Numerical simulation of transients in single-reed woodwind instruments
Sebastian SCHMUTZHARD(1) , Montserrat PÀMIES-VILÀ(2) , Alex HOFMANN(3) , Vasilis CHATZIIOANNOU(4)(1) Acoustics Research Institute, Austrian Academy of Sciences, Austria, sebastian.schmutzhard@oeaw.ac.at(2) Department of Music Acoustics, University of Music and Performing Arts Vienna, Austria, pamies-vila@mdw.ac.at(3) Department of Music Acoustics, University of Music and Performing Arts Vienna, Austria, hofmann-alex@mdw.ac.at(4) Department of Music Acoustics, University of Music and Performing Arts Vienna, Austria, chatziioannou@mdw.ac.at

Abstract
In this work, we discuss a mathematical model of the sound production in single-reed woodwind instruments
focusing on player-instrument interactions. The model consists of a coupled system of an ordinary and a partial
differential equation. The motion of the reed is described by a nonlinear harmonic oscillator, where the nonlinearity stems from the collision of the reed with the mouthpiece lay and the coupling between air flow andpressure. The dynamics of air pressure and air velocity inside the instrument are described by a boundary valueproblem for the wave equation. We introduce a method to model the interaction between the player’s tongue andthe reed, which leads to a further nonlinearity in the reed equation. Moreover, we study strategies to estimate
the model parameters relevant for the production of realistic tones, as well as for realistic transients in between
tones. Finally, using the proposed model, recordings of real players are resynthesised.
Keywords: Single-reed, Transients, Articulation

1INTRODUCTION

In order to synthesise sounds of musical instruments, numerical simulations of the physical processes that generate the respective sounds have been developed in the past decades [1, 13, 14]. Besides obtaining realistic
sound waves, such an approach also allows to study the sound excitations mechanism, in terms of the underlying physics. In this framework, the player-instrument interaction can be seen as a further component that isincluded in the physical model. The scope of this paper is to incorporate player control in physical modelsof single-reed woodwind instruments. One main part of the player-instrument interaction takes place in theplayer’s mouth by either changing the embouchure or by directly altering the vibration of the reed with thetongue [6, 11]. In this work, we describe a method to include the tongue-reed interaction into the physics-based
synthesis of sounds of single-reed woodwind instruments. In an attempt to resynthesise recordings from realplayers, an inverse modelling approach is applied.

2PHYSICAL MODEL OF A SINGLE-REED WOODWIND INSTRUMENT

2.1 Vibrations of a reed
We identify the whole reed with its tip, and model the vibration of the reed as a harmonic oscillator for thedisplacement y(t) of the reed tip. See Figure 1 for a sketch of the system. During motion, the reed collides withthe mouthpiece lay. The collision force is modelled following the Hunt-Crossley impact model [7]. Denoting
the reed area by Sr and assuming an effective reed mass m, damping γ and stiffness k, the pressure differencep∆ = pb − pin across the reed drives the reed tip y(t) according todydyd2 y= Sr p∆ .
(1)
m 2 + mγ + ky + klay by − ylay cαlay 1 + rlay
dt
dt
dt
The parameters klay , rlay and αlay denote the contact stiffness, contact damping and the collision exponent of
the Hunt-Crossley model for the collision between the reed and the mouthpiece lay. Furthermore, byc = yH(y),5515

Figure 1. Single-reed – mouthpiece system. y(t) denotes the reed displacement, yc the displacement when
the reed closes the mouthpiece, pb the blowing pressure, pin the pressure at the beginning of the tube, uf theBernoulli flow and ur the reed induced flow.
where H denotes the Heaviside step function. The collision force becomes active whenever y > ylay . Two typesof air flow into the mouthpiece are considered. First, the motion of the reed induces the air flow ur = Sr dy
dt .
The q
pressure difference p∆ introduces air flow into the mouthpiece according to Bernoulli’s principle, uf =σ λ h 2|pρ∆ | . Here, σ = sign p∆ , λ denotes the reed width and h = byc − yc denotes the reed opening, hence theproduct λ h is the cross-sectional area of the opening. The total air flow into the mouthpiece is therefore givenbysdy
2 |p∆ |
uin = ur + uf = Sr + σ λ h,(2)dtρ
where ρ denotes air density.
2.2 Dynamics of pressure and particle velocity inside a tube
The corpus of the single-reed woodwind instrument is assumed to be a tubular duct of length L and crosssectional area S(x), 0 ≤ x ≤ L. The spatio-temporal behaviour of the pressure p and particle velocity v inside
the tube is modelled by [2, 12]∂ p(x,t)∂ v(x,t)+ρ+ zv ∗ v(x,t) = 0,∂x∂t∂ (S(x)v(x,t)) S(x) ∂ p(x,t)+ 2+ S(x)yθ ∗ p(x,t) = 0,∂xρc∂t(3a)(3b)where c is the speed of sound. The convolutions zv ∗ v and yθ ∗ p are performed with respect to the time variablet and model viscous and thermal losses, respectively, inside the tube. The impedance zv and the addmitance yθare defined in the frequency-domain, see [8]. To complete the model, boundary conditions have to be imposed.
At the radiating end x = L, a stipulated radiation impedance zr leads to the boundary conditionp(L,t) = S(L)zr ∗ v(L,t).
Specific forms of zr can be found in [1, Section 9.1.3].

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(4)

2.3 Reed-tube coupling
At the excitation end x = 0, the tube is coupled to the motion of the reed by the assumption that the air flow
is continous, hence the air flow at the beginning of the tube is equal to the airflow coming into the tube,
S(0)v(0,t) = uin .

(5)

Furthermore the pressure difference p∆ is related to p(0,t) by pin = p(0,t), i.e.
p∆ = pb − p(0,t),(6)

where pb is the blowing pressure.

3PHYSICAL MODEL OF THE TONGUE - REED INTERACTION

We aim at modelling the interaction of a player’s tongue with the reed. For certain articulation techniques, theplayer stops and releases the vibrating reed with the tongue [11]. For this work, we assume that the tongue
motion is perpendicular to the reed surface, as indicated in Figure 1. In order to model this phenomenon, weinclude an extra collision term in the equation of the reed movement according to the Hunt-Crossley impact
model, such that the reed equation (1) now reads asdydyd2 yαlay(7a)m 2 + mγ + ky + klay by − ylay c
1 + rlaydtdtdt
d(ytg − y)− ktg bytg − ycαtg 1 + rtg= Sr p∆ .
(7b)
dt
Again, ktg , rtg denote contact stiffness and damping, and αtg denotes the collision exponent. As opposed to thedy
mouthpiece lay, the tongue is moving itself, hence we also have to take the velocity of the tongue dttg intoaccount.
For synthesising the sound of an actual single-reed woodwind instrument, the model given by equations (3),(4), (5), (6) and (7) is discretised using a finite difference method that satisfies an energy balance, see forexample [3, 5]. The special treatment of the visco-thermal losses related convolutions in Equation (3) is discussed in [12]. The finite difference method gives approximations pnm , vnm and yn of the pressure, the particle velocity and the reed displacement, respectively at discrete timesteps t n = n∆t , n = 0, 1, 2 . . . and positionx = m∆x , m = 0, 1, . . . , M, with L = M∆x . A more detailed description together with a discussion of a finite
difference method for the numerical implementation can be found in [4].

4EXPERIMENTAL SETUP

In order to obtain reference measurements regarding the clarinet playing technique, a professional player fromthe Vienna Symphonic Orchestra was invited to perform musical excerpts using an equipped clarinet. The
experimental setup consists of measuring the acoustic pressure in the mouthpiece of the clarinet and in themouth of the player, together with the reed oscillation. For the mouthpiece pressure, a pressure transducer(Endevco 8507C-2) is inserted into the mouthpiece through a drill-hole at 7.5 cm from the mouthpiece tip. An
identical pressure transducer is placed at the side of the mouthpiece and it remains inside the player’s mouth
during playing. In addition to that, we obtain information regarding the reed bending by attaching a strain-gauge
on the surface of the reed. This signal allows to identify the tongue-reed contact, which is detected when thereed stops vibrating and closes against the lay of the mouthpiece. During the steady part of the signal, i.e.
where there is no interaction between the player’s tongue and the reed, the measured bending signal can beconverted to reed displacement, as described in [10].

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5PARAMETER ESTIMATION

Our aim is to resynthesise an excerpt from the Clarinet Concerto No. 2, by Carl Maria von Weber (see Figure2), based on the signals recorded with real players.

Figure 2. Excerpt from the Clarinet Concerto No. 2, by Carl Maria von Weber. The analysed part is highlighted.
To do that, we estimate the model parameters by comparing recorded mouthpiece pressure and reed displacement (obtained from the recorded reed bending) with their synthesised counterparts. In this work, we apply
the Levenberg-Marquardt method [9, Chapter 10], which is an iterative optimisation technique that interpolates
between Gauß-Newton’s method and the gradient descent method.
In a first step, we obtain the effective tube length L for each single note. Since we are not modelling toneholes,
the effective tube length might be smaller than the actual tube length. This depends on the clarinet register,with the discrepancy being insignificant in the lower register. A steady part of the note is first considered andthe Levenberg-Marquardt method is applied to minimise the objective functions ky(t) − ŷkkp(0.02,t) − p̂kfobj =+,(8)ky(t)kkp(0.02,t)kwhere ŷ is the reed signal computed from the bending measurement, p̂ is the measured mouthpiece pressure, yis the modelled reed tip displacement and p here denotes the modelled pressure, obtained close to the excitationend. For the minimisation, the synthesised pressure is evaluated at x = 0.02m, i.e. two cm from the effective
beginning of the tube. The symbol k · k denotes the `2 -norm with respect to the discrete time samples underconsideration.
Other reed parameters such as mass m, damping γ and stiffness k, or the reed-lay collision parameters klay , rlay
and αlay do not have a direct physical counterpart, hence they are of effective nature as well. One reason forthat is that the reed is modelled by a simple harmonic oscillator that aims to capture the effect of the player’sembouchure. Therefore, having found the effective length L, we perform a second run on the same steady partof the signal in order to estimate the remaining reed-related parameters.
We model the motion of the tongue as a piecewise linear function, i.e. we assume that the tongue moves withconstant velocity. During steady tones, the tongue rests at a resting position, where it does not interact withthe reed. For each transient, the tongue starts moving upwards towards the reed at a time t1 linearly from its
resting position to its maximal value ytg,max , which it attains at time t2 , and leaves at time t3 in order to fallback to its resting position at time t4 . See Figure 3 for an illustration. The resting position is kept fixed. Weobtain the parameters related to the motion of the tongue t1 , t2 , t3 , t4 and ytg,max and the parameters related tothe collision of the tongue with the reed, ktg , rtg and αtg by applying the Levenberg-Marquardt method on thetransient parts of the signal.
Since the reed displacement obtained from the bending measurement is not reliable during reed - tongue contact,
we modify our objective function in a way that it ignores the reed signal during contact, i.e.
s kH(y − ytg )(y(t) − ŷ)kkp(0.02,t) − p̂kfobj =+.
(9)kH(y − ytg )y(t)kkp(0.02,t)kBehold that H(y − ytg ) is 0 whenever y < ytg , i.e. when the tongue is in contact with the reed, and 1 otherwise.

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Figure 3. Illustration of the modelled motion of the tongue.
Figure 4 shows the estimated tongue movement together with the synthesised reed displacement and the synthesised mouthpiece pressure corresponding to the highlighted phrase of the excerpt in Figure 2. The synthesised
signals are qualitatively similar to the reference signals in Figure 5.

Figure 4. Top: Synthesised reed displacement (solid blue) and tongue displacement (dashed red) in mm, for thenote transitions highlighted in Figure 2. Bottom: Synthesised mouthpiece pressure.

6CONCLUSION

This work discusses a physical model for the tongue-reed interaction while simulating the sound generation ofa single-reed woodwind instrument. This is established by including a further collision term into the equations
that describe the oscillation of the reed in the case of tongue interference. Using the recorded pressure andreed displacement signals obtained from measurements with real players, an optimisation procedure finds pa-5519

Figure 5. Signals obtained from measurements with a professional clarinettist. Top: Reference reed displacement. Bottom: Reference mouthpiece pressure.
rameters for which the physical model synthesises pressure and reed signals that are qualitatively similar to therecordings.

ACKNOWLEDGEMENTS
This research is supported by the Austrian Science Fund (FWF): P28655-N32

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