A Digital Synthesis Model of Double-Reed Wind Instruments

Home , Double reed, Embouchure, Reed (mouthpiece)

EURASIP Journal on Applied Signal Processing 2004:7, 990–1000 c 2004 Hindawi Publishing Corporation

Ph. Guillemain LaboratoiredeM´ecanique et d’Acoustique, Centre National de la Recherche Scientiﬁque, 31 chemin Joseph-Aiguier, 13402 Marseille cedex 20, France Email: [email protected]

Received 30 June 2003; Revised 29 November 2003

We present a real-time synthesis model for double-reed wind instruments based on a nonlinear physical model. One specificity of double-reed instruments, namely, the presence of a confined air jet in the embouchure, for which a physical model has been proposed recently, is included in the synthesis model. The synthesis procedure involves the use of the physical variables via a digital scheme giving the impedance relationship between pressure and flow in the time domain. Comparisons are made between the behavior of the model with and without the confined air jet in the case of a simple cylindrical bore and that of a more realistic bore, the geometry of which is an approximation of an oboe bore. Keywords and phrases: double-reed, synthesis, impedance.

1. INTRODUCTION The physical model is first summarized in Section 2.In order to obtain the synthesis model, a suitable form of the The simulation of woodwind instrument sounds has been in- flow model is then proposed, a dimensionless version is writ- vestigated for many years since the pioneer studies by Schu- ten and the similarities with single-reed models (see, e.g., macher [1] on the clarinet, which did not focus on digital [7]) are pointed out. The resonator model is obtained by as- sound synthesis. Real-time-oriented techniques, such as the sociating several elementary impedances, and is described in famous digital waveguide method (see, e.g., Smith [2]and terms of the acoustic pressure and flow. Valim¨ aki¨ [3]) and wave digital models [4] have been intro- ffi Section 3 presents the digital synthesis model, which re- duced in order to obtain e cient digital descriptions of res- quires first discrete-time equivalents of the reed displacement onators in terms of incoming and outgoing waves, and used and the impedance relations. The explicit scheme solving the to simulate various wind instruments. nonlinear model, which is similar to that proposed in [6], is The resonator of a clarinet can be said to be approxi- then briefly summarized. mately cylindrical as a first approximation, and its embou- In Section 4, the synthesis model is used to investigate the chure is large enough to be compatible with simple airflow effects of the changes in the nonlinear characteristics induced models. In double-reed instruments, such as the oboe, the by the confined air jet. resonator is not cylindrical but conical and the size of the air jet is comparable to that of the embouchure. In this case, the dissipation of the air jet is no longer free, and the jet remains 2. PHYSICAL MODEL confined in the embouchure, giving rise to additional aero- The main physical components of the nonlinear synthesis dynamic losses. model are as follows. Here, we describe a real-time digital synthesis model for (i) The linear oscillator modeling the first mode of reeds double-reed instruments based on one hand on a recent vibration. study by Vergez et al. [5], in which the formation of the con- (ii) The nonlinear characteristics relating the flow to the fined air jet in the embouchure is taken into account, and on pressure and to the reed displacement at the mouth- the other hand on an extension of the method presented in piece. [6] for synthesizing the clarinet. This method avoids the need (iii) The impedance equation linking pressure and flow. for the incoming and outgoing wave decompositions, since it deals only with the relationship between the impedance vari- Figure 1 shows a highly simplified embouchure model for an ables, which makes it easy to transpose the physical model to oboe and the corresponding physical variables described in a synthesis model. Sections 2.1 and 2.2. A Digital Synthesis Model of Double-Reed Wind Instruments 991

The relationship between the mouth pressure pm and the y/2 pressure of the air jet pj (t) and the velocity of the air jet vj (t) p p v p q m H j , j r , and the volume flow q(t), classically used when dealing with y/2 single-reed instruments, is based on the stationary Bernoulli Reeds Backbore Main bore equation rather than on the Backus model (see, e.g., [10]for justification and comparisons with measurements). This re- Figure 1: Embouchure model and physical variables. lationship, which is still valid here, is 1 pm = pj (t)+ ρvj (t)2, 2.1. Reed model 2 (4) Although this paper focuses on the simulation of double- q(t) = Sj (t)vj (t) = αSi(t)vj (t), reed instruments, oboe experiments have shown that the dis- placements of the two reeds are symmetrical [5, 8]. In this where α, which is assumed to be constant, is the ratio be- case, a classical single-mode model seems to suffice to de- tween the cross section of the air jet Sj (t) and the reed open- scribe the variations in the reed opening. The opening is ing Si(t). based on the relative displacement y(t) of the two reeds when It should be mentioned that the aim of this paper is to adifference in acoustic pressure occurs between the mouth propose a digital sound synthesis model that takes the dis- pressure pm and the acoustic pressure pj (t) of the air jet sipation of the air jet in the reed channel into account. For formed in the reed channel. If we denote the resonance fre- a detailed physical description of this phenomenon, readers quency, damping coefficient, and mass of the reeds ωr , qr and can consult [5], from which the notation used here was bor- µr , respectively, the relative displacement satisfies the equa- rowed. tion 2.2.2. Flow model d2 y t dy t pm − pj t ( ) ( ) 2 ( ) + ωr qr + ωr y(t) =− . (1) In the framework of the digital synthesis model on which dt2 dt µr this paper focuses, it is necessary to express the volume flow q(t) as a function of the difference between the mouth pres- Based on the reed displacement, the opening of the reed sure pm and the pressure at the entrance of the resonator channel denoted Si(t) is expressed by pr (t). From (4), we obtain Si(t) = Θ y(t)+H × w y(t)+H ,(2) 2 w H v t 2 = p − p t where denotes the width of the reed channel, denotes the j ( ) ρ m j ( ) ,(5) distance between the two reeds at rest (y(t)andpm = 0) and Θ is the Heaviside function, the role of which is to keep the q2(t) = α2Si(t)2vj (t)2. (6) opening of the reeds positive by canceling it when y(t)+H< 0. Substituting the value of pj(t)givenby(3) into (5)gives

2.2. Nonlinear characteristics 2 q(t)2 vj (t)2 = pm − pr (t) − Ψ . (7) ρ S2 2.2.1. Physical bases r In the case of the clarinet or saxophone, it is generally rec- Using (6), this gives ognized that the acoustic pressure pr (t)andvolumevelocity v t r ( ) at the entrance of the resonator are equal to the pressure 2 q(t)2 p t v t q2(t) = α2Si(t)2 pm − pr (t) − Ψ ,(8) j ( )andvolumevelocity j ( ) of the air jet in the reed chan- ρ S2 nel (see, e.g., [9]). In oboe-like instruments, the smallness of r the reed channel leads to the formation of a confined air jet. from which we obtain the expression for the volume flow, According to a recent hypothesis [5], pr (t)isnolongerequal namely, the nonlinear characteristics in this case to pj (t), but these quantities are related as follows q(t) = sign pm − pr (t) 1 q(t)2 pj (t) = pr (t)+ ρΨ ,(3) S2 2 ra αSi(t) 2 (9) × pm − pr (t) . Ψα2S t 2/S2 ρ where Ψ is taken to be a constant related to the ratio between 1+ i( ) r the cross section of the jet and the cross section at the entrance of the resonator, q(t) is the volume flow, and ρ is the 2.3. Dimensionless model mean air density. In what follows, we will assume that the The reed displacement and the nonlinear characteristics are area Sra, corresponding to the cross section of the reed chan- converted into the dimensionless equations used in the syn- nel at the point where the flow is spread over the whole cross thesis model. For this purpose, we first take the reed displace- section, is equal to the area Sr at the entrance of the resonator. ment equation and replace the air jet pressure pj (t) by the 992 EURASIP Journal on Applied Signal Processing expression involving the variables q(t)andpr (t)(equation In addition to the parameter ζ, two other parameters βx (3)), and βu depend on the height H of the reed channel at rest. Although, for the sake of clarity in the notations, the vari- d2 y t dy t p − p t q t 2 ( ) ( ) 2 m r ( ) ( ) able t has been omitted, γ, ζ, βx,andβu are functions of time + ωr qr + ωr y(t) =− + ρΨ . dt2 dt µr 2µr Sr2 (but slowly varying functions compared to the other vari- (10) ables). Taking the difference between the jet pressure and the resonator pressure into account results in a flow which is no On similar lines to what has been done in the case of single- longer proportional to the reed displacement, and a reed dis- y t reed instruments [11], ( ) is normalized with respect to the placement which is no longer linked to pe(t) in an ordinary p p = Hω2µ static beating-reed pressure M defined by M r r . linear differential equation. We denote by γ the ratio, γ = pm/pM and replace y(t)by x(t), where the dimensionless reed displacement is defined 2.4. Resonator model x t = y t /H γ by ( ) ( ) + . We now consider the simplified resonator of an oboe-like in- With these notations, (10)becomes strument. It is described as a truncated, divergent, linear conical bore connected to a mouthpiece including the backbore 1 d2x(t) qr dx(t) pr (t) ρΨ q(t)2 + + x(t) = + (11) to which the reeds are attached, and an additional bore, the ωr2 dt2 ωr dt pM 2pM Sr2 volume of which corresponds to the volume of the missing and the reed opening is expressed by part of the cone. This model is identical to that summarized in [12]. Si(t) = Θ 1 − γ + x(t) × wH 1 − γ + x(t) . (12) 2.4.1. Cylindrical bore Likewise, we use the dimensionless acoustic pressure The dimensionless input impedance of a cylindrical bore pe(t) and the dimensionless acoustic flow ue(t)definedby is first expressed. By assuming that the radius of the bore p t ρc q t is large in comparison with the boundary layers thick- p t = r ( ) u t = ( ) nesses, the classical Kirchhoff theory leads to the value of e( ) p , e( ) S p , (13) M r M the complex wavenumber for a plane wave k(ω) = ω/c − / / (i3 2/2)ηcω1 2,whereη is a constant depending on the radius where c is the speed of the sound. / R of the bore η = (2/Rc3 2)( lv +(cp/cv − 1) lt). Typical val- With these notations, the reed displacement and the non- − ues of the physical constants, in mKs units, are lv = 4.10 8, linear characteristics are finally rewritten as follows, − lt = 5.6.10 8, Cp/Cv = 1.4 (see, e.g., [13]). The transfer function of a cylindrical bore of infinite length between 1 d2x(t) qr dx(t) + + x(t) = pe(t)+Ψβuue(t)2 (14) x = 0andx = L, which constitutes the propagation filter ω2 dt2 ωr dt r associated with the Green formulation, including the prop- and using (9)and(12), agation delay, dispersion, and dissipation, is then given by F(ω) = exp(−ik(ω)L). ue(t) = Θ 1 − γ + x(t) sign γ − pe(t) Assuming that the radiation losses are negligible, the di- ζ 1 − γ + x(t) mensionless input impedance of the cylindrical bore is clas- × γ − pe(t) 2 (15) sically expressed by 1+Ψβx 1 − γ + x(t) C ω = i k ω L . = F x(t), pe(t) , ( ) tan ( ) (18) C ω where ζ, βx and βu are defined by In this equation, ( ) is the ratio between the Fourier transforms Pe(ω)andUe(ω) of the dimensionless variables √ ρ cαw α2w2 ω2µ p t u t ζ = H 2 β = H2 β = H r r . e( )and e( )definedby(13). The input admittance of the , x , u −1 µr Sr ωr Sr2 2ρc2 cylindrical bore is denoted by C (ω). (16) Adifferent formulation of the impedance relation of a cylindrical bore, which is compatible with a time-domain This dimensionless model is comparable to the model implementation, and was proposed in [6], is used and ex- described, for example, in [7, 9] in the case of single-reed in- tended here. It consists in rewriting (18)as struments, where the dimensionless acoustic pressure pe(t), the dimensionless acoustic flow ue(t), and the dimensionless 1 exp − 2ik(ω)L C(ω) = − . (19) reed displacement x(t) are linked by the relations 1+exp − 2ik(ω)L 1+exp − 2ik(ω)L

1 d2x(t) qr dx(t) Figure 2 shows the interpretation of (19)intermsof + + x(t) = pe(t), ω2 dt2 ωr dt looped propagation ﬁlters. The transfer function of this r model corresponds directly to the dimensionless input ue(t) = Θ 1 − γ + x(t) sign γ − pe(t) (17) impedance of a cylindrical bore. It is the sum of two parts. ×ζ 1 − γ + x(t) γ − pe(t) . The upper part corresponds to the ﬁrst term of (19) and the A Digital Synthesis Model of Double-Reed Wind Instruments 993

2.4.3. Oboe-like bore

pe(t) The complete bore is a conical bore combined with a mouth- − exp − 2ik(ω)L ue(t) piece. The mouthpiece consists of a combination of two bores, − exp − 2ik(ω)L (i) a short cylindrical bore with length L1,radiusR1,sur- face S1, and characteristic impedance Z1. This is the backbore to which the reeds are attached. Its radius is small in comparison with that of the main conical Figure 2: Impedance model of a cylindrical bore. bore, the characteristic impedance of which is denoted Z2 = ρc/Sr ,and L ue(t) x pe(t) (ii) an additional short cylindrical bore with length 0,ra- e D c dius R0,surfaceS0, and characteristic impedance Z0. Its radius is large in comparison with that of the backbore. This role serves to add a volume corresponding to the truncated part of the complete cone. This − C 1(ω) −1 makes it possible to reduce the geometrical dispersion responsible for inharmonic impedance peaks in the Figure 3: Impedance model of a conical bore. combination backbore/conical bore.

The impedance C1(ω) of the short cylindrical backbore is based on an approximation of i tan(k (ω)L )withsmall lower part corresponds to the second term. The filter having 1 1 values of k (ω)L . It takes the dissipation into account and the transfer function −F(ω)2 =−exp(−2ik(ω)L) stands for 1 1 neglects the dispersion. Assuming that the radius R is large the back and forth path of the dimensionless pressure waves, 1 in comparison with the boundary layers thicknesses, using with a sign change at the open end of the bore. (19), C (ω) is first approximated by Although k(ω) includes both dissipation and dispersion, 1 √ the dispersion is small (e.g., in the case of a cylindrical bore 1 − exp − η c ω/2L exp − 2iωL /c with a radius of 7 mm, η = 1.34.10−5), and the peaks of the C (ω) 1 √ 1 1 , (21) 1 1+exp − η c ω/2L exp − 2iωL /c input impedance of a cylindrical bore can be said to be nearly 1 1 1 harmonic. In particular, this intrinsic dispersion can be ne- which, since L1 is small, is finally simplified as glected, unlike the dispersion introduced by the geometry of √ the bore (e.g., the input impedance of a truncated conical − − η c ω/ L − iωL /c C ω 1 exp 1 2 √1 1 2 1 . bore cannot be assumed to be harmonic). 1( ) (22) 1+exp − η1c ω/2L1 √ 2.4.2. Conical bore G ω = − −η c ω/ L / By noting√ ( ) (1 exp( 1 2 1)) (1 + From the input impedance of the cylindrical bore, the di- exp(−η1c ω/2L1)), and H(ω) = (L1/c)(1 − G(ω)), the mensionless input impedance of the truncated, divergent, expression of C1(ω)reads conical bore can be expressed as a parallel combination of a cylindrical bore and an “air” bore, C1(ω) = G(ω)+iωH(ω). (23)

1 This approximation avoids the need for a second delay line S2(ω) = , (20) 1/ iωxe/c +1/C(ω) in the sampled formulation of the impedance. The transmission line equation relates the acoustic pres- x where e is the distance between the apex and the input. It is sure pn and the flow un at the entrance of a cylindrical bore θ expressed in terms of the angle of the cone and the input (with characteristic impedance Zn,lengthLn,andwavenum- R x = R/ θ/ radius as e sin( 2). ber kn) to the acoustic pressure pn+1 and the flow un+1 at The parameter η involved in the definition of C(ω)in the exit of a cylindrical bore. With dimensioned variables, (20), which depends on the radius and characterizes the it reads losses included in k(ω), is calculated by considering the ra- p ω = k ω L p ω iZ k ω L u ω dius of the cone at (5/12)L. This value was determined em- n( ) cos n( ) n n+1( )+ n sin n( ) n n+1( ), pirically, by comparing the impedance given by (20)withan i un(ω) = sin kn(ω)Ln pn+1(ω)+cos kn(ω)Ln un+1(ω), input impedance of the same conical bore obtained with a se- Zn ries of elementary cylinders with different diameters (stepped (24) cone), using the transmission line theory. Denoting by D the differentiation operator D(ω) = iω yielding and rewriting (20) in the form S2(ω) = D(ω)(xe/c)/(1 + D ω x /c C−1 ω pn(ω) pn (ω)/un (ω)+iZn tan kn(ω)Ln ( )( e ) ( )), we propose the equivalent scheme in = +1 +1 . (25) Figure 3. un(ω) 1+(i/Zn)tan kn(ω)Ln pn+1(ω)/un+1(ω) 994 EURASIP Journal on Applied Signal Processing

pe(t) C ω Z ω 1( ) Z1 e( ) ue(t) 1 pe(t) Z2 H, pm S ω 2( ) Z2 Reed −V ζ,βx,βu,γ model D(ω) ρc2 x t u t ( ) e( ) f p t Figure 4: Impedance model of the simpliﬁed resonator. e( )

Figure 5: Nonlinear synthesis model. Using the notations introduced in (20)and(23), the input impedance of the combination backbore/main conical bore reads p ω Z S ω Z C ω 2.5. Summary of the physical model 1( ) 2 2( )+ 1 1( ) = , (26) The complete dimensionless physical model consists of three u1(ω) 1+ Z2/Z1 S2(ω)C1(ω) equations, which is simplified as p1(ω)/u1(ω) = Z2S2(ω)+Z1C1(ω), 2 q since Z1 Z2. 1 d x(t) r dx(t) + + x(t) = pe(t)+Ψβuue(t)2, (31) In the same way, the input impedance of the whole bore ω2 dt2 ωr dt r reads ζ − γ x t 1 + ( ) ue(t) = p (ω) p (ω)/u (ω)+iZ tan k (ω)L 2 0 = 1 1 0 0 0 , (27) 1+Ψβx 1 − γ + x(t) u (ω) 1+(i/Z )tan k (ω)L p (ω)/u (ω) 0 0 0 0 1 1 × Θ 1 − γ + x(t) sign γ − pe(t) × γ − pe(t) , which, since Z Z , is simplified as 0 1 (32) p0(ω) p1(ω)/u1(ω) P ω = Z ω U ω . = . (28) e( ) e( ) e( ) (33) u0(ω) 1+(i/Z0)tan k0(ω)L0 p1(ω)/u1(ω) These equations enable us to introduce the reed and the Since L is small and the radius is large, the losses in- 0 nonlinear characteristics in the form of two nonlinear loops, cluded in k0(ω) can be neglected, and hence k0(ω) = ω/c as shown in Figure 5. The first loop relates the output pe to and tan(k0(ω)L0) = (ω/c)L0. Under these conditions, the in- the input ue of the resonator, as in the case of single-reed put impedance of the bore is given by instruments models. The second nonlinear loop corresponds u2 x p (ω) 1 to the e -dependent changes in . The output of the model is 0 = p u x u (ω) 1/ p (ω)/u (ω) + iω/c L /Z given by the three coupled variables e, e,and . The control 0 1 1 0 0 (29) parameters of the model are the length L of the main conical 1 = . bore and the parameters H(t)andpm(t)fromwhichζ(t), / Z S ω Z C ω iω/c L S /ρc 1 2 2( )+ 1 1( ) + 0 0 βx(t), βu(t), and γ(t) are calculated. In the context of sound synthesis, it is necessary to calcu- If we take V to denote the volume of the short addi- late the external pressure. Here we consider only the propa- tional bore V = L S and rewrite (29) with the dimension- 0 0 gation within the main “cylindrical” part of the bore in (20). less variables Pe and Ue (Ue = Z u ), the dimensionless in- 2 0 Assuming again that the radiation impedance can be ne- put impedance of the whole resonator relating the variables glected, the external pressure corresponds to the time deriva- Pe(ω)andUe(ω)becomes tive of the flow at the exit of the resonator pext(t) = dus(t)/dt. P ω Using the transmission line theory, one directly obtains Z ω = e( ) e( ) U ω e( ) U ω = − ik ω L P ω U ω . /Z (30) s( ) exp ( ) e( )+ e( ) (34) = 1 2 . iωV/ ρc2 / Z C ω Z S ω +1 1 1( )+ 2 2( ) From the perceptual point of view, the quantity exp(−ik(ω)L) can be left aside, since it stands for the After rearranging (30), we propose the equivalent scheme in losses corresponding to a single travel between the em- Figure 4. bouchure and the open end. This simplification leads to the It can be seen from (30) that the mouthpiece is equivalent following expression for the external pressure to a Helmholtz resonator consisting of a hemispherical cavity 3 with volume V and radius Rb such that V = (4/6)πRb,con- d L p t = pe t ue t . nected to a short cylindrical bore with length 1 and radius ext( ) dt ( )+ ( ) (35) R1. A Digital Synthesis Model of Double-Reed Wind Instruments 995

3. DISCRETE-TIME MODEL 6

In order to draw up the synthesis model, it is necessary to 5 use a discrete formulation in the time domain for the reed displacement and the impedance models. The discretization schemes used here are similar to those described in [6]for 4 the clarinet, and summarized in [12] for brass instruments and saxophones. 3

3.1. Reed displacement 2 We take e(t) to denote the excitation of the reed e(t) = pe(t)+Ψβuue(t)2. Using (31), the Fourier transform of the 1 ratio X(ω)/E(ω) can be readily written as

2 0 X(ω) ωr = . (36) 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 E(ω) ωr2 − ω2 + iωqr ωr Hz

An inverse Fourier transform provides the impulse response Figure 6: Approximated (solid line) and exact (dotted line) reed h(t) of the reed model frequency response with parameter values fr = 2500 Hz, qr = 0.2, and fe = 44.1kHz. 2ωr 1 1 h(t) = exp − ωr qr t sin 4 − qr2ωr t . (37) 4 − qr2 2 2 stability condition makes this discretization scheme unsuit- Equation (37) shows that h(t)satisfiesh(0) = 0. This prop- able for use at low sampling rates, but in practice, at the CD erty is most important in what follows. In addition, the range quality sample rate, this problem does not arise for a reed res- q onance frequency of up to 5 kHz with a quality factor of up to of variations allowed for r is ]0, 2[. . The discrete-time version of the impulse response uses 0 5. For a more detailed discussion of discretization schemes, two centered numerical differentiation schemes which pro- readers can consult, for example, [14]. vide unbiased estimates of the first and second derivatives The bilinear transformation does not provide a suitable when they are applied to sampled second-order polynomi- discretization scheme for the reed displacement. In this case, als the impulse response does not satisfy the property of the con- tinuous model h(0) = 0. f iω e z − z−1 , 3.2. Impedance 2 (38) 2 2 −1 −ω fe z − 2+z , A time domain equivalent to the inverse Fourier transform of impedance Ze(ω)givenby(30)isnowrequired.Herewe z = iω ω = ω/ f f where exp( ˜), ˜ e,and e is the sampling fre- express pe(n) as a function of ue(n). quency. The losses in the cylindrical bore element contributing to With these approximations, the digital transfer function the impedance of the whole bore are modeled with a digi- of the reed is given by tal low-pass filter. This filter approximates the back and forth F ω 2 = − ik ω L X z losses described by ( ) exp( 2 ( ) ) and neglects the ( ) = (small) dispersion. So that they can be adjusted to the ge- E z ( ) ometry of the resonator, the coefficients of the filter are ex- z−1 pressed analytically as functions of the physical parameters, − − , fe2/ωr2 + feqr / 2ωr −z 1 2 fe2/ωr2 −1 −z 2 feqr / 2ωr −fe2/ωr2 rather than using numerical approximations and minimiza- (39) tions. For this purpose, a one-pole filter is used, yielding a difference equation of the type b0 exp(−iωD˜ ) F˜(ω˜) = , (41) 1 − a1 exp(−iω˜) x n = b e n − a x n − a x n − . ( ) 1a ( 1) + 1a ( 1) + 2a ( 2) (40) where ω˜ = ω/ fe,andD = 2 fe(L/c) is the pure delay corre- This difference equation keeps the property h(0) = 0. sponding to a back and forth path of the waves. Figure 6 shows the frequency response of this approxi- The parameters b0 and a1 are calculated so that mated reed model (solid line) superimposed with the exact |F(ω)2|2 =|F˜(ω˜)|2 for two given values of ω,andareso- one (dotted line). lutions of the system This discrete reed model is stable under the condi- 2 tion ωr

The acoustic pressure and flow in the mouthpiece at sam- 2 pling time n are then finally obtained by the sequential cal- V x n W u n culation of ˜ with (46), ( )with(47), with (50), e( ) 1.5 with (51), and pe(n)with(48). Theexternalpressurep (n) is calculated using the dif- ext 1 ference between the sum of the internal pressure and the flow 0 500 1000 1500 2000 2500 3000 3500 4000 at sampling time n and n − 1. Hz

(b) 4. SIMULATIONS The eﬀects of introducing the conﬁned air jet into the non- Figure 8: (a) represents impedance (dotted line) and ratio between the spectra of pe and ue (solid line), while (b) represents reed trans- linear characteristics are now studied in the case of two dif- 2 fer (dotted line) and ratio of spectra between x and pe +Ψβuue (solid ferent bore geometries. In particular, we consider a cylindri- line). cal resonator, the impedance peaks of which are odd harmonics, and a resonator, the impedance of which contains all the harmonics. We start by checking numerically the va- lidity of the resolution scheme in the case of the cylindrical −10 bore. (Sound examples are available at http://omicron.cnrs- mrs.fr/∼guillemain/eurasip.html.) −8 4.1. Cylindrical resonator

We first consider a cylindrical resonator, and make the pa- −6 Ψ rameter vary linearly from 0 to 4000 during the sound kHz synthesis procedure (1.5 seconds). The transient attack cor- −4 responds to an abrupt increase in γ at t = 0. During the decay phase, starting at t = 1.3 seconds, γ decreases linearly towards zero. Its steady-state value is γ = 0.56. The other −2 − parameters are constant, ζ = 0.35, βx = 7.5.10 4, βu = − 6.1.10 3. The reed parameters are ωr = 2π.3150 rad/second, 0 qr = 0.5. The resonator parameters are R = 0.0055 m, 00.20.40.60.811.21.4 L = 0.46 m. s Figure 8 shows superimposed curves, in the top figure, the digital impedance of the bore is given in dotted lines, Figure 9: Spectrogram of the external pressure for a cylindrical bore and the ratio between the Fourier transforms of the sig- and a beating reed where γ = 0.56. nals pe(n)andue(n) in solid lines; in the bottom figure, the digital reed transfer function is given in dotted lines, and the ratio of the Fourier transforms of the signals x(n)and pe(n)+Ψ(n)βuue(n)2 (including attack and decay transients) 4.1.1. The case of the beating reed in solid lines. The first example corresponds to a beating reed situation, As we can see, the curves are perfectly superimposed. which is simulated by choosing a steady-state value of γ There is no need to check the nonlinear relation between greater than 0.5 (γ = 0.56). ue(n), pe(n), and x(n), which is satisfied by construction Figure 9 shows the spectrogram (dB) of the external pres- since ue(n) is obtained explicitly as a function of the other sure generated by the model. The values of the spectrogram variables in (51). In the case of the oboe-like bore, the re- are coded with a grey-scale palette (small values are dark and sults obtained using the resolution scheme are equally accu- high values are bright). The bright horizontal lines corre- rate. spond to the harmonics of the external pressure. 998 EURASIP Journal on Applied Signal Processing

×10−2 ×10−2 18 18 −10 16 16 14 14 12 12 −8 10 10 8 8 6 6 − 4 4 6 2 2 kHz 0 0 −8 −6 −4 −20 2 4 6 −8 −6 −4 −20 2 4 6 −4 ×10−1 ×10−1

(a) (b) −2

Figure 10: ue(n)versuspe(n): (a) t = 0.25 second, (b) t = 0.5 second. 0 00.20.40.60.811.21.4 s

×10−2 ×10−2 Figure 12: Spectrogram of the external pressure for a cylindrical 16 14 bore and a nonbeating reed where γ = 0.498. 14 12 12 10 10 8 − − 8 ×10 2 ×10 2 6 6 16 16 4 4 14 14 2 2 12 12 0 0 10 10 −8 −6 −4 −20 2 4 6 −8 −6 −4 −20 2 4 6 8 8 − − ×10 1 ×10 1 6 6 4 4 (a) (b) 2 2 0 0 −5 −4 −3 −2 −1012345 −5 −4 −3 −2 −1012345 Figure 11: ue(n)versuspe(n): (a) t = 0.75 second, (b) t = 1 second. ×10−1 ×10−1

(a) (b) Ψ ff Increasing the value of mainly a ects the pitch and u n p n t = . t = . ff Figure 13: e( )versus e( ): (a) 0 25 second, (b) 0 5 only slightly a ects the amplitudes of the harmonics. In par- second. ticular, at high values of Ψ, a small increase in Ψ results in a strong decrease in the pitch. A cancellation of the self-oscillation process can be ob- 4.1.2. The case of the nonbeating reed served at around t = 1.2 seconds, due to the high value of Ψ, since it occurs before γ starts decreasing. The second example corresponds to a nonbeating reed situa- Odd harmonics have a much higher level than even har- tion, which is obtained by choosing a steady-state value of γ monics as occuring in the case of the clarinet. Indeed, the smaller than 0.5 (γ = 0.498). even harmonics originate mainly from the flow, which is Figure 12 shows the spectrogram of the external pressure taken into account in the calculation of the external pressure. generated by the model. Increasing the value of Ψ results in However, it is worth noticing that the level of the second har- a sharp change in the level of the high harmonics at around monic increases with Ψ. t = 0.4 seconds, a slight change in the pitch, and a cancella- t = . Figures 10 and 11 show the flow ue(n) versus the pressure tion of the self-oscillation process at around 0 8 seconds, Ψ pe(n), obtained during a small number (32) of oscillation pe- corresponding to a smaller value of than that observed in riods at around t = 0.25 seconds, t = 0.5 seconds, t = 0.75 the case of the beating reed. seconds and t = 1 seconds. The existence of two different Figure 13 shows the flow ue(n) versus the pressure pe(n) paths, corresponding to the opening or closing of the reed, is at around t = 0.25 seconds and t = 0.5 seconds. Since the due to the inertia of the reed. This phenomenon is observed reed is no longer beating, the whole path remains continu- also on single-reed instruments (see, e.g., [14]). A disconti- ous. The changes in its shape with respect to Ψ are smaller nuity appears in the whole path because the reed is beating. than in the case of the beating reed. This cancels the opening (and hence the flow) while the pressure is still varying. 4.2. Oboe-like resonator The shape of the curve changes with respect to Ψ. This Inordertocomparetheeffects of the confined air jet with the shape is in agreement with the results presented in [5]. geometry of the bore, we now consider an oboe-like bore, A Digital Synthesis Model of Double-Reed Wind Instruments 999

0.4 0.2 −10 0 −0.2 −0.4 −8 00.511.5 s −6 (a) kHz 0.2 −4 0.1 0 −2 −0.1 −0.2 0 500 1000 1500 2000 2500 3000 3500 4000 0 samples 00.20.40.60.811.21.4 s (b)

0.1 Figure 15: Spectrogram of the external pressure for an oboe-like γ = . 0.05 bore where 0 4. 0 −0.05 −0.1 ×10−2 ×10−2 0 500 1000 1500 2000 2500 3000 3500 4000 18 18 samples 16 16 14 14 12 12 (c) 10 10 8 8 Figure 14: (a) represents external acoustic pressure, and (b), (c) 6 6 4 4 represent attack and decay transients. 2 2 0 0 −16 −12 −8 −40 4−14 −10 −6 −22 ×10−1 ×10−1

(a) (b) the input impedance, and geometric parameters of which correspond to Figure 7. The other parameters have the same u n p n t = . t = . values as in the case of the cylindrical resonator, and the Figure 16: e( )versus e( ): (a) 0 25 second, (b) 0 5 second. steady-state value of γ is γ = 0.4. Figure 14 shows the pressure pext(t). Increasing the effect Ψ of the air jet confinement with , and hence the aerodynam- ×10−2 ×10−2 ical losses, results in a gradual decrease in the signal ampli- 18 16 tude. The change in the shape of the waveform with respect 16 14 14 to Ψ can be seen on the blowups corresponding to the attack 12 12 10 and decay transients. 10 8 Figure 15 shows the spectrogram of the external pressure 8 6 6 generated by the model. 4 4 Since the impedance includes all the harmonics (and not 2 2 0 0 only the odd ones as in the case of the cylindrical bore), −12 −8 −40 4−10 −8 −6 −4 −20 2 4 the output pressure also includes all the harmonics. This ×10−1 ×10−1 makes for a considerable perceptual change in the timbre (a) (b) in comparison with the cylindrical geometry. Since the input impedance of the bore is not perfectly harmonic, it is u n p n t = . t = not possible to determine whether the “moving formants” Figure 17: e( )versus e( ): (a) 0 75 second, (b) 1 second. are caused by a change in the value of Ψ or by a “phasing effect” resulting from the slight inharmonic nature of the impedance. Figures 16 and 17 show the flow ue(n) versus the pressure Increasing the value of Ψ affects the amplitude of the har- pe(n)ataroundt = 0.25 seconds, t = 0.5 seconds, t = 0.75 monics and slightly changes the pitch. In addition, as in the seconds, and t = 1 seconds. The shape and evolution with Ψ case of the cylindrical bore with a nonbeating reed, a large of the nonlinear characteristics are similar to what occurs in value of Ψ brings the self-oscillation process to an end. the case of a cylindrical bore with a beating reed. 1000 EURASIP Journal on Applied Signal Processing

5. CONCLUSION [7] J. Kergomard, “Elementary considerations on reed-instrument oscillations,” in Mechanics of Musical Instruments, The synthesis model described in this paper includes the for- A. Hirschberg, J. Kergomard, and G. Weinreich, Eds., mation of a confined air jet in the embouchure of double- Springer-Verlag, New York, NY, USA, 1995. reed instruments. A dimensionless physical model, the form [8] A. Almeida, C. Vergez, R. Causse,´ and X. Rodet, “Physical of which is suitable for transposition to a digital synthesis study of double-reed instruments for application to sound- model, is proposed. The resonator is modeled using a time synthesis,” in Proc. International Symposium in Musical Acous- domain equivalent of the input impedance and does not re- tics, pp. 221–226, Mexico City, Mexico, December 2002. [9] A. Hirschberg, “Aero-acoustics of wind instruments,” in Me- quire the use of wave variables. This facilitates the model- chanics of Musical Instruments, A. Hirschberg, J. Kergomard, ing of the digital coupling between the bore, the reed and and G. Weinreich, Eds., Springer-Verlag, New York, NY, USA, the nonlinear characteristics, since all the components of the 1995. model use the same physical variables. It is thus possible to [10] S. Ollivier, Contribution al’` étude des oscillations des instru- obtain an explicit resolution of the nonlinear coupled sys- ments avent` a` anche simple, Ph.D. thesis, l’Universitedu´ tem thanks to the specific discretization scheme of the reed Maine, France, 2002. model. This is applicable to other self-oscillating wind in- [11] T. A. Wilson and G. S. Beavers, “Operating modes of the clarinet,” Journal of the Acoustical Society of America, vol. 56, no. struments using the same flow model, but it still requires to 2, pp. 653–658, 1974. be compared with other methods. [12] Ph. Guillemain, J. Kergomard, and Th. Voinier, “Real-time This synthesis model was used in order to study the in- synthesis models of wind instruments based on physical mod- fluence of the confined jet on the sound generated, by carry- els,” in Proc. Stockholm Music Acoustics Conference,Stock- ing out a real-time implementation. Based on the results of holm, Sweden, 2003. informal listening tests with an oboe player, the sound and [13] A. D. Pierce, Acoustics—An Introduction to Its Physical Prin- ciples and Applications, McGraw-Hill, New York, NY, USA, dynamics of the transients obtained are fairly realistic. The 1981, reprinted by Acoustical Society of America, Woodbury, simulations show that the shape of the resonator is the main NY, USA, 1989. factor determining the timbre of the instrument in steady- [14] F. Avanzini and D. Rocchesso, “Efficiency, accuracy, and sta- state parts, and that the confined jet plays a role at the con- bility issues in discrete time simulations of single reed instru- trol level of the model, since it increases the oscillation step ments,” JournaloftheAcousticalSocietyofAmerica, vol. 111, and therefore plays an important role mainly in the transient no. 5, pp. 2293–2301, 2002. parts. [15] G. Borin, G. De Poli, and D. Rocchesso, “Elimination of delay- free loops in discrete-time models of nonlinear acoustic sys- tems,” IEEE Trans. Speech and Audio Processing, vol. 8, no. 5, ACKNOWLEDGMENTS pp. 597–605, 2000. The author would like to thank Christophe Vergez for helpful discussions on the physical flow model, and Jessica Blanc for Ph. Guillemain was born in 1967 in Paris. reading the English. Since 1995, he has been working as a full time researcher at the Centre National de la Recherche Scientifique (CNRS) in Mar- REFERENCES seille, France. 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