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Page 00000001 Comparison of Real Trumpet Playing, Latex Model of Lips and Computer Model C. VERGEZ AND X. RODET vergezOircam.fr,firstname.lastname@example.org IRCAM,1 place Igor-Stravinsky, 75004 PARIS, FRANCE Abstract For several years, we have been developing and studying computer-simulated models of brass instruments including the player's lips. We demonstrate here such a physical model. It is played in real-time with MIDI events. We present recent developments concerning both physical functioning and real-time controls. They result in great improvements of sound quality and musical expressivity. To better understand the physics of real lips, we have built a latex model of the lips. We show that this artificial mouth can play a trumpet like a real trumpet player. We analyse lips behavior and acoustic measurements. We make comparisons between naturally blown trumpet signals, artificially blown trumpet signals, and simulations with our computer model. 1 Introduction In [Rod95], we explained and demonstrated a simple real-time 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 certain class of models, we can study analytically and really understand the functioning of the corresponding system. In [RV96], we showed that it is possible to predict the behavior of a simple trumpet model. By applying the Hopf theorem to the nonlinear dynamical system formed by our model, we proved existence, uniqueness and stability of the oscillating solution in the vicinity of the fixed point. We also showed that frequency and amplitude of oscillation can be precisely predicted. In the present paper, we start with briefly describing the proposed basic model. Then we present several improvements, aimed at getting a better approximation of natural instrument sound production, and at improving its phrasing, its articulation qualities and its expressivity. We first study the influence of a more realistic bore. We explain how we compute a reflection function from the input impedance measurements which we have made. Then we present a simplification of the reflection function which results in large gains of CPU time. Moreover, it allows a low cost implementation of valves. The same idea can be extended to add a slide. We also propose a more realistic closure model of the lips. Each time we improve our basic model toward a more complex one, we take care to keep the model behavior under control. The stability results obtained with the basic model are extended to the improved model. We have found that a physical model exhibiting most of the properties of trumpet like instruments, can be studied in the scope of the Hopf theorem. A real-time simulation system has been built. We describe it as well as the playing interface. This interface allows the musician to play the model in a very expressive and intuitive way. Another aspect of our research concerns the building of a latex model of the lips. They are inserted in a steel mouth supplied with pressured air. We present the model as well as the possibilities that it provides. By varying the lips' characteristics, we show that the latex model is able to play from the pedal note up to the eighth mode and produces very realistic sounds. The latex model is used to estimate the nonlinearity at the opening of the lips. This estimation is used to improve the simulated model. 2 The Basic Model A schematic of our basic model [Rod95] is displayed in figure 1. Visualisations by [Mar42] have shown that the lower lip displacement is often negligible in regard to the amplitude of the upper lip oscillation. Therefore, we limit ourselves to the oscillation of the upper lip. This lip is modelled as a single parallelepi
Page 00000002 22 0O lv m" P3s _ _ Figure 1: A schematic of our basic model of trumpetlike irtstrumertts pedic oscillatirtg mass m, the stiffrtess artd the dampirtg coefficiertt of which are k artd r respectively. Irt the most basic model, the resortator is simplified irtto art ideal straight lossless tube with cross-sectiort A. We suppose there is art air jet betweert the lips. The mouth pressure p, is cortsidered to be steady, whereas the air flow ill the mouth is rteglected. The upper lip displacemertt, the pressure artd the air flow urtder the upper lip are desigrtated by x, p artd a respectively. Several forces apply ort the mass: the weight of the mass, the sprirtg force, the dampirtg force, artd the aero-acoustics forces applied ort each face of the lip. Irt [RDFF9O], the weight of the mass t~ ml; is showrt to be rtegligible. The projectiort ort the vertical axis, of the furtdamerttal equatiort of dyrtamics leads to: m~(t) + r~t) + kx(t) A, cosct(p8 - p(t)) + FB6rn where A, is the trartsversal area of the vibratirtg lip artd a~ is the artgle betweert the lip side artd the vertical axis. FB6rn is the Berrtoulli force actirtg ort the bottom side of the lip. Whert the lips close, we add art extra stiffrtess coefficiertt artd art extra dampirtg coefficiertt to the vibratirtg lip. As a first approximatiort, we cortsider ortly lirtear wave propagatiort irt the trumpet bore, sirtce this is valid whert acoustic pressure is rtot too high. Moreover, whert trartsversal dimertsiorts are small compared to the wave lertgth, bore acoustics cart be described withirt the framework of plartar wave approximatiort. The tube acoustics cart thert be fully described by its impulse resportse t) p(t) - (g *t u)(t) where * is the cortvolutiort operator. We first cortsider art ideal straight lossless tube. We suppose that the irtcomirtg wave is ortly reflected artd damped at the opert ertd of the tube, i.e. it is multiplied by a rtegative coefficiertt greater thart -1. The sirtgle mass lirtear oscillator artd the lirtear model of the bore are rtortlirtearly coupled by the air flow. Sirtce the mouth cavity is big ertough, we cart rteglect the air flow irtside it. The statiortary Berrtoulli theorem applied betweert the mouth artd the mouthpiece leads, for x(t) > 0 artd p, > p(t), to: u(t) yix(t) p~- p3(t) where Thi - 1, 1 is the lip lertgth artd p is air dertsity. This air flow model is rto more valid whert p, < p(t) artd x(t) > 0. Irt fact, irt this case, the air flow goes irtside the mouth. Therefore, we will cortsider irt this basic model that ps < p(t) artd x(t) > o result irt u(t) 0. Of course, for x(t) < 0, we also take u(t)= 0. The system described irt this sectiort has beert implemertted irt C. It rurts irt HTM, the CNMAT realtime toolbox created by Adriart Freed. Irt spite of the crude approximatiorts we made, sortic results show that some of the esserttial characteristics of brass irtstrumertts seem to be captured irt this basic model. Irt fact, trartsitiorts betweert acoustic modes of the tube are irtduced by chartges of the visco-elastic properties of the lips. These trartsitiorts are heard as clearly cortrected with the furtctiortirtg of real brass irtstrumertts. This result is all the more irtterestirtg as our model is very simple but rtevertheless captures art importartt part of brass basic furtctiortirtg. Moreover, we have proved irt [RV9G] that the behavior of this simple model cart be predicted by usirtg rtortlirtear dyrtamical systems theory. 3 Improvements of the Sound Production M/echanismn We have presertted here above a reasortable example of the basic furtctiortirtg of brass irtstrumertts. The rtext step is to refirte the model with a better approximatiort of a rtatural irtstrumertt sourtd productiort. 3.1 A measured reflection function The pressure artd the flow wave irt the bore are cortsidered as the sum of art irtcomirtg artd art outgoirtg wave. This leads to the use of the reflectiort furtctiort of the bore irt place of its impulse resportse. The most precise lirtear model of the bore is probably orte which uses the measured reflectiort furtctiort of a real trumpet. With the help of R. Causs6 artd N. Misdariis, we have measured the complex irtput
Page 00000003 impedance of several trumpets. In the frequency domain, this can be written: R Z((f) - ZC Z(f) + Zc where R(f) is the Fourier transform of the reflection function, Z(f) the input impedance of the instrument and Zc the characteristic impedance at the input of the trumpet. We can then calculate the time domain reflection function by computing the inverse Fourier transform of R(f). However, this is not as straightforward as theory suggests, since distortions and ripples may alter the reflection function. To minimize these alterations, we follow the procedure described in [GGA95]. It first includes a phase rotation on the input impedance in order to compensate for the measurement done through a capillary fiber. Then the input impedance is interpolated between OHz and the first measured frequency. A hermitian symmetrical reconstruction is done. The center of symmetry is chosen to fulfill continuity conditions at best. Finally we apply a phase rotation to the reflection function spectrum in order to prevent ripples. The resulting reflection function is presented in figure 2. The use of this precise reflection function largely improves the sound quality of the model, but it also largely increases its computational cost. Since we want to keep with real-time, we had to make some drastic simplifications of the reflection function. As a first option, we neglect the reflection function after the main peak representing the bell (around sample 400 in figure 2). The result is presented in figure 3. This decreases the computational cost of the program, but is not without drawbacks. Since we neglect the reflection function after about 10ms, the amplitude of the impedance peaks in the vicinity of 1/0.01 = 100Hz, i.e the amplitude of the lowest impedance peaks, are under-estimated (see figure 4) compared to the measured impedance (see figure 5. It is then obvious that the model has difficulties to be played on the first two acoustic modes. We propose several solutions to this problem in section 3.2. 3.2 Strengthening of the first impedance peaks As seen in section 3.1, the system has some difficulties to lock on the first two modes, since the bore doesn't reflect enough energy at the corresponding frequencies, with respect to the lips' magnitude impedance. A first solution is to decrease the lips' impedance peak by increasing their viscous damping. The inverse solution consists in artificially increasing the acoustic load of the bore. In both cases, the aim is 0.03 0.025 0.02 yi- 0.015 0.01 < 0.005 S 0 0-5 < -0.005 -0.01 -0.015 I; - 0 200 400 600 800 1000 1200 1400 1600 1800 Time in samples Figure 2: Begining of the reflection function of a Bb trumpet, calculated from the measured impedance. Sampling rate is 44,1 kHz. 0.025 0.02 0.015 D 0.01 E 0.005 < -0.005 -0.01 -0.015 -0.02 0 50 100 150 200 250 300 350 400 450 500 Time in samples Figure 3: Reflection function of Figure 2 truncated after the first reflection at the bell. 500 1000 1500 Frequency (Hz) Figure 4: Modulus of the reconstructed input impedance according to the simplified reflection function.
Page 00000004 500 1000 Frequency (Hz) Figure 5: Modulus of the measured input impedance of a Bb trumpet. Convolution by the reflexion function Incoming An order two Reflected wave recursive filter wave> An order two recursive filter Figure 6: The incoming wave is convolved with the reflection function and filtered with the aim to strengthen the first two peaks of the input impedance to tune energy exchanges between excitator and resonator. Although they allow the model to play the first modes, these solutions influence the general behavior of the model. To avoid this infuence, it is then necessary to swap from a set of parameters to another set depending on the acoustic mode to be locked on. A third possility is to strengthen the concerned impedance peaks without influencing the other ones. This is done by using an order 2 recursive filter for each peak which we want to strengthen. Each filter is simply added in parallel with the reflection function convolution (cf. figure 6). The frequency, the amplitude and the phase of each filter can be adjusted in order to obtain a resulting impedance as close to the measured impedance as needed. Our experiments show that two recursive filters are sufficient for easy playing of the two under-estimated modes. 3.3 Different valve positions In order to simulate the different valve positions, eight reflection functions are needed. Storing them in memory could cause problems, for example with cache memory. The solution follows from the next remark. The consequence of the various valve positions is to add cylindrical tubes in the middle of the acoustic resonator. Since these added tubes are cylindrical, they cause nearly no reflection. Thus, the only difference between two reflection functions corresponding to two different valve positions is the length of the nearly null central portion. We therefore simplify the reflection function by only keeping the first pattern due to the mouthpiece and the last pattern due to the bell. The middle portion is replaced by a delay. We modify the delay duration in order to simulate different valve positions. This decreases the computational cost of the program, with nearly no loss in sound quality. It is obvious that an instantaneous change from a delay duration to another is unrealistic and does not take into account the internal acoustic state in the bore. A valve movement takes a certain duration. In our simplified implementation, during such a transitional period, two convolutions are performed at the same time, one with the previous and one with the new reflection function. A weighted sum of the two convolution results is used as the reflected pressure and the weights vary from 0 to 1 and 1 to 0 respectively during the transition. Moreover, since a new valve position is accompanied by a modification of lips stiffness, we also interpolate lips stiffness during the transition. Sonic results are very good even though the simplicity of this transition model. 3.4 A slide trumpet We can as well add a slide to our trumpet. In a glissando of trombon, the eigenfrequency of the player's lips are adjusted with the resonant frequency of the bore, so that the lips excite the same acoustic resonator mode, the frequency of which varies in time. In our model, it is straightforward to simulate a slide by changing the bore length progressively, i.e. by varying the delay duration (cf. section 3.3). We modify the lips' eigenfrequency in the same proportion in order to stay on the same acoustic mode. Note that, in our real-time system, this can be done in a rather intuitive way with a MIDI pedal. When controlled by MIDI, pitch changes are not exactly continuous, since they are discretized by the 8 bits MIDI messages. However, for reasonable ranges, a glissando sounds continuous. As an example we can play a glissando from 450 Hz to 650Hz which sounds very realistic without perceptible steps. Finally, in order to get an exactly continuous slide whatever the range of the glissando, we plan to interpolate the
Page 00000005 MIDI values and to implement fractional delays (cf [DT96]). 3.5 A refined closure model Let k and r be the stiffness and the viscous damping coefficients, respectively, when the lips are opened. When the lips are closed, we add an extra stiffness factor 3k as well as an extra damping factor 4r. To avoid abrupt changes of parameter values at the lips' closure time, we interpolate the damping and the stiffness coefficients from the opened state to the closed state and vice versa at the lips' opening time. In fact, if we consider real lips during oscillation, they are not lumped elements. They close progressively, beginning by the corners. Thus, we can suppose that stiffness and damping are modified progressively in the case of real lips. Moreover, for the same reason, we apply an interpolation when the opening between the lips is inferior to a certain threshold. At closure and opening, our expression of the flow between the lips is modified. If not, the Bernoulli pressure force dramatically increases just before the closure time, and is set to zero at the exact closure time. This results in unrealistic high frequencies in the sound. Note that in previous models we were neglecting limit layers. The smaller the lips opening, the less justified it is. Roughly speaking, in the limit layers, the flow decreases due to viscosity effects. In order to take this phenomenon into account in a simple manner, we smoothly damp the flow when the lips' opening is smaller than a chosen threshold. Simulated sounds are more realistic when this closure model is applied. 3.6 Control of the model behavior In [RV96] we have shown that, as long as the underlying structure of the model does not change, the Hopf theorem can be applied. It has been proven that the final model which we present, including minor simplifications, fulfills the conditions required to apply the Hopf theorem. Thus, exact conditions of oscillation can be found analytically. Existence, uniqueness, stability, frequency and amplitude of the oscillation can also be obtained near the fixed point. 4 Implementation and Playing Interface The overall system is composed of different parts which are distributed over three desktop workstations. This has several advantages among which are easier development and a smaller CPU load for each workstation as required for a real-time use. The numerical model itself is calculated on an SGI Indigo2 R10000 workstation. The graphical and MIDI interfaces run on an SGI Indy R4400 workstation. The parameter values are sent through ethernet from the Indy to the Indigo in UDP messages. MIDI messages are sent to a Macintosh PowerPC, in order to be recorded in a sequencer, and to the Indy in order to be interpreted. When we replay a sequence from the sequencer, MIDI messages are directly sent from the Macintosh to the Indy. One of the main interest of physical models is that they offer great control possibilities. When developing our interface, we tried different manners of varying parameters, in order to find the most intuitive way of playing the model. We have written a graphical interface and a MIDI interface which are coupled and can be used simultaneously. The graphical interface is made of buttons, sliders and signal and spectrum displays. The player can use the mouse in order to modify the parameter values. The MIDI interface interpretes messages from an electronic keyboard, a MIDI sax Yamaha WX7, three foot pedals as well as the Opcode StudioVision Pro MIDI sequencer ([Stu95]). By using MIDI controllers, the musician can modify, in real-time, mouth pressure, stiffness of the lips, viscous damping, as well as valve positions. In the trombone mode, the slide length is also controlled in real-time with a pedal. We can record in the sequencer the evolution of parameter values sent by MIDI devices. This offers the possiblity to replay a sequence by sending exactly the same parameter values. Furthermore, one can edit and modify at will the parameter values in StudioVision. Both possibilities are very usefull to test and study our simulation models. For instance, we plan to reproduce, at best, trumpet phrases played by a professional trumpet player, in order to find out how close the simulated model can be from a real trumpet player. Moreover, we can use the MIDI sequencer as a graphical help to study the MIDI device messages and the phrasing of musical sentences. The use of the playing interface offers great control possibilities and underlines the musical qualities of our model. Composers working at IRCAM have shown a great interest in our model because it does react like an acoustic instrument and allows subtle or larges changes of sound quality. For research also, real-time control is very important. First of all, it provides an easy and quick way of testing new improvements of the simulation program. Moreover, the possibilities and limits of the instrument are determined all the better as one can freely explore the parameter space.
Page 00000006 5 A Latex Model of The Lips Details of aero-acoustic and acoustic phenomena around the lips are not well known. To continue the improvement of the model in these domains, we felt the need for precise experiments and measurements on a real instrument. Therefore, we have built a latex model of lips inserted in a tube representing the mouth and pressed against the mouthpiece of a real trumpet (cf. figure 7). The tube is connected to compressed air. The mouth project has been remarkably carried out by Benoit Govignon [Gov97]. The beautiful artificial mouth has been conceived and constructed by Alain Terrier who also sculpted the plexiglass mouthpiece (see section 5.2). 5.1 Benefits of an artificial mouth The main nonlinearity in our simulated model is located at the lips. The flow behind the lips is calculated according to the stationary Bernoulli theorem without losses. This theorem applies to incompressible stationary flows. As the lips close hundreds of times per second, this condition is probably never fulfilled. Numerical flow simulation could provide some insight into what is really happening. However, threedimensional simulation taking into account moving lips and mouthpiece geometry seems out of reach today. Only simplified-shape fixed-lips two-dimensional simulations could reasonably be undertaken. The mouthpiece after the lips, instead of a free space, drastically increases theoretical difficulties as well as flow simulation cost. Another way of obtaining more knowledge on the aero-acoustic phenomena occurring in the vicinity of the lips, is to make measurements on a real trumpet player. However, reproducibility is difficult when human factors are involved. More disturbingly, some measurements are impossible with human beings. Introducing several experimental devices into the trumpet player's mouth might affect the quality of the playing. The use of a stroboscope is not easy with human players, because they have to keep a stable playing frequency during the recording, which is especially difficult for high notes. Note that recent visualisations have been carried out with a trombone (cf [Jor95] and [CS96]). On the other hand, an artificial mouth and lips device solves these difficulties and offers new possibilities for experiments and measurement. Latex lip I I / Mouthpiece Teeth channel Figure 7: A longitudinal section of the steel mouth with the two latex lips 5.2 Description of the artificial mouth The mouth is a steel tube, the volume of which is chosen to match the one of a real mouth (cf. figure 7). In order to approach the visco-elastic properties of real lips, we tested different elastic materials. We got advice from plastic surgeons familiar with human face reconstruction. Finally, our artificial lips are made of latex and filled with water. We can modify the stiffness of the lips by varying the volume of water inside the latex envelope, or by varying the latex thickness. On one side, the lips rest on a steel part which represents the teeth. The opening between the steel teeth has been set according to the average position of a trumpet player's lower jaw. We use a plexiglass mouthpiece sculpted according to the measures of the widely used Bach 1 1/2 C mouthpiece. A planar side of the plexiglass mouthpiece has been cut with an angle calculated so as to reduce optic distortion and enlarge the vision of the lips. This facilitates stroboscopic visualisations and video recording. The trumpet is fixed on a support. The mouth is attached to a micrometric mechanism which allows subtle adjustment in two directions. Firstly, the mouthpiece pressure on the lips can be adjusted. Secondly, we can set the angle between the lips and the mouthpiece. These two controls of the mouthpiece position, with respect to the lips, are crucial according to trumpet players. This has been confirmed by our tests on the device. Experimental conditions are therefore reproducible. The supplied air pressure is measured with a manometer and can be adjusted. The volume of water injected into the latex envelopes is controlled by two leakless graduated syringes and is used to vary lips' stiffness. Finally the mouthpiece position is set by the micrometric mechanism.
Page 00000007 ~iiiiiiiiiiiiiiiiiiiiiiiiiiiii~ w i ii a~ /i n.'g i / K j a?II 1: i, I -. " ii ~ n ii iI ii I e i 'i I itl.iiII u:r 1,i 1~ " P I P IS S 6S f~P` LI a II S?L~~Y illpl an ~;~~~uJ ~ r ~srra~r~ re i;si~a~sa~ s~r~ ws.a;ruaa ~_______________________________________ ________________________________________ V.i I ' ' ' ' ' ' ' ' ' I ' ' ' ' ' ' ' ' ' I ' ' ' ' ' ' ' ' ' I 0.962 0.965 0.967 0.970 1""'""1""'""1 ~I sec~nds 0.972 0.975 0.977 Figure 8: Comparisou betweeu a humau blowu aud au artificially blowu souud. Spectra aud siguals. 5.3 Measurements and recordings We measure the lips opertirg, the pressure irtside the mouth, the pressure ill the cup of the mouthpiece artd the pressure irtside the bore. A light source is irtserted ill the mouth. Art optic LED placed ill the mouthpiece measures the light irttertsity comirtg through the vibratirtg lips artd gives art estimate of the lips opertirtg. Lips oscillatiort is filmed through the plartar side of the trartsparertt mouthpiece, with stroboscopic illumirtatiort. Films show the real movemertt of the lips artd will be presertted at the cortferertce. However, sirtce oscillatiort filmirtg is rtot easy to do artd to irtterpret, it is used esserttially as a meart of computirtg lips opertirtg area as a furtctiort of the LED output sigrtal. Cortsequerttly, the LED sigrtal really gives art easy access to the lips opertirtg, exactly syrtchrortised with the other sigrtals. 5.4 RLesults The artificial mouth that we built produces excellertt sortic results, very close to humart blowrt trumpet sourtds (Figure 8). By varyirtg the supplyirtg air pressure, the lips stiffrtess artd the mouthpiece positiort, we cart obtairt a large rartge of rtotes, from the pedal rtote to the eighth acoustic mode. The artificial mouth behaves artd reacts like a real orte. For example, we have to irtcrease the supplyirtg pressure artd to press the lips ort the mouthpiece harder irt order to lock ort the highest acoustic modes. W~le wartt to estimate the rtortlirtear relatiort be tweert the lips opertirtg area, the flow betweert the lips, artd the differertce of the pressure applied ort each side of the lips. The ortly urtkrtowrt is the flow. The acoustic flow is calculated as the cortvolutiort of the mouthpiece pressure artd the trumpet measured admittartce. Therefore, we cart get art estimate of the rtortlirtearity at the lips artd compare it with the rtortlirtearity expressiort derived from the Berrtoulli theorem which is used irt our simulatiorts. Results will be presertted at the cortferertce, as well as comparisorts of sourtds from the latex model, the rtumerical model, artd a humart blowrt trumpet. 6 Conclusion We started from the simplest model of trumpet-like irtstrumertts. Irt spite of its simplicity, simulatiorts show that some characteristic of brass behavior are captured. Moreover, irt [RV9G] we artalytically derived the exact cortditiorts artd characteristics of the oscillatiorts. We have exposed recertt improvemertts cortcerrtirtg both the physical furtctiortirtg artd the real-time corttrol of the model. We have replaced the reflectiort furtctiort of the straight lossless bore by a measured reflectiort furtctiort. Irt order to implemertt a slide artd to keep simulatiorts irt real-time, we have proposed simplicatiorts of the reflectiort furtctiort. The cortsequertce of these simplificatiorts of the model behavior has beert studied. We have proposed solutiorts to compertsate for the resultirtg urtderestimatiort of the first impedartce peaks. It is possible to strertgthert
Page 00000008 the underestimated peaks without significant increase in the CPU cost. We have then presented a low-cost implementation of the valves. We have extended this method to propose a slide trumpet. Finally a more refined lips closure model has been implemented. The final model does react like a brass instrument. In order to highlight its musical qualities, we wrote a playing interface which allows one to feel, in real-time, the model behavior. The result is very interesting considering the sound quality, but also because of the strong interactive relation between the player and the virtual trumpet. Stability results are still guaranteed since the Hopf theorem can be applied. A more faithful understanding of aero-acoustic phenomena occurring around the lips is necessary for the development of better models. We have chosen to build a latex model of the lips. We have shown that the model can produce very realistic sounds on a wide range of frequencies. We have made measurements and films of lips oscillation to better understand the nonlinear behavior of lips and air flow. Future developments include more measurements on the artificial mouth. We would like to study the transient phenomena occurring when a valve is pressed or when the system jumps from an acoustic mode to another. We want to specify the effect of nonlinear propagation in the bore ([JG96] and [TMDD97]) and implement a nonlinearity of propagation in our model. Another direction considered is automatic optimisation of parameter values in order to make the model match musical phrases recorded by professional trumpet players. This method should make explicit the musical possibilities and the limits of our model. Automatic estimation of the parameters of a physical model would also open fascinating sound possibilities for musical applications. Acknowledgements We thank very much B. Govignon and A. Terrier for their very valuable work on the artificial mouth and R. Causse and N. Misdariis for their help with impedance measurements. References [CS96] D. C. Copley and W. J. Strong. A stroboscopic study of lip vibrations in a trombone. JACOS, 99(2):1219-1226, Feb 1996. [DT96] P. Depalle and S. Tassart. Fractional delay lines using lagrange interpolators. [GGA95] [Gov97] In Proceedings ICMC'96. ICMC, 1996. Hong-Kong. B. Gazengel, J. Gilbert, and N. Amir. Time domain simulation of single reed wind instrument. from the measured input impedance to the synthesis signal. where are the traps? ACTA, 1995. B. Govignon. Construction of an artificial mouth with latex lips coupled to a real trumpet. Master's thesis, DEA ATIAM, 1997. To appear. [JG96] J. F. Petiot J. Gilbert. Nonlinearites dans les instruments de type cuivre: resultats experimentaux. In Actes de Colloques du quatrieme Congres Francais d'Acoustique, Marseille, 1996. [Jor95] [Mar42] D. Jorno. Etude theorique et experimentale de l'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, 1995. D. W. Martin. Lip vibrations in a cornet mouthpiece. JACOS, 13:305-308, 1942. [RDFF90] X. Rodet, P. Depalle, G. Fleury, and F.Lazarus. Modiles de signaux et modales physiques d'instruments: etudes et comparaisons. In Actes du Colloque Modeles Physiques de Grenoble, Grenoble, 1990. Colloques Modiles Physiques. [Rod95] X. Rodet. One and two mass model oscillation for voices and instruments. In ICMC proceedings, Banff, Canada, 1995. ICMC. [RV96] X. Rodet and C. Vergez. Physical models of trumpet-like instruments. detailed behavior and model improvements. In ICMC96 proceedings, Hong-Kong, August 1996. ICMC. [Stu95] MIDI Reference Manual for Vision and Studio Vision Pro, opcode systems edition, 1995. [TMDD97] S. Tassart, R. Msallam, Ph. Depalle, and S. Dequidt. A fractional delay application: time-varying propagation speed in waveguides. In Proceedings of ICMC97, 1997.