</td><td>0.267</td><td>0.23</td><td>31.03 Hz</td><td>5.7 Hz</td></tr><tr><td>Trumpet 2</td><td>0.094</td><td>0.07</td><td>25.63 Hz</td><td>14.9 Hz</td></tr></tbody></table>

## 3. CONCLUSIONS

We presented a "hybrid" numerical approach based on numerical continuation and constraints coming from playing measurements, that allows variation of lip parameters to be computed along the solution branch. Considering a two d.o.f lip model and a nonlinear constraint we could find a periodic solution that matches relatively well the performance of the player on two trumpets. In this first attempt of trumpet comparisons, the variations of the lip parameters along the branch suggest that the two trumpets have different requirements in terms of embouchure adjustment in order to perform a crescendo at constant frequency.

<footnote n="1">¹ ManLab. http://manlab.lma.cnrs-mrs.fr/spip/. Accessed: 2023-04-17</footnote>
Furthermore, the player reported experiencing more difficulties in maintaining constant pitch with trumpet 2. In light of the numerical results, we may then make the hypothesis that $\Delta f'_x$ (variation of $f_x$ above 1.6 kPa) may be associated to the feeling of embouchure adjustment reported by the player. This link between constrained continuation results and blowing perception should be studied in a more systematic manner in the future in order to make it a trustworthy prediction tool.

## 4. REFERENCES

[1] S. Adachi, N. Sato. Trumpet sound simulation using a two-dimensional lip vibration model. *J. Acoust. Soc. Am.*, 99(2):1200–1209, 1996.

[2] V. Fréour, L. Guillot, H. Masuda, C. Vergez, B. Cochelin. Parameter identification of a physical model of brass instruments by constrained continuation. *Acta Acustica*, 6(9), 2022.

[3] B. Cochelin, C. Vergez. A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions. *Journal of Sound and Vibration*, 243:262–222, 2009.