# Air Flow Related Improvements for Basic Physical Models of Brass Instruments

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Page 00000001 AIR FLOW RELATED IMPROVEMENTS FOR BASIC PHYSICAL MODELS OF BRASS INSTRUMENTS Ch. VERGEZ vergez@ircam.fr X. RODET rod@ircam.fr IRCAM,1 place Igor-Stravinsky 75004 PARIS, FRANCE Abstract Physical models of trumpet-like instruments have been studied for several years. Though simple models have proved to produce realistic sounds, some points still have to be improved. In this paper, volume flow related improvements are presented. They allow to avoid unnatural high frequencies in the sound, which occur otherwise. 1 INTRODUCTION The general behavior of brass instruments is well understood since a few decades ([Backus, 1963], [Benade, 1973], [Elliott and Bowsher, 1982]). Lots of physical models have been proposed in order to imitate this behavior. We have experienced that quite a simple model controlled in real-time can produce surprisingly realistic results ([Vergez and Rodet, 1997]). Such a model is briefly described in section 2. However, a detailed analysis of the simulation results reveals two unrealistic features (cf. section 3) which are mainly responsible for annoying high frequencies in the sound produced: a discontinuous volume flow derivative at lips closure, and a rapid alternance of positive and negative volume flow just after lips opening. It is shown that these two features correspond to two oversimplifications in the physical model: a parallelepipedic lips model and a viscousless air flow in the mouth of the player and between the lips. These two oversimplifications are corrected in section 4 and 5 respectively. Since it is important to keep a real-time functionning, attention is paid to design improvements which do not increase dramatically the computational cost. Thus, though the improvements proposed in this paper have a great influence on sonic results, they don't preclude a real-time fuctionning on midrange linux PCs. 2 A SIMPLE MODEL OF BRASS INSTRUMENTS A simple physical model of trumpet-like instruments has first been developped. It is derived from the typical single mass lips model, nonlinearly coupled with a linear model for the bore (cf. [Elliott and Bowsher, 1982]). The lips model (see figure 1) includes a single parallelepipedic mass m attached to a spring k and a damper r. The mass moves along the "x" axis. The acoustic pressure field inside the bore is decomposed into an outgoing and an incoming travelling waves po and pi. The bore of the trumpet is modelled by its time-domain reflection function r' (t). This reflection function is derived from complex r k pm Mouth Lips Instrument channel Figure 1: Model for the lips of the trumpet player: a single mass m attached to a spring k and a damper r. input impedances measured on real trumpets in an anechoic room ([Vergez and Rodet, 1997]). The air flow is supposed laminar in the mouth and in the lips channel. An air jet formed after the lips, is supposed to dissipate all its kinetic energy in the cup of the mouthpiece through turbulence without pressure recovery ([Hirschberg, 1995]). Therefore, nonlinear coupling between the lips model and the model of the bore is represented by the Bernoulli equation which connects volume flow between the lips u(t), lips aperture x(t) and pressure difference between the mouth and the mouthpiece (p, - p(t)) (air velocity in the mouth is neglected). Finally, this basic physical model is described by the following system of equations for positive x(t): m((t) + rx(t) + kx(t) = E Faero-acoustics Pi(t) = (r' *o)(t) (1) u(t) = x(t)sgn(pm - p(t)) Pm-pt) (t) = po(t) + Pi(t) 1u(t) = (po(t) - pi(t)) where 1 is the length of the mass in the transversal dimension, Zc = PC is the characteristic impedance at the entry of the mouthpiece (p is the air density, c the sound velocity

Page 00000002 1 I5 =0. S.^... i.............. volume flow Sderivative C a0. 0 -0. -0. 4 Zoom on the volume flow deri | Zoom on the volume flow derivative discontinuity -n 40 60 80 100 120 samples 40 60 80 100 120 samples 40 42 44 46 48 50 52 samples Figure 2: Volume flow calculated with a constant mass length I = 10-2m: a discontinuous derivative at exact lips closure is observed. and Acup is the cross section area of the mouthpiece entry). When lips are closed (x(t) < 0) the lower lip is taken into account by additional stiffness and damping coefficients 3k and 4r respectively. The volume flow is set to zero. Before a numerical simulation can be performed, system (1) has to be solved. Successive calculations applied at each time step are described below: * Knowing p (calculated at the end of the previous time step), x is found by solving numerically the differential equation (first equation of system (1)). * pi is found by the second equation of system (1) (the current value of po is not yet known, but the first coefficient of r' is null). * Knowing pi and x, the analytical solution of third to fifth equations of system (1) is used to find next values of p and u (cf. [McIntyre et al., 1981]): P = Ph - -A(x) [A(x) - A2(x) +4|pm (2) 2 withph = 2pi, ( = sgn(pm - Ph), A(x) = lZcz 1 1 u = 2Z A(x) [A(x) - +AA2) + 4|pm -Ph I(3) 2Zc 3 ANALYSIS OF NUMERICAL SIMULATION RESULTS This physical model has been implemented in the IRCAM realtime environment jMax ([Tisserand, 1999]) and is controlled by a MIDI playing interface ([Vergez and Rodet, 1997]). Sonic results are realistic but unnatural high frequencies are generated. The cause for these high frequencies may be highlighted by an analysis of volume flow signals in the time domain (see figure 2). In fact, two features may correspond to the high frequencies observed: * Firstly, the discontinuous time derivative of the volume flow signal at exact lips closure. * Secondly, just after the lips opening, the rapid alternance of a strongly positive and negative volume flow. It is shown in paragraphs 4 and 5 that these two features result from two oversimplifications in the physical model. Improvements are also proposed in order to reach a better sound quality, without precluding the real-time functionning. 4 COMPENSATING FOR THE BIDIMENSIONAL LIPS MODEL AT SMALL OPENINGS Let us consider equation (3). For "'" small enough (close to closure), we can write A2 (x) < pm - Ph. Then, considering |pm - Ph locally constant (in time) allows us to calculate: 9u 1 dA(x) r d9x 2 Z, dx -4pm-Ph (4) Since A(x) = lZ,x, equation (4) shows that the volume flow derivative tends toward a non zero value for small openings and is abruptly set to 0 at exact closure. The discontinuity may be observed (in the time domain) in central or right pictures of figure 2. Obviously a discontinuous volume flow derivative has no physical meaning and results from an oversimplification in the modelling. A simple solution to cope with this problem is to take into account the fact that lips begin to close by the corners. Then the length of the mass I should depend on x for small openings. In fact, experiments with a trombone player have shown that the ratio of the lips' length over the lips opening is close to 5 ([Jorno, 1995]). This has been used in a physical model of the trombone by [Msallam, 1998]. In order to have l/x = b (b is a constant) for small openings, while keeping I independant of x for large openings, we propose to replace 1 by 10 (x) with ~01(x) = tanh( z). Then the air flow partial derivative becomes continous since: { ' = 0 when x < 0 Ox Pm- (5) O x 20b x -|pm - Ph | for small openings

Page 00000003 .3 fl -0.2 -0.4 Zoom on the volume flow derivative -0.5 144 146 148 150 152 154 156 samples no more discontinuity. -1.5 I I I / I I I 80 100 120 140 160 samples 140 160 180 200 220 samples Figure 3: Introducing a varying lips length I tanh( zx) allows to keep a continuous air flow derivative at lips closure. The resulting time domain signal is shown in figure 3 with b = 6. A significant part of the disturbing high frequencies in the sound is eliminated. The replacement of 1 by 10Q (x) may be seen as a way to take into account three-dimensional effects in a twodimensional lips model. 5 VISCO-THERMAL LOSSES IN THE AIR FLOW BETWEEN THE LIPS We cope in this section with the rapid alternance of a positive and a negative volume flow, just after the mass opening (see left picture of figure 2). Though to our knowledge, none experimental measurements is available, such an alternance doesn't seem realistic. This alternance is possible in the model because viscothermal losses and inertia are neglected in the air flow. A viscousless approximation involves a tangential slip velocity of air along the mass. On the contrary, experiments have shown that air sticks to the mass ([Hirschberg, 1995]). In a thin zone called "boundary layer" air velocity goes from the mass velocity in the flow direction (which is null here) to the main flow velocity. For small apertures, the boundary layers thickness is no more negligible. Then, influence of boundary layers in the mass channel (i.e. the region under the mass) should become particularly perceptible. The hypothesis of a viscous flow (following the Poiseuille law) under the mass has been studied (inertial effects are neglected). Considering the bidimensional geometry of the model and the coupling between the mouth and the bore, the analytical expression for u is given (after some calculus) by equation (6). -= -ui+ u + 4m -ph|- (6) 1(6) 2Alip Alip with ul ( )2Ae + Zc where Ali = Ix, vr is air dynamic viscosity and e is the width of the mass. This equation is far more complex than equation (3) and increases the computational cost. However viscous effects are only significant for small apertures. An approximation of equation (6) for small x leads to: 1 u ~X= O I(Pm - ph)X3 12rle (7) For large x, equation (3) is a good approximation since viscosity can be neglected. Therefore, we propose to switch smoothly between equation (7) and (3): -11 u = 02 x (A(x) - A2x)+4|pm-ph [A(x)- (x)+ m-PIj 2 Zc (8) with * 02 = tanh (alX2 (Pm - h)05) * ai- (= 0(105)) * 02 = 2 Through our own experience, it appears that it is interesting to let the player modify al and a2 around these values. This allows a finer control on the timbre of the sound. For example, figure 4 shows the influence of a1 (a2 being set to 3). The smaller a1, the softer the sound. 6 CONCLUSION A simple physical model of trumpet-like instruments has been built. Though realistic behaviors are produced by this model, volume flow related improvements are necessary to reach an even more realistic synthesis. The first improvement is related to the volume flow control by both the lip's aperture and the lip's width (and not only by the lip's aperture). Otherwise, the volume flow derivative is discontinuous at exact lips closure. The second improvement consists in deriving a new model for the volume flow calculation. This model is equivalent to the bidimensional Poiseuille model when boundary layers influence becomes important, and equivalent to a viscousless air flow model otherwise. This second improvement prevents the high amplitude altemance of a positive and a negative air flow after lips opening.

Page 00000004 10,1.5 1 0.5 I 1.5 1 0.5 -0.5 1. 0 CO Q~0. a i 50 100 150 200 samples 0 50 100 150 200 samples samples Figure 4: Volume flow calculated according through equation (8) with c2 = 3. From left to right at = 106, Ca = 2.105, al - 105. Since care has been taken not to increase dramatically the computational cost, the improved model easily runs in real-time on a PC. Other refinements have been taken into account in the model which will be demonstrated at the conference, like the nonlinear propagation in the bore of the trumpet, responsible for brassy sounds at high playing levels ([Vergez and Rodet, 2000]). This physical model (detailed in [Vergez, 2000]) is implemented in real-time under jMax on a Linux PC. References [Backus, 1963] Backus, J. (1963). Small vibration theory of the clarinet. J. Acoust. Soc. Amer., 35, 305. and Erratum (61) [1977], 1381. [Benade, 1973] Benade, A. H. (1973). The Physics of Brasses. Sci. Am., 299. [Elliott and Bowsher, 1982] Elliott, S. J. and Bowsher, J. M. (1982). Regeneration in Brass Wind Instruments. Journal of Sound and Vibration, 83(2):181-217. [Hirschberg, 1995] Hirschberg, A. (1995). Mechanics of Musical Instruments (Chap. 7), chapter Aero-acoustics of wind instruments. Springer Verlag. [Jorno, 1995] Jorno, D. (1995). Etude theorique et experimentale de 1'auto-oscillation des levres en presence d'un couplage acoustique. application aux instruments a anches lippales. Master's thesis, Universite Paris 6, DEA ATIAM, ICP. [McIntyre et al., 1981] McIntyre, M. E., Schumacher, R. T., and Woodhouse, J. (1981). Aperiodicities in Bowed-String Motion. ACUSTICA - acta acustica, 49:13-32. [Msallam, 1998] Msallam, R. (1998). Modeles et simulations numeriques de l'acoustique non lineaire dans les conduits. Application a l'etude des effets non lineaires dans le trombone et a la synthese sonore par modele physique. PhD thesis, Paris 6. [Tisserand, 1999] Tisserand, P. (1999). Implementa-tion de modeles physiques sous environnement temps-reel jMax. Master's thesis, ENSM Nancy-IRCAM. [Vergez, 2000] Vergez, C. (2000). Trompette et trompettiste: un systeme dynamique non lineaire analyse modelise et simule dans un contexte musical. PhD thesis, Universite Paris 6. [Vergez and Rodet, 1997] Vergez, C. and Rodet, X. (1997). Comparison of real trumpet playing, latex model of lips and computer model. In Procedings ICMC'97, Thessalonike. [Vergez and Rodet, 2000] Vergez, C. and Rodet, X. (2000). New algorithm for nonlinear propagation of a sound wave. application to a physical model of a trumpet. Journal of Signal Processing, 4(1):79-87. Special issue on nonlinear signal processing.