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."The fundamental equation governing the action of the reed is continuity of volumevelocity, i.e.,(10.75)where(10.76)and(10.77)is the volume velocity corresponding to the incoming pressure wave and outgoingp+bpressure wave pb.(The physical pressure in the bore at the mouthpiece is of course+pb = + p.) The wave impedance of the bore air-column is denoted Rb (computablepb bas the air density times sound speed c divided by cross-sectional area).In operation, the mouth pressure pm and incoming traveling bore pressure arep+bgiven, and the reed computation must produce an outgoing bore pressure pb whichsatisfies (10.75), i.e., such that(10.78) PRINCIPLES OF DIGITAL WAVEGUIDE MODELS OF MUSICAL INSTRUMENTS459Solving for p is not immediate because of the dependence of R on p which,m "bin turn, depends on pb.A graphical solution technique was proposed [Friedlander,1953, Keller, 1953, McIntyre et al., 1983] which, in effect, consists of finding theintersection of the two terms of the equation as they are plotted individually on the samegraph, varying pb.This is analogous to finding the operating point of a transistor byintersecting its operating curve with the  load line determined by the load resistance.It is helpful to normalize (10.78) as follows: Defineand note thatwhere Then (10.78) can be multiplied+through by Rb and written as 0 = G(p" ) + p"  p , or"(10.79)+The solution is obtained by plotting G(x ) and p  x on the same graph, finding"the point of intersection at (x, y) coordinates (p , G(p )), and computing finally the" "outgoing pressure wave sample as(10.80)An example of the qualitative appearance of G(x) overlaying p+  x is shown in"Fig.10.19.Figure 10.19 Normalized reed impedanceoverlaid with the  bore load line APPLICATIONS OF DSP TO AUDIO AND ACOUSTICS460Scattering-Theoretic Formulation of the Reed.Equation (10.78) can be solved forpb to obtain(10.81)(10.82)(10.83)where(10.84)We interpret Á(p" ) as a signal-dependent reflection coefficient.Since the mouthpiece of a clarinet is nearly closed, R » R which implies r H" 0m band Á H" 1.In the limit as R goes to infinity relative to Rb , (10.82) reduces to them+simple form of a rigidly capped acoustic tube, i.e., p = p.bbp+Computational Methods.Since finding the intersection of G(x) and  x re-"quires an expensive iterative algorithm with variable convergence times, it is not wellsuited for real-time operation.In this section, fast algorithms based on precomputednonlinearities are described.Let h denote half-pressure p/2, i.e., and Then (10.83)becomes(10.85)+Subtracting this equation from p givesb(10.86)The last expression above can be used to precompute Á as a function of+ +pb = pm /2  pb.Denoting this newly defined function as(10.87)(10.85) becomes(10.88) PRINCIPLES OF DIGITAL WAVE GUIDE MODELS OF MUSICAL INSTRUMENTS461This is the form chosen for implementation in Fig.10.17.The control variable is mouth++half-pressure h , and h = h  p is computed from the incoming bore pressurem m" busing only a single subtraction.The table is indexed by h+ , and the result of the lookup"is then multiplied by h+ Finally, the result of the multiplication is subtracted from"h to give the outgoing pressure wave into the bore.The cost of the reed simulationmis only two subtractions, one multiply, and one table lookup per sample.Because the table contains a coefficient rather than a signal value, it can be moreheavily quantized both in address space and word length than a direct lookup of a+p"signal value such as p ( ) or the like.A direct signal lookup, though requiring"much higher resolution, would eliminate the multiply associated with the scatteringcoefficient.Figure 10.20 Simple, qualitatively chosen reed table for the digital waveguide clarinet.In the field of computer music, it is customary to use simple piecewise linearfunctions for functions other than signals at the audio sampling rate, e.g., for amplitudeenvelopes, FM-index functions, and so on [Roads and Strawn, 1985, Roads, 1989].Along these lines, good initial results were obtained using the simplified qualitativelychosen table(10.89)cdepicted in Fig.10.20 for m = 1/(hc + 1).The corner point h" is the smallest"pressure difference giving reed closure.7 Embouchure and reed stiffness correspondcto the choice of offset h and slope m.For simplicity, an additive offset for shifting"the curve laterally is generally used as an embouchure parameter.Brighter tones are APPLICATIONS OF DSP TO AUDIO AND ACOUSTICS462obtained by increasing the curvature of the function as the reed begins to open; forexample, one can use for increasing k e" 1.Another approach is to replace the table-lookup contents by a piecewise polynomialapproximation.While less general, good results have been obtained in practice [Cook,1992, Cook, 1996].For example, one of the SynthBuilder clarinet patches employsthis technique using a cubic polynomial [Porcaro et al., 1995].An intermediate approach between table lookups and polynomial approximationsis to use interpolated table lookups.Typically, linear interpolation is used, but higherorder polynomial interpolation can also be considered [Laakso et al., 1996].Practical Details.To finish off the clarinet example, the remaining details of theSynthBuilder clarinet patch  Clarinet2.sb are described.The input mouth pressure is summed with a small amount of white noise, corre-sponding to turbulence.For example, 0.1% is generally used as a minimum, and largeramounts are appropriate during the attack of a note [ Pobierz caÅ‚ość w formacie PDF ]

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