Review

© The Author(s), Published by EDP Sciences, 2025

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.

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 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 2, these two flow rates are equal to U_tb and U_R respectively. The goal is to modify the radiated airflow as

$$U_{\mathrm{R}}=H_{\text{VT}}·U_{\text{tb}},$$(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 loudspeaker behaves like an electrical current to acoustical 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 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 isobar 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 $Z_{\mathrm{R}}=\frac{P_{\text{out}}}{U_{\text{out}}}$ is modelled in the Laplace domain by

$$Z_{\mathrm{R}}(s)=\frac{{\rho }_0{c}_0}{S_0}\tilde{Z}\left(\frac{s}{{\omega }_0}\right)\hspace{0.5em}\mathrm{with}\hspace{0.5em}{\omega }_0=\frac{2\pi c_0{\nu }_c({\theta }_0)}{r_0},$$(2a)

$$\tilde{Z}(\tilde{s})=\frac{\alpha ({\theta }_0)\tilde{s}+{\tilde{s}}^2}{1+2\xi ({\theta }_0)\tilde{s}+{\tilde{s}}^2}\hspace{0.5em}\hspace{0.5em}(\mathrm{adimensioned}),$$(2b)

where (see Tab. 1) ρ₀ and c₀ are the air density and the speed of sound, r₀ is the sphere radius, θ₀ is the cap half-angle ($r_0=\frac{r_{\text{bell}}}{sin{\theta }_0}$), and where optimal parameters α, ξ and ν_c are given by the following functions, defined for all ${\theta }_0\in [0,\frac{\pi }{2}]$, $\xi ({\theta }_0)=0.0207{\theta }_0^4-0.144{\theta }_0^3+0.221{\theta }_0^2+0.0799{\theta }_0+0.72\ge 0$, $\alpha ({\theta }_0)=(0.1113{\theta }_0^5-0.636{\theta }_0^4+1.162{\theta }_0^3-1.242{\theta }_0^2+1.083{\theta }_0+0.8788{)}^{-1}\ge 0$ 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 $\overline{{ℂ}_0^{+}}$ (denoting ${ℂ}_0^{+}=\{s\in ℂ\mathrm{s}.\mathrm{t}.\mathfrak{R}e(s)>0\}$) and it is proved to define a causal, stable, passive system [26, 28] which therefore meets the expectations formulated in (H1), Section 2.3. Figure 5 shows the values of module and phase of $\tilde{Z}$ for several values of θ₀.

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 β=1+α²−2ξ². For typical values measured on a real trombone bell, θ₀=72.4° and r₀=8.39 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 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 $Z_{\mathrm{R}}^{\mathrm{VT}}$, so that

$$\left[\begin{array}{c}\hfill P_{\mathrm{in}}\hfill \\ \hfill U_{\mathrm{in}}\hfill \end{array}\right]=\underset{\left[\begin{array}{c}\hfill *\hfill \\ \hfill H_{\mathrm{VT}}^{-1}\hfill \end{array}\right]}{\underbrace{T_{\mathrm{VT}}\left[\begin{array}{c}\hfill Z_{\mathrm{R}}^{\mathrm{VT}}\hfill \\ \hfill 1\hfill \end{array}\right]}}{U}_{\mathrm{out}}.$$(3)

Figure 7 shows the Bode diagram of H_VT computed for the vowel [a] described by the area profile in Figure 8 and choosing¹ a_ext=38.2 cm².

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 vowel². Subsequently, the robustness of vowel rendering is evaluated. Initially, a basic test examines deviations due to an angle error θ₀ in the radiation impedance model. Following this, a sensitivity analysis with respect to θ₀ 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 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,

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

$$P_{\mathrm{tb}}=P_{\mathrm{ac}}=P_{\mathrm{R}}\hspace{0.5em}\hspace{0.5em}(:=P).$$(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_VTU_tb) rather than the trombone flow itself (U_tb), that is (see Fig. 9)

$$U_{\mathrm{R}}=H_{\mathrm{VT}}{U}_{\mathrm{tb}}.$$(5)

Note that, from the trombone viewpoint, this amounts to loading its bell extremity by the modified radiation impedance H_VTZ_R, denoted $Z_{\mathrm{R}}^c$ below.

Then, modelling the acoustic control by its admittance Y_ac=U_ac/P_ac, it follows from (4a) that, at the acoustic junction,

$$U_{\mathrm{tb}}+Y_{\mathrm{ac}}{P}_{\mathrm{ac}}=Z_{\mathrm{R}}^{-1}{P}_{\mathrm{R}}.$$(6a)

The expected target (U_R=H_VTU_tb) is achieved if $U_{\mathrm{tb}}=H_{\mathrm{VT}}^{-1}{U}_{\mathrm{R}}=H_{\mathrm{VT}}^{-1}{Z}_{\mathrm{R}}^{-1}{P}_{\mathrm{R}}$, so that (6a) becomes

$$H_{\mathrm{VT}}^{-1}{Z}_{\mathrm{R}}^{-1}{P}_{\mathrm{R}}+Y_{\mathrm{ac}}{P}_{\mathrm{ac}}=Z_{\mathrm{R}}^{-1}{P}_{\mathrm{R}}.$$(6b)

Finally, from (4b), the target acoustic control admittance is

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

leading to the equivalent formula for the vocal tract filter

$$H_{\mathrm{VT}}={\left(1-Z_{\mathrm{R}}{Y}_{\mathrm{ac}}\right)}^{-1}.$$(7b)

4.2 Qualitative robustness test

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

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

$$H_{\mathrm{VT}}^{\mathrm{err}}=\frac{1}{1-\left(Z_{\mathrm{R}}^{†}/Z_{\mathrm{R}}^{\mathrm{err}}\right)\left(1-1/H_{\mathrm{VT}}^{†}\right)}·$$(8)

Following this equation, the relative errors

$${ϵ}_{\mathrm{H}}:=\frac{H_{\mathrm{VT}}^{\mathrm{err}}-H_{\mathrm{VT}}^{†}}{H_{\mathrm{VT}}^{†}}\hspace{0.5em}\mathrm{and}\hspace{0.5em}{ϵ}_{\mathrm{Z}}:=\frac{Z_{\mathrm{R}}^{\mathrm{err}}-Z_{\mathrm{R}}^{†}}{Z_{\mathrm{R}}^{†}},$$(9a)

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

$${ϵ}_{\mathrm{H}}={ϵ}_{\mathrm{Z}}\frac{1-H_{\mathrm{VT}}^{†}}{1+{ϵ}_{\mathrm{Z}}{H}_{\mathrm{VT}}^{†}}·$$(9b)

A robustness test is presented for $H_{\mathrm{VT}}^{†}$ given in Figure 7 where $Z_{\mathrm{R}}^{†}$, with parameter ${\theta }_0^{†}=72.4^{\circ }$ (see (2a), (2b) and Tab. 1), is erroneously replaced by $Z_{\mathrm{R}}^{\mathrm{err}}$, with the same parameters except a deviation of +5% on the angle ${\theta }_0^{\mathrm{err}}=1.05{\theta }_0^{†}\approx 76.{02}^{\circ }$ (see Fig. 10).

The vocal tract filter $H_{\mathrm{VT}}^{\mathrm{err}}$ resulting from (8) is compared to $H_{\mathrm{VT}}^{†}$ 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,θ₀), assuming that:

1. a target control $Y_{\mathrm{ac}}^{†}$ is pre-computed from (7a) with fixed parameters T^†=300 K, ${\theta }_0^{†}=72.4^{\circ }$,

2. the radiation impedance is subject to small variations in its parameters a around $a^{†}=(T^{†},{\theta }_0^{†})$, and is renoted Z_R(s;a).

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

$$H_{\mathrm{VT}}(s;a)={\left(1-Z_{\mathrm{R}}(s;a)Y_{\mathrm{ac}}^{†}\right)}^{-1}.$$(10a)

Then, the first order expansion of H_VT(s;a^†+δa) with respect to δa₁=δT and δa₂=δθ₀ leads to

$$H_{\mathrm{VT}}=H_{\mathrm{VT}}^{†}+S_{\mathrm{T}}\delta T+S_{{\theta }_0}\delta {\theta }_0+\mathcal{O}\left(\delta T^2+\delta {\theta }_0^2\right),$$(10b)

introducing $H_{\mathrm{VT}}^{†}(s)=H_{\mathrm{VT}}(s;a^{†})$ and the sensitivity functions for a₁=T and a₂=θ₀ as

$$S_{{\text{a}}_{\mathrm{i}}}(s;a)=\frac{\partial H_{\mathrm{VT}}(s;a)}{\partial a_{\mathrm{i}}}·$$(10c)

From (10a), it follows that

$$S_{{\text{a}}_{\mathrm{i}}}=\frac{Y_{\mathrm{ac}}^{†}}{{\left(1-Y_{\mathrm{ac}}^{†}{Z}_{\mathrm{R}}\right)}^2}\frac{\partial Z_{\mathrm{R}}}{\partial a_{\mathrm{i}}},$$(11a)

where $\frac{\partial Z_{\mathrm{R}}}{\partial a_{\mathrm{i}}}$ is derived from the chain rule applied to (2a), (2b) in which ρ₀, c₀, S₀, ω₀ are the a-dependent functions

$${\rho }_0(T)=\frac{P_{0}M}{\mathit{RT}},$$(12a)

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

$$S_0({\theta }_0)=2\pi {\left(\frac{r_{\mathrm{bell}}}{sin({\theta }_0)}\right)}^2\left(1-cos({\theta }_0)\right),$$(12c)

$${\omega }_0({\theta }_0)=\frac{2\pi c_0(T){\nu }_c({\theta }_0)}{r_0({\theta }_0)},\hspace{0.5em}\mathrm{with}\hspace{0.5em}{r}_0({\theta }_0)=\frac{r_{\mathrm{bell}}}{sin({\theta }_0)},$$(12d)

and α(θ₀), ξ(θ₀) and ν_c(θ₀) 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

$$Z_c\left(T,{\theta }_0\right)=\frac{{\rho }_0(T)c_0(T)}{S_0({\theta }_0)}=\sqrt{\frac{\gamma M}{\mathit{RT}}}\frac{P_0}{\pi r_{\mathrm{bell}}^2}{cos}^2\left(\frac{{\theta }_0}{2}\right)$$(13)

involved in (2a) yields

$$\frac{\partial Z_c}{\partial {\theta }_0}=-\sqrt{\frac{\gamma M}{\mathit{RT}}}\frac{P_0}{2\pi r_{\mathrm{bell}}^2}sin({\theta }_0),$$(14a)

$$\mathrm{and}\hspace{0.5em}\hspace{0.5em}\frac{\partial Z_c}{\partial T}=-\sqrt{\frac{\gamma M}{R{T}^3}}\frac{P_0}{2\pi r_{\mathrm{bell}}^2}{cos}^2\left(\frac{{\theta }_0}{2}\right)·$$(14b)

Figure 12 reveals that variations in θ₀ and T have a significant impact on the position and amplitude of vocal tract resonances, potentially modifying the radiated vowel. In particular, overestimation of θ₀ 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 θ₀ has the opposite effect, fusing resonances 1 & 2 when θ₀=80 °. Estimation errors on T cause resonance frequencies to move in the same direction as estimation errors on θ₀, 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 electro-acoustic 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^micro where V^micro=K_micP denotes the voltage delivered by a pressure microphone of ideal gain K_mic. The pressure-to-current gain is then described by I_LS=K_totP with K_tot=K·K_mic.

The transfer function K_tot is the electronic control to be determined in Section 5.2, in order to realise the acoustic target Y_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]:

$$v_{\mathrm{LS}}(t)=R_{\mathrm{e}}{i}_{\mathrm{LS}}(t)+L_{\mathrm{e}}\frac{\mathrm{d}{i}_{\mathrm{LS}}}{\mathrm{d}t}+\mathit{Bl}\frac{\mathrm{d}z}{\mathrm{d}t},$$(15a)

$$M_{\mathrm{ms}}\frac{{\mathrm{d}}^{2}z}{\mathrm{d}t}=\mathit{Bl}{i}_{\mathrm{LS}}(t)-R_{\mathrm{ms}}\frac{\mathrm{d}z}{\mathrm{d}t}-\frac{1}{C_{\mathrm{ms}}}z(t)-S_{\mathrm{d}}P,$$(15b)

where M_ms is the mass of the driver diaphragm and voice-coil assembly, R_ms and C_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:

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

$$\mathrm{with}\hspace{0.5em}u=\left[\begin{array}{c}\hfill v_{\mathrm{LS}}\hfill \\ \hfill P\hfill \end{array}\right],\hspace{0.5em}x=\left[\begin{array}{c}\hfill i_{\mathrm{LS}}\hfill \\ \hfill z\hfill \\ \hfill \dot{z}\hfill \end{array}\right],\hspace{0.5em}y=\left[\begin{array}{c}\hfill i_{\mathrm{LS}}\hfill \\ \hfill U_{\mathrm{LS}}\hfill \end{array}\right],$$(16b)

$$A=\left[\begin{array}{ccc}\hfill -\frac{1}{{\tau }_{\mathrm{e}}}\hfill & \hfill 0\hfill & \hfill -\mathit{Bl}{\beta }_{\mathrm{e}}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \mathit{Bl}{\beta }_{\mathrm{m}}\hfill & \hfill -{\omega }_{\mathrm{m}}^2\hfill & \hfill -\frac{1}{{\tau }_{\mathrm{m}}}\hfill \end{array}\right],\hspace{0.5em}B=\left[\begin{array}{cc}\hfill {\beta }_{\mathrm{e}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -S_{\mathrm{d}}{\beta }_{\mathrm{m}}\hfill \end{array}\right],$$(16c)

$$C=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill S_{\mathrm{d}}\hfill \end{array}\right].$$(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·s) in the Laplace domain (for zero initial conditions), it straightforwardly follows from (15b) that

$$\begin{array}{cc}\hfill U_{\mathrm{LS}}(s)& =A_{\mathrm{I}}(s)I_{\mathrm{LS}}(s)+A_{\mathrm{P}}(s)P(s),\hfill \end{array}$$(17a)

$$\begin{array}{cc}\hfill A_{\mathrm{I}}(s)& =\frac{s{S}_{\mathrm{d}}\mathit{Bl}}{M_{\mathrm{ms}}{s}^2+R_{\mathrm{ms}}s+\frac{1}{C_{\mathrm{ms}}}}\hfill \\ \hfill & =\frac{S_{\mathrm{d}}\mathit{Bl}{\beta }_{\mathrm{m}}s}{s^2+\frac{s}{{\tau }_{\mathrm{m}}}+{\omega }_{\mathrm{m}}^2},\hfill \end{array}$$(17b)

$$\begin{array}{cc}\hfill A_{\mathrm{P}}(s)& =\frac{-s{S}_d^2}{M_{\mathrm{ms}}{s}^2+R_{\mathrm{ms}}s+\frac{1}{C_{\mathrm{ms}}}}\hfill \\ \hfill & =\frac{-s{S}_{\mathrm{d}}^2{\beta }_{\mathrm{m}}}{s^2+\frac{s}{{\tau }_{\mathrm{m}}}+{\omega }_{\mathrm{m}}^2}·\hfill \end{array}$$(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:

$$U_{\mathrm{LS}}=\left(A_{\mathrm{I}}K+A_{\mathrm{P}}\right)P.$$(18)

Finally, from (4a) with U_ac=U_LS, the radiated flow is given by $U_{\mathrm{R}}=U_{\mathrm{tb}}+\left(A_{\mathrm{I}}K+A_{\mathrm{P}}\right)Z_{\mathrm{R}}{U}_{\mathrm{R}}$, so that by identification with (5), the controller transfer function is:

$$K=A_{\mathrm{I}}^{-1}\left[\frac{1}{Z_{\mathrm{R}}}\left(1-H_{\mathrm{VT}}^{-1}\right)-A_{\mathrm{P}}\right].$$(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 ϕ_i:Bl, ω_m,τ_m,β_m,S_d. From (19) it follows that:

$$H_{\mathrm{VT}}={\left[1-Z_{\mathrm{R}}\left(A_{\mathrm{I}}K+A_{\mathrm{P}}\right)\right]}^{-1},$$(20)

in which H_VT, A_I and A_P depend on ϕ_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:

$$\begin{array}{cc}\hfill & \frac{\partial H_{\mathrm{VT}}}{\partial {\varphi }_{\mathrm{i}}}=S_{{\varphi }_i}\hfill \\ \hfill & =Z_{\mathrm{R}}\left[\frac{\partial A_{\mathrm{I}}}{\partial {\varphi }_{\mathrm{i}}}K+\frac{\partial A_{\mathrm{P}}}{\partial {\varphi }_{\mathrm{i}}}\right]{\left[1-Z_{\mathrm{R}}\left(A_{\mathrm{I}}K+A_{\mathrm{P}}\right)\right]}^{-2}.\hfill \end{array}$$(21)

Then:

$$H_{\mathrm{VT}}=H_{\mathrm{VT}}^{†}+{\displaystyle \sum _i}{S}_{{\varphi }_i}\delta {\varphi }_i+{\displaystyle \sum _i}\mathcal{O}\left(\delta {\varphi }_i^2\right),$$(22)

where the sensitivity functions $S_{{\varphi }_i}$ are calculated from nine terms ∂A_I/∂ϕ_i,ϕ_i∈{Bl,β_m,S_d,ω_m,τ_m} and ∂A_P/∂ϕ_i,ϕ_i∈{β_m,S_d,ω_m,τ_m} (A_P does not depend on Bl). These terms can be grouped into two categories:

• the derivatives with respect to the parameters Bl, β_m, and S_d of the numerator, which are proportional to A_I and A_P:

  $$\begin{array}{cc}\hfill & \frac{\partial A_{\mathrm{I}}}{\partial \mathit{Bl}}=\frac{A_{\mathrm{I}}}{\mathit{Bl}},\hfill \\ \hfill & \frac{\partial A_{\mathrm{I}}}{\partial S_{\mathrm{d}}}=\frac{A_{\mathrm{I}}}{S_{\mathrm{d}}}\hspace{0.5em}\mathrm{and}\hspace{0.5em}\frac{\partial A_{\mathrm{P}}}{\partial S_{\mathrm{d}}}=\frac{2A_{\mathrm{P}}}{S_{\mathrm{d}}},\hfill \\ \hfill & \frac{\partial A}{\partial {\beta }_m}=\frac{A}{{\beta }_m}\hspace{0.5em}\mathrm{with}\hspace{0.5em}A=A_i\hspace{0.5em}\mathrm{or}\hspace{0.5em}{A}_p\hfill \end{array}$$

• the derivatives with respect to the parameters ω_m and τ_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 ω_m:

  $$\begin{array}{cc}\hfill & \frac{\partial A}{\partial {\omega }_{\mathrm{m}}}=-\frac{2{\omega }_{\mathrm{m}}}{s^2+\frac{s}{{\tau }_{\mathrm{m}}}+{\omega }_{\mathrm{m}}^2}A,\hfill \\ \hfill & \frac{\partial A}{\partial {\tau }_{\mathrm{m}}}=\frac{s/{\tau }_{\mathrm{m}}^2}{s^2+\frac{s}{{\tau }_{\mathrm{m}}}+{\omega }_{\mathrm{m}}^2}A,\hfill \end{array}$$

  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 β_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 ω_m and τ_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.

$$\hat{A}(z)=\frac{1+2z^{-1}+z^{-2}}{\begin{array}{c}\hfill [Y_0+2f_s{C}_a+2f_s{Y}_0{R}_a{C}_a+4f_s^2{Y}_0{C}_{{\mathit{aM}}_{}}]\hfill \\ \hfill +z^{-1}[2Y_0-8Y_0{f}_s^2{C}_a{M}_a]\hfill \\ \hfill +z^{-2}[Y_0-2f_s{C}_a-2f_s{Y}_0{R}_a{C}_a+4Y_0{f}_s^2{C}_a{M}_a]\hfill \end{array}}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}$$(23a)

$$\hat{B}(z)=\frac{\begin{array}{c}\hfill [Y_0-2f_s{C}_a+2f_s{Y}_0{R}_a{C}_a+4f_s^2{Y}_0{C}_a{M}_a]\hfill \\ \hfill +z^{-1}[2Y_0-8Y_0{f}_s^2{C}_a{M}_a]\hfill \\ \hfill +z^{-2}[Y_0+2f_s{C}_a-2f_s{Y}_0{R}_a{C}_a+4Y_0{f}_s^2{C}_a{M}_a]\hfill \end{array}}{\begin{array}{c}\hfill [Y_0+2f_s{C}_a+2f_s{Y}_0{R}_a{C}_a+4f_s^2{Y}_0{C}_{{\mathit{aM}}_{}}]\hfill \\ \hfill +z^{-1}[2Y_0-8Y_0{f}_s^2{C}_a{M}_a]\hfill \\ \hfill +z^{-2}[Y_0-2f_s{C}_a-2f_s{Y}_0{R}_a{C}_a+4Y_0{f}_s^2{C}_a{M}_a]\hfill \end{array}}$$(23b)

$$\hat{C}(z)=\frac{2Y_0+4Y_0{z}^{-1}+2Y_0{z}^{-2}}{\begin{array}{c}\hfill [Y_0+2f_s{C}_a+2f_s{Y}_0{R}_a{C}_a+4f_s^2{Y}_0{C}_{{\mathit{aM}}_{}}]\hfill \\ \hfill +z^{-1}[2Y_0-8Y_0{f}_s^2{C}_a{M}_a]\hfill \\ \hfill +z^{-2}[Y_0-2f_s{C}_a-2f_s{Y}_0{R}_a{C}_a+4Y_0{f}_s^2{C}_a{M}_a]\hfill \end{array}}$$(23c)

$$\hat{D}(z)=\frac{\begin{array}{c}\hfill [1+R_a{Y}_0+2f_s{M}_{{\mathit{aY}}_{}}]\hfill \\ \hfill +z^{-1}[1+2R_a{Y}_0]\hfill \\ \hfill +z^{-2}[1+R_a{Y}_0-2f_s{M}_{{\mathit{aY}}_{}}]\hfill \end{array}}{\begin{array}{c}\hfill [Y_0+2f_s{C}_a+2f_s{Y}_0{R}_a{C}_a+4f_s^2{Y}_0{C}_{{\mathit{aM}}_{}}]\hfill \\ \hfill +z^{-1}[2Y_0-8Y_0{f}_s^2{C}_a{M}_a]\hfill \\ \hfill +z^{-2}[Y_0-2f_s{C}_a-2f_s{Y}_0{R}_a{C}_a+4Y_0{f}_s^2{C}_a{M}_a]\hfill \end{array}}$$(23d)

$$\hat{R}(z)=\frac{\begin{array}{c}\hfill [Z_c{Y}_2^c(2\alpha \kappa f_s+4{\kappa }^2{f}_s^2)-(1+4\xi \kappa f_s+4{\kappa }^2{f}_s^2)]\hfill \\ \hfill +z^{-1}[-8{\kappa }^2{f}_s^2{Z}_c{Y}_2^c-(2-8{\kappa }^2{f}_s^2)]\hfill \\ \hfill +z^{-2}[Z_c{Y}_2^c(-2\alpha \kappa f_s+4{\kappa }^2{f}_s^2)-(1-4\xi \kappa f_s+4{\kappa }^2{f}_s^2)]\hfill \end{array}}{\begin{array}{c}\hfill [Z_c{Y}_2^c(2\alpha \kappa f_s+4{\kappa }^2{f}_s^2)+1+4\xi \kappa f_s+4{\kappa }^2{f}_s^2]\hfill \\ \hfill +z^{-1}[-8{\kappa }^2{f}_s^2{Z}_c{Y}_2^c+2-8{\kappa }^2{f}_s^2]\hfill \\ \hfill +z^{-2}[Z_c{Y}_2^c(-2\alpha \kappa f_s+4{\kappa }^2{f}_s^2)+1-4\xi \kappa f_s+4{\kappa }^2{f}_s^2]\hfill \end{array}}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}$$(23e)

$$\hat{T}(z)=Z_c\frac{\begin{array}{c}\hfill [4\alpha Y_2^c\kappa f_s+8Y_2^c{\kappa }^2{f}_s^2]\hfill \\ \hfill +z^{-1}[-16Y_2^c{\kappa }^2{f}_s^2]\hfill \\ \hfill +z^{-2}[-4\alpha Y_2^c\kappa f_s+8Y_2^c{\kappa }^2{f}_s^2]\hfill \end{array}}{\begin{array}{c}\hfill [Z_c{Y}_2^c(2\alpha \kappa f_s+4{\kappa }^2{f}_s^2)+1+4\xi \kappa f_s+4{\kappa }^2{f}_s^2]\hfill \\ \hfill +z^{-1}[-8{\kappa }^2{f}_s^2{Z}_c{Y}_2^c+2-8{\kappa }^2{f}_s^2]+z^{-2}\hfill \\ \hfill [Z_c{Y}_2^c(-2\alpha \kappa f_s+4{\kappa }^2{f}_s^2)+1-4\xi \kappa f_s+4{\kappa }^2{f}_s^2]\hfill \end{array}}·$$(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 ${\hat{U}}_{\mathrm{in}}$ at the input of the trombone model. The outputs of the discrete-time model, namely the radiated pressure ${\hat{P}}_{\mathrm{out}}$ and the mouthpiece pressure ${\hat{P}}_{\mathrm{in}}$ 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 ${\hat{U}}_{\mathrm{tbn}}$ 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 ${\hat{U}}_{\mathrm{ac}}$ component provided by the loudspeaker. Finally, the radiation impedance and the input impedance are estimated by the ratios of the discrete Fourier Transforms ${\hat{P}}_{\mathrm{out}}/{\hat{U}}_{\mathrm{tbn}}$, and ${\hat{P}}_{\mathrm{in}}/{\hat{U}}_{\mathrm{in}}$ 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^∅): 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_VTZ_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.

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)

$$C_{\mathrm{a}}=\frac{V_{\mathrm{m}}}{{\rho }_0{c}_0^2},M_{\mathrm{a}}=\frac{{\rho }_0{L}_{\mathrm{s}}}{\pi r_{\mathrm{s}}^2},R_{\mathrm{a}}=\frac{8\mu L_{\mathrm{s}}}{\pi r_{\mathrm{s}}^4},$$(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 μ is the air viscosity (cf. [16]).

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

$$P_{\mathrm{in}}(s)=s{M}_{\mathrm{a}}{U}_0(s)+R_{\mathrm{a}}{U}_0(s)+P_0(s),$$(A.2a)

$$U_0(s)=U_{\mathrm{in}}(s)-s{C}_{\mathrm{a}}{P}_{\mathrm{in}}(s),$$(A.2b)

from which the deduced acoustic transfer relation is

$$\begin{array}{cc}\hfill & \underset{X_{\mathrm{in}}(s)}{\underbrace{\left[\begin{array}{cc}{P}_{\mathrm{in}}(s)\hfill & \hfill \\ U_{\mathrm{in}}(s)\hfill & \hfill \end{array}\right]}}\hfill \\ \hfill & =\underset{T_{\mathrm{a}}(s)}{\underbrace{\left[\begin{array}{cc}1\hfill & R_{\mathrm{a}}+s{M}_{\mathrm{a}}\hfill \\ s{C}_{\mathrm{a}}\hspace{0.5em}\hfill & 1+s{R}_{\mathrm{a}}{C}_{\mathrm{a}}+s^2{C}_{\mathrm{a}}{M}_{\mathrm{a}}\hfill \end{array}\right]}}\underset{X_0(s)}{\underbrace{\left[\begin{array}{cc}{P}_0(s)\hfill & \hfill \\ U_0(s)\hfill & \hfill \end{array}\right]}},\hfill \end{array}$$(A.3)

and where matrix T_a takes the following concise form

$$T_{\mathrm{a}}=\left[\begin{array}{cc}1\hfill & \alpha \hfill \\ \beta \hfill & 1+\alpha \beta \hfill \end{array}\right]\hspace{0.5em}\mathrm{with}\hspace{0.5em}\{\begin{array}{cc}\alpha (s)\hfill & =R_{\mathrm{a}}+s{M}_{\mathrm{a}},\hfill \\ \beta (s)\hfill & =s{C}_{\mathrm{a}}.\hfill \end{array}$$(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≤n≤N (see Fig. B.1) localised for ℓ∈(L_n−1,L_n) with L₀=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 $X_n=\left[\begin{array}{cc}\hfill P_n& \hfill \\ \hfill U_n& \hfill \end{array}\right]$ at ℓ=L_n to X_n−1 at ℓ=L_n−1 as (see e.g., [39])

$$\begin{array}{cc}\hfill X_{n-1}& =T_n{X}_n\hfill \\ \hfill \hspace{0.5em}\mathrm{with}\hspace{0.5em}{T}_n(s)& =\left[\begin{array}{cc}\hfill cosh\frac{s{l}_n}{c_0}& \hfill Z_{n}sinh\frac{{\mathit{sl}}_n}{c_0}\\ \hfill Z_n^{-1}sinh\frac{{\mathit{sl}}_n}{c_0}& \hfill cosh\frac{{\mathit{sl}}_n}{c_0}\end{array}\right]\hfill \end{array}$$(B.1)

for all Laplace variable $s\in {ℂ}_0^{+}=\{s\in ℂ\mathrm{s}.\mathrm{t}.\mathfrak{R}e(s)>0\}$, where Z_n=ρ₀c₀/A_n and A_n denote the characteristic impedance and the cross section area of the nth cylinder respectively. The acoustics of the complete pipe is then described by

$$X_0=T_{\mathrm{pipe}}{X}_N\hspace{0.5em}\mathrm{with}\hspace{0.5em}{T}_{\mathrm{pipe}}=T_1{T}_2\dots T_N,$$(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 $\pi r_{\mathrm{b}}^2$, 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_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 l₁ 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∈ [6, 40]), to those of a real bass trombone (F_k,k∈ [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.

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_bell=450·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_pipe) and the radiation load (Z_R) leads to the transfer relation (denoting $X_{\mathrm{out}}=\left[\begin{array}{c}{P}_{\mathrm{out}}\hfill \\ U_{\mathrm{out}}\hfill \end{array}\right]=X_N$), from (B.2) and (A.3):

$$X_{\mathrm{in}}=T_{\mathrm{tb}}{X}_{\mathrm{out}},$$(B.1a)

$$\hspace{0.5em}\mathrm{with}\hspace{0.5em}{T}_{\mathrm{tbn}}=T_{\mathrm{a}}{T}_{\mathrm{pipe}}\hspace{0.5em}\mathrm{and}\hspace{0.5em}{X}_{\mathrm{out}}=\left[\begin{array}{c}{Z}_{\mathrm{R}}\hfill \\ 1\hfill \end{array}\right]U_{\mathrm{out}}.$$(C.1b)

The input impedance Z_in=P_in/U_in is then given by

$$Z_{\mathrm{in}}=\frac{T_{\mathrm{tb}}^{11}{Z}_{\mathrm{R}}+T_{\mathrm{tb}}^{12}}{T_{\mathrm{tb}}^{21}{Z}_{\mathrm{R}}+T_{\mathrm{tb}}^{22}}·$$(C.2)

Figure C.1 compares Z_in with the measured input impedance $Z_{\mathrm{in}}^{†}$ 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_in than in $Z_{\mathrm{in}}^{†}$, 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_in and the measurement $Z_{\mathrm{in}}^{†}$.

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 $P_n^{\mathrm{l}\pm }=(P_{n-1}\pm Z_n{U}_{n-1})/2$, those at its right side are $P_n^{\mathrm{r}\pm }=(P_n\pm Z_n{U}_n)/2$, and they are related by a pure delay, called D_n (see Fig. D.1):

$$\left[\begin{array}{c}\hfill P_n^{\mathrm{r}+}\\ \hfill P_n^{\mathrm{l}-}\end{array}\right]=D_n\left[\begin{array}{c}\hfill P_{\mathrm{n}}^{\mathrm{l}+}\\ \hfill P_{\mathrm{n}}^{\mathrm{r}-}\end{array}\right]\hspace{0.5em}\mathrm{with}\hspace{0.5em}{D}_{\mathrm{n}}(s)={\mathrm{e}}^{-\frac{sd}{2c}}.$$(D.1)

Between pipes n and n+1, expressing $P_{n+1}^{\mathrm{l}+}$ and $P_n^{\mathrm{r}-}$ as functions of $P_n^{\mathrm{r}+}$ and $P_{n+1}^{\mathrm{l}-}$ 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

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

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

$$\underset{X_{\mathrm{in}}}{\underbrace{\left[\begin{array}{c}\hfill P_{\mathrm{in}}\hfill \\ \hfill U_{\mathrm{in}}\hfill \end{array}\right]}}=T_{\mathrm{a}}\left[\begin{array}{c}\hfill P_0\hfill \\ \hfill U_0\hfill \end{array}\right]=\underset{M}{\underbrace{T_{\mathrm{a}}\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill Y_1\hfill & \hfill -Y_1\hfill \end{array}\right]}}\left[\begin{array}{c}\hfill P_1^{\mathrm{l}+}\hfill \\ \hfill P_1^{\mathrm{l}-}\hfill \end{array}\right],$$(D.3)

with, from (A.3),

$$M=\left[\begin{array}{cc}\hfill 1+\alpha Y_1\hfill & \hfill 1-\alpha Y_1\hfill \\ \hfill \beta +(1+\alpha \beta )Y_1\hfill & \hfill \beta -(1+\alpha \beta )Y_1\hfill \end{array}\right].$$(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 $P_1^{\mathrm{l}+}$ as functions of U_in and $P_1^{\mathrm{l}-}$. Computations yield:

$$\begin{array}{cc}\hfill A& =\frac{1}{M_{21}},\hspace{0.5em}B=-\frac{M_{22}}{M_{21}},\hspace{0.5em}C=-\frac{\text{det}(M)}{M_{21}},\hspace{0.5em}\hfill \\ \hfill D& =\frac{M_{11}}{M_{21}},\hfill \end{array}$$(D.5)

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

At the trombone output, P_N=Z_RU_N so that signals $P_N^{\mathrm{r}\pm }=(P_N\pm Z_N{U}_N)/2$ equal (Z_RU_N±Z_NU_N)/2. Then, the reflection function $R=P_N^{\mathrm{r}-}/P_N^{\mathrm{r}+}$ in Figure 14 is given by

$$R(s)=\frac{Z_{\mathrm{R}}(s)-Z_N}{Z_{\mathrm{R}}(s)+Z_N},$$(D.6a)

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

$$T(s)=\frac{2Z_{\mathrm{R}}(s)}{Z_{\mathrm{R}}(s)+Z_N},$$(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:

$$\{\begin{array}{cc}{P}_n=P_{n+1}^{l+}+P_{n+1}^{l-}\hfill & \hfill \\ Z_{n+1}{U}_n=P_{n+1}^{l+}-P_{n+1}^{l-}.\hfill & \hfill \end{array}$$(E.1)

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

$$\{\begin{array}{cc}{P}_n=P_n^{r+}+P_n^{r-}\hfill & \hfill \\ Z_n{U}_n=P_n^{r+}-P_n^{r-}.\hfill & \hfill \end{array}$$(E.2)

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

$$\begin{array}{cc}\hfill P_n^{r-}& =P_n-P_n^{r+}\hfill \\ \hfill & =P_{n+1}^{l+}+P_{n+1}^{l-}-P_n^{r+}\hfill \\ \hfill & =Z_{n+1}{U}_n+2P_{n+1}^{l-}-P_n^{r+}\hfill \\ \hfill & =\frac{Z_{n+1}}{Z_n}\left(P_n^{r+}-P_n^{r-}\right)+2P_{n+1}^{l-}-P_n^{r+},\hfill \end{array}$$

by regrouping the following terms,

$$\begin{array}{cc}\hfill \iff P_n^{r-}& \left(1+\frac{Z_{n+1}}{Z_n}\right)=\left(\frac{Z_{n+1}}{Z_n}-1\right)P_n^{r+}+2P_{n+1}^{l-}\hfill \\ \hfill \iff P_n^{r-}& =\frac{Z_{n+1}-Z_n}{Z_{n+1}+Z_n}{P}_n^{r+}+\frac{2Z_n}{Z_{n+1}+Z_n}{P}_{n+1}^{l-}.\hfill \end{array}$$

As a result, a k_n coefficient is expressed,

$$P_n^{r-}=k_n{P}_n^{r+}+\left(1-k_n\right)P_{n+1}^{l-},$$(E.3)

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

Appendix F Optimisation

The length of the resonator L_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) $\frac{c_0}{4L_{\mathrm{tot}}}$. 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_tot=561 (this approximation gives f₁=111.92 Hz, which corresponds to a relative error of 0.018%). As N_tot, N_bell must be an integer, i.e., $2f_s{L}_{\mathrm{bell}}/c_0\in \mathbb{N}$. Moreover, the optimisation algorithm is based on a target of a 15-radius model. For this reason, N_bell must therefore be an integer and a multiple of 15. These conditions led us to choose N_bell=30×15=450. Table F.1 summarises the values chosen to the algorithmic optimisation.

References

All Tables

All Figures