EURASIP Journal on Applied Signal Processing 2004:7, 990–1000 c 2004 Hindawi Publishing Corporation A Digital Synthesis Model of Double-Reed Wind Instruments 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 en- trance 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 .