J.-B Doc

C Vergez

J Hannebicq

The objective of this work is to estimate by inverse problem lip parameters values of trumpet model so that the oscillation thresholds for successive playing registers occur for the same blowing pressure as the one measured on several trumpet players. The lips vibration is modeled through an oscillator including unknown parameters such as: resonance frequency, quality factor, surface mass, stiffness and opening at rest of the lips. The oscillation threshold is calculated through linear stability analysis of the outward-striking model including the nonlinear coupling with the bore of the trumpet.

It appears that many combinations of parameter values are suitable to obtain the same blowing pressure at threshold as in the experiments. According to the analysis of the possible parameter values, some hypotheses are formulated about the playing strategies used by the trumpeter to select the different registers of the instrument.

In addition to the lips resonance frequency, the control of the lips opening at rest appears as a viable strategy to match experimental oscillation thresholds in terms of blowing pressure. A set of numerical values for the lip parameters is given and allows through sound synthesis to play the successive registers of the trumpet.

INTRODUCTION

Digital models of brass instruments are commonly used to simulate [START_REF] Bilbao | Chick: Finite difference time domain simulation for the brass instrument bore[END_REF][START_REF] Fréour | Acoustical interaction between vibrating lips, downstream air column, and upstream airways in trombone performance[END_REF][START_REF] Gazengel | From the measured input impedance to the synthesis signal: where are the traps?[END_REF][START_REF] Velut | Measurements and time-domain simulations of multiphonics in the trombone[END_REF] the sound produced or to study the dynamic behaviour of the instrument [START_REF] Vergez | Dynamical Systems and Physical Models of Trumpet-like Instruments. Analytical Study and Asymptotical Properties[END_REF][6][START_REF] Fréour | Numerical continuation of a physical model of brass instruments: Application to trumpet comparisons[END_REF][START_REF] Boutin | Relationships between pressure, flow, lip motion, and upstream and downstream impedances for the trombone[END_REF] . Models used to describe the lips motion may seem oversimplified regarding the complexity of a human lip. However, taking into account biomechanical details is probably unreachable. Moreover, even a simplistic modeling of this vibration mechanism involves a large number of parameters [START_REF] Campbell | The Science of Brass Instruments[END_REF][START_REF] Adachi | Trumpet sound simulation using a two-dimensional lip vibration model[END_REF] . The simplest model for the lips vibration relies on a single degree of freedom oscillator including several unknowns parameters: resonance frequency, quality factor, surfacic mass, stiffness and opening at rest of the lips [START_REF] Velut | Physical modelling of trombone mutes, the pedal note issue[END_REF][START_REF] Velut | How Well Can Linear Stability Analysis Predict the Behaviour of an Outward-Striking Valve Brass Instrument Model?[END_REF] .

Limited knowledge is available concerning the values of these parameters. A bibliographical study is carried out in [START_REF] Velut | How Well Can Linear Stability Analysis Predict the Behaviour of an Outward-Striking Valve Brass Instrument Model?[END_REF] . In most studies, these values are chosen a posteriori so that the model considered produces self-sustained oscillation. Moreover they are given for a note, but no information is given on how to play the different registers (i.e. the notes playable with this fingering). This is due to the fact that direct estimation during the playing is not achievable through the experimental methods available. Some results based on experimental observations are however given for a two-mode lip's model by analyzing frequency responses of an artificial lip 6, [START_REF] Richards | Modelling the lip reed: computational and experimental investigations of two-mode inward/outward striking behaviour[END_REF] or by using a high-speed camera to record the motion of a brass player's lips during playing [START_REF] Bromage | Open areas of vibrating lips in trombone playing[END_REF] . Considering in addition to high-speed imaging an instrumented mouthpiece, many parameters are estimated and discussed in [START_REF] Boutin | Lipping down on the trombone: phases of lip motion and pressures[END_REF] . More recently, in the case of the trumpet [START_REF] Fréour | Numerical continuation of a physical model of brass instruments: Application to trumpet comparisons[END_REF] , some lip parameters, like the lip position at rest, are tuned by hand so that for the note B 4, the blowing pressure at the oscillation threshold is not unre-alistically high while keeping a low value for the quality factor, as estimated by [START_REF] Newton | Mechanical response measurements of real and artificial brass player's lips[END_REF] on human lips. Most often in the previous studies, it seems difficult whatever the model considered to provide a complete set of parameters values.

In this paper, a methodology is proposed to choose parameter values according to an expected behavior of the model. More precisely, an inverse problem is posed to tune the parameters of the lip model so that the minimal blowing pressure that allows the emergence of sound is the same as the minimal blowing pressure measured on trumpet player to play the same note.

On the experimental side, the minimal blowing pressures are estimated through in-vivo measurements with trumpeters playing an instrumented mouthpiece (see section II). On the numerical side, the model considered is presented in section III. For a given set of lip parameter values, the minimal blowing pressure is defined as the value at which a Hopf bifurcation occurs. The following inverse problem is then studied in the rest of section IV: for different registers of the open-valve fingering, find the parameters value of the lip model so that the minimal blowing pressure is the same numerically and experimentally. Results are analyzed in regard to possible playing strategies used by trumpeters to select the different registers of the instrument. A summary table is given that provides convenient lip parameters value estimated by the inverse problem. It is demonstrated in section V using time domain simulation [START_REF] Silva | MoReeSC: A Framework for the Simulation and Analysis of Sound Production in Reed and Brass Instruments[END_REF] that these parameters can be used to make the model play on the expected registers even far from the oscillation threshold.

The minimal blowing pressure used by musicians to produce a sound for a given note is measured. This is done for the registers 2 to 6 of the open valve position of the trumpet.

Register 1, which corresponds to the pedal tone is rarely used by trompet players and is not considered in this work. The objective of these measurements is to provide input data for the inverse method of estimating lip parameters. Therefore, these measurements are only intended to provide a realistic order of magnitude for the threshold blowing pressures for each register considered.

A. Experimental device

The trumpet used for this experiment is a Yamaha YTR 1335 and the mouthpiece is a Bach 1-1/2 C. The instrumented mouthpiece developed in [START_REF] Velut | Simultaneous and in vivo measurements of control parameters used in trumpet performance[END_REF] for a more general purpose has been used. A short and thin tube is inserted in the mouth of trumpet players. This tube is connected to an Endevco 8510B-5 pressure sensor outside the mouth (see figure 1).

Since we are only interested in quasi-static blowing pressure, this tube does not influence the measurement. Acoustic pressure inside the mouthpiece is also recorded thanks to a pressure sensor of the same model.

B. Experiment Protocol

Before each measurement, an unsupervised free play warm-up phase takes place so that the musicians can get used to the instrumented mouthpiece and bring the instrument up to temperature. The participants in the study are experienced amateur musicians. Trumpet players are asked to play crescendo and decrescendo on regimes 2 to 6 of a B trumpet, which correspond to notes C3 to B 4 (written note). More precisely they are given the score which is shown in figure 2.

The "ppp"/"fff" indications correspond to the lowest/highest playing nuance that the musician can achieve. The metronome is set to 40 light pulses per minute, and each note should last 8 pulses. This exercise can be repeated many times in order to have several threshold pressures values per musician and per register in the estimation of the final pressure threshold. Trumpet players are informed that musicality is not an objective to be achieved during this exercise.

The measurements presented hereafter are therefore marred by an uncertainty linked to the measurement protocol and the musician's instrumentation. This does not call into question the inverse method of estimating the lip parameters presented in this article. The influence of the uncertainty of the blowing pressure thresholds on the final results of the method are discussed in Section V B.

C. Results

A typical output of the experiment is displayed in Fig. 3, with the blowing pressure and the mouthpiece pressure synchronized. Playing at threshold is not a musical task. Therefore a musician can't be ask to tell when he/she plays at threshold. Hence, the blowing pressure threshold is determined by an analysis of the recordings displayed in Fig. 3: it is defined as the minimum air pressure measured in the mouth of the musician that allows to sustain a sound. It is represented by a red point in Fig. 3.

The results obtained by gathering the measurements of four amateur trumpeters are considered. For each register, some outliers points are eliminated. They correspond to unreliable threshold values due to material problems or failed musical attempts. The boxplot representation of the threshold pressures is shown in Fig. 4. The red "+" signs correspond to the median value for each register. The numerical values are indicated in table I. Fig. 4 shows for each register the significant variation in threshold pressure produced by the musicians. To provide input data for the inverse method, the median of all the values retained is calculated. Generally a steady increase of the blowing pressure threshold is observed with the register index. This is consistent with the experience of trumpet players. Very few experimental data are available in the litterature that can be directly compared with. However, in [START_REF] Fletcher | Blowing pressure, power, and spectrum in trumpet playing[END_REF] , Fletcher and Tarnopolsky measured the threshold blowing pressures for three players and give values of the same order.

The question explored throughout this paper is: to what extent is it possible to tune the unknown lip's parameters of a physical model of trumpet so that the minimal blowing pressure required to play the oscillation regimes 2 to 6 is the same as the values shown in table I ? The physical model considered is described in the next section.

III. TRUMPET PHYSICAL MODEL CONSIDERED

Sound production in brass instruments relies on the coupling of a mechanical oscillator (the lips of the player) and an acoustical resonator (the air column inside the instrument). The height between the lips is h(t), the airflow between the lips is u(t) and the pressure in the cup of the mouthpiece is p(t).

The air flow blown into the instrument through the lips results the coupling mechanism between these two oscillators [START_REF] Eliott | Regeneration in brass wind instruments[END_REF][START_REF] Helmholtz | On the sensations of tone[END_REF][START_REF] Mcintyre | On the oscillations of musical instruments[END_REF] .

A. Lips model

A common modelling approach consists in considering a single mechanical mode for the lips [START_REF] Fletcher | Autonomous vibration of simple pressure-controlled valves in gas flows[END_REF][START_REF] Yoshikawa | Acoustical behavior of brass player's lips[END_REF] . The lips are supposed in this paper to open when the pressure difference across the lips is positive, which corresponds to the so-called outward-striking valve model (as opposed to the inward-striking model widely used to model single-reed instruments [START_REF] Wilson | Operating modes of the clarinet[END_REF]26 ).

The relevance of this choice has been demonstrated for common playing situations [START_REF] Velut | How Well Can Linear Stability Analysis Predict the Behaviour of an Outward-Striking Valve Brass Instrument Model?[END_REF] .

The outward-striking valve model 9 gives a relation between the height h of the channel between the lips, its velocity ∂ t h, its acceleration ∂ 2 t h and the pressure difference between the mouth p b and the cup of the mouthpiece p (see figure 5):

∂ 2 t h(t) + ω l Q l ∂ t h(t) + ω 2 l (h(t) -h 0 ) = 1 µ (p b -p(t)) (1)
where ω l = 2πf l (rad.s -1 ) is the lip resonance angular frequency; Q l the dimensionless quality factor of the lips; h 0 the value of h(t) at rest; µ a lip surface mass (kg.m -2 ). These parameters are considered independent.

B. Model for the air column

As far as the air column is concerned, a few acoustical modes are considered. At least one mode is necessary to model a resonance peak of the input impedance of the air column.

Nonlinear effects in the resonator, responsible for brassy sounds at medium/high playing levels [START_REF] Hirschberg | Shock waves in trombones[END_REF] , should be taken into account for a more accurate modelling [START_REF] Myers | Effects of nonlinear sound propagation on the characteristic timbres of brass instruments[END_REF] . However, the primary goal of this work is to adjust the model parameters according to measurements carried out at low levels. Therefore the acoustic propagation along the bore is considered linear in this study.

The input impedance of the B trumpet used in this study is measured with the sensor presented in [START_REF] Macaluso | Trumpet with near-perfect harmonicity: Design and acoustic results[END_REF] . The result is shown in Fig. 6 and is fitted by a sum of complex modes:

Z(ω) = Z c N n=1 C n jω -s n + C * n jω -s * n , (2)
dp n dt = Z c C n u(t) + s n p n , ∀n ∈ [1, N ]. (3)

C. Model for the air flow blown into the instrument

The two mechanical and acoustical oscillators presented above are nonlinearly coupled by the airflow through the lip channel. The air flow is assumed laminar in the lip channel, but turbulent in the mouthpiece. Moreover, it is supposed that the turbulent mixing dissipates all the kinetic energy of the jet without pressure recovery [START_REF] Elliott | Regeneration in brass wind instruments[END_REF][START_REF] Giordano | Force on the lips of a trumpet player[END_REF] . The following expression of the flow between the lips is then obtained and commonly used in models of brass and reed instruments [START_REF] Eliott | Regeneration in brass wind instruments[END_REF][START_REF] Wilson | Operating modes of the clarinet[END_REF][START_REF] Hirschberg | Mechanics of musical instruments[END_REF] :

u(t) = 2 ρ L max(0, h(t)) |p b -p(t)|sign(p b -p(t)), (4)
with u(t) being the airflow rate (m 3 s -1 ), ρ = 1.19kg.m -3 the density of the air at 20 • C and L the width of the lip channel (m).

IV. INVERSE PROBLEM BY LINEAR STABILITY ANALYSIS OF THE PRES-SURE THRESHOLD

The aim of this study is to estimate values of the lip parameters for which the blowing pressure at threshold is the same as the pressure threshold measurements given in Table I.

This inverse problem is based on systematic linear stability analysis for a large range of values of the outward-striking valve model parameters. The range of parameters studied are arbitrary and do not take into account the practical aspect of the musician's playing (morphology, playing confort, air noise...).

A. Linear stability analysis method

The well-known Linear Stability Analysis method (LSA) is commonly used to analyse the behaviour of dynamical systems around equilibrium points and has already been applied to physical models of woodwind and brass instruments. It provides the stability information of the equilibrium (non oscillating solution). Although limited to the equilibrium points, LSA can predict important features of periodic regimes produced by brass instruments [START_REF] Velut | How Well Can Linear Stability Analysis Predict the Behaviour of an Outward-Striking Valve Brass Instrument Model?[END_REF] . LSA method works on the nonlinear coupled-system, described by Eqs (1),( 3) and ( 4), rewritten into a state-space representation Ẋ = F (X), where F is a nonlinear vector function and X the state vector:

X = h(t), dh dt , {p n (t), n ∈ [1, N ]} t . (5)
Note that F contains the parameters of the lips model and the poles and residues result from the modal decomposition of the input impedance of the trumpet.

For a given set of lips parameters, the LSA method is used to determine the minimum blowing pressure required to produce sound at small amplitude on the different registers of the trumpet. To do this, the equilibrium solution of the coupled system is calculated and the system is linearized in the vicinity of this solution. The loss of stability of the equilibrium solution, through the crossing of the imaginary axis by a pair of complex conjugate eigenvalues of the linearized system, implies the emergence of a periodic solution. Under the hypothesis of a direct bifurcation, the emergent periodic oscillation is stable. Thus, the threshold pressure value corresponds to the blowing pressure above which the equilibrium solution is unstable and the system admits a self-sustaining oscillation regime (at least in a neighborhood of the bifurcation point). When the equilibrium solution becomes unstable, it is also possible to calculate the frequency f th of the solutions of the linearized system and thus to estimate the pulsation of the emerging self-oscillation.

This work is based on the formulation of the LSA method proposed in article [START_REF] Velut | How Well Can Linear Stability Analysis Predict the Behaviour of an Outward-Striking Valve Brass Instrument Model?[END_REF] which has demonstrated its relevance in highlighting several behaviours of an outward-striking valve model. People interested in the details of the calculation steps can refer to [START_REF] Velut | How Well Can Linear Stability Analysis Predict the Behaviour of an Outward-Striking Valve Brass Instrument Model?[END_REF] .

Here is a summary of the computation method of threshold pressures and frequencies at the pressure threshold with the LSA method. For a given set of lips parameters (ω l , Q l , µ, h 0 and L), the equilibrium solutions of the model are calculated iteratively for an increasing blowing pressure p b from 0Pa to 15kPa with an interval of 400Pa. A first approximation of the threshold pressure is calculated as the first pressure value that makes the equilibrium solution unstable. To ensure a more accurate estimate, the equilibrium solutions are recalculated on a blowing pressure grid with a much finer step of 0.1Pa around the first approximate threshold.

Fig. 7 shows an example of threshold pressures (top) and frequencies at the pressure threshold (bottom) obtained using the LSA method. The upper curve corresponds to the blowing pressure limit between non-oscillating and oscillating regimes. Hence each point of this curve is a Hopf bifurcation point. Each U-shaped portion is related to a register. The identification of the register number is done by comparing the oscillation frequency at the threshold with the resonance frequencies of the instrument (represented in horizontal dotted lines on the bottom Fig. 7). For the lip parameter values chosen in this calculation, the first eight registers of the trumpet can be produced over a wide lip resonance range. Between the 1 st and 2 nd register the curve is interrupted because the equilibrium solution is always stable whatever p b for these lip resonance frequency values.

The evolution of the oscillation frequencies at threshold as a function of the lip resonance frequency also shows that the oscillation frequencies thresholds are always higher than the instrument's resonance frequencies. This behaviour has been classically reported for the outward-striking valve [START_REF] Velut | How Well Can Linear Stability Analysis Predict the Behaviour of an Outward-Striking Valve Brass Instrument Model?[END_REF] . It is also observed that for each register, the minimum of the related U-shaped portion (marked by a red point) corresponds to an oscillation frequency at threshold close to the resonance frequency of the lips (dotted line of unit slope on bottom Fig. 7).

B. The least effort strategy Fig. 7 shows that when Q l , µ, h 0 and L are fixed, the oscillation threshold for each register is not defined uniquely: on the contrary, each pair of values (p b , f l ) lying on the U-shape related to a register corresponds to a possible oscillation threshold. In this work we follow Velut et al. [START_REF] Velut | How Well Can Linear Stability Analysis Predict the Behaviour of an Outward-Striking Valve Brass Instrument Model?[END_REF] who consider that for each register "the optimal playing configuration for a human performer" corresponds to the minimum of the related U-shape portion. This defines a unique pair of value (p * b , f * l ) for each register. As commented in [START_REF] Velut | How Well Can Linear Stability Analysis Predict the Behaviour of an Outward-Striking Valve Brass Instrument Model?[END_REF] , "this hypothesis is in line with what musicians report, i.e. they develop a strategy to minimize the effort to produce a sound on a given regime".

C. Coincidence of model pressure thresholds with targets values

For each set of values of the lips parameters Q l , µ, h 0 and L, the linear stability analysis (see section IV A) completed by the assumption that the player seeks for least effort at threshold (see section IV B) allows to identify for each register a pair of values (p * b , f * l )

corresponding to the oscillation threshold. From a practical point of view, in order to accelerate the computation, LSA is parallelized on several processors.

to 6 the same blowing pressure at threshold p * b as the ones measured with musicians p exp m th .

We focus on registers 2 to 6, since register 1 is peculiar (pedal note) and registers above 6 correspond to high-pitches note out of the scope of this study. To expand the study and highlight the versatility of the approach, the values of p * b are now computed by varying two parameters: Q l (as above) and the lips surfacic mass µ. The aim is then to find, if any, sets of values of the pair (Q l , µ) for which numerical and experimental thresholds are equal (same blowing pressure). Results can be presented in a two-dimensional lip parameter space like in Fig. 9. For each register a curve with a different color represents values of (Q l , µ) for which numerical and experimental thresholds match, i.e. occur for the same blowing pressure.

To draw the curve for each register, the identification of admissible values of (Q l , µ) is carried out on a regular mesh. The range of both parameters Q l and µ is discretized on 30 values, which means that 900 LSA are calculated to draw the figure. Then, in order to avoid a higher computational cost, an interpolation of the computed thresholds is performed to determine precisely pairs (Q l , µ) ensuring the coincidence between the experimental and numerical thresholds. The main observation is that there is no intersection between the curves obtained in Fig. 9, which means that there is no value of the pair of parameters (Q l , µ) which can be used to reproduce with the model the experimental thresholds for more than one register. The linear evolution of the curves of Fig. 9 shows that a realistic production of the pressure thresholds with the model imposes a balance between stiffness and vibrating mass of the lips. Thus, choosing a large lip mass µ in the model implies a high quality factor Q l .

V. ESTIMATION OF LIPS PARAMETERS ASSURING THE REGISTERS SE-

LECTION

The previous section highlighted the importance of the values of the lip damping and surfacic mass parameters (Q l , µ) to produce sound on the different trumpet registers. In fact, two other parameters are known to vary to play successive registers in brass instruments 14 :

the width L and height h 0 of the lips opening at rest. These parameters are also included in the outward-striking valve lip model. The estimation by inverse problem of admissible values of lips parameters in order to produce the trumpet registers is now extended to a 3D parameter space. The methodology remains the same, but the exploration of coincidence between simulated and measured pressure thresholds is extended to the width L or the height of the lips opening at rest h 0 .

A. Coincidence of simulated and measured threshold pressures in a 3D lips parameter space

The correspondence between the measured and simulated blowing pressure thresholds is represented in a 3D parameter space by a hyperplane for each register. These hyperplanes indicate the lips parameter values that allow the measured threshold pressures to be reproduced with the outward-striking valve model. As with 2D plans (see Section IV C), interpolation between experimental and simulated data is necessary to obtain their coincidence with a reduced discretization of the parameter space. It still takes about 22000 parameter combinations to converge to the solution of the inverse problem (about 12 hours with a multiprocessor desktop computer).

Fig. 10 shows that the opening at rest of the lips h 0 has a great importance on the selection on the produced registers. The hyperplanes are overlaid on each other. The registers are ordered from the lowest to the highest when h 0 decreases. This is relevant with the experimental observations of playing strategies employed by musicians [START_REF] Bromage | Open areas of vibrating lips in trombone playing[END_REF] . It is also observed that the spacing between the hyperplanes gradually tightens as h 0 decreases. part V C. In practice, the musician having to increase the resonance frequency of their lips, it is probably oversimplified to consider a constant vibrating mass of the lips regardless of the register played. Many playing strategy -possibly corresponding to different playerscan be proposed as a progression through one hyperplane to another in Fig. 10. So, the successive selection of registers can be done with multiple combinations of parameters.

Another way to represent the evolution of the lip parameters in order to select a particular register of the instrument is to explore the coincidences between simulation and threshold pressure measurements by varying the width of the lips opening L (the height at rest is then constant). Fig. 11 presents the result of the inverse method in the space of the parameters (Q L , µ and L) for an intermediate value of opening height of the lips at rest h 0 = 0.5 mm. Hyperplanes have a more complex shape while still being overlaid on each other. By comparison between Figs. 10 and 11, it can be observed that for the opening surfaces at rest of the lips (S 0 = Lh 0 ) the shape of the hyperplanes is different. This comes from the motion equation ( 1) which depends on the lip-opening at rest h 0 and not on the width L. Thus, it is possible to observe that depending on the opening section of the air flow at rest, the parameters of the lips Q L and µ can be quite different for the production of the same register. This can be seen as a way to render the wide variety of musician morphologies. Also, on Fig. 11 we can propose a register selection scenario as a trajectory in parameter space crossing the normal to the surface of each hyperplane.

The resolution of the inverse problem in a three-dimensional parameter space presents trends that are consistent with current knowledge on the mechanisms of sound production by brass instruments. We will therefore use in Section V C the results presented in Figs. 10 and 11 to extract numerical values that can be used with the outward-striking trumpet model.

B. Impact of measurement uncertainty on estimated lips parameters

The results presented above are based on threshold blowing pressure values estimated experimentally. The instrumentation on musicians and the associated measurement protocol mar these values with uncertainty. In order to study the influence of the uncertainty on the estimation of the lip parameters, the simulations carried out in Part V A are reinterpreted with a tolerance of 10 percent on the threshold blowing pressure values p exp m th considered in this study (see Table I). In the 3D space of the lip parameters (Q l , µ, h 0 ), Fig. 12 shows that a variation of the threshold pressure considered on each register gives the same tendencies as those described in Fig. 10 for all registers considered. The hyperplanes remain overlaid on each other according to the lip-opening at rest h 0 . A 10 percent increase in the threshold pressure corresponds to slightly greater opening values h 0 . Similarly, a 10 percent decrease in the threshold pressure results in slightly lower opening values h 0 . This confirms that the uncertainty relating to the measurement of the blowing pressure threshold does not call into question the methodology presented in this article. The inverse method presented in this article is based on a static consideration of the acoustic quantities of the outward-striking trumpet model through the linear stability analysis employed (see Section IV A). In order to check that the estimated lip parameter values can be useful for the sound production simulation of brass instrument, the sound synthesis method called MoReeSC 17 is used to reproduce a rise of the register 2 to 6 of the trumpet.

This method relies on the modal expansion of the input impedance of the trumpet used in this study associated with the outward-striking valve model (see Fig. 6). By using finitedifference scheme, MoReeSC method provides ab-initio discrete-time computation of the self-oscillations controlled by time-varying parameters, including the mouth pressure and the lips parameters. For the following simulations, a sample rate of 44100Hz is used.

As shown in Fig. 10, the lip-opening at rest h 0 is a parameter which allows the model to select registers for fixed values of the surface mass, the quality factor and the width of II.

VI. CONCLUSION

With the inverse problem presented in this paper, the unknown lip parameters of the outward-striking model are estimated with respect to the measured blowing pressure. To the authors' knowledge, this is the first time that the values of these lip parameters are chosen according to an experimentally observed behavior on an auto-oscillating system. Although the method relies on a linearized analysis of the model, it is interesting that the simulated nonlinear dynamics of the model far from the threshold confirms the robustness of the parameter choices in terms of register selection. This opens perspectives for the control of physical models with sound synthesis purpose. Another perspective is to use the

Figure 1 .

 1 Figure 1. (Color online) Measurement of the mouth pressure through a thin tube inserted at the

Figure 2 .

 2 Figure 2. Instructions given to trumpeters in the form of a score. The "ppp"/"fff" indications

Figure 3 .

 3 Figure 3. (Color online) A typical example of raw data obtained for one trumpeter playing the note

Figure 4 .

 4 Figure 4. (Color online) Blowing pressure thresholds measured on trumpet players and represented

Figure 5 .

 5 Figure 5. (Color online) Sketch of the mouth and lips of the musician and the instrument mouth-

Figure 6 .

 6 Figure 6. (Color online) Measured input impedance (modulus and phase) of the B trumpet with

Figure 7 .

 7 Figure 7. (Color online) Threshold pressures (top) and frequencies at the pressure threshold (bot-

Fig. 7

 7 Fig. 7 indicates by the red circles the points (p * b , f * l ) for each register. From a practical

Figure 8 .

 8 Figure 8. (Color online) Computed blowing pressure thresholds p * b with respect to the dimensionless

Fig. 8 Fig. 8 .

 88 figure, horizontal dotted lines are plotted that correspond to the minimal pressure threshold

Figure 9 .

 9 Figure 9. (Color online) Values of (Q l , µ) for which the blowing pressure p * b at oscillation threshold given by the model through LSA is the same as the experimental value p exp m th obtained with trumpet

Fig. 10 illustrates

 10 Fig. 10 illustrates several scenarios for selecting instrument registers only by modifying

Figure 10 .

 10 Figure 10. (Color online) Coincidence between blowing pressure thresholds measured p exp m th and

Figure 11 .

 11 Figure 11. (Color online) Coincidence between pressure thresholds measured p exp m th and calculated p * b

Figure 12 .

 12 Figure 12. (Color online) Effect of an uncertainty of ten percents on the measured pressure threshold

  the lip. Thus, we set the parameters L = 12mm, µ = 9kg.m -2 and Q l = 20 in order to extract on the hyperplanes of each register the corresponding values of lip-opening h 0 . For each of these values we recover in the simulated data the associated resonance frequencies f l of the lip. The blowing pressure values considered for successive registers are the threshold values p exp m th obtained experimentally increased by 20% to go well beyond the emergence of self-oscillations.

Figure 13

 13 Figure13shows a spectrogram of the internal pressure p(t) simulated by sound synthesis.

Figure 13 .

 13 Figure 13. (Color online) Spectrogram of the internal acoustic pressure p(t) simulated by sound

J. Cullen, J. Gilbert, D. M. Campbell: Brass instruments: linear stability analysis and experiments with an artificial mouth. Acta Acust united Acustica, 86 (2000),

704-724.

ACKNOWLEDGMENTS

Authors would like to thanks Lionel Velut, Joël Gilbert, and Patrick Sanchez for their contribution to the experimental work with the instrumented trumpet during the PhD of Lionel Velut.

 II. Values of lip parameters allowing the selection of registers 2 to 6 estimated by the inverse method and used in sound synthesis with L = 12mm, Q l = 20 and µ = 9 (see Fig. 13).

experimentally based parameter values given in this article to carry out the study of the nonlinear dynamics of the model, including the calculation of its bifurcation diagram.

While examplified on a simple one-degree-of-freedom outward-striking lip model, the approach proposed may be easily extended to other lip models. It can even be applied to any model of sound production, provided its linear stability analysis is carried out and experimental data at the oscillation threshold are available.

Concerning the experiments with trumpet players, they provide an order of magnitude of the minimum blowing pressures for playing registers 2 to 6. Of course these measurements could be generalized to a larger number of musicians in order to refine the information provided. One could also imagine, on the contrary, considering individual measurements in order to fit parameters that make the oscillation thresholds of the model correspond to those of a particular musician. A first step perhaps towards the notion of a digital twin within the context of sound synthesis.


New approach of Dbar method for discontinuous conductivity

Toufic El Arwadi

To cite this version:

Toufic El Arwadi. New approach of Dbar method for discontinuous conductivity. 2009. hal-00384367v2