© V. Fréour et al., Published by EDP Sciences, 2022

1 Introduction

Physical modeling is a valuable strategy in order to study numerically the response of a musical instruments [1]. In the case of brass instruments, various studies have focused on the modeling and simulation of the brass player and his/her instrument [2–10]. Some techniques such as numerical continuation are particularly relevant in order to compute periodic solution branches of wind instrument models [11]. Such a technique has been applied to woodwind instruments [12] and brass instruments [13, 14], and recently to the objective comparison of trumpets, through the extraction of descriptors calculated from the solution branches obtained by numerical continuation [15].

Nevertheless, the question of the values of the reed parameters (lip parameters for brass instruments) remains particularly sensitive. In brass instruments, the lip-reed system consists in a sophisticated biological system made of several layers of biological tissues, whose mechanical properties can be controlled by the player by activating various orofacial muscles. The question of the equivalent parameters of a biomechanical lip-reed model is made complicated by the difficulty in estimating these parameters experimentally, as well as by the uncertainty on the variations of these parameters in the course of a given musical task.

Some experimental studies on artificial player systems [6, 16], or human performers [2, 17–20], have shed some light on the dynamics of the lip-reed valve and proposed some values for the mechanical parameters of the lips, which are derived from frequency response measurements. Other strategies, based on numerical optimization techniques applied to the internal pressure, have been proposed in woodwind instruments in order to derive the mechanical parameters of a reed model from experimental measurements [21, 22]. In physical models of brass instruments, the mechanical parameters of the lip-reed model are often assumed constant for a given note. Velut et al. [23] proposed a review on the values used in different studies focused primarily on the trombone.

In this context, the question of matching the output of a physical model with experimental measurements becomes an important topic of interest. Furthermore, the strategy of numerical continuation applied to a physical model of brass instrument offers a global view of the response of the system (evolution of the state of the system with respect to one parameter of the model), likely to be used as a basis for fitting the model with experimental observations. The idea we propose in this study then comes from a dynamical system approach, where the modeled system can be augmented with additional equations that impose some constraints on the outputs of the model. In this article, these constraints are established from experimental observations on a musician. Adding these constraints then requires relaxation of some parameters of the model. The parameters which values and evolution are relatively uncertain along a periodic branch will be prioritized: the mechanical parameters of the lip model in the case of brass instruments.

This approach has two great benefits: 1) it enables identification of the evolution of the lip parameter values along a solution branch, based on some constraints established from experimental observations. This then provides information about the “gesture” required by the player’s model to fulfill the additional constraints. 2) It provides new information about the instrument, by allowing the variation of lip parameters required to achieve a given task (defined by the constraints) to be quantified. This is then suitable for instrument comparison, which is one of the motivations behind this work.

In this paper, application of constrained continuation to a physical model of the trumpet is presented, where the constraints are established from measurements on a trumpet player: frequency and/or amplitude of the sound produced during a crescendo. This paper is organized as follows: the physical model and basic continuation results are recalled in Section 2, measurements on a trumpet player are described in Section 3, the constrained continuation approach is then detailed in Section 4. Results and conclusions are presented in Sections 5 and 6.

2 Numerical continuation of physical model of brass instruments

2.1 Model and dimensionless equations

We consider a one-dimensional lip model, coupled to the resonator impedance described by a series of complex modes similar to what is proposed in [15]. The coupling between the mechanical oscillator and the acoustic resonator is achieved by a stationary Bernoulli flow equation, considering turbulent mixing in the mouthpiece with no pressure recovery. The mechanical and acoustic equations are given in System 1, where y is the vertical lip position (y₀ is the lip position at rest), ω_l, Q_l, μ_l and b the lip mechanical parameters (resonance angular frequency, quality factor, mass per surface area and lip opening width respectively), s_k and C_k with k ∈ [1, N] the modal parameters (poles and residues respectively) of the N resonances of the acoustic impedance of the instrument, Z_c the characteristic impedance, u the volume flow, p the downstream pressure at the input of the instrument (in the mouthpiece), and p₀ the upstream (mouth) static pressure:$$\{\begin{array}{l}\ddot{y}\left(t\right)+\frac{{\omega }_l}{Q_l}\dot{y}\left(t\right)+{{\omega }_l}^2\left(y\left(t\right)-y_0\right)=\frac{1}{{\mu }_l}\left(p_0-p\left(t\right)\right)\\ {\dot{p}}_k\left(t\right)=Z_c{C}_{k}u\left(t\right)+s_k{p}_k\left(t\right),\forall k\in \left[1,N\right],\\ \end{array}$$(1)with $p\left(t\right)=2{\sum }_{k=1}^N\mathrm{}\mathfrak{R}\left(p_k\right(t\left)\right)$ and $u=\sqrt{\frac{2|p_0-p|}{\rho }}b\cdot \mathrm{sign}(p_0-p)\cdot \theta \left(y\right)$, where $\theta \left(y\right)=\frac{\left|y\right|+y}{2}$, b is the lip width and ρ is the air density.

The case of a negative opening of the lips is managed by introducing the function θ(y) which enforces u = 0 if y < 0. The modal parameters of the N modes of the impedance are extracted from the measured input impedance, corrected to 27 °C [24], using the high resolution method ESPRIT [25]. We remind that nonlinear propagation phenomena that may originate from large pressure levels (at high mouth pressure values) are not taken into account in this model.

Figure 1 represents a reconstruction of the input impedance of a B♭ trumpet in open fingering (no valve pressed) from the superposition of 11 complex modes, against the measured input impedance. Despite some discrepancies at some anti-resonances, this representation shows overall a very satisfactory match in both magnitude and phase between the two curves. The corresponding poles and residues values are given in Table 1.

Figure 1 Measured input impedance (dashed black) and reconstructed input impedance from 11 complex modes (red) for a B♭ trumpet with open fingering.

The choice is made to work with the Asymptotic Numerical Method (ANM) implemented in the software MANLAB [26]. Recently, this method has been associated with the Harmonic Balance Method (HBM) for the search of periodic solutions of oscillating systems [27]. One requirement of MANLAB relies on the recast of nonlinearities of the model into, at most, quadratic nonlinearities. This system of equations can be made dimensionless by introducing the following variables:$$(\begin{array}{l}x=\frac{y}{y_0}\\ P_M={\mu }_l{{\omega }_l}^2{y}_0\\ \gamma =\frac{p_0}{P_M}\\ \tilde{p}=\frac{p}{P_M}\\ {\tilde{R}}_k=\frac{R_k}{P_M}\\ {\tilde{I}}_k=\frac{I_k}{P_M}\\ \tilde{u}=u\frac{Z_c}{P_M}\\ \tilde{v}=\frac{v}{\sqrt{2P_M/\rho }}\\ \zeta =Z_{c}b{y}_0\sqrt{\frac{2}{\rho P_M}}\\ \tilde{t}=t\mathfrak{I}\left(s_1\right)\\ {\tilde{\omega }}_l=\frac{{\omega }_l}{\mathfrak{I}\left(s_1\right)}\\ {\tilde{C}}_k=\frac{C_k}{\mathfrak{I}\left(s_1\right)}\\ {\tilde{s}}_k=\frac{s_k}{\mathfrak{I}\left(s_1\right)},\end{array}$$(2)with R_k and I_k the real and imaginary parts of the pressure components p_k with k ∈ [1, N]. The complete quadratic dimensionless model can then be written as follows:$$(\begin{array}{l}{\widetilde{\stackrel{̇}{R}}}_k=\mathfrak{R}\left({\tilde{C}}_k\right)\tilde{u}+\mathfrak{R}\left({\tilde{s}}_k\right){\tilde{R}}_k-\mathfrak{I}\left({\tilde{s}}_k\right){\tilde{I}}_k,\forall k\in [1,N]\\ {\widetilde{\stackrel{̇}{I}}}_k=\mathfrak{I}\left({\tilde{C}}_k\right)\tilde{u}+\mathfrak{I}\left({\tilde{s}}_k\right){\tilde{R}}_k+\mathfrak{R}\left({\tilde{s}}_k\right){\tilde{I}}_k,\forall k\in [1,N]\\ \widetilde{\stackrel{̇}{x}}={\tilde{\omega }}_l\tilde{z}\\ \widetilde{\stackrel{̇}{z}}={\tilde{\omega }}_l(1-x-\frac{1}{Q_l}z+\gamma -\tilde{p}),\end{array}$$(3)with the auxilary equations:$$(\begin{array}{l}0=2\sum _{k=1}^N\mathrm{}{\tilde{R}}_k-\tilde{p}\\ 0=x^2+{ϵ}_x-s^2\\ 0=\gamma -\tilde{p}-\tilde{v}w\\ 0={\tilde{v}}^2+{ϵ}_v-w^2\\ 0=\zeta \frac{(s+x)}{2}\tilde{v}-\tilde{u},\end{array}$$(4)with 0 < ϵ_v ≪ 1 and 0 < ϵ_x ≪ 1, the regularization constants such as ϵ_v = ϵ_x = 10⁻³. The reader is invited to consult [15] for more details.

2.2 Contiuation by ANM

The dynamical system described by equations (3) and (4) can be analyzed by numerical continuation, especially using the Asymptotic Numerical Method (ANM) [28] implemented in the software MANLAB [26]. This method is based on the expansion of the solutions under the form of truncated Taylor series, providing analytical approximate formulations of the branch of solution. Associated with the HBM, the ANM allows for search of periodic solutions of the dynamical system, the unknowns being the Fourier coefficients of each variable [29, 30] and the oscillation frequency. For more details, the reader is invited to refer to the specific literature on the subject [31, 32].

Figure 2 shows the results of continuation of the system described by equations (3) and (4): it represents the bifurcation diagram (peak to peak amplitude of p with respect to p₀) of the branch of periodic solution corresponding to a B♭4 . This result is obtained with the lip parameters given in Table 2.

Figure 2 Bifurcation diagram (peak-to-peak amplitude of p with respect to p0) of the periodic branch of solution for a B♭4 (470 Hz). The dotted line indicates the unstable part of the solution branch, while the solid line indicates the stable part of the branch. ○: Hopf bifurcation (pH point), ∇: fold, Δ: |p| at the Hopf bifurcation, □: |p| when p0 = 5 kPa.

The lip natural frequency is set by Linear Stability Analysis (LSA) [15, 23] to f_l = ω_l/2π = 382.18 Hz, so that the playing frequency is closed to a B♭4 (f₀ ≃ 470 Hz).

The bifurcation diagram obtained in Figure 2 sheds light on some behaviors that brass players are familiar with, and that have been described in previous studies [13, 15]. These features include: 1) an inverse bifurcation and Hopf point around p₀ = 2.3 kPa; 2) an unstable section oriented towards the left until it reaches a fold; 3) at the fold, a change of stability towards a stable branch where the amplitude of the internal pressure p increases with the mouth pressure p₀. Note that for the lip parameters chosen (Table 2), the order of magnitude of p₀ is coherent with experimental observations in the trumpets [33].

3 Measurements on trumpet player

Whereas the numerical results obtained by continuation seem coherent with the behavior of a brass instrument, it is then relevant to compare more accurately these numerical outputs with some experimental data obtained on brass players. For this purpose, experimental bifurcation diagrams were collected by measurements conducted on a trumpet player using the trumpet whose impedance was measured in order to implement the model described in the previous section. For these measurements, the tuning slide was set to its default setting, as in the impedance measurements, in order to use exactly the same resonator in both conditions.

3.1 Experimental setup and protocol

A National Instrument acquisition card and two ENDEVCO pressure sensors (model 8510B) are used to measure the upstream (mouth) pressure ${p_0}_e$ and downstream (mouthpiece) pressure p_e respectively (the sampling rate is set to f_s = 51.2 kHz). The mouthpiece sensor is set up so that the extremity is mounted flush in the mouthpiece cup. The mouth sensor is connected to a small tube that the musician is asked to insert in the mouth. An illustration of the experimental setup is shown in Figure 3.

Figure 3 Trumpet player during experiments: two miniature pressure sensors are used to measure the upstream (mouth) and downstream (mouthpiece) pressures during performance.

In order to collect experimental data comparable as much as possible with a bifurcation diagram obtained numerically, the player is instructed to produce a slow crescendo-decrescendo from ppp to ff (see Fig. 4). The objective is to be as close as possible to the hypothesis of a quasi-static variation of the mouth pressure. The player is then instructed to avoid using the tongue at the note onset (no tonguing at the attack) and to make the crescendo-decrescendo as regular and as slow as possible (recommended tempo: 70 bpm). No particular instructions were given to the players with regards to the stability of the fundamental frequency.

Figure 4 Musical task asked to the player. Recommended tempo: 70 bpm. Note that the musical staff shows written pitch C4, a tone higher than sounding pitch B♭4.

The collected data are treated as follows: a moving window is applied to the data (window size 20 ms with no overlap). The peak-to-peak amplitude of the mouthpiece pressure |p_e| is extracted for each window, as well as the mean of the low-pass filtered mouth pressure ${{\overline{p}}_0}_e$. The duration of the crescendo and decrescendo phases are then extracted from the peak-to-peak envelope (Fig. 5). In order to impose a relatively low variability between the tasks used for analysis, only the maneuvers where the duration of the crescendo and decrescendo are within 3.5 and 4.5 seconds are considered. Out of eight repetitions of the task, three maneuvers are then discarded. The time-evolution of |p_e| and ${{\overline{p}}_0}_e$ for the five selected repetitions of the crescendo-decrescendo task is shown in Figure 6.

Figure 5 Crescendo-decrescendo maneuvers performed by the trumpet player. Mouthpiece pressure (gray), mouth pressure (blue) and peak-to-peak mouthpiece pressure envelope (red). The crosses indicate the onset, maximum amplitude and sound extinction locations. The green area corresponds to the crescendo phase, the magenta area to the decrescendo phase.

Figure 6 Five crescendo-decrescendo maneuvers performed by the trumpet player. Top: mouth pressured measured (black) and low-pass filtered (red). Bottom: mouthpiece pressure (black) and peak-to-peak envelope (red).

The experimental bifurcation diagrams (|p_e| with respect to ${{\overline{p}}_0}_e$, and f₀ with respect to ${{\overline{p}}_0}_e$) associated with the data represented in Figure 6 are then represented in Figure 7.

Figure 7 Experimental bifurcation diagrams obtained from one player playing a Bb4 with slow crescendo-decrescendo. Top: |pe| with respect to ${{\overline{p}}_0}_e$. Bottom: f0 with respect to ${{\overline{p}}_0}_e$.

The results of Figure 7 show some common features with the numerical diagram of Figure 2: 1) a minimum blowing pressure in the same pressure range as calculated by continuation, 2) an inverse bifurcation that appears under the form of an hysteresis in the experimental data, 3) a relatively linear increase of |p_e| with ${{\overline{p}}_0}_e$ as observed in numerical results, 4) relatively small variations of f₀ along the maneuver. These results then support our hypothesis regarding the ability of the 1-mode outward-striking lip model to generate behaviors that seem relatively coherent with human behaviors.

Figure 8 shows the five bifurcation diagrams of Figure 7 overlapped on the same plot. This representation highlights the relatively high repeatability of the player across the five iterations. In the case of |p_e|, the average of the linear regression coefficients of each bifurcation diagram is calculated for ${{\overline{p}}_0}_e>1.5$ kPa (plotted as a red line in Fig. 8, top). In the case of f₀, the average of f₀ values for ${{\overline{p}}_0}_e>1.5$ kPa is calculated (plotted as a red line in Fig. 8, bottom).

Figure 8 Experimental bifurcation diagrams obtained from one player playing a Bb4 with slow crescendo-decrescendo (same data as in Fig. 7). Top: |pe| with respect to ${{\overline{p}}_0}_e$. Bottom: f0 with respect to ${{\overline{p}}_0}_e$. The top red line is obtained by averaging the linear regression coefficients of each diagram for ${{\overline{p}}_0}_e>1.5$ kPa. The bottom red line is obtained by averaging f0 values for ${{\overline{p}}_0}_e>1.5$ kPa.

3.2 Comparison with numerical solutions

Although some important similarities can be found between the bifurcation diagrams obtained numerically (Fig. 2) and experimentally (Fig. 8), it seems very difficult, with constant lip parameters, to generate a numerical bifurcation diagram with a stable branch of similar slope as the measured diagrams. Figure 9 shows results of 100 bifurcation diagrams calculated from random variations of the lip parameters. More precisely, the parameters are varied by ±50% around the values given in Table 2 for Q_l and b, and by ±90% for μ_l and y₀. For each calculation, the lip natural frequency f_l is obtained from LSA. In this figure, the amplitude of p is estimated by the L² norm, which choices is motivated by the mathematical form of this norm, as it will be explained in the next section.

Figure 9 Left: 100 bifurcation diagrams obtained from random variations of the lip parameters around the values given in Table 2, the plain red lines are the targets obtained from human performance. The dashed line on the bottom plot corresponds to the modified target. Top right: mouth pressure at Hopf bifurcation and slope calculated from 100 linear regressions performed on the stable sections of the 100 bifurcation diagrams, the red diamond corresponds to the red line in the left plot. Bottom right: Δf0 and f0 obtained from the 100 bifurcation diagrams, the plain red diamond correspond to the red line in the left plot. The empty red diamond corresponds to the modified target.

Linear regression applied to the stable part of the calculated diagrams (${\Vert p\Vert }_{L^2}$ w.r.t. p₀) allows extraction of a slope value for each diagram. On the top right plot of Figure 9, these slope values and the mouth pressure at the Hopf point, p_H, are represented for each diagram, along with the slope and intercept with the horizontal axis of the red line obtained from experimental data and represented on the top left plot of Figure 9. This intercept with the horizontal axis corresponds indeed to the Hopf point of the target solution. Despite large variations of the lip parameters across calculations, the slopes of the calculated stable branches are lower than the slope of the experimental diagrams. Furthermore, on the bottom right plot, f₀ values, as well as the variations of f₀ along the solution branch (Δf₀ = max(f₀₎ − min(f₀)), are represented for all diagrams, along with the values obtained from the experimental data. It can be noticed from Figure 8 that Δf₀ is associated with an overall decrease of f₀ with p₀. The calculated fundamental frequencies are significantly above the values observed experimentally, which is usually expected with an outward striking lip model [34].

These observations suggest that it might be relatively difficult to find a combination of lip parameters allowing experimental diagrams to be replicated numerically. In the following, an approach is proposed to study if a numerical bifurcation diagram closer to the experimental one can be reached if some parameter values are allowed to vary with respect to the bifurcation parameter p₀.

More precisely, by enabling some lip parameters to vary along the solution branch, and by imposing some constraints to the solution, the periodic solutions of this new extended system are derived, as well as the evolution of the relaxed parameters along the solution branch.

4 Constrained continuation

In order to generate numerical diagrams with similar features as the one obtained with the trumpet player, some constraints should then be defined from the bifurcation diagrams obtained experimentally.

4.1 Constraints

Two constraints are defined (red lines of Fig. 9), which represent the result of the musician’s action. The first constraint concerns the relationship between the L² norm of $\tilde{p}$ (${\Vert \tilde{p}\Vert }_{L^2}$) and the dimensionless mouth pressure γ, along the stable part of the branch of the bifurcation diagram:$${\Vert \tilde{p}\Vert }_{L^2}=\mathrm{S\gamma }+I,$$(5)where γ = p₀/P_M is the dimensionless mouth pressure with $P_{\mathrm{M}}={\mu }_l{{\tilde{\omega }}_l}^2{y}_0$, $S$ and $I$ are constant values, and ${\Vert \tilde{p}\Vert }_{L^2}=2{\Vert {\sum }_{k=1}^N\mathrm{}\mathfrak{R}\left({\tilde{p}}_k\right)\Vert }_{L^2}=2\sqrt{{\sum }_{k=1}^N\mathrm{}\mathfrak{R}({\tilde{p}}_k{)}^2}$. The value of ${\tilde{\omega }}_l$ is calculated by LSA as in Section 2 such as $f_l = {\tilde{\omega }}_l/2\pi = 382.18 \mathrm{Hz}$. The choice of the L² norm is motivated by the quadratic form of equation (5) w.r.t. the unknowns when elevated at the power of 2, quadratic nonlinear equations being a prerequisite of the ANM.

The second constraint simply writes as follows:$$\tilde{\omega }=F,$$(6)where F is a constant value.

4.2 Relaxed parameters and new system to solve

Equations (5) and (6) are added to the original system (Systems 3 and 4). Adding two equations to the system has to be balanced by the introduction of two new unknowns. Q_l and $\zeta =Z_{c}b{y}_0\sqrt{\frac{2}{\rho P_M}}$ are chosen as the relaxed parameters. The parameter ζ can be interpreted as an “embouchure” parameter as it depends only on lip and mouthpiece parameters, and in particular on the lip width b that appears exclusively in the expression of ζ.

This requires the system of equations to be recast in order to preserve the quadratic property of the model. The main equations are unchanged compared to equation (3):$$(\begin{array}{l}{\widetilde{\stackrel{̇}{R}}}_k=\mathfrak{R}\left({\tilde{C}}_k\right)\tilde{u}+\mathfrak{R}\left({\tilde{s}}_k\right){\tilde{R}}_k-\mathfrak{I}\left({\tilde{s}}_k\right){\tilde{I}}_k,\forall k\in \left[1,N\right]\\ {\widetilde{\stackrel{̇}{I}}}_k=\mathfrak{I}\left({\tilde{C}}_k\right)\tilde{u}+\mathfrak{I}\left({\tilde{s}}_k\right){\tilde{R}}_k+\mathfrak{R}\left({\tilde{s}}_k\right){\tilde{I}}_k,\forall k\in \left[1,N\right]\\ \widetilde{\stackrel{̇}{x}}={\tilde{\omega }}_l\tilde{z}\\ \widetilde{\stackrel{̇}{z}}={\tilde{\omega }}_l\left(1-x-q_{l}z+\gamma -\tilde{p}\right),\end{array}$$(7)with q_l = 1/Q_l. Two equations are introduced to account for the constraints:$$(\begin{array}{l}0=(\mathrm{S\gamma }+I{)}^2-\sum _{k=1}^N\mathrm{}{{\tilde{R}}_k}^2\\ 0=2\pi F-\tilde{\omega }.\end{array}$$(8)

One additional auxiliary variable m, as well as the corresponding equation are introduced in order to preserve the quadratic property of the model. Indeed, since ζ is now an unknown, the last equation of System 4 is no longer quadratic. The system of auxiliary equations then becomes:$$(\begin{array}{l}0=2\sum _{k=1}^N\mathrm{}{\tilde{R}}_k-\tilde{p}\\ 0=x^2+{ϵ}_x-s^2\\ 0=\gamma -\tilde{p}-\tilde{v}w\\ 0={\tilde{v}}^2+{ϵ}_v-w^2\\ 0=\frac{\left(s+x\right)}{2}\tilde{v}-m\\ 0=\zeta m-\tilde{u}.\\ \end{array}$$(9)

The system formed by equations (7), (8) and (9) is dimensionless and quadratic. It is then compatible with the application of the Asymptotic Numerical Method (ANM) using MANLAB.

The following vector of unknowns is then considered:$$X=\left[R_1,I_1,R_2,I_2,\dots ,R_N,I_N,x,z,\tilde{p},s,\tilde{v},w,m,\tilde{u},\zeta ,q_l,\tilde{\omega }\right].$$(10)

5 Results of constrained continuation

5.1 Values of the constraints

The two constraints are defined based on the experimental observations of Section 3:

1. The values of S and I are calculated from the average of the linear regressions applied to the musician’s data: S = 1.5461 and I = −0.9724. In Figure 9, the red line crosses the beam of blue diagrams for ${\Vert p\Vert }_{L^2}$ (top left). This suggests that it should be possible to find a solution with relaxed parameters that will follow the red constraint.

2. In Figure 9, although the red line crosses the beam of blue diagrams for ${\Vert p\Vert }_{L^2}$ (top left), it is not the case for f₀ (bottom left), where the plain red line that comes from experimental results does not share any point with the numerical results. This mismatch is due to the outward striking lip model which, by construction, oscillates at frequencies slightly above human lips. This observation implies that no solution with relaxed parameter should be able to reach measured f₀ values. The value of F is then set to 480 Hz, slightly above the averaged value measured on the player, as shown in Figure 9. This frequency remains in the vicinity of B♭4: 42.8 cents above the theoretical B♭4 at 468.28 Hz (with 442 Hz A4 tuning reference).

5.2 Initial conditions

In order to initialize the calculation with constraints, an initial point should be defined, preferably not too far from the target solution branch. To this end, among the 100 bifurcation diagrams generated in Section 3, one diagram, and the associated lip parameters, is selected, so that it crosses the target diagram within the p₀ range of interest (between 1 kPa and 4 kPa). This diagram is represented in Figure 10 and the associated lip parameters are given in Table 3. A point near the intersection with the target (around p₀ = 2.5 kPa) is extracted and is used as the initial point for the constrained continuation. Priority is given to the intersection with the ${\Vert p\Vert }_{L^2}$ target since, after several trials, it appears to be more critical than f₀ for the initialization of the calculation and for the convergence towards the constrained solution.

Figure 10 Initial bifurcation diagram (blue) and target diagram (dashed red).

5.3 Results

Figure 11 shows the results of continuation with the constraints defined above. It can be seen that the constrained continuation can be performed successfully over a large range of p₀ (up to about 5 kPa). The variations of ζ ans Q_l can be observed along the solution branch. Within the p₀ range covered by the human player (up to about 3 kPa), these variations are such as 0.04 < ζ < 0.1 and 2.9 < Q_l < 5. For Q_l, this interval falls within expected values for a brass player lip Q factors [23]. For ζ, this is equivalent to a variation by about 60% of the maximum value. Among the lip parameters, the lip width b is the only parameter that appears only in the expression of ζ, and not in the definition of P_M (System 2). Furthermore, b is proportional to ζ, assuming other parameters constant. The variations of ζ can then be interpreted as variations of b by 60% of its maximum value, which we believe to be quite realistic: it is physically acceptable to consider that a player may vary the lip width by such amount within a crescendo-decrescendo maneuver, although this should be eventually confirmed by experimental measurements if possible.

Figure 11 Bifurcation diagram with constraints and evolution of the relaxed parameters ζ and Ql with respect to p0. The target diagram is represented by the red dashed line.

Figures 12 (left plot) shows the waveform of p, obtained from experimental measurements with the trumpet player, for three values of p₀: 1 kPa, 2 kPa and 3 kPa. In addition, the waveform of p obtained from the bifurcation diagram of Figure 11 is represented for the same p₀ values in Figure 12 (right plot). Despite some differences in peak-to-peak amplitudes (${\Vert p\Vert }_{L^2}$ is not the peak-to-peak amplitude of p), both figures show a clear, and relatively similar, evolution of the waveform amplitude and shape with increase in p₀. This observation confirms the relevance of the model and of the constrained continuation in reproducing the behavior monitored on a human player.

Figure 12 Waveform of p obtained from experimental measurements on a trumpet player (left) and from constrained continuation (right), at p0 = 1 kPa (solid line), p0 = 2 kPa (dashed-dotted line) and p0 = 3 kPa (dashed line).

To illustrate this statement, two-second duration sounds corresponding to the waveforms of Figure 12 are generated (after normalization and application of a linear enveloppe at attack and release), from the experimental measurements for p₀ = 1 kPa , p₀ = 2 kPa , and p₀ = 3 kPa , as well as from the bifurcation diagram for p₀ = 1 kPa , p₀ = 2 kPa , and p₀ = 3 kPa . Despite some differences in intonation between the model and the experiment (as expected with the outward striking lip model), these sound excerpts confirm a similar evolution of the timbre with p₀ between the model and the musician’s recording.

In order to further investigate the nature of the calculated solution, the constrained bifurcation diagram, along with a number of bifurcation diagrams where the lip parameters are set constant and taken at various points of the constraint solution, are represented in Figure 13. The intersection points (circle markers) highlight the path followed by the constraint solution across constant-parameters solution branches. Note that to be considered as an intersection point with a constant-parameter diagram, both f₀ and ${\Vert p\Vert }_{L^2}$ diagrams should cross a same constant-parameter diagram. In this figure, it appears that all the calculated diagrams are characterized by inverse bifurcations, and that below a certain p₀ value (around 1.4 kPa), the constrained solution crosses unstable branches only. In other words, the constrained solution is formed by an ensemble of points that belong to bifurcation diagrams which emerge from inverse Hopf bifurcations. Some of these points are then stable and other are unstable, which makes the constrained solution stable above p₀ = 1.4 kPa, and unstable below this value. Note also that the constrained solution thus does not emerge from any Hopf bifurcation, especially as it is produced through the joint modification of three parameters (ζ, Q_l and p₀) while a Hopf bifurcation is defined trough the variation of a single parameter (p₀ in our case). That being said, the question whether this stability result depends on the initial conditions (lip parameters at the initial calculation point) remains to be clarified: it is possible that another set of lip parameter values at the initial calculation point will produce a different stability result.

Figure 13 Bifurcation diagram with constraints (blue), overlapped with constant-parameter bifurcation diagrams where ζ and Ql are set to values obtained from constrained continuation. Circle markers indicate intersection points between the diagram with constraints and the constant-parameters diagrams.

6 Conclusions

In this article, a method of continuation where constraints are included in the system of equations, and parameters of the model are relaxed, is proposed in order to force the model to follow a solution branch close to a human performance. Contrary to an optimization strategy, this method provides the solution of the constrained problem without any need of defining a cost function nor exploring the parameter space. Within some limitations regarding the fundamental frequency, that we may assume to be linked to the properties of the outward striking lip model, this method was successfully applied to constraints established from measurements on a trumpet player. The obtained trajectories of the relaxed parameters reveal significant but realistic variations along the solution branch, which confirms the great potential of the one-degree-of-freedom lip model in producing results close to human performances, as well as the importance to make lip model parameters vary to produce these outputs. Nevertheless, the obtained results also show some limitations of the present model in producing oscillations in the same range as human players, particularly a stable branch down to ${\Vert p\Vert }_{L^2}=0$. Modifying the lip model parameters that define the initial conditions, may result in different behaviors (e.g. direct bifurcations), then allowing to compute stable constrained solutions down to lower pressure values. Moreover, applying the constraint from a given point of the initial bifurcation diagram (from the fold point for instance) may contribute to generate a constrained solution closer to experimental bifurcation diagrams. Closer investigations at the influence of the initial conditions on the constrained continuation results, as well as at the possibility to apply the constraints from specific landmarks of an initial solution, should be the objects of future investigations.

In the future, we also plan to apply this method to more sophisticated lip models, although uncertainties on some parameter values of the model (second mode, contact parameters) should be addressed carefully. More advanced constraints should also be defined, in order to attempt a better match with experimental observations.

Regarding potential applications, this method opens some interesting perspectives for the comparison of musical instruments, by providing new indicators related to the control of sound production. For instance, the variations of the lip parameters along the solution branch may provide some useful indications on corrections needed at the level of the embouchure in order to produce a constant-pitch crescendo. It may then bring some new basis for objective comparisons of brass instruments using physical modeling and numerical continuation.

Data availability statement

This article includes audio files embedded in the article. The sound files associated with this article are also available in https://medihal.archives-ouvertes.fr/hal-03545981 [35].

Conflict of interest

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

Acknowledgments

The authors would like to thank the Yamaha employees who participated in the trumpet playing measurements.

References

All Tables

All Figures

	Figure 1 Measured input impedance (dashed black) and reconstructed input impedance from 11 complex modes (red) for a B♭ trumpet with open fingering.
In the text

	Figure 2 Bifurcation diagram (peak-to-peak amplitude of p with respect to p0) of the periodic branch of solution for a B♭4 (470 Hz). The dotted line indicates the unstable part of the solution branch, while the solid line indicates the stable part of the branch. ○: Hopf bifurcation (pH point), ∇: fold, Δ: |p| at the Hopf bifurcation, □: |p| when p0 = 5 kPa.
In the text

	Figure 3 Trumpet player during experiments: two miniature pressure sensors are used to measure the upstream (mouth) and downstream (mouthpiece) pressures during performance.
In the text

	Figure 4 Musical task asked to the player. Recommended tempo: 70 bpm. Note that the musical staff shows written pitch C4, a tone higher than sounding pitch B♭4.
In the text

	Figure 5 Crescendo-decrescendo maneuvers performed by the trumpet player. Mouthpiece pressure (gray), mouth pressure (blue) and peak-to-peak mouthpiece pressure envelope (red). The crosses indicate the onset, maximum amplitude and sound extinction locations. The green area corresponds to the crescendo phase, the magenta area to the decrescendo phase.
In the text

	Figure 6 Five crescendo-decrescendo maneuvers performed by the trumpet player. Top: mouth pressured measured (black) and low-pass filtered (red). Bottom: mouthpiece pressure (black) and peak-to-peak envelope (red).
In the text

	Figure 7 Experimental bifurcation diagrams obtained from one player playing a Bb4 with slow crescendo-decrescendo. Top: |pe| with respect to ${{\overline{p}}_0}_e$. Bottom: f0 with respect to ${{\overline{p}}_0}_e$.
In the text

Figure 8 Experimental bifurcation diagrams obtained from one player playing a Bb4 with slow crescendo-decrescendo (same data as in Fig. 7). Top: |pe| with respect to ${{\overline{p}}_0}_e$. Bottom: f0 with respect to ${{\overline{p}}_0}_e$. The top red line is obtained by averaging the linear regression coefficients of each diagram for ${{\overline{p}}_0}_e>1.5$ kPa. The bottom red line is obtained by averaging f0 values for ${{\overline{p}}_0}_e>1.5$ kPa.

In the text

Figure 9 Left: 100 bifurcation diagrams obtained from random variations of the lip parameters around the values given in Table 2, the plain red lines are the targets obtained from human performance. The dashed line on the bottom plot corresponds to the modified target. Top right: mouth pressure at Hopf bifurcation and slope calculated from 100 linear regressions performed on the stable sections of the 100 bifurcation diagrams, the red diamond corresponds to the red line in the left plot. Bottom right: Δf0 and f0 obtained from the 100 bifurcation diagrams, the plain red diamond correspond to the red line in the left plot. The empty red diamond corresponds to the modified target.

In the text

	Figure 10 Initial bifurcation diagram (blue) and target diagram (dashed red).
In the text

	Figure 11 Bifurcation diagram with constraints and evolution of the relaxed parameters ζ and Ql with respect to p0. The target diagram is represented by the red dashed line.
In the text

	Figure 12 Waveform of p obtained from experimental measurements on a trumpet player (left) and from constrained continuation (right), at p0 = 1 kPa (solid line), p0 = 2 kPa (dashed-dotted line) and p0 = 3 kPa (dashed line).
In the text

	Figure 13 Bifurcation diagram with constraints (blue), overlapped with constant-parameter bifurcation diagrams where ζ and Ql are set to values obtained from constrained continuation. Circle markers indicate intersection points between the diagram with constraints and the constant-parameters diagrams.
In the text