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Page 00000001 A GENERALIZED PARAMETRIC REED MODEL FOR VIRTUAL MUSICAL INSTRUMENTS Tamara Smyth School of Computing Science Simon Fraser University Jonathan Abel Universal Audio Santa Cruz, CA, US A Julius O. Smith CCRMA, Dept. of Music Stanford University A reed, or more generally, a pressure-controlled valve, is the primary resonator for many wind instruments and vocal systems. In physical modeling synthesis, the method used for simulating the reed typically depends on whether an additional upstream or downstream pressure causes the corresponding side of the valve to open or close further. In this work, a generalized and configurable model of a pressure controlled valve is presented, allowing the user to design a reed simply by setting the model parameters. The parameters are continuously variable, and may be configured to produce blown closed models (like woodwinds or reed-pipes), blown open models (as in simple lip-reeds, the human larynx, harmonicas and harmnioniums) and symmetric "swinging door" models. This generalized virtual reed affords the musician the ability to produce a wide variety of sounds which would otherwise only be obtained with several reed instruments. 1. INTRODUCTION There are several examples of musical instruments (e.g. woodwind and brass) and vocal systems (e.g. the human vocal tract and the avian syrinx) where air pressure from the lungs, or other source, controls the oscillation of a valve by changing the pressure across the valve's reed or membrane to create a constriction through which air flows. Sound sources of this kind are referred to as pressure-controlled valves and they have been simulated in various ways to synthesize virtual musical instruments. The similarities and differences among various valve geometries lend themselves quite nicely to a single generalized parametric model-one that is completely configurable as determined by the needs of the musician. The generalized model of the valve presented here, and the acoustic tube to which it is connected, is implemented using numerical methods and waveguide synthesis, and runs in the real-time programming environment Pd . We begin by describing the three classes of valves and then discuss how the valve dynamics are generalized to produce a single parametric model. Finally, the musical effects produced by modifying different parameters and changing valve configurations are examined. 2. THE PRESSURE-CONTROLLED VALVE Pressure-controlled valves are classified according to the effect of an additional pressure applied to the upstream or downstream side of the valve [3, 4]. Fletcher uses the couplet (cr1, -2) to describe the valve behaviour, with cr -i-1 signifying an opening of the valve, and o-i::::::: - -----. signimfying a closing of the valve, in response to an upstream (i = 1) or downstream (i 2) pressure increase. U P2 P2.. U Figure 1. Simplified models of common configurations of the pressure-controlled valve as seen in . 1) (-, +) defines a valve that is blown closed, and is typical of woodwind instruments. 2) (+, -) defines a valve that is blown open, and is exemplified by brass and other lip-reed instruments as well as the human larynx. 3) (+, +) is the principle configuration of the avian syrinx, where an overpressure applied to either side of the valve will cause it to open. This construction is very useful when evaluating the force driving a mode of the vibrating valve. Consider Fletcher's generalized double reed in a blown open configuration, as shown in Figure 2. In this case, surface S1 sees an input or upstream pressure p 1, surface S2 sees the downstream pressure p2 after flow separation, and surface S3 sees the flow at the interior of the valve channel and the resulting Bernoulli pressure. With these areas and the corresponding geomnetric couplet defined, the motion of the
Page 00000002 a) a)..A - Pm Pb b () U-m, Am-~ Pm (A b: b: Ir.ark4N-t Figure 2. Geometry of a blown-open pressure-controlled valve showing, effective areas S, S2, S3 . valve opening r(t) is governed by &2 ndt2 2ny + k(x - >o) =o'ip (S1 +Sa) +s2p2S2 where y is the damping coefficient, xo the equilibrium position of the valve opening in the absence of flow, k the valve stiffness, and at the reed mass [3, 4]. The motion equation (1) intentionally does not take into account the force applied by flow for the purpose of simplification. 3. THE G3ENERALIZED PARAMETRIC MODEL The generalized parametric model of a pressure controlled valve described below can be configured to operate in any number of ways, allowing the musician the benefit of producing a range of musical effects. We see fromn (1) that the behavior of the valve is governed by two features: its dynamics (i.e., how it responds to applied forces), and the manner in which upstream and downstream pressures exert force on the valve. As we will see in (3.2, flow through the valve depends on the valve opening area as a function of time. To develop a genera'ized pressure controlled valve, therefore, it is desirable to independently control the valve dynamics, the effect of upstream and downstream forces, and the valve area as a function of the valve state. 3.1. Valve dynamics Figure 3 illustrates one mode of oscillation for each of three possible generalized valve configurations. The displacement of the valve is given by its angle 0 fromn the vertical axis. The configuration of the valve is determined in part by the initial position of the valve 00 (its equilibrium position in the absence of flow), and in part by the use of a stop-a numerical limit placed at the center vertical axis which prevents the valve from swinging beyond the point 0 0 and into the shaded region of Figure 3 b) and c). If no stop is placed, as shown in Figure 3 a) the valve is free to swin i across this center boundary and the model pro vide: a:ymmetric (+, -+) ty'pe o f model, that is, an additional pre:s:ure fromr either side of the valve xwill cause c) PmI Pb 0 Figure 3. Configurations of the generalized parametric valve, where H_(0) is a function which determines the height of the valve channel. In configuration a), no stop is specified and the valve swings freely producing a (+, ) symmetric "swinging-door"' model. In configurations b) and c), a stop prevents the valve from swinging beyond 0 0 and into the shaded regions, creating a (-, +) blow n closed model and (~, -) blown open model respectively. it to open further. If a stop is placed in the channel, the configuration is further determined by the initial equilibrium position of the valve 00. If the valve's initial position is to the left of the center (00 < 0), a pressure increase from the air source will cause the valve to close further and a pressure increase from the bore will cause it to open further. bhis creates a (-, +) blown closed model similar to woodwind instruments and the valve shown in Figure 1. b). Contrarily, if the valve's initial position is to the right of the stop point (00 > 0), a pressure increase from the air source will cause the valve to open further and a pressure increase from the bore will cause it to close further. This creates a (+, -) blown open model similar to lip-reeds and shown in Figure 1, c). Once the valve is set into motion, the value for 0 is determined, for small displacements, by the familiar second order differential equation d 20 dO nTY dt2 r m2y k(0 - 00) dt2 (it (2) where k is the stiffness of the reed, - and mTt are defined as above, and F is the overall driving force acting on the reed. The fundamental fr equency of valve vibration (resonance frequency) is given by c,, /. In the generalized model, the displacement of the valve is determined
Page 00000003 by first considering the force F in (2). Let us assume the valve reed is hinged as in Figure 3. Let A, be the effective length of the valve which sees the mouth pressure p,, Ab be the length of the valve which sees the bore pressure Pb, and p the length of the valve that sees the flow. There is a force in the positive 0 direction, on the surface area A,,w, given by Fe = wiV7Pm., (3) where wt is the width of the valve channel. There is also a force on the bore side of the reed away from the jet, given by Fb = --UAbPb, (4) forcing the valve in the negative 0 direction when pb > 0. The force applied by the flow (which also forces the reed open) is found by integrating the pressure along the flow and is given by The dynamic model, which replaces the static table with a differential equation for volume flow, was presented in [5, 6] and permitted the development of the feathered reed-a smoothing of the volume flow cutoff between open and closed valve states. Within the context of the generalized valve, the dynamic model now also has the added benefit of allowing for valve modifications in real-time. The flow is in contact with the surface of the reed for a distance p, beyond which it is assumed that the flow separates and forms a jet. For this reason, we are interested in the differential flow before that point. The force on a thin slice dy along the valve channel is given by F = A(y; 0)Ap(y). (7) where A(y; 0) is the cross section area of the valve channel at a point y along the channel for the generalized reed at an angle 0, and where Ap(y) is the pressure drop across this section of the reed. The force is applied to a volume of air A(y; 0)dy having mass pA(y; 0)dy, (8) where p is the air density. Newton's second law-force is the product of mass and acceleration-can then be applied to (7) and (8) to obtain A(y; )Ap(y) = pA(y; O)dy (9) where acceleration is given by the time derivative of the particle velocity, dv/idt, and is assumed constant over this section dy of the reed. We can then substitute particle velocity for volume flow scaled by area and integrate over the length of the channel to obtain.Fi = sign(0)toi, p 2.2 (5) where A is the cross-sectional area of the valve channel given by toH, H is the opening of the valve (see Figure 3) calculated from 0 and the geometry of the valve, and sign(0) determines the direction of the force: if 0 > 0, the force acts in the positive 0 direction and if 0 < 0 the force acts in the negative 0 direction. The force F acting on the valve is then obtained by summing (3), (4) and (5) and is given by F::::::to w:,iPm, - sign(0)towAbb --- wP, Pm. A y - ) (6) To specify the valve classification, the musician need only specify the equilibrium position O0 and whether the valve should be limited by 0 = 0 (for the blown open and closed cases). It may also be desirable to create a stop point that is valid only under certain conditions: overblowing, for example, could cause the valve to beat against the stop with enough force to push it past the limit, effectively blowing the valve into a new configuration. This and other variations could be implemented by making the reed stiffness k a function of valve angle 0. 3.2. Volume flow Many valve models (and particularly clarinet reeds) are implemented using a lookup table which matches the value for flow with the pressure drop across the valve [2, 4]. This is known as the quasi-static, Bernoulli-flow model because the value of flow U is established by relating the pressure difference and the volume flow under constantflow conditions. Though this implementation has produced satisfactory sound at low computational cost, it is not suitable for a generalized model as it is not physically accurate and does not provide access to certain desired parameter values. p(0) - p(A) = p dUl /v='* c/t 4=0 dyc/A(y: 0'), (10) where y = 0 is the channel entrance and y = p, is the point of flow separation. Bernoulli's equation is used to calculate the pressure entering the valve, po, and then the pressure at flow separation p(p) is replaced with the bore pressure Pb to obtain the differential equation governing volume flow dU 4(x) S-(p -Pb) A i t (11) 2pA(0)' where the flow is assumed to be in contact with the reed for a distance of p. The singularity in (11) as the valve opening approaches zero makes clear the need for a feathered valve, a method fully developed for the avian syrinx  and the clarinet reed  using the small area solution for dU(t)/dt. The update governing air flow is given by U(I 14A(t)T U(t)2'l U(t + 1) -= -(t) p, - pb) P 1)P 2p/A(t) --+ U(t)T' (12) where A(t) is the valve channel area at time t, and T is the sampling interval.
Page 00000004 mouth pressure reed frequency bore frequency valve type 220 1 0 0: blown closed 1: blown open 2: swinging nominal pos jet width flow on reed Figure 4. Pd object showing input parameters. It is clear that the valve channel area is critical to the volume flow and the sound of the instrument. As the reed angle 0 changes, the valve opening area changes according to the changing valve channel height, A(0) w- u.H(0). As illustrated in Figure 3, any number of channel area functions are possible by choice of channel profile, and, in particular, by the channel height function H(O). Setting H(0) = sin 09 for example, approximates the channel area of a clarinet reed, whereas the function H(0) 1 - cos 0 approximates that of a lip reed. 4. EXAMPLE AND CONCLUSIONS The model was implemented in Pd  and takes arguments as shown in Figure 4. Figure 5 shows spectrograms of the sound produced in response to a burst of mouth pressure for blown-closed (top) and blown-open (bottom) valves attached to identical bores. In both cases, the reed and resonant frequencies, f, and fb respectively, were quite different with fr > fb as is typically the case. As expected, in the blown closed case the reed frequency fr has very little effect on the sounding frequency which, as seen by the spectrogram, is closer to the frequency of the bore fb. Also as anticipated, in the blown open case the sounding frequency is closer to the reed resonant frequency f,.. The generalized pressure controlled valve presented here is capable of expressing any of the three valve classesblown open, blown closed, and a symmetric model-by changing the model parameters. The classes have significantly different pitch and timbre characteristics, made even more so by channing certain of the model's geometric quantities. This model contributes a very useful tool for creating, and blenlding among, a variety of instrument sounds under real time control. 5.REFERENCES  Pd. http://www.pure-data.org.  Jean-Pierre Dalmont, Jo6l Gilbert, and Sdbastien Oliver. Nonlinear characteristics of single-reed instruments: Quasi-static volume flow and reed open Blown Closed: fr=620 Hz, fb=220 Hz 150( N a1 > 100( 0 C S50( LL 150( N I " 50( LL Time (s) Blown Open: fr=420 Hz, fh=140 Hz U U.5 2.5 Time (s) Figure 5. In the blown closed case, top, the sounding frequency, as seen by the spectrogram, is closer to the frequency of the bore fb. In the blown open case, bottom, the sounding frequency is closer to the frequency of the reed fr. ing measurements. Journal of the Acoustical Society of America, 114(4):2253-2262, October 2003.  Neville H. Fletcher. Autonomous vibration of simple pressure-controlled valves in gas flows. Jounal of the Acoustical Society of America, 93(14):2172-2180, April 1993.  Neville H. Fletcher and Thomas D. Rossing. The Physics of Musical Instruments. Springer-Verlag, 1995.  Tamara Smyth, Jonathan Abel, and Julius 0. Smith. Discrete-time simulation of air-flow cut-off in pressure-controlled valves. In Proceedings of the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WA4SPAA'03), New Paltz, New York, October 2003.  Tamara Smyth, Jonathan Abel, and Julius O. Smith. The feathered clarinet reed. In Proceedings of the International Conference on Digital Audio Effects (DIAFx'04), Naples, Italy, October 2004.