Page  448 ï~~Physical Models of Trumpet-like Instruments Detailed Behavior and Model Improvements X. RODET C. VERGEZ IRCAM,1 place Igor-Stravinsky 75004 PARIS, FRANCE Abstract A simple physical model of trumpet-like instruments is studied. Its functioning is analysed from the point of view of the theory of nonlinear dynamical systems. Using Hopf theorem, it is proved that the system has a unique stable periodic orbit in the vicinity of the principal fixed point. Then the model is improved with a better approximation of a natural instrument sound production. This is done while carefully preserving the possibility of understanding and controlling the behavior of the model. 1 Introduction In [Rodet, 1995], we explained and demonstrated a simple model of trumpet-like instruments. The aim was to find the simplest structure which would exhibit the basic properties of this class of instruments. Furthermore, as long as we keep to a very simple model, we can study analytically and really understand the functioning of the corresponding system. Knowing the exact conditions and characteristics of the oscillations is important for a musical use of the model of an instrument. Firstly, it provides the performer with parameter regions in which the model will sound. Moreover, it allows the performer to know in advance the behavior which the system will exhibit in this or that region. In the present paper, we start with the analytical study of this model. For this we look for and examine the fixed point which corresponds to the non-oscillating position of the natural instrument. We find the stability condition of this fixed point. Then we look for a proof of the hypothesis that oscillation occurs as a Hopf bifurcation of the fixed point when a parameter, such as blowing pressure, becomes greater than a critical value. The system which describes our model, is autonomous, nonlinear, and includes a delay and the corresponding type of equation is very complex to cope with. As proposed in [Rodet, 1993] we show that the Hopf theorem can be applied to the bifurcation of our system. It is a powerful method for studying periodic solutions in nonlinear autonomous systems. It allows us to prove the existence, uniqueness and stability of an oscillation around the fixed point under study. Moreover, the frequency and amplitude of the oscillation can be forecast as a function of the bifurcation parameter value. The experimental results obtained by numerical simulations are found to be in good agreement with the theoretical predictions. After the analysis of the behavior, we present several improvements of our basic model aimed at getting a better approximation of the natural instrument sound production and at improving its phrasing and articulation qualities and its expressivity. A more precise description of the bore is first taken into account. Then we design a better model of the forces acting on the lip. We also discuss the attack transients. Improvement are done taking care not to eliminate the possibility of understanding the model and applying the Hopf theorem. Thus we can keep the behavior of the model under control. 2 The Basic Model Our basic model [Rodet, 1995] is displayed in figure 1. We consider the oscillation of the upper lip only. The lip is modelled as a single mass oscillator m whose stiffness and damping coefficients are k and r. The resonator is simplified into a straight tube of cross section A. We suppose an air jet under the lip. The mouth pressure p, is considered to be steady, whereas the air flow in the mouth is neglected. The lip displacement, the pressure and the air flow under the lip, are respectively designated by x, p and u. Let's call g(t) the impulse response of the bore, defined as the inverse Fourier transform of the input impedance of the instrument. This physical model is described by the following system of equations (1) for positive x: { + rx(t) + kx(t)- =-Y66scale(ps p(t) = g(t) * u(t) e(t) s wt e isgn(p - p(t))epos - p(t)i Rodet & Vergez 448 ICMC Proceedings 1996

Page  449 ï~~x 0 Figure 1: ments yr PS P0A p u Our basic model of trumpet-like instru lossless tube as: +CO g(i) = Z(5(i) + 2Zc 3 reflecti S(t - iT) i=1 where Zc - is the characteristic impedance of A the tube and reflect a negative coefficient greater than -1. 3 Theoretical study To analyse the dynamical behavior of the model, we first examine the fixed points. Then the Hopf theorem gives us the circumstances under which instability results in oscillations around a fixed point. The characteristics of the periodic solution are also specified. Calculations are made with the MAPLE symbolic calculation software. Finally, theoretical results are compared with numerical simulations. 3.1 Calculation of Fixed Points We consider the previous system (1) where we set = x= 0. Let's write (1) with x, p and u for the steady values of x, p and u respectively: Figure 2: The nonlinear multiple feedback loop representation which is required to apply the Hopf theorem 3.2 The Hopf theorem 3.2.1 Presentation The Hopf theorem is a powerful tool which permits, for a certain class of nonlinear autonomous systems, to study oscillations near equilibrium when a parameter p is modified. It is based on the calculation of the eigenvectors of the linearised equations defining the system, and of certain derivatives of these equations. In the light of this theorem, we can conclude on the existence of a periodic solution, its unicity, and its stability. Furthermore we can obtain some insight into the frequency, the amplitude of the periodic solution and into the order of the error. We choose the frequential and graphical approach of the Hopf theorem [Mees and Chua, 1979], because of its intuitive use and large computational advantage. Note that a delay system implies an infinite dimension state space matrix in the time domain [Dieckmann et al., 1995]. On the contrary, in the frequency domain, our system leads to matrices of dimension two at most. From now on, we follow the method presented by [Mees and Chua, 1979]. 3.2.2 Conditions Required For the Hopf theorem to apply, the system has to be separable into a linear and a nonlinear part, where the linear part G can include delays and the nonlinear part f, has to be a C4 instantaneous function. In this case the system can be rewritten according to Figure 2. Finally a bifurcation parameter p is chosen. The behavior of the system is studied as a function of this parameter p. The value Po of the parameter p beyond which the stability of the fixed point changes, i.e. at which bifurcation occurs, is named the critical value. Let's call Ak(jw, pt) the eigenvalues of the matrix G(jw, p))J(p), where J(pi) is the Jacobian of the nonlinear part evaluated at the fixed point. Let w,.(ii) be the frequency at which Ak (jWr, it) crosses the real axis. If it exists p. =-l'o such that exactly one of the Ak loci, called A, passes through -1l+0j (2) kx= Y6Y6scale (Ps -jp) = (Zc+2Z _reflecti) 1Z - refiecf 1 -reflect u = 71x sgn(p -i3) Ip --iI The system (2) is easily solved in k with the help of MAPLE. In solving this, we use the hypothesis that x is positive and that ps8- > 0. This will be checked later. Three solutions in jp are found. Two of them are ignored. One is disregarded because it is greater than p3. Another one is disregarded because it leads to a negative x. The last solution in p is retained and used to find x and ui from (2). It is obvious that it would be possible to find the regions of stability and instability of the fixed point by computing the eigenvalues of the Jacobian matrix of the open loop. But stability will be determined in the scope of the Hopf theorem ICMC Proceedings 1996 449 Rodet & Vergez

Page  450 ï~~for a frequency wro, then the stability of the fixed point is altered. To specify what is happening for p lower than o and p greater than 0, we have to apply what is strictly the Hopf theorem. 3.2.3 Graphical Approach For p not too far from 0, we compute a quantity (wr,p) ([Mees and Chua, 1979], p. 248). Its expression depends on the successive derivatives, up to the third one, of the nonlinearity, and on the eigenvectors of the matrix G(jw, )J( ) associated with A. If the segment [-1 + 0j; (Wr,ip)] crosses the locus A transversally, and if there is no intersection between the loci of the other eigenvalues and this segment, then the system has a periodic solution. We will see a successful application of this approach to our model in the section 3.3. Having found a periodic solution, we then determine its frequency, its amplitude, and associated properties such as uniqueness, stability, and persistence. 3.2.4 Periodic Solution Let M=A(w, p) be the first intersection point between the locus A and [-1 + Oj; (wr,,p)]. Then the system has a periodic solution with frequency Ws. Moreover, let 0 be the value such that M - -1 + 02 (Wr). This value is associated with the amplitude of the periodic orbit so that the amplitude of e1, e2 (defined in figure 3) and x can be derived very simply from 0. Consider now the point M + &(wr, p), where the positive number 5 can be chosen as small as needed to insure that the vector [-1 + 0j; M + & (wr, p)] has no other intersection with A than M. If M + 5 (w,., p) is encircled anticlockwise by the loci of the eigenvalues a number of times equal to the number of poles with positive real part of the corresponding Ak (jw, p), the periodic orbit is an attractor. 3.3 Application to our model 3.3.1 Conditions fulfillment We now check that our system agrees with the conditions presented in paragraph 3.2.2 in order to apply the Hopf theorem. Let us introduce the reduced variables, t5 and i. Suppose that p - p > 0. The system (1) becomes (3): Figure 3: The nonlinear multiple feedback loop representation of our system according to the Hopf formalism After a Laplace-transform, (3) can be represented by figure 3. We choose ps as the parameter 1 It is clear that the nonlinearity is instantaneous, but note that it is not C4 at the point elo = /I-eI. But we are studying oscillations near an equilibrium solution which is far from the point E0. Therefore, the portion of the nonlinear function we are working with is C4. This completes the fulfillment of the necessary conditions for application of the graphical version of the Hopf theorem to our basic model. 3.3.2 Theoretical Results Because of very bad conditioning of the matrix G(jw,p)J(p), MAPLE had difficulties in numerically computing the vector (wr,,p), even with a huge number of digits. Nevertheless, the small dimension of the matrix allows us to derive the eigenvectors formally by hand. For example, we choose the following values of paramaters, given in USI: m = 1.5 10-4, k = 1087, r =0.14, y6,o,, = 2, 76 = 5 10-5, Z= - 4.78 106, y1 = 1.325 10-2, r - 7 10-3 and reflect = -0.97. Figure 4 shows the locus A in the Nyquist plane for p = 12.9. We find that a bifurcation takes place at p0=12.891, since the locus A passes through the point -1 + Oj for this value. For p < 110, A does not intersect the segment [-1 + Oj; (wr,P)], therefore the fixed point is stable. When p becomes greater than po (Cf. figure 5), the segment [-1 +0j; (wr, 1P)] transversally intersects A and there is no intersection between [-1 +Oj; M] and the loci of the other eigenvalues. Therefore, a locally unique periodic orbit appears, the frequency and amplitude of which are computed and given in table 1. We verify that M + 5 (wr,1u) is encircled anticlockwise zero times by the loci of the eigenvalues and that none of the eigenvalues has poles with a positive real part. Therefore the periodic orbit is an attractor. { mg(t) Â~ rx(t) + k), (t) = -yY6scaeP(t) A )= g(t) * ii(t) Rodet & Vergez 450 ICMC Proceedings 1996

Page  451 ï~~P T T value of w (Hz) ampl of p amplof u (10- ) Value of the parameter i 13 13.5 14 0 499.1 500.06 0 6.01 9.85 0 4.02 6.84 1 Figure 4: Locus A in the Nyquist plane 0.0015 0.001 0.0005 0 -1.0-0.999 -0.0005 -0.001 -0.0015 Figure 5: A zoom of figure 4 around -1+Oj shows the intersection, for p=12.9, between A and the segment [-1 + 0j; (wor, u)] In addition we calculate an approximate solution in i, and x. More precisely, [Mees and Chua, 1979) shows that 0 and w can be seen as the solution of an order two harmonic balance. Finally, we check the hypothesis we made concerning x and p, i.e. x > 0 and p -p > 0 near equilibrium. 3.3.3 Numerical Simulation A discrete model is obtained by using the most simple and inexpensive backward Euler finite difference scheme. A numerical simulation is obtained by the iteration of the nonlinear discrete system. We then compare the theoretical results and the numerical simulation. It is well known that numerical simulation introduces distortions related to the discretisation scheme, sampling rate and finite precision. We first noticed that single precision floating point Table 2: Simulation results at 48kHz arithmetic introduces parasitic oscillations before criticality. This difficulty was essentially cured by use of double precision floating point arithmetic. The simple discretisation scheme induces a bias of the damping of the mass. This was alleviated by adjusting the damping coefficient of the discrete model, in order to match the impulse response of both systems. Various sampling rates have been tested. 3.3.4 Theory Versus Experimentation The results presented in table 2 have been obtained with a sampling rate of 48kHz. As expected, the simulation results match the forecasts all the more closely as the sampling rate is high. Although the simulation is quite good at 48kHz, a high sampling rate leads to a more precise approximation of the continuous case near,u0. As an example figure 6 shows the predicted amplitude of the pressure p (noted x), compared with the measured amplitude of p in a simulation at 1GHz (noted +). The two sets of points are in good agreement. However, it is known that the Euler scheme is not the best. A better scheme, such as Runge-Kutta, could be used in order to obtain an efficient simulation at a lower sampling rate. Nevertheless, the inexpensive Euler scheme seems to retain the behavior of the continuous model and to be totally satisfying for musical use. The Hopf theorem only makes local predictions, close to the equilibrium solution. Just how near to criticality p has to be, is not determined. It only gives an estimate of the order of the error: the error in magnitude is O(Ip- p02), whereas the error in frequency is O(Ip - poI4). However, experimentation shows that the prediction remains qualitatively correct even when the system is very far from bifurcation. We choose p to be the blowing pressure because it seems to be natural, but other choices could be made. Value of the parameter p 13 13.5 j 14 value ofw (Hz) 501.29 501.31 501.33 ampl of p 3.54 8.49 11.57 ampl of u (10-8) 2.4 5.9 8.1 4 Improvements We present the influence of various improvements which preserve the conditions required to apply the Hopf theorem. ___! Table 1: Theoretical forecasts ICMC Proceedings 1996 451 Rodet & Vergez

Page  452 ï~~11 13.5 11 11.5 IS Figure 6: Amplitude of the oscillation in p versus mouth pressure pu = ps. Points noted x represent theoretical results while points noted + represent experimental results at 1GHz 4.1 A More Precise Model of the Bore Our model of a trumpet-like resonator is based on its reflection function. The most precise linear model of the bore is probably the one using the measured impulse response of a real trumpet. Following [Schumacher, 1981]: p -) ZT f - )+ p(t -T)) r(T) dT So we can write in the frequency domain: R(f) - Z(f) -z Z(f) +z where R(f) is the Fourier transform of the reflection function, and Z(f) is the input impedance of the instrument. We can then calculate the time domain reflection function by computing the inverse Fourier transform of R(f). The use of this precise reflection function largely improves the sound quality of the model. But it also largely increases its computational cost. Since the middle part of the bore is cylindrical, the central portion of the reflection function is practically flat, between a first pattern due to the mouthpiece, and a last pattern due to the bell (Cf. figure 7). In consequence, the measured reflection function can be simplified, keeping only its two most important patterns and replacing the central part by a delay line with delay T'. This decreases the computational cost of the program, with nearly no loss in sound quality. Moreover, when a valve is pressed, the main effect on the reflection function is that the duration of the flat portion is increased while the shape of the first and last patterns remain essentially unchanged. Therefore, by changing the duration of the delay r', different valve positions can be simulated at a low computational cost. The last step is to model the above mentioned patterns with recursive filters. The computational Figure 7: Simplified reflection function cost is reduced again, facilitating real time simulation at a higher sampling rate. Finally, note that all the proposed improvements of the model of the bore are implemented in terms of a reflection function r(t). Since the impedance of the bore is easily computed from r(t), we can still apply the Hopf theorem. 4.2 Refined Lip Dynamic The expression of the Bernoulli pressure has been slightly modified. Before this modification, when playing the model, the transitions between successive modes had a noisy character. This unrealistic feature has now completely disappeared and the transitions are more realistic. A second significant modification is linked with the shape of the nonlinearity. We introduced an artificial coefficient which controls the smoothness of the nonlinear function. By adjusting this coefficient, the player can modify important characteristics of the timbre in real time, as will be demonstrated at the conference. This modification does not hinder the use of the Hopf theorem since the nonlinearity is still instantaneous and C4 near the equilibrium solution. 4.3 The Attack Transients We noticed a lack of realism in attack transients. This may be due to our rough modelling of the lip. To tackle the problem, we think we need a better understanding of the way in which the variables x, p and u evolve for the instrument and for our model during the first instants. Therefore we have studied the numerical system in the 3D state space (x,p, u). We display the trajectory of our model from the beginning of the attack up to the steady state. We then modify initial conditions or vary some parameters during the attack and evaluate their influence on the global behavior. By this procedure, the attack mechanism can be better understood. Among different modifications aimed at improving attack transients, it could be necessary to con sider a more complex lip dynamic with two degrees of freedom. Models with two degrees of freedom have been proposed by [Rodet et al., 1990] and Rodet & Vergez 452 ICMC Proceedings 1996

Page  453 ï~~more recently by [Adachi and Sato, 1996]. However, as noted in [Rodet and Depalle, 1992], such models seem rather too complicated to be understood analytically. 5 Playing Interface One of the main interest of physical models is to offer great control possibilities. Real time execution of our model on a desktop workstation (SGI Indy) renders control possibilities even more interesting. Mouse and graphic sliders are useful and convenient for research, but combining the evolution of several parameters simultaneously is necessary for musical use and interpretation, as done naturally by the trumpet player. Moreover, the possibilities and limits of the instrument are determined all the better as the musician can freely explore the parameter space. For musical control, we have written a MIDI interface connected to an electronic keyboard, a breath controller, and a foot pedal. The instrument and its real-time control will be demonstrated at the conference. 6 Conclusion We started from the simplest model of trumpetlike instruments where the vibrating lip is modelled by a single mass and the bore by a straight tube. We then studied the delayed nonlinear equation describing this physical system. In the light of the graphical version of the Hopf theorem, we were able to obtain important results. The regions of stability and of oscillation have been determined. In the case of oscillation, amplitude, frequency, uniqueness, stability and persistence of the periodic solution were obtained. The numerical simulations we made showed the great precision of our theoretical forecasts near the equilibrium solution. Several improvements were then added to the basic model: a more realistic bore, a more precise lip model, and a nonlinearity with a user controlled smoothness. These improvements did not preclude the use and results of the Hopf theorem. To highlight the sounds possibilities of the new model, we wrote a MIDI interface which offers an interesting and musical control of our virtual trumpet. Future developments include more measurements of natural trumpet sound production and comparison with the sounds of our physical model. In particular, the attack mechanism will be studied thoroughly. Noise components in trumpet sounds also are under study since they play an important role in the sound quality of the instrument. On the one hand, noise generation in acoustic tubes and in the trumpet is studied both from the theoretical and the experimental point of view [Verge, 1995], [Bru, 1996]. On the othei hand, noise components are extracted from trumpet recordings. The modulation and spectrum of these components are studied in different playing conditions. The noise components can then be modelled as white noise filtered and modulated by the air flow at the lips. Acknowledgements We would like to thanks S. Tassart for his precious help with MAPLE. References [Adachi and Sato, 1996] Adachi, S. and Sato, M. (1996). Trumpet sound simulation using a twodimensional lip vibration model. J. Acoust. Soc. Amer., pages 1200-1209. [Bru, 1996] Bru, O. (1996). Etude du bruit de turbulence dans les instruments a vent. Master's thesis, Ecole Centrale Lyon, IRCAM. To appear. [Dieckmann et al., 1995] Dieckmann, 0., van Gils, S. A., Lunel, S. M. V., and Walther, H. O. (1995). Delay equations. Functional, Complex and nonlinear analysis, volume 110 of Applied mathematical sciences. Springer-Verlag. [Mees and Chua, 1979] Mees, A. I. and Chua, L. 0. (1979). The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems. IEEE Trans. Circuits and Systems, cas-26(4):235-254. [Rodet, 1993] Rodet, X. (1993). Flexible yet controllable physical models: a nonlinear dynamics approach. In 1CMC proceedings, Tokyo, Japan. [Rodet, 1995] Rodet, X. (1995). One and two mass model oscillation for voices and instruments. In ICMC proceedings, Banff, Canada. [Rodet and Depalle, 1992] Rodet, X. and Depalle, P. (1992). A physical model of lips and trumpet. In ICMC proceedings, San Jose, California. [Rodet et al., 1990] Rodet, X., Depalle, P., Fleury, G., and F.Lazarus (1990). Mod~les de signaux et modules physiques d'instruments: 6tudes et comparaisons. In Actes du Colloque Mod~les Physiques de Grenoble. [Schumacher, 1981] Schumacher, R. T. (1981). Ab initio calculations of the oscillations of a clarinet. Acustica, 48:71-85. [Verge, 1995] Verge, M. P. (1995). Aeroacoustics of confined jets, with applications to the physical modeling of recorder-like instruments. PhD thesis, Technische Universiteit Eindhoven. ICMC Proceedings 1996 453 Rodet & Vergez