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# Electro-acoustic control of radiation impedance for brass instrument timbre shaping: design of a vocalizing mute.

Vincent Martos, Henri Boutin, Thomas Hélie, Brigitte d'Andrea-Novel

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Vincent Martos, Henri Boutin, Thomas Hélie, Brigitte d'Andrea-Novel. Electro-acoustic control of radiation impedance for brass instrument timbre shaping: design of a vocalizing mute.. Acta Acustica, 2025, 9, pp.40. 10.1051/aacus/2025020. hal-05169384

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# Electro-acoustic control of radiation impedance for brass instrument timbre shaping: design of a vocalizing mute

Vincent Martos<fnref n="1" />,*, Henri Boutin<fnref n="1" />, Thomas Hélie<fnref n="2" />, and Brigitte d'Andréa-Novel<fnref n="3" />

Received 20 August 2024, Accepted 10 May 2025

**Abstract** – The application of active control to musical instruments brings many benefits to composers and performers, by expanding their sound possibilities. This paper addresses the active control of a brass instrument to design a vocalizing mute. To this end, a sensor (pressure transducer) and an actuator (loudspeaker) with a feedback loop are placed at the bell extremity. A single-input single-output controller is designed to simulate the insertion of a flow-to-flow vocal filter, upstream of the natural radiation impedance load. The vocal transfer function with its target resonances is basically derived using the transfer matrix method for a vocal tract composed of concatenated acoustic cylinders. The loudspeaker model is based on the Thiele and Small description. Numerical experiments are presented on the controller, for a simplified trombone model that admits a Kelly-Lochbaum structure (mouthpiece, bore, bell and radiation are cascaded using the transfer matrix method). Finally, the sensitivity of the control to air temperature, bell opening angle and loudspeakers parameters is studied.

**Keywords.** Active control, Augmented brass instrument, Formants

## 1 Introduction

Active control serves as a versatile technique for modifying the vibrations of a system through the use of transducers. It was first used in noise reduction (cf. Lueg patent to suppress pure tones in pipes [1], Fogel for noise reduction in aircraft cockpits [2]). It has also been used for several years on musical instruments enabling the creation of so-called augmented instruments. The first piano with an active control system was patented in 1893 [3] by Eisenmann, on which a controller extended the string vibration using electromagnets. Feedback controllers based on the same principle have been developed more recently for electric guitar strings [4, 5]. In the family of percussion instruments, active modal control has also been developed to control the vibration modes of a xylophone bar [6]. Concerning wind instruments, active control has been used to modify the input impedance of a clarinet [7] and to study nonlinear couplings with its resonator [8]. This paper is part of the general study of active control applied to brass instruments.

Playing vowels with instruments can produce a spectacular effect, for example with hand movements in front of a trombone bell in the manner of a Kempelen

machine [9], or with dedicated mutes, as in the case of S. Turre or P. Eötvös in the piece *Snatches of Conversation*, who uses an association of Pixie and Plunger mutes. Also, using active control to modify the sound of brass instruments has a number of advantages. Thus, during performance, a brass player naturally controls his instrument through auditory feedback (radiated sound) and haptic feedback (felt through the vibration of his lips), which are correlated. By using active control, which allows the instrument to produce new sounds by modifying its own vibration (that of its air column), this correlation is preserved. This is in contrast to usual sound effects, which act on recorded sound and then play it back through loudspeakers. For these two reasons, this study aims to design a vocalizing mute using active control. To do that, we propose to modify the radiation impedance of a trombone, in order to simulate the presence of a human vocal tract downstream of its resonator. The choice of the geometry of the simulated vocal tract will make it possible to produce new trombone sounds by applying a spectral envelope similar to that of human vowel sounds. This article follows on from [10], which presents the first theoretical results. The simulations are improved using a Kelly-Lochbaum structure, and an analysis of the controller sensitivity is added.

<footnote n="1">¹S3AM-STMS-Sorbonne Université-Ircam, 1, Place Igor Stravinsky, 75004 Paris, France</footnote>

<footnote n="2">²S3AM-STMS-CNRS-Ircam, 1, Place Igor Stravinsky, 75004 Paris, France</footnote>

<footnote n="3">³Mines Paris – PSL, Center of Robotics, 60 bd Saint-Michel, 75006 Paris, France</footnote>

<footnote>*Corresponding author: vincentmartos@hotmail.fr</footnote>
The paper is organised as follows. Section 2 presents a global block diagram of the instrument (from the performer's mouth cavity to the acoustic radiation), introduces the control block inserted to achieve the objective, and sets out the working hypotheses on which the control law will be designed. Section 3 details basic passive acoustic modellings of radiation impedance and of the vocal tract to be virtualised. They are presented in the spectral domain (Laplace and Z). Section 4 is devoted to the design of the acoustic control law that achieves the target behaviour. It also examines its robustness through a sensitivity analysis. Then, Section 5 focuses on the sensors and actuators. It introduces their modelling with parameters and then derives the electro-acoustic feedback law to be implemented between these transducers. Finally, Section 6 presents numerical experiments and discusses the impact of the active control method developed in this article on the waveform and spectrum of the radiated sound.

## 2 Problem statement

This section sets out the work basis chosen in this paper to modify a trombone and make it radiate vowels.

It describes the played instrument in the form of a block diagram (Sect. 2.1) used to formulate the objective (Sect. 2.2) and the working hypotheses (Sect. 2.3). It then presents the control structure (Sect. 2.4), on the basis of which the control law will be designed (Sect. 2.5).

### 2.1 Structure of a played trombone and acoustic elements

Addressing the control of the trombone under realistic playing conditions requires consideration of several working constraints, such as:

(C1) complex aero-acoustic coupling with lips [11, 12],

(C2) complex modelling of visco-thermal losses inside the pipe [13–17],

(C3) both longitudinal and transverse modes above approximately 1250 Hz (see [18, Fig. 4]), which overlap the frequency range of the targeted application (the first four vowel resonances typically lying between 250 Hz and 4 kHz),

(C4) time-varying length due to slide movements when playing,

(C5) nonlinear propagation at high playing levels (see [19, 20] for modelling and [21] for sound synthesis).

This is witnessed by many studies and a large literature; see, e.g., [22] and the references therein.

To address these constraints and prepare an effective control strategy, the instrument is first decomposed into several blocks, as shown in Figure 1:

**Block A** represents the complex dynamical system composed of the performer excitation coupled to the

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Block diagram of a played trombone in its environment. First, the pipe is described by a quadripole. The two inputs are the mouthpiece volumetric flow rate, denoted $U_{in}$ and the pressure at the bell, denoted $P$. The two outputs are the flow rate at the bell, $U_{tb}$ and the mouthpiece pressure $P_{in}$. Second, the radiation is represented by a linear impedance load $Z_R$ (equal to $P/U_{tb}$ in the Laplace domain). And the transfer impedance $T_R(r)$ between $U_{tb}$ and the pressure $P_{out}(r)$ at a point $r$. Note that in all the paper, flow rates are algebraic quantities counted positively in the direction of $e_\ell$ (e.g., $U_{in} = U_{in}e_\ell$).</figcaption></figure>

acoustic bore until the acoustic state (pressure, airflow) at the bell extremity,

**Block B** represents the radiation load experienced by this state at the bell extremity,

**Block C** represents how this state propagates into the room, including directivity effects.

Figure 1 also introduces notations and conventions used within this paper.

### 2.2 Objective

Our objective is to develop a control method that operates effectively and independently of constraints (C1)–(C5) to make the trombone radiates vowels.

**Strategy.** We choose to work only on block B, by introducing a controller with a feedback loop, as proposed in Figure 2: the natural bell flow $U_{tb}$ is augmented by an additional component $U_{ac}$ for reshaping the radiation load, using a control, to meet a target. In the block diagram, this insertion is acoustically implemented at the isobar located at the bell extremity (see hypotheses below). It operates as a conservative junction with three ports, all of which share the same pressure and effectively balances all incoming flow rates. Following the orientation convention for flow rates, this balance equation reads $U_R = U_{ac} + U_{tb}$.

**Target.** In this paper, active control is used to simulate the presence of a vocal tract downstream of the trombone resonator, an organ that, in humans, extends from the glottis to the bucconasal cavity. Here, the nasal cavity is assumed to be closed by the soft palate, which allows only nonnasal vowels to be produced. The vocal tract is therefore considered an unbranched tube with a single input, the incoming flow rate on the glottis side, and a single output, the outgoing flow rate on the lips side. As the simulated vocal tract is placed at the output of the trombone resonator, according to the notations shown in
<figure><img src="image_4.png" /><figcaption>Figure 2. Block diagram of the musical instrument equipped with active control. Note that with no control ($U_{ac} = 0$), we recover the Figure 1 ($U_R = U_{tb}$).</figcaption></figure>

Figure 2, these two flow rates are equal to $U_{tb}$ and $U_R$ respectively. The goal is to modify the radiated airflow as

$$U_R = H_{VT} \cdot U_{tb}, \quad (1)$$

where $H_{VT}$ represents the vocal tract transfer function in the Laplace domain, that relates the airflow at the glottis to the airflow at the lips for a given vowel.

## 2.3 Hypotheses

The following assumptions are made, focusing exclusively on Block B in alignment with the strategy and objective:

(H1) **Radiation load:** approximated by a linear time-invariant passive impedance on a spherical cap (model chosen in Sect. 3.1);

(H2) **Vocal tract (target):** modeled by a linear time-invariant transfer function that relates the airflow at the glottis to the airflow at the lips (model chosen in Sect. 3.2);

(H3) **Controller:** linear time invariant feedback-loop between transducers (sensors and actuators) also assumed to be linear (design in Sects. 2.4 and 5.2). In this theoretical study, transducers are supposed to be co-located and distribute a homogeneous acoustic control over the spherical cap.

## 2.4 Control structure

The overall structure of the controller is shown in Figure 3. As explained in the objective, in Section 2.2, it takes the bell pressure as its only input. According to hypothesis of co-located control (H3), the loudspeaker generates a contribution $U_{ac}$ at the bell, which is added to the outgoing flow rate of the trombone $U_{tb}$, so that the total flow rate is equal to $U_{tb}+U_{ac}$ on the spherical isobar where the microphone is placed. As a consequence, this spherical isobar can be seen as a junction linking the two systems {Trombone Resonator} and {Controller}, which are fed by the same input, and whose outputs add up to each other.

For sake of simplicity, the microphone is described as a linear pressure-to-voltage converter with unit gain. The

<figure><img src="image_5.png" /><figcaption>Figure 3. Block diagram in the Laplace domain of the overall control structure. It is composed of a microphone (transfer function between acoustic pressure $P$ and voltage $V^{micro}$), a controller (cascaded transfer functions $K$ and $Ampli$ between voltage $V^{micro}$) and current $I_{LS}$ and a loudspeaker LS (quadripole with two inputs – pressure $P$ and current $I_{LS}$ – and two outputs – voltage $V_{LS}$ and flow rate $U_{ac}$).</figcaption></figure>

loudspeaker behaves like an electrical current to acoustic flow converter. Thus, between these two transducers, the controller designed to impose the desired vocal tract transfer function between $U_{tb}$ and $U_R$, must convert a voltage into a current. For later experimental implementation, the controller transfer function is separated into two cascaded parts:

- the one, called $Ampli$ in Figure 3, converts a voltage into a current with unit gain. An electronic solution of design has been proposed by Mc Pherson, based on a feedback loop circuit including an operational amplifier [23].

- the other, called $K$, which must therefore supply an electrical voltage, can thus be implemented in a conventional digital signal processor. Its expression is derived in Section 5.

## 2.5 Summary

In summary, this paper proposes the design of an active control system, localised at the bell, that operates as a targetable mute. Note that, as standard mutes do, such control impacts the entire instrument and the acoustic boundary conditions experienced by the musician's lips (the input impedance in the linear time-invariant case of a static bore at low amplitude levels). In this context, the active control is not merely an electronic vocoder effect played back through external loudspeakers.

# 3 Radiation impedance $Z_R$ and vocal tract $H_{VT}$ modeling

This section presents models of the components required for calculating the acoustic control law (transfer function $K$). These models refer to the components of
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Radiation model of the bell of a wind instrument. The purple spherical cap $S_0$ is the uniformly animated part and the orange cap $S - S_0$ is motionless (see details in [26]).</figcaption></figure>

Block B presented in Figure 2 (see Sect. 2.2): the radiation impedance $Z_R$ (in Sect. 3.1) and the vocal tract transfer function $H_{VT}$ (in Sect. 3.2). A particular attention is paid to the choice of its modelling, which results from a compromise between realism (consideration of non-planar wavefronts) and simplicity.

## 3.1 Radiation

Several acoustic radiation models are available (see e.g., [24, 25] for a review of classical models) for the 1D modelling of musical instruments. A first interest of the work proposed in [26] is to account for the effect of iso-bar curvature at the bell outlet, by assuming that the bell radiates similarly to a pulsating spherical cap (see Fig. 4). A second interest is to approximate the impedance formula (decomposed over spherical harmonics) with optimised simpler differential models. The model chosen here is the second-order high-pass filter described below, which exhibits better results than classical 1D planar models (see [27, 28]).

This radiation impedance $Z_R = \frac{P_{out}}{U_{out}}$ is modelled in the Laplace domain by

$$Z_R(s) = \frac{\rho_0 c_0}{S_0} \tilde{Z}\left(\frac{s}{\omega_0}\right) \quad \text{with} \quad \omega_0 = \frac{2\pi c_0 \nu_c(\theta_0)}{r_0}, \quad (2a)$$

$$\tilde{Z}(\tilde{s}) = \frac{\alpha(\theta_0) \tilde{s} + \tilde{s}^2}{1 + 2\xi(\theta_0) \tilde{s} + \tilde{s}^2} \quad (\text{adimensioned}), \quad (2b)$$

where (see Tab. 1) $\rho_0$ and $c_0$ are the air density and the speed of sound, $r_0$ is the sphere radius, $\theta_0$ is the cap half-angle ($r_0 = \frac{r_{bell}}{\sin\theta_0}$), and where optimal parameters $\alpha$, $\xi$ and $\nu_c$ are given by the following functions, defined for all $\theta_0 \in [0, \pi/2]$, $\xi(\theta_0) = 0.0207\theta_0^4 - 0.144\theta_0^3 + 0.221\theta_0^2 + 0.0799\theta_0 + 0.72 \ge 0$, $\alpha(\theta_0) = (0.1113\theta_0^5 - 0.636\theta_0^3 + 1.162\theta_0^2 - 1.242\theta_0^2 + 1.083\theta_0 + 0.8788)^{-1} \ge 0$ and $\nu_c(\theta_0) = (-0.198\theta_0^5 + 0.2607\theta_0^4 - 0.424\theta_0^3 - 0.07946\theta_0^2 + 4.704\theta_0 + 0.022)^{-1} \ge 0$.

The Laplace region of convergence of $Z_R$ includes $C_0^+$ (denoting $C_0^+ = \{s \in \mathbb{C}\text{ s.t. } \Re e(s) > 0\}$) and it is proved to define a causal, stable, passive system [26, 28] which therefore meets the expectations formulated in

<table><caption>**Table 1.** Numerical values of parameters used in the model of $Z_R$. $r_{bell}$ is measured on a trombone *Jupiter* JVL 528. The value of $\theta_0$ comes from [26].</caption><thead><tr><th>Symbol</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>ρ<sub>0</sub></td><td>1.2</td><td>kg.m<sup>-3</sup></td></tr><tr><td>c<sub>0</sub></td><td>340</td><td>m.s<sup>-1</sup></td></tr><tr><td>r<sub>bell</sub></td><td>8.0 · 10<sup>-2</sup></td><td>m</td></tr><tr><td>θ<sub>0</sub></td><td>72.4</td><td>°</td></tr><tr><td>r<sub>0</sub> = <sup>r<sub>bell</sub></sup>&frasl;<sub>sin(θ<sub>0</sub>)</sub></td><td>8.39 · 10<sup>-2</sup></td><td>m</td></tr><tr><td>S<sub>0</sub> = 2πr<sub>0</sub><sup>2</sup>(1 - cos(θ<sub>0</sub>))</td><td>3.09 · 10<sup>-2</sup></td><td>m<sup>2</sup></td></tr><tr><td>f<sub>c</sub></td><td>893</td><td>Hz</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Bode diagram of adimensioned radiation impedance $\tilde{\omega} \mapsto \tilde{Z}(i\tilde{\omega})$. Note that the adimensioned angular frequency is $\tilde{\omega} = \omega/\omega_0 = 2\pi f/\omega_0$.</figcaption></figure>

(H1), Section 2.3. Figure 5 shows the values of module and phase of $\tilde{Z}$ for several values of $\theta_0$.

Note that the cutoff frequency (for which $|\tilde{Z}|^2 = \frac{1}{2}$) is given by $f_c = \frac{c_0}{r_0}\nu_c[[1 + \beta^2]^{\frac{1}{2}} - \beta]^{\frac{1}{2}}$ with $\beta = 1 + \alpha^2 - 2\xi^2$. For typical values measured on a real trombone bell, $\theta_0 = 72.4^\circ$ and $r_0 = 8.39 \text{ cm}$, $f_c$ is equal to 893 Hz.

## 3.2 Vocal tract

To design the vocal effect subsequently applied by active control to the radiated impedance of the trombone, we consider a static vocal tract (producing one vowel, see (H2)). Then, we need a model of its flow-to-flow filter (see (1)), the resonances of which are related to formants and characterise the vowel. This linear time-invariant causal filter is derived by considering $N_{VT}$ cascaded straight pipes of equal length and of cross-section area $a_n$ (see Fig. 6), radiating in a semi-infinite end pipe of cross-section area $a_{ext}$. As proposed by Mathur et al. [29], their length, equal to 3.968 mm, is slightly greater than the distance covered in half a sampling period. This allows the vocal tract to vary in length and leads to a more realistic synthesis. This filter admits a standard Kelly-Lochbaum structure [30, 31] with relevant computational
<figure><img src="image_8.png" /><figcaption>Figure 6. Modelling the vocal tract.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7. Bode diagram of the transfer function $H_{VT}(s = 2i\pi f)$ of a vocal tract for the vowel profile [a] (see Fig. 8), loaded by $Z_R^{VT} = \rho_0 c_0 / a_{ext}$ with $a_{ext} = 38.2 \text{ cm}^2$: (top) modulus in dB; (bottom) phase in radians.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 8. Area profile of the vocal tract for vowel [a] extracted from [31, Table 3] ($N_{VT} = 44$ cascaded pipes, total length $L_{VT} = 17.46 \text{ cm}$, so that $L_{VT}/N_{VT} = 3.968 \text{ mm}$).</figcaption></figure>

details provided in Appendix D to ensure self-consistency (see Fig. D.1 and Eqs. (D.1), (D.2)).

The vocal tract transfer function $H_{VT} = U_{out}/U_{in}$ is derived from the transfer matrix $T_{VT}$ of the cascaded pipes loaded by a radiation impedance $Z_R^{VT}$, so that

$$ \begin{bmatrix} P_{in} \\ U_{in} \end{bmatrix} = T_{VT} \underbrace{\begin{bmatrix} Z_R^{VT} \\ 1 \end{bmatrix}}_{*} U_{out}. \quad (3) $$

<figure><img src="image_11.png" /><figcaption>Figure 9. Block diagram of the target radiation impedance in the Laplace domain. Here, the vocal tract $H_{VT}$ is placed between the pipe and the radiation such that the new radiation flow rate $U_R$ is now equal to $U_R = H_{VT}U_{tb}$.</figcaption></figure>

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

<figure><img src="image_13.png" /><figcaption>Figure 10. Bode diagrams: (a) radiation impedances $Z_R^\dagger$ (exact, in red) corresponding to $\theta_0^\dagger = 72.4^\circ$ and $Z_R^{\text{err}}$ (errorous, in blue) corresponding to $\theta_0^{\text{err}} = 76.02^\circ$ (deviation of 5%); (b) error characterisation. (a) radiation impedances: (bottom) phase in radians. (b) error: (top) relative error modulus $|\epsilon_Z|$ (log scale); (bottom) phase deviation $\Delta\phi = \phi^{\text{err}} - \phi^\dagger$.</figcaption></figure>

Figure 7 shows the Bode diagram of $H_{VT}$ computed for the vowel [a] described by the area profile in Figure 8 and choosing<fnref n="1" /> $a_{ext} = 38.2 \text{ cm}^2$.

<footnote n="1">¹ This corresponds to a typical quality factor $Q = 10$ for the first resonance of a straight pipe with the cross-section area $3 \text{ cm}^2$ (approximate averaged value of cross-section area based on Fig. 8).</footnote>
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 11.** Bode diagrams: (a) vocal tract filters $H^\dagger$ (exact, in red) and $H^{\text{err}}$ (erroneous, in blue); (b) error characterisation. (a) vocal tract filters: (top) modulus in dB; (bottom) phase in radians. (b) error: (top) relative error modulus $|\epsilon_H|$ (log scale); (bottom) phase deviation $\Delta\phi = \phi^{\text{err}} - \phi^\dagger$.</figcaption></figure>

# 4 Target acoustic control

This section introduces the control law designed within the acoustic domain to achieve the desired behaviour. This law relies solely on the radiation impedance $Z_R$ and the transfer function $H_{VT}$ of the target vowel<fnref n="2" />. Subsequently, the robustness of vowel rendering is evaluated. Initially, a basic test examines deviations due to an angle error $\theta_0$ in the radiation impedance model. Following this, a sensitivity analysis with respect to $\theta_0$ and temperature $T$ is presented.

## 4.1 Control model

As mentioned in Section 1, the target application is to produce a vocalised version of the radiated sound. This amounts to applying resonances (through $H_{VT}$) to the

<table><caption>**Table 2.** Comparison of the frequencies $F_k$ and amplitudes $A_k$ of the first five peaks of $H^\dagger$ and $H^{\text{err}}$.</caption><thead><tr><th>k</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td>$F_k^\dagger$ [Hz]</td><td>785</td><td>1158</td><td>2795</td><td>3323</td><td>4434</td></tr><tr><td>$A_k^\dagger$ [dB]</td><td>31.4</td><td>24.0</td><td>29.2</td><td>45.8</td><td>23.9</td></tr><tr><td>$F_k^{\text{err}}$ [Hz]</td><td>710</td><td>1224</td><td>2775</td><td>3369</td><td>4400</td></tr><tr><td>$A_k^{\text{err}}$ [dB]</td><td>18.6</td><td>31.4</td><td>23.7</td><td>36.0</td><td>21.2</td></tr><tr><td>$\frac{F_k^{\text{err}} - F_k^\dagger}{F_k^\dagger}$</td><td>-9.6%</td><td>+5.7%</td><td>-0.7%</td><td>+1.4%</td><td>-0.8%</td></tr><tr><td>$\frac{A_k^{\text{err}} - A_k^\dagger}{A_k^\dagger}$</td><td>-77%</td><td>+134%</td><td>-46%</td><td>-67.6%</td><td>-26.7%</td></tr></tbody></table>

<table><caption>**Table 3.** Numerical values of physical parameters. The value of $\theta_0^\dagger$ is taken from [26].</caption><thead><tr><th>Symbol</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>$\gamma$</td><td>7/5</td><td>Ø</td></tr><tr><td>$R$</td><td>8.314</td><td>J.K<sup>-1</sup>.mol<sup>-1</sup></td></tr><tr><td>$M$</td><td>29</td><td>g.mol<sup>-1</sup></td></tr><tr><td>$P_0$</td><td>1</td><td>bar</td></tr><tr><td>$\theta_0^\dagger$</td><td>72.4</td><td>°</td></tr><tr><td>$T^\dagger$</td><td>300</td><td>K</td></tr></tbody></table>

radiated pressure (originally generated by the trombone flow rate $U_{tb}$). To this end, the principle consists of adding a controlled acoustic flow rate ($U_{ac}$) to that of the trombone ($U_{tb}$) at the bell extremity in order to form a modified version of the flow rate ($U_R = U_{tb} + U_{ac}$) experienced by the natural acoustic radiation. This is implemented by adding a flow source, assumed to be ideal in the low frequency range (H1), so that it defines an ideal conservative junction with three ports, that balances flows and equalises pressures (see Fig. 2), that is,

$$U_{\text{tb}} + U_{\text{ac}} = U_{\text{R}}, \quad (4a)$$

$$P_{\text{tb}} = P_{\text{ac}} = P_{\text{R}} (:= P). \quad (4b)$$

Then, the control is designed such that the flow experienced by the radiation ($U_R$) becomes a vocalised version of the trombone flow ($H_{VT}U_{tb}$) rather than the trombone flow itself ($U_{tb}$), that is (see Fig. 9)

$$U_R = H_{VT}U_{tb}. \quad (5)$$

Note that, from the trombone viewpoint, this amounts to loading its bell extremity by the modified radiation impedance $H_{VT}Z_R$, denoted $Z_R^c$ below.

Then, modelling the acoustic control by its admittance $Y_{ac} = U_{ac}/P_{ac}$, it follows from (4a) that, at the acoustic junction,

$$U_{\text{tb}} + Y_{\text{ac}} P_{\text{ac}} = Z_{\text{R}}^{-1} P_{\text{R}}. \quad (6a)$$

The expected target ($U_R = H_{VT}U_{tb}$) is achieved if $U_{tb} = H_{VT}^{-1}U_R = H_{VT}^{-1}Z_R^{-1}P_R$, so that (6a) becomes

$$H_{\text{VT}}^{-1} Z_{\text{R}}^{-1} P_{\text{R}} + Y_{\text{ac}} P_{\text{ac}} = Z_{\text{R}}^{-1} P_{\text{R}}. \quad (6b)$$

<footnote n="2">² This means that the law is independent of the trombone configuration and the played note, but it is valid only for a static vowel.</footnote>
Finally, from (4b), the target acoustic control admittance is

$$Y_{ac} = (1 - 1/H_{VT})/Z_{R}, \quad (7a)$$

leading to the equivalent formula for the vocal tract filter

$$H_{VT} = (1 - Z_{R}Y_{ac})^{-1}. \quad (7b)$$

## 4.2 Qualitative robustness test

Denote $H_{VT}^{\dagger}$ the target vocal filter. Denote $Z_{R}^{\dagger}$ the exact (but unknown) radiation impedance and $Z_{R}^{\text{err}}$ the (possibly) erroneous model.

Following (7a), the acoustic control we compute for $H_{VT}^{\dagger}$ and $Z_{R}^{\text{err}}$ leads to $Y_{ac}^{\text{err}} = (1 - 1/H_{VT}^{\dagger})/Z_{R}^{\text{err}}$. Using this control $Y_{ac}^{\text{err}}$, the vocal filter produced in reality by the exact radiation $Z_{R}^{\dagger}$ is, following (7b) $H_{VT}^{\text{err}} = (1 - Z_{R}^{\dagger}Y_{ac}^{\text{err}})^{-1}$, that is

$$H_{VT}^{\text{err}} = \frac{1}{1 - (Z_{R}^{\dagger}/Z_{R}^{\text{err}})(1 - 1/H_{VT}^{\dagger})}. \quad (8)$$

Following this equation, the relative errors

$$\epsilon_H := \frac{H_{VT}^{\text{err}} - H_{VT}^{\dagger}}{H_{VT}^{\dagger}} \quad \text{and} \quad \epsilon_Z := \frac{Z_{R}^{\text{err}} - Z_{R}^{\dagger}}{Z_{R}^{\dagger}}, \quad (9a)$$

committed respectively on the vocal filter and on the radiation impedance, are related by

$$\epsilon_H = \epsilon_Z \frac{1 - H_{VT}^{\dagger}}{1 + \epsilon_Z H_{VT}^{\dagger}}. \quad (9b)$$

A robustness test is presented for $H_{VT}^{\dagger}$ given in Figure 7 where $Z_{R}^{\dagger}$, with parameter $\theta_{0}^{\dagger} = 72.4^{\circ}$ (see (2a), (2b) and Tab. 1), is erroneously replaced by $Z_{R}^{\text{err}}$, with the same parameters except a deviation of +5% on the angle $\theta_{0}^{\text{err}} = 1.05 \theta_{0}^{\dagger} \approx 76.02^{\circ}$ (see Fig. 10).

The vocal tract filter $H_{VT}^{\text{err}}$ resulting from (8) is compared to $H_{VT}^{\dagger}$ in Figure 11: the peaks are modified (see Tab. 2) but are still qualitatively representative of a vowel [a]. In the following section, we complete this test by a sensitivity analysis.

## 4.3 Sensitivity analysis

In this section, we analyse the sensitivity of $H_{VT}$ to the parameters $a = (T, \theta_{0})$, assuming that:

(i) a target control $Y_{ac}^{\dagger}$ is pre-computed from (7a) with fixed parameters $T^{\dagger} = 300$ K, $\theta_{0}^{\dagger} = 72.4^{\circ}$,

(ii) the radiation impedance is subject to small variations in its parameters $a$ around $a^{\dagger} = (T^{\dagger}, \theta_{0}^{\dagger})$, and is renoted $Z_{R}(s; a)$.

In this case, the dependency of the vocal tract filter to parameters $a$ reads, from (7b),

$$H_{VT}(s; a) = (1 - Z_{R}(s; a)Y_{ac}^{\dagger})^{-1}. \quad (10a)$$

Then, the first order expansion of $H_{VT}(s; a^{\dagger} + \delta a)$ with respect to $\delta a_{1} = \delta T$ and $\delta a_{2} = \delta \theta_{0}$ leads to

$$H_{VT} = H_{VT}^{\dagger} + S_{T} \delta T + S_{\theta_{0}} \delta \theta_{0} + \mathcal{O}(\delta T^{2} + \delta \theta_{0}^{2}), \quad (10b)$$

introducing $H_{VT}^{\dagger}(s) = H_{VT}(s; a^{\dagger})$ and the sensitivity functions for $a_{1} = T$ and $a_{2} = \theta_{0}$ as

$$S_{a_i}(s; a) = \frac{\partial H_{VT}(s; a)}{\partial a_i}. \quad (10c)$$

From (10a), it follows that

$$S_{a_i} = \frac{Y_{ac}^{\dagger}}{(1 - Y_{ac}^{\dagger} Z_{R})^2} \frac{\partial Z_{R}}{\partial a_i}, \quad (11a)$$

where $\frac{\partial Z_{R}}{\partial a_{i}}$ is derived from the chain rule applied to (2a), (2b) in which $\rho_{0}$, $c_{0}$, $S_{0}$, $\omega_{0}$ are the $a$-dependent functions

$$\rho_0(T) = \frac{P_0 M}{RT}, \quad (12a)$$

$$c_0(T) = \sqrt{\frac{\gamma M}{M}}, \quad (12b)$$

$$S_0(\theta_0) = 2\pi \left( \frac{r_{\text{bell}}}{\sin(\theta_0)} \right)^2 (1 - \cos(\theta_0)), \quad (12c)$$

$$\omega_0(\theta_0) = \frac{2\pi c_0(T) \nu_c(\theta_0)}{r_0(\theta_0)}, \quad \text{with } r_0(\theta_0) = \frac{r_{\text{bell}}}{\sin(\theta_0)}, \quad (12d)$$

and $\alpha(\theta_{0})$, $\xi(\theta_{0})$ and $\nu_{c}(\theta_{0})$ are detailed in Section 3.1.

The physical values of parameters are given in Table 3.

Note that detailing the expression of the sensitivity functions of the characteristic impedance

$$Z_c(T, \theta_0) = \frac{\rho_0(T)c_0(T)}{S_0(\theta_0)} = \sqrt{\frac{\gamma M}{RT}} \frac{P_0}{\pi r_{\text{bell}}^2} \cos^2\left(\frac{\theta_0}{2}\right) \quad (13)$$

involved in (2a) yields

$$\frac{\partial Z_c}{\partial \theta_0} = -\sqrt{\frac{\gamma M}{RT}} \frac{P_0}{2\pi r_{\text{bell}}^2} \sin(\theta_0), \quad (14a)$$

$$\text{and } \frac{\partial Z_c}{\partial T} = -\sqrt{\frac{\gamma M}{RT^3}} \frac{P_0}{2\pi r_{\text{bell}}^2} \cos^2\left(\frac{\theta_0}{2}\right). \quad (14b)$$

Figure 12 reveals that variations in $\theta_{0}$ and $T$ have a significant impact on the position and amplitude of vocal tract resonances, potentially modifying the radiated vowel. In particular, overestimation of $\theta_{0}$ tends to separate resonances 1 & 2 on the one hand, and 3 & 4 on the other, and to reduce the 5th resonance frequency. Underestimation of $\theta_{0}$ has the opposite effect, fusing resonances 1 & 2 when $\theta_{0} = 80^{\circ}$. Estimation errors on $T$ cause resonance
<table><caption>**Figure 12.** Bode diagram (modulus and phase) of the vocal tract filter $H_{VT}$. The variation in the value of the $a_i$ parameters is studied in order to observe the changes in the transfer function. The errors $E_{a_i}$ and sensitivities $S_{a_i}$ are presented for several variations of $a_i$.</caption><thead><tr><th>Parameters a<sub>i</sub></th><th>θ<sub>0</sub></th><th>T</th></tr></thead><tbody><tr><td>H<sub>VT</sub></td><td></td><td></td></tr><tr><td>E<sub>a<sub>i</sub></sub> = H<sub>VT</sub> - H<sup>†</sup></td><td></td><td></td></tr><tr><td>S<sub>a<sub>i</sub></sub> = ∂H<sub>VT</sub>/∂a<sub>i</sub></td><td></td><td></td></tr></tbody></table>
<table><caption>**Table 4.** Physical, Thiele and Small and reduced parameters of the speaker SICA 3L 0.8 SL 8Ω mode Z000900, given by the manufacturer. *Note:* Greyed-out parameters in the first 2 lines are not used in the expression of the controller *K*, because it is designed to control the loudspeaker in current.</caption><thead><tr><th rowspan="2"></th><th colspan="2">Physical parameters</th><th colspan="2">Reduced parameters</th></tr><tr><th>Symbol</th><th>Value</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td rowspan="2">Electrical</td><td>R<sub>e</sub></td><td>8Ω</td><td>τ<sub>e</sub> = L<sub>e</sub>/R<sub>e</sub></td><td>1.38 · 10<sup>-5</sup> s</td></tr><tr><td>L<sub>e</sub></td><td>0.11 mH</td><td>β<sub>e</sub> = 1/L<sub>e</sub></td><td>9.09 · 10<sup>3</sup> H<sup>-1</sup></td></tr><tr><td>Electro/Mechanical Coupling</td><td colspan="4">Bl = 2.8 N.A<sup>-1</sup></td></tr><tr><td rowspan="3">Mechanical</td><td>C<sub>ms</sub></td><td>602 µm.N<sup>-1</sup></td><td>β<sub>m</sub> = 1/M<sub>ms</sub></td><td>5.26 · 10<sup>2</sup> kg<sup>-1</sup></td></tr><tr><td>R<sub>ms</sub></td><td>4.06 · 10<sup>-1</sup> N.m<sup>-1</sup>.s</td><td>τ<sub>m</sub> = M<sub>ms</sub>/R<sub>ms</sub></td><td>4.68 · 10<sup>-3</sup> s</td></tr><tr><td>M<sub>ms</sub></td><td>1.9 · 10<sup>-3</sup> kg</td><td>ω<sub>m</sub> = 1/√C<sub>ms</sub>M<sub>ms</sub></td><td>9.35 · 10<sup>2</sup> rad.s<sup>-1</sup></td></tr><tr><td>Mechanical/Acoustic Coupling</td><td colspan="4">S<sub>d</sub> = 30.2 cm<sup>2</sup></td></tr></tbody></table>

frequencies to move in the same direction as estimation errors on $\theta_0$, but to a lesser extent, at usual temperatures. Indeed, previous measurements carried out with real players [32] show that the temperature in the bore generally rises while playing from room temperature to around 37.7 °C, the temperature of the human body, inducing typical variations of the order of 10 °C. Consequently, an incorrect temperature estimate should have a negligible effect on the vocal tract resonances applied to the radiated sound.

## 5 Electro-acoustic implementation

The active control solution implemented uses a loudspeaker at the end of the bell as an actuator. Section 5.1 presents a brief reminder on its standard linear electroacoustic modelling with Thiele and Small parameters [33, 34]. In this paper the loudspeaker is driven by a current source assumed to be ideal on the frequency range of interest. Note that the natural causal electric input of a loudspeaker is the voltage (and not the current). This means that the current source requires the use of a dedicated feedback-loop circuit involving an operational amplifier (see e.g., [23]). The output current (see Fig. 3) is governed by $I_{LS} = K V^{\text{micro}}$ where $V^{\text{micro}} = K_{\text{mic}} P$ denotes the voltage delivered by a pressure microphone of ideal gain $K_{\text{mic}}$. The pressure-to-current gain is then described by $I_{LS} = K_{\text{tot}} P$ with $K_{\text{tot}} = K \cdot K_{\text{mic}}$.

The transfer function $K_{\text{tot}}$ is the electronic control to be determined in Section 5.2, in order to realise the acoustic target $Y_{\text{ac}}$ given the loudspeaker model.

### 5.1 Loudspeaker modelling

A basic linear description of the loudspeaker (LS) is modelled by the following electrical and mechanical equations, cf. [33, 34]:

$$v_{\text{LS}}(t) = R_e i_{\text{LS}}(t) + L_e \frac{di_{\text{LS}}}{dt} + Bl \frac{dz}{dt}, \quad (15a)$$

$$M_{\text{ms}} \frac{d^2 z}{dt^2} = Bl i_{\text{LS}}(t) - R_{\text{ms}} \frac{dz}{dt} - \frac{1}{C_{\text{ms}}} z(t) - S_d P, \quad (15b)$$

where $M_{\text{ms}}$ is the mass of the driver diaphragm and voice-coil assembly, $R_{\text{ms}}$ and $C_{\text{ms}}$ the mechanical resistance and compliance (suspension, spider, acoustic chamber) and $S_d$ the equivalent surface area of the cone. $Bl$ is the coefficient of the magnetic force induced by the current passing through the voice-coil of length $l$. This model admits the following input (u)–state (x)–output (y) representation:

$$\dot{x} = Ax + Bu, \quad y = Cx, \quad (16a)$$

$$\text{with } u = \begin{bmatrix} v_{\text{LS}} \\ P \end{bmatrix}, \quad x = \begin{bmatrix} i_{\text{LS}} \\ z \\ \dot{z} \end{bmatrix}, \quad y = \begin{bmatrix} i_{\text{LS}} \\ U_{\text{LS}} \end{bmatrix}, \quad (16b)$$

$$A = \begin{bmatrix} -\frac{1}{\tau_e} & 0 & -Bl\beta_e \\ 0 & 0 & 1 \\ Bl\beta_m & -\omega_m^2 & -\frac{1}{\tau_m} \end{bmatrix}, \quad B = \begin{bmatrix} \beta_e & 0 \\ 0 & 0 \\ 0 & -S_d\beta_m \end{bmatrix}, \quad (16c)$$

$$C = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & S_d \end{bmatrix}. \quad (16d)$$

Parameters used for simulations are given in Table 4.

### 5.2 Controller expression

The state-space representation recalled in Section 5.1 describes a causal stable system. However, as mentioned above, we consider here that the loudspeaker is controlled by a current source ($I_{LS}$). In this case, expressing the position $Z(s)$ as $U_{LS}(s)/(S_d \cdot s)$ in the Laplace domain (for zero initial conditions), it straightforwardly follows from (15b) that

$$U_{\text{LS}}(s) = A_{\text{I}}(s) I_{\text{LS}}(s) + A_{\text{P}}(s) P(s), \quad (17a)$$

$$\begin{align}
A_{\text{I}}(s) &= \frac{sS_{\text{d}}Bl}{M_{\text{ms}}s^2 + R_{\text{ms}}s + \frac{1}{C_{\text{ms}}} \nonumber \\
&= \frac{S_{\text{d}} Bl \beta_{\text{m}} s}{s^2 + \frac{s}{\tau_{\text{m}}} + \omega_{\text{m}}^2}, \tag{17b}
\end{align}$$
<table><caption>**Figure 13.** Bode diagram (modulus and phase) of the vocal tract $H_{VT}$. The variation in the value of the $\phi_i$ parameters is studied in order to observe the changes in the transfer function. The errors $E_{\phi_i}$ and sensitivities $S_{\phi_i}$ are also presented for variations of $\phi_i$ between -10% and +10%. The variations in $Bl$, $S_d$ and $\beta_m$ (left-hand column) considerably impact the amplitudes and positions of the formants. In contrast, variations in $\omega_m$ and $\tau_m$ (right-hand column) have negligible effect on $H_{VT}$.</caption><thead><tr><th>Parameters φ<sub>i</sub></th><th>Bl, S<sub>d</sub>, β<sub>m</sub></th><th>ω<sub>m</sub>, τ<sub>m</sub></th></tr></thead><tbody><tr><td>H<sub>VT</sub></td><td></td><td></td></tr><tr><td>E<sub>φ<sub>i</sub></sub> = H<sub>VT</sub> - H<sup>†</sup></td><td></td><td></td></tr><tr><td>S<sub>φ<sub>i</sub></sub> = ∂H<sub>VT</sub>/∂φ<sub>i</sub></td><td></td><td></td></tr></tbody></table>
<figure><img src="image_16.png" /><figcaption>**Figure 14.** Description of the numerical testbed use for the trombone in the continuous and discrete time. Transfer functions A, B, C, D, T and R are given in equations X, from which transfer functions $\hat{A}$, $\hat{B}$, $\hat{C}$ and $\hat{D}$, $\hat{T}$ and $\hat{R}$ are deduced using the bilinear transform: $\forall z \in \mathbb{C}$ s.t. $|z| < 1$, $s = 2f_s \frac{1-z^{-1}}{1+z^{-1}}$, and given in equations (23a)–(23f), using $\kappa = \frac{r_0}{2\pi\nu_c(\theta_0)c_0}$.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 15.** Transfer functions $H^i = Z_{in}, Z_{R}$ in the Fourier domain, with ($i = c$) control and without ($i = \emptyset$) control. The amplitude differences and frequency deviations of resonance peaks are noted in Table 5.</figcaption></figure>
<table><caption>**Table 5.** Comparison of the first five frequencies peaks $F_k$ and amplitudes $A_k$ of the first five peaks of the input impedance without control $Z_{in}^0$ and with control $Z_{in}^c$.</caption><thead><tr><th>k</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td>$F_k^0$ [Hz]</td><td>109</td><td>173</td><td>237</td><td>288</td><td>351</td></tr><tr><td>$A_k^0$ [dB]</td><td>192</td><td>177</td><td>178</td><td>139</td><td>164</td></tr><tr><td>$F_k^c$ [Hz]</td><td>108</td><td>172</td><td>234</td><td>287</td><td>349</td></tr><tr><td>$A_k^c$ [dB]</td><td>183</td><td>159</td><td>170</td><td>145</td><td>163</td></tr><tr><td>$\frac{F_k^c - F_k^0}{F_k^0}$</td><td>-0.91%</td><td>-0.58%</td><td>-1.12%</td><td>-0.34%</td><td>-0.56%</td></tr></tbody></table>

$$A_P(s) = \frac{-s S_d^2}{M_{ms}s^2 + R_{ms}s + \frac{1}{C_{ms}}} = \frac{-s S_d^2 \beta_m}{s^2 + \frac{s}{\tau_m} + \omega_m^2} \quad (17c)$$

In practice, the current generator is built according to the Howland model [23]. Here, we consider that its cut-off frequency is higher than the frequency range of interest. As a result, the chain composed of the microphone, the current source and the amplifier is a constant gain arbitrarily set at 1 to simplify the following calculations. Then, the transfer function from $P$ to $I_{LS}$, shown in Figure 3, is equal to $K$. It follows from (17a) that:

$$U_{LS} = (A_I K + A_P) P. \quad (18)$$

Finally, from (4a) with $U_{ac} = U_{LS}$, the radiated flow is given by $U_R = U_{tb} + (A_I K + A_P) Z_R U_R$, so that by identification with (5), the controller transfer function is:

$$K = A_I^{-1} \left[ \frac{1}{Z_R} (1 - H_{VT}^{-1}) - A_P \right]. \quad (19)$$

### Passivity of the controller

According to hypothesis (H3) (cf. Sect. 2.3) and to the properties of $Z_R$ (cf. Sect. 3.1), $K$ is passive as a multiplication of passive elements.

## 5.3 Sensitivity to the Thiele and Small parameters

In the same way as in Section 4.3, we study the controller sensitivity to the following five reduced parameters noted $\phi_i: Bl, \omega_m, \tau_m, \beta_m, S_d$. From (19) it follows that:

$$H_{VT} = [1 - Z_R(A_I K + A_P)]^{-1}, \quad (20)$$

in which $H_{VT}$, $A_I$ and $A_P$ depend on $\phi_i$. Note that $Z_R$ is independent of the loudspeaker and $K$ is assumed to be fixed.

We use the chain rule on (20) to obtain several functions of sensitivity:

$$\frac{\partial H_{VT}}{\partial \phi_i} = S_{\phi_i}$$

<figure><img src="image_18.png" /><figcaption>**Figure 16.** Normalized acoustic pressure waveforms in the trombone mouthpiece $\hat{P}_{in}$ and at the bell $\hat{P}_{out}$, simulated from a measurement of acoustic flowrate in a real trombone mouthpiece</figcaption></figure>

$$= Z_R \left[ \frac{\partial A_I}{\partial \phi_i} K + \frac{\partial A_P}{\partial \phi_i} \right] [1 - Z_R(A_I K + A_P)]^{-2}. \quad (21)$$

Then:

$$H_{VT} = H_{VT}^{\dagger} + \sum_{i} S_{\phi_i} \delta \phi_i + \sum_{i} \mathcal{O}(\delta \phi_i^2), \quad (22)$$

where the sensitivity functions $S_{\phi_i}$ are calculated from nine terms $\partial A_I / \partial \phi_i, \phi_i \in \{Bl, \beta_m, S_d, \omega_m, \tau_m\}$ and $\partial A_P / \partial \phi_i, \phi_i \in \{\beta_m, S_d, \omega_m, \tau_m\}$ ($A_P$ does not depend on $Bl$). These terms can be grouped into two categories:

- the derivatives with respect to the parameters $Bl, \beta_m$, and $S_d$ of the numerator, which are proportional to $A_I$ and $A_P$:

$$\begin{aligned} \frac{\partial A_I}{\partial Bl} &= \frac{A_I}{Bl}, \\ \frac{\partial A_I}{\partial S_d} &= \frac{A_I}{S_d} \quad \text{and} \quad \frac{\partial A_P}{\partial S_d} = \frac{2A_P}{S_d}, \\ \frac{\partial A}{\partial \beta_m} &= \frac{A}{\beta_m} \quad \text{with } A = A_i \text{ or } A_P \end{aligned}$$

- the derivatives with respect to the parameters $\omega_m$ and $\tau_m$ of the denominator, which multiply $A_I$ and $A_P$ by a low-pass filter and a band-pass filter respectively, of natural frequency equal to $\omega_m$:

$$\begin{aligned} \frac{\partial A}{\partial \omega_m} &= - \frac{2\omega_m}{s^2 + \frac{s}{\tau_m} + \omega_m^2} A, \\ \frac{\partial A}{\partial \tau_m} &= \frac{s/\tau_m^2}{s^2 + \frac{s}{\tau_m} + \omega_m^2} A, \end{aligned}$$

with $A = A_I$ or $A_P$.

Figure 13 shows the vocal tract transfer functions calculated for different values of the loudspeaker parameters, their difference with the target vocal tract filter and the functions of sensitivity. It appears that the first category of parameters $Bl, S_d$ and $\beta_m$, which has a strong impact on the derivatives of $A_I$ and $A_P$, significantly modifies the
resonance amplitudes and frequencies of $H_{VT}$. Overestimating them by 10% even causes the first two formants to merge around 1 kHz, which affects the vowel produced. These parameters must therefore be determined experimentally as accurately as possible in order to build the desired target in the controller. In contrast, for the second category of parameters $\omega_m$ and $\tau_m$, similar variations have negligible effect on the vocal tract transfer function.

# 6 Numerical experiments

In this section, we design a numerical testbed to test and examine the control of the trombone. Section 6.1 derives the numerical testbed. Section 6.2 presents the numerical experiments and discusses the results on the controller.

## 6.1 Numerical testbed

In the trombone modelling, we prioritise simplicity and ease of simulation in the discrete-time domain over incorporating the refinements (C1)–(C5) listed in Section 2.1, since the control law is independent of block A in Figure 2.

The trombone is decomposed into several passive components: a dissipative mouthpiece (described in Appendix A) and a trombone resonator composed of concatenated straight pipes (described in Appendix B). So that this approach yields a Kelly-Lochbaum structure like that used for the vocal tract system, because its passivity is preserved when it is discretized by bilinear transform.

All these steps are detailed in Appendix D and the full process is summarized in Figure 14.

$$ \hat{A}(z) = \frac{1 + 2z^{-1} + z^{-2}}{[Y_0 + 2f_s C_a + 2f_s Y_0 R_a C_a + 4f_s^2 Y_0 C_a M_a] + z^{-1}[2Y_0 - 8Y_0 f_s^2 C_a M_a] + z^{-2}[Y_0 - 2f_s C_a - 2f_s Y_0 R_a C_a + 4Y_0 f_s^2 C_a M_a]} \quad (23a) $$

$$ [Y_0 - 2f_s C_a + 2f_s Y_0 R_a C_a + 4f_s^2 Y_0 C_a M_a] + z^{-1}[2Y_0 - 8Y_0 f_s^2 C_a M_a] $$

$$ \hat{B}(z) = \frac{+z^{-2}[Y_0 + 2f_s C_a - 2f_s Y_0 R_a C_a + 4Y_0 f_s^2 C_a M_a]}{[Y_0 + 2f_s C_a + 2f_s Y_0 R_a C_a + 4f_s^2 Y_0 C_a M_a] + z^{-1}[2Y_0 - 8Y_0 f_s^2 C_a M_a] + z^{-2}[Y_0 - 2f_s C_a - 2f_s Y_0 R_a C_a + 4Y_0 f_s^2 C_a M_a]} \quad (23b) $$

$$ \hat{C}(z) = \frac{2Y_0 + 4Y_0 z^{-1} + 2Y_0 z^{-2}}{[Y_0 + 2f_s C_a + 2f_s Y_0 R_a C_a + 4f_s^2 Y_0 C_a M_a] + z^{-1}[2Y_0 - 8Y_0 f_s^2 C_a M_a] + z^{-2}[Y_0 - 2f_s C_a - 2f_s Y_0 R_a C_a + 4Y_0 f_s^2 C_a M_a]} \quad (23c) $$

$$ [1 + R_a Y_0 + 2f_s M_a Y_0] + z^{-1}[1 + 2R_a Y_0] $$

$$ \hat{D}(z) = \frac{+z^{-2}[1 + R_a Y_0 - 2f_s M_a Y_0]}{[Y_0 + 2f_s C_a + 2f_s Y_0 R_a C_a + 4f_s^2 Y_0 C_a M_a] + z^{-1}[2Y_0 - 8Y_0 f_s^2 C_a M_a] + z^{-2}[Y_0 - 2f_s C_a - 2f_s Y_0 R_a C_a + 4Y_0 f_s^2 C_a M_a]} \quad (23d) $$

$$ \hat{R}(z) = \frac{[Z_c Y_2^c (2\alpha\kappa f_s + 4\kappa^2 f_s^2) - (1 + 4\xi\kappa f_s + 4\kappa^2 f_s^2)] + z^{-1}[-8\kappa^2 f_s^2 Z_c Y_2^c - (2 - 8\kappa^2 f_s^2)] + z^{-2}[Z_c Y_2^c (-2\alpha\kappa f_s + 4\kappa^2 f_s^2) - (1 - 4\xi\kappa f_s + 4\kappa^2 f_s^2)]}{[Z_c Y_2^c (2\alpha\kappa f_s + 4\kappa^2 f_s^2) + 1 + 4\xi\kappa f_s + 4\kappa^2 f_s^2] + z^{-1}[-8\kappa^2 f_s^2 Z_c Y_2^c + 2 - 8\kappa^2 f_s^2] + z^{-2}[Z_c Y_2^c (-2\alpha\kappa f_s + 4\kappa^2 f_s^2) + 1 - 4\xi\kappa f_s + 4\kappa^2 f_s^2]} \quad (23e) $$

$$ \hat{T}(z) = Z_c \frac{[4\alpha Y_2^c \kappa f_s + 8Y_2^c \kappa^2 f_s^2] + z^{-1}[-16Y_2^c \kappa^2 f_s^2]}{[Z_c Y_2^c (2\alpha\kappa f_s + 4\kappa^2 f_s^2) + 1 + 4\xi\kappa f_s + 4\kappa^2 f_s^2] + z^{-1}[-8\kappa^2 f_s^2 Z_c Y_2^c + 2 - 8\kappa^2 f_s^2] + z^{-2}[Z_c Y_2^c (-2\alpha\kappa f_s + 4\kappa^2 f_s^2) + 1 - 4\xi\kappa f_s + 4\kappa^2 f_s^2]} \quad (23f) $$

## 6.2 Numerical experiments on the controller: results and discussion

Two numerical experiments are implemented, aiming at simulating the model of trombone described in Section 3 in the absence and in the presence of the model of controller described in Section 4. The first experiment aims to validate the operation of the control structure proposed in this paper and the expression of the controller model given in (19). To this end, a swept sine covering the frequency range of interest and sampled at 44.1 kHz is synthesised and used as the discrete-time mouthpiece flow $\hat{U}_{in}$ at the input of the trombone model. The outputs of the discrete-time model, namely the radiated pressure $\hat{P}_{out}$ and the mouthpiece pressure $\hat{P}_{in}$ are provided by the Kelly-Lochbaum representation of the trombone and the expressions $\hat{A}, \hat{B}, \hat{C}, \hat{D}, \hat{R}$ and $\hat{T}$ defined in Figure 14 and given by (23a)–(23e). The numerical scheme is detailed in Appendix E. The $\hat{U}_{tnb}$ component of the discrete-time radiated flowrate at the bell is then approximated by $\frac{1}{Z_n}(P_N^{r+} - P_N^{r-})$. In the presence of the controller, it is added to the $\hat{U}_{ac}$ component provided by the loudspeaker. Finally, the radiation impedance and the input impedance are estimated by the ratios of the discrete Fourier Transforms $\hat{P}_{out}/\hat{U}_{tnb}$, and $\hat{P}_{in}/\hat{U}_{in}$ respectively. Figure 15 shows these ratios in the absence (left-hand column) and in the presence (right-hand column) of a controller.

In the absence of the controller ($H^0$): the estimate of $Z_{in}$ is similar to the trombone input impedance given in (C.2) and shown in Figure C.1. In particular, the first 5 peaks corresponding to the 5 lowest notes played on the trombone with the slide in first position (excluding the pedal note) are superimposed (see Tabs. 5 and C.1). As exposed in Appendix C, the $Z_{in}$ spectrum shows a decay of 20 dB per decade in the higher range, due to the mouthpiece acoustic compliance in parallel with the trombone resonator (see Appendix A). Then, the estimate of $Z_R$, (see Fig. 15), is also similar to the radiation impedance model described in Section 3.1 and shown in Figure 9.

In the presence of the controller ($H^c$): $Z_{in}$ is slightly modified. As shown in Table 5, the first 5 resonance peaks are lowered in frequency, but by less than -1.12%, i.e. about one sixth of a semi-tone. The peak amplitudes are
also modified, particularly in the vicinity of the formant frequencies, i.e. where $H_{VT}$ is large. Also, as expected, the spectral envelope of the radiation impedance $Z_R$ is close to the multiplication $H_{VT}Z_R$. As a result, the presence of the controller has an effect on both the radiation and the input impedance of the trombone model. This means that the performer, while playing, should be able to feel the presence of the controller at the mouthpiece.

The second experiment aims to simulate the acoustic pressure in the mouthpiece and at the bell, in particular to listen to the controller effect on the sound of the trombone. For this purpose, the trombone model input is an acoustic flow waveform, extracted from previous measurements [35], with a duration of 184 ms and sampled at 44.1 kHz. The discrete-time acoustic pressure is calculated in the mouthpiece and at the bell using the Kelly-Lochbaum representation described in Figure D.1 in Appendix D. The vowel effect [a] applied to the trombone sound by active control can be appreciated by playing, listening and comparing the acoustic pressures $P_{in}$ and $P_{out}$ whose normalized waveforms are plotted in Figure 16. Their periods are similar, and approximately equal to 8.6 ms, which corresponds to a playing frequency of 116.28 Hz.

# 7 Conclusion

This analysis is a continuation of work begun previously [10]. By considering a simple but effective model of the instrument, the numerical results showed that it is possible to radiate a vowel through the trombone using active control. Sensitivity studies on the physical parameters and on the loudspeaker parameters allow us to envisage physical experiments.

Further studies will aim to develop a radiation-independent control, so that the control can be adapted to any playing situation. All the hypotheses cited above are discussed in perspective of an experimental realization: choice and co-location of microphones, pressure-to-voltage converter, loudspeakers.

The application of this work to a real trombone will be the object of a next publication.

# Conflicts of interest

The authors declare that they have no conflicts of interest in relation to this article.

# Data availability statement

The data are available from the corresponding author on request.

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**Cite this article as:** Martos V. Boutin H. Hélie T. & d'Andréa-Novel B. 2025. Electro-acoustic control of radiation impedance for brass instrument timbre shaping: design of a vocalizing mute. Acta Acustica, 9, 40. https://doi.org/10.1051/aacus/2025020.
<figure><img src="image_19.png" /><figcaption>Figure A.1. Mouthpiece modelling.</figcaption></figure>

**Table A.1.** Numerical values chosen for the physical parameters of the mouthpiece. The radius $r_m$ and the length $l_m$ are based on the values of the Breslmaier trombone models. The volume $V_m$ is approximated to that of half an ellipsoid. The length value $L_s$ is taken from Mignot [16]. The radius $r_s$ is approximated to be equal to $r_b$ for the sake of continuity.

<table><thead><tr><th>Symbol</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>μ</td><td>1.8 · 10<sup>-5</sup></td><td>kg.m<sup>-1</sup>.s<sup>-1</sup></td></tr><tr><td>r<sub>m</sub></td><td>13 · 10<sup>-3</sup></td><td>m</td></tr><tr><td>l<sub>m</sub></td><td>25 · 10<sup>-3</sup></td><td>m</td></tr><tr><td>V<sub>m</sub> = <sup>2</sup>&frasl;<sub>3</sub>πr<sub>m</sub><sup>2</sup>l<sub>m</sub></td><td>8.84 · 10<sup>-6</sup></td><td>m<sup>3</sup></td></tr><tr><td>L<sub>s</sub></td><td>1 · 10<sup>-2</sup></td><td>m</td></tr><tr><td>r<sub>s</sub></td><td>6.7 · 10<sup>-3</sup></td><td>m</td></tr></tbody></table>

## Appendix A Mouthpiece

The mouthpiece of a trombone is composed of two parts: a cavity (bowl) and a straight tube (tail). For wavelengths greater than its characteristic dimensions, the acoustic behaviour can be approximated by a lumped parameter model, as a Helmholtz resonator (see [36]): an acoustic compliance ($C_a$) for the bowl, in parallel with an acoustic mass-damper system ($M_a, R_a$) for the tail (see Fig. A.1 for an electronic analogy).

Parameters $C_a, M_a$ and $R_a$ are given by (see Tab. A.1 for typical values)

$$C_a = \frac{V_m}{\rho_0 c_0^2}, \quad M_a = \frac{\rho_0 L_s}{\pi r_s^2}, \quad R_a = \frac{8\mu L_s}{\pi r_s^4}, \quad (A.1)$$

where $V_m$ is the volume of the bowl, $r_s$ and $L_s$ are the radius and length of the tail and $\mu$ is the air viscosity (cf. [16]).

Using the Kirchhoff's laws on the equivalent circuit yields, for all $s \in \mathbb{C}_0^+$,

$$P_{\text{in}}(s) = s M_a U_0(s) + R_a U_0(s) + P_0(s), \quad (A.2a)$$

$$U_0(s) = U_{\text{in}}(s) - s C_a P_{\text{in}}(s), \quad (A.2b)$$

from which the deduced acoustic transfer relation is

$$\underbrace{\begin{bmatrix} P_{\text{in}}(s) \\ U_{\text{in}}(s) \end{bmatrix}}_{X_{\text{in}}(s)}$$

<figure><img src="image_20.png" /><figcaption>Figure B.1. Trombone modelling using concatenation of conservative straight pipes with $L_0 = 0$ and $L_n = L_{n-1} + l_n$, $1 \le n \le N$.</figcaption></figure>

$$= \underbrace{\begin{bmatrix} 1 & R_a + s M_a \\ s C_a & 1 + s R_a C_a + s^2 C_a M_a \end{bmatrix}}_{T_a(s)} \underbrace{\begin{bmatrix} P_0(s) \\ U_0(s) \end{bmatrix}}_{X_0(s)}, \quad (A.3)$$

and where matrix $T_a$ takes the following concise form

$$T_a = \begin{bmatrix} 1 & \alpha \\ \beta & 1+\alpha\beta \end{bmatrix} \quad \text{with} \quad \begin{cases} \alpha(s) = R_a + s M_a, \\ \beta(s) = s C_a. \end{cases} \quad (A.4)$$

## Appendix B Conservative bore and bell

This section describes the acoustic propagation in the bore and bell of the trombone, using a very simple model that is suitable for real-time simulations. The use of more refined models, in particular by considering viscothermal losses, see [37], is beyond the scope of this paper.

The pipe (straight and curved parts) is modelled by concatenating elementary 1D waveguides, so that it admits a Kelly-Lochbaum representation [30]. Although such representations are available for several piecewise-defined descriptions (e.g., conservative conical pipes [38], dissipative conical or curved pipes [16]), the choice adopted here for sake of simplicity is to concatenate elementary conservative straight pipes of length $l_n$ and cross-section area $A_n$, for $1 \le n \le N$ (see Fig. B.1) localised for $\ell \in (L_{n-1}, L_n)$ with $L_0 = 0$, and $L_n = L_{n-1} + l_n$.

The acoustics of each elementary cylinder $n$ is described by its transfer matrix $T_n(s)$ in the Laplace domain. It relates the acoustic states $X_n = \begin{bmatrix} P_n \\ U_n \end{bmatrix}$ at $\ell = L_n$ to $X_{n-1}$ at $\ell = L_{n-1}$ as (see e.g., [39])

$$X_{n-1} = T_n X_n \\ \text{with } T_n(s) = \begin{bmatrix} \cosh \frac{s l_n}{c_0} & Z_n \sinh \frac{s l_n}{c_0} \\ Z_n^{-1} \sinh \frac{s l_n}{c_0} & \cosh \frac{s l_n}{c_0} \end{bmatrix} \quad (B.1)$$

for all Laplace variable $s \in \mathbb{C}_0^+ = \{s \in \mathbb{C} \text{ s.t. } \Re(e(s)) > 0\}$, where $Z_n = \rho_0 c_0 / A_n$ and $A_n$ denote the characteristic impedance and the cross section area of the $n$th cylinder respectively.
<table><caption>**Table B.1.** Characteristics of the trombone resonator used for simulation. These values are taken from the optimisation calculation [41], see Appendix F.</caption><thead><tr><th>Name</th><th>Symbol</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>Bore radius</td><td>r<sub>b</sub></td><td>6.7 &middot; 10<sup>-3</sup></td><td>m</td></tr><tr><td>Bore section</td><td>S<sub>b</sub> = &pi;r<sub>b</sub><sup>2</sup></td><td>1.57 &middot; 10<sup>-4</sup></td><td>m<sup>2</sup></td></tr><tr><td>Segment length</td><td>d</td><td>3.85 &middot; 10<sup>-3</sup></td><td>m</td></tr><tr><td>Bore length</td><td>l<sub>1</sub> = 141 &middot; d</td><td>0.5435</td><td>m</td></tr><tr><td>Bell length</td><td>l<sub>bell</sub> = 450 &middot; d</td><td>1.7347</td><td>m</td></tr></tbody></table>

<figure><img src="image_21.png" /><figcaption>**Figure B.2.** Profile of the bell.</figcaption></figure>

The acoustics of the complete pipe is then described by

$$X_0 = T_{\text{pipe}}X_N \text{ with } T_{\text{pipe}} = T_1 T_2 \dots T_N, \quad (\text{B.2})$$

the Kelly-Lochbaum structure of which is recalled below in Appendix D.

In a real trombone, the cylindrical bore is followed by the flare, almost conical part, and then the bell, more rapidly flaring part. Accordingly, in our model, the bore is modelled as a straight tube ($n = 1$) with a constant cross section area equal to $\pi r_b^2$, see Figure B.1. For the sake of simplicity, the flared part of the resonator hereafter called “the bell” is composed of a succession of $N_{\text{bell}}$ short straight tubes subsequently called segments of equal length $d$.

Note that for the following simulations: (i) $d$ is chosen so that the propagation time in each tube is half a sampling period [14]; (ii) the length of the bore $l_1$ is a multiple of $d$. In addition, the tubes radii in the resonator are chosen such that their rate of increase along the bore exceeds -13.64%, minimum spatial derivative of the radius reported in [40] on a measured trombone bore profile (Courtois TB 14, France).

These parameters result from an optimisation procedure [41], which minimises the distance between the input impedance peaks 2 to 6 of our model ($F_k, k \in [6, 40]$), to those of a real bass trombone ($F_k, k \in [6, 40]$) with the slide in first position, measured in a previous study [35] (Yamaha YBL 321, Hamamatsu, Japan). The optimisation criterion does not concern the first impedance peak, which is generally not played because its harmonics do not coincide with the following peaks.

<figure><img src="image_22.png" /><figcaption>**Figure C.1.** Modulus (top) and phase (bottom) of measured (red) and modelled (blue) trombone input impedance. The mouthpiece effect appears in the higher frequency range, when the modulus tends to -20 dB/dec asymptotic curve (green) and the phase tends to -π/2.</figcaption></figure>

Table B.1 gives the numerical values of the above-mentioned physical parameters of the resonator. The resulting profile of the bell is shown in Figure B.2: the 450 segments of equal length $d$ verifying the relationship $l_{\text{bell}} = 450 \cdot d$ are represented. Their radii take on 15 different values, each repeated 30 times to facilitate optimisation. Radius values are reported in Table B.2.

## Appendix C Global transfer functions

Assembling the mouthpiece ($T_a$), the pipe ($T_{\text{pipe}}$) and the radiation load ($Z_R$) leads to the transfer relation (denoting

$$X_{\text{out}} = \begin{bmatrix} P_{\text{out}} \\ U_{\text{out}} \end{bmatrix} = X_N), \text{ from (B.2) and (A.3):}$$

$$X_{\text{in}} = T_{\text{tb}} X_{\text{out}}, \quad (\text{C.1a})$$

$$\text{with } T_{\text{tbn}} = T_a T_{\text{pipe}} \text{ and } X_{\text{out}} = \begin{bmatrix} Z_R \\ 1 \end{bmatrix} U_{\text{out}}. \quad (\text{C.1b})$$

The input impedance $Z_{\text{in}} = P_{\text{in}}/U_{\text{in}}$ is then given by

$$Z_{\text{in}} = \frac{T_{\text{tb}}^{11} Z_{\text{R}} + T_{\text{tb}}^{12}}{T_{\text{tb}}^{21} Z_{\text{R}} + T_{\text{tb}}^{22}}. \quad (\text{C.2})$$

Figure C.1 compares $Z_{\text{in}}$ with the measured input impedance $Z_{\text{in}}^{\dagger}$ of the real bass trombone considered for the optimisation in Section B (see [35]).

It appears that the Q-factors of the resonances are greater in $Z_{\text{in}}$ than in $Z_{\text{in}}^{\dagger}$, as the model does not take viscothermal losses into account. Above the cut-off frequency, the impact of mouthpiece compliance on the input impedance is therefore lower in the model than in the measurement. This explains why the resonance Q-factors are greater in the model, even in the higher frequency range.

Table C.1 presents the amplitude and frequency differences of peaks 2 to 6 between the model $Z_{\text{in}}$ and the measurement $Z_{\text{in}}^{\dagger}$.
<table><caption>**Table B.2.** Comparison of the values of the 15 radius $R_k$ and their rate of increase. As shown in Figure B.2, the radius profile increases overall. A previous measurement [40] report a comparable radius profile, ranging from 6.9 mm to 101.8 mm with a rate of increase, that can be negative, but always bigger than -13.64%.</caption><thead><tr><th>k</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>13</th><th>14</th><th>15</th></tr></thead><tbody><tr><td>R<sub>k</sub> [mm]</td><td>7.07</td><td>7.15</td><td>7.89</td><td>7.10</td><td>7.21</td><td>6.92</td><td>13.55</td><td>4.85</td><td>10.52</td><td>12.30</td><td>6.22</td><td>16.72</td><td>7.59</td><td>36.66</td><td>110.30</td></tr><tr><td>R<sub>k</sub> - R<sub>k-1</sub> / d</td><td>×</td><td>+0.07%</td><td>+0.64%</td><td>-0.68%</td><td>+0.09%</td><td>-0.25%</td><td>+5.74%</td><td>-7.52%</td><td>+4.90%</td><td>+1.54%</td><td>-5.26%</td><td>+9.49%</td><td>-7.44%</td><td>+24.28%</td><td>+63.71%</td></tr></tbody></table>

<table><caption>**Table C.1.** Comparison of the frequencies $F_k$ and amplitudes $A_k$ of the first five peaks $Z_{in}^\dagger$ and $Z_{in}$.</caption><thead><tr><th>k</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td>$F_k^\dagger$ [Hz]</td><td>112.376</td><td>169.574</td><td>226.772</td><td>286.661</td><td>341.84</td></tr><tr><td>$A_k^\dagger$ [dB]</td><td>147.867</td><td>146.212</td><td>146.63</td><td>147.294</td><td>145.778</td></tr><tr><td>$F_k^\dagger - F_{k-1}^\dagger$ [Hz] ×</td><td></td><td>57.198</td><td>57.198</td><td>59.889</td><td>55.179</td></tr><tr><td>$F_k$ [Hz]</td><td>108.339</td><td>172.939</td><td>228.117</td><td>290.698</td><td>343.858</td></tr><tr><td>$A_k$ [dB]</td><td>171.039</td><td>174.805</td><td>172.307</td><td>152.902</td><td>182.28</td></tr><tr><td>$F_k^\dagger - F_{k-1}^\dagger$ [Hz] ×</td><td></td><td>64.6</td><td>55.178</td><td>62.581</td><td>53.16</td></tr><tr><td>$\frac{F_k - F_k^\dagger}{A_k - A_k^\dagger}$</td><td>-3.59%</td><td>+1.99%</td><td>+0.59%</td><td>+1.41%</td><td>+0.59%</td></tr><tr><td>$\frac{A_k - A_k^\dagger}{A_k^\dagger}$</td><td>+15.67%</td><td>+19.56%</td><td>17.51%</td><td>+3.81%</td><td>+25.04%</td></tr></tbody></table>

## Appendix D Kelly-Lochbaum structure

The Kelly-Lochbaum structure governs travelling wave variables. Consider the nth elementary straight pipe with characteristic impedance $Z_n = \rho_0 c_0 / A_n = Y_n^{-1}$. The travelling wave signals at its left side are $P_n^{\dagger} = (P_{n-1} \pm Z_n U_{n-1})/2$, those at its right side are $P_n^{\text{r}\pm} = (P_n \pm Z_n U_n)/2$, and they are related by a pure delay, called $D_n$ (see Fig. D.1):

$$ \begin{bmatrix} P_n^{\text{r}+} \\ P_n^{\text{l}+} \end{bmatrix} = D_n \begin{bmatrix} P_n^{\text{l}+} \\ P_n^{\text{r}+} \end{bmatrix} \quad \text{with } D_n(s) = e^{-\frac{s}{2c}}. \quad \text{(D.1)} $$

Between pipes n and n + 1, expressing $P_{n+1}^{\text{l}+}$ and $P_n^{\text{r}+}$ as functions of $P_n^{\text{r}+}$ and $P_{n+1}^{\text{l}+}$ yields the structure of junction $J_n$ detailed in Figure D.1 and involved in Figure 14, where the constant reflection coefficients $k_n$ are

$$ k_n = \frac{Z_{n+1} - Z_n}{Z_{n+1} + Z_n}. \quad \text{(D.2)} $$

At the trombone input, the relation between $X_{in}$ and $P_1^{\pm}$ is described by

$$ \underbrace{\begin{bmatrix} P_{in} \\ U_{in} \end{bmatrix}}_{X_{in}} = T_a \begin{bmatrix} P_0 \\ U_0 \end{bmatrix} = \underbrace{T_a \begin{bmatrix} 1 & 1 \\ Y_1 & -Y_1 \end{bmatrix}}_{M} \begin{bmatrix} P_1^{\text{l}+} \\ P_1^{\text{l}-} \end{bmatrix}, \quad \text{(D.3)} $$

with, from (A.3),

$$ M = \begin{bmatrix} 1 + \alpha Y_1 & 1 - \alpha Y_1 \\ \beta + (1 + \alpha \beta) Y_1 & \beta - (1 + \alpha \beta) Y_1 \end{bmatrix}. \quad \text{(D.4)} $$

The transfer functions A, B, C, D involved in Figure 14 are derived from the elements of M, by expressing $P_{in}$ and $P_1^{\text{l}+}$ as functions of $U_{in}$ and $P_1^{\text{l}-}$. Computations yield:

$$ \begin{align} A &= \frac{1}{M_{21}}, & B &= -\frac{M_{22}}{M_{21}}, & C &= -\frac{\det(M)}{M_{21}}, \nonumber \\ D &= \frac{M_{11}}{M_{21}}, & & & \tag{D.5} \end{align} $$

<figure><img src="image_23.png" /><figcaption>**Figure D.1.** Kelly-Lochbaum structure of a segment, composed of two delays $D_n(s)$, and a junction $J_n$, that governs travelling wave signals (see Appendix E for detailed computations). On the left: the schematic diagram of the junction; on the right: the equivalent way in which the junction is implemented.</figcaption></figure>

<table><caption>**Table D.1.** Expressions of the transfer functions A, B, C and D defined in (D.5) for the mouthpiece (left-hand column) and the vocal tract (right-hand column). These transfer functions give the pressure travelling waves at the inlet to the first tube of the Kelly-Lochbaum structure in Figure 14, with respect to the input pressure and flow.</caption><thead><tr><th></th><th>Mouthpiece (Sect. D)</th><th>Vocal Tract (Sect. 3.2)</th></tr></thead><tbody><tr><td>A</td><td>$\frac{1}{\beta+(1+\alpha\beta)Y_1}$</td><td>$Z_1$</td></tr><tr><td>B</td><td>$-\frac{\beta-(1+\alpha\beta)Y_1}{\beta+(1+\alpha\beta)Y_1}$</td><td>1</td></tr><tr><td>C</td><td>$\frac{2Y_1}{\beta+(1+\alpha\beta)Y_1}$</td><td>2</td></tr><tr><td>D</td><td>$\frac{1+\alpha Y_1}{\beta+(1+\alpha\beta)Y_1}$</td><td>$Z_1$</td></tr></tbody></table>

the detailed expressions of which are given in the left-hand column of Table D.1.

At the trombone output, $P_N = Z_R U_N$ so that signals $P_N^{\text{r}\pm} = (P_N \pm Z_N U_N)/2$ equal $(Z_R U_N \pm Z_N U_N)/2$. Then,
the reflection function $R = P_N^{l-}/P_N^{r+}$ in Figure 14 is given by

$$R(s) = \frac{Z_R(s) - Z_N}{Z_R(s) + Z_N}, \qquad (\text{D.6a})$$

which is similar to (D.2) but not constant, and the transmission function $T = P_N/P_N^{r+} = (P_N^{r+} + P_N^{r-})/P_N^{r+} = 1 + R$ is given by

$$T(s) = \frac{2Z_R(s)}{Z_R(s) + Z_N}, \qquad (\text{D.6b})$$

## Appendix E Coefficient relations

We note the demonstration of the coefficient relations exposed in Figure D.1. Pressure in nth segment $P_n$ is computed from the left-progressive one of the (n + 1)th segment noted $P_{n+1}^{l+}$ and from the left-regressive one of the (n + 1)th segment noted $P_{n+1}^{l-}$. Flow rate is deduced thanks to the impedance relation. We obtain:

$$\begin{cases} P_n = P_{n+1}^{l+} + P_{n+1}^{l-} \\ Z_{n+1}U_n = P_{n+1}^{l+} - P_{n+1}^{l-}. \end{cases} \qquad (\text{E.1})$$

The same relation can be exposed on the right-side of the segment,

$$\begin{cases} P_n = P_n^{r+} + P_n^{r-} \\ Z_n U_n = P_n^{r+} - P_n^{r-}. \end{cases} \qquad (\text{E.2})$$

Using (E.1) and (E.2), we can express $P_n^{r-}$ as a function of $P_n^{r+}$ and $P_{n+1}^{l-}$,

$$\begin{aligned} P_n^{r-} &= P_n - P_n^{r+} \\ &= P_{n+1}^{l+} + P_{n+1}^{l-} - P_n^{r+} \\ &= Z_{n+1}U_n + 2P_{n+1}^{l-} - P_n^{r+} \\ &= \frac{Z_{n+1}}{Z_n} (P_n^{r+} - P_n^{r-}) + 2P_{n+1}^{l-} - P_n^{r+}, \end{aligned}$$

by regrouping the following terms,

$$\begin{align*} \Leftrightarrow P_n^{r-} \left(1 + \frac{Z_{n+1}}{Z_n}\right) &= \left(\frac{Z_{n+1}}{Z_n} - 1\right) P_n^{r+} + 2P_{n+1}^{l-} \\ \Leftrightarrow P_n^{r-} &= \frac{Z_{n+1} - Z_n}{Z_{n+1} + Z_n} P_n^{r+} + \frac{2Z_n}{Z_{n+1} + Z_n} P_{n+1}^{l-}. \end{align*}$$

As a result, a $k_n$ coefficient is expressed,

$$P_n^{r-} = k_n P_n^{r+} + (1 - k_n) P_{n+1}^{l-}, \qquad (\text{E.3})$$

with $k_n = \frac{Z_{n+1} - Z_n}{Z_{n+1} + Z_n}$.

<table><caption>**Table F.1.** List of the chosen parameters for simulations. The choices of $L_{\text{bell}}$ and $N_{\text{bore}}$ are explained below.</caption><thead><tr><th colspan="2">Resonator</th></tr></thead><tbody><tr><td>$f_0$</td><td>37.3 Hz</td></tr><tr><td>$f_p$</td><td>(2p + 1) <sup>c<sub>0</sub></sup>&frasl;<sub>4L<sub>tot</sub></sub></td></tr><tr><td>$d = \frac{c}{2f}$</td><td>3.8548 mm</td></tr><tr><td>$L_{\text{tot}} = (2p + 1) \frac{c_0}{4f_p}$</td><td>2.2768 m</td></tr><tr><td>$N_{\text{tot}} = \frac{L_{\text{bore}}}{d} = \frac{3f_p}{2f}$</td><td>561</td></tr><tr><td colspan="2">Bell</td></tr><tr><td>$L_{\text{bell}}$</td><td>1.7347 m</td></tr><tr><td>$N_{\text{bell}} = \frac{L_{\text{bell}}}{d}$</td><td>450</td></tr><tr><td colspan="2">Bore</td></tr><tr><td>$L_{\text{bore}} = L_{\text{tot}} - L_{\text{bell}}$</td><td>0.5435 m</td></tr><tr><td>$N_{\text{bore}} = \frac{L_{\text{bore}}}{d}$</td><td>141</td></tr></tbody></table>

## Appendix F Optimisation

The length of the resonator $L_{\text{tot}}$ is chosen to fix the first harmonic peak at 37 Hz. As an open/close pipe, its resonance frequencies follow the relation $f_p = (2p + 1) \frac{c_0}{4L_{\text{tot}}}$. The length of a straight pipe is equal to $d$ (see Sect. 3.2). So, the number of straight pipes must be equal to 560.6, approximating to $N_{\text{tot}} = 561$ (this approximation gives $f_1 = 111.92$ Hz, which corresponds to a relative error of 0.018%). As $N_{\text{tot}}$, $N_{\text{bell}}$ must be an integer, i.e., $2f_s L_{\text{bell}}/c_0 \in \mathbb{N}$. Moreover, the optimisation algorithm is based on a target of a 15-radius model. For this reason, $N_{\text{bell}}$ must therefore be an integer and a multiple of 15. These conditions led us to choose $N_{\text{bell}} = 30 \times 15 = 450$. Table F.1 summarises the values chosen to the algorithmic optimisation.