Inverting the Clarinet

Chatziioannou, V., & Van Walstijn, M. (2009). Inverting the Clarinet. 133-138. Paper presented at Proc. 12th Int.Conference on Digital Audio Effects (DAFx-09), Como, Italy.

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Proc. of the 12th Int. Conference on Digital Audio Effects (DAFx-09), Como, Italy, September 1-4, 2009

INVERTING THE CLARINET

Vasileios Chatziioannou and Maarten van Walstijn

Sonic Arts Research CentreSchool of Electronics, Electrical Engineering and Computer Science

Queen’s University Belfast{vchatziioannou01,m.vanwalstijn}@qub.ac.uk

ABSTRACT

Physical-modelling based sound resynthesis is considered byestimating physical model parameters for a clarinet-like system.Having as a starting point the pressure and flow signals in themouthpiece, a two-stage optimisation routine is employed, in or-der to estimate a set of physical model parameters that can be usedto resynthesise the original sound. Tested on numerically gener-ated signals, the presented inverse-modelling method can almostentirely resynthesise the input sound. For signals measured un-der real playing conditions, captured by three microphones em-bedded in the instrument bore, the pressure can be successfullyreproduced, while uncertainties in the fluid dynamical behaviourreveal that further model refinement is needed to reproduce theflow in the mouthpiece.

1. INTRODUCTION

Resynthesis of realistic sounds using physical models can be achie-ved by estimating the value of the parameters that govern the in-strument oscillations. Even thoughthis is possible by carrying outa system identification process that estimates fitted parameters [1],this study aims to estimate parameters that have a direct physicalinterpretation; this is a pre-requisite for achieving the long termaim of the authors to perform physically-based transformation byaltering these parameters. Since musical instruments behave non-linearly, a physical model may fail to inherit all the aspects of thereal sound-production mechanism [2]. In the clarinet mouthpiecetwo types of non-linearities manifest themselves. A mechanicalone, due to the interaction of the reed with the mouthpiece lay,and one attributed to fluid dynamical effects. The former can inprinciple be incorporated in a lumped reed model by estimatingits parameters using a mechanical description of the system [3],or by using quasi-static measurements [4, 5]. However, under realplaying conditions, the system may exhibit dynamic behaviour thatcannot be captured by quasi-static analysis. The present study es-timates lumped reed model parameters from oscillations generatednaturally by an instrumentalist and investigates to what extent thismodel can follow such measurement signals.

Focusing on the clarinet, this paper presents an inverse mod-elling procedure that takes as input the pressure and flow signalsinside the clarinet mouthpiece and estimates the physical modelparameters needed to resynthesise a sound as close as possible tothe one performed by the real player. This procedure consists of atwo-step routine; two different optimisation methods are used, the

This work has been supported by the EPSRC, grant numberEP/D074983/1

�

�

forward model

inverse model

Ka

Sr

ym

p

u

physical modelparameters

signals insidethe mouthpiece

Figure 1: Forward and inverse modelling of the reed-mouthpiecesystem, where Ka is the effective stiffness, Sr the effective reedsurface, ym the closing position of the reed and p and u the pres-sure and flow inside the mouthpiece.

first one to bring the problem to a good starting point, and the sec-ond to minimise the difference of the real and the estimated signalsin the mouthpiece. Rather than using a perceptual criterion for theobjective function [2], we intend to study to what extent the modelwill achieve to reproduce the original waveforms, since with per-ceptual criteria it is not guaranteed that a physically meaningfulset of parameters is extracted.

To simulate a sustained clarinet tone, a non-linear excitationmechanism (reed-mouthpiece-lip system) is coupled to a linearresonator (bore). Using a set of parameters for the physical model,the pressure and flow signals in the mouthpiece can be generatednumerically (i.e. a forward model). These signals can also bemeasured experimentally using wave separation techniques [6, 7].The process presented in the current paper uses the signals in themouthpiece as an input to estimate the parameters (i.e. an inversemodel), as depicted in Figure 1. Section 2 describes the lumpedreed model that interacts with the resonator to generate the os-cillations of the instrument and Sections 3 and 4 present the twooptimisation methods as applied to numerically generated signals.The experimental setup used to obtain the signals under real play-ing conditions is described in Section 5, and the application of thewhole optimisation routine based on the measured signals is dis-cussed in Section 6. Finally, the results of the process and futureobjectives are summarised in Sections 7 and 8.

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2. A LUMPED REED MODEL

The reed oscillation is simulated using a lumped model of the reed-mouthpiece-lip system [3, 8]. In previous studies the mechanicalparameters of this model were estimated from a two-dimensionaldistributed model of a vibrating reed clamped to the mouthpiece[9]. The resulting lumped model follows the mechanical behaviourof the distributed model, taking into account the effect of the play-ers lips, as well as the interaction of the reed with the mouthpiecelay. It can be argued that keeping the effective mass and dampingof the reed constant in a lumped model formulation captures mostof the dynamics of the system, at least for small amplitude oscil-lations [10]. Hence, the equation of motion for the lumped reedmodel is

md2y

dt2+ g

dy

dt+ Ka(∆p)y = ∆p, (1)

wherey is the reed displacement,m the mass per unit area andg the damping per unit area. The effective stiffness per unit area,Ka, is treated as a function of∆p, the pressure difference acrossthe reed, thus rendering the model able to incorporate the quasi-static mechanical non-linear behaviour of the system.

Concerning the flow inside the mouthpiece, an air jet witha varying height is formed in the reed channel, as predicted byboundary layer flow theory. Ifα is the “vena contracta” factor andSf the opening surface , it can be assumed that [8]

αSf ≈ λh, (2)

whereλ is the effective width of the reed andh the reed opening.The flow (u) inside the reed channel is expressed by Bernoulli’sequation for ideal fluid flow [11]

1

2ρ|u|2 + p = const. (3)

wherep is the pressure in the mouthpiece, and the flow induced bythe oscillation of the reed is

ur =dy

dtSr, (4)

with Sr the effective moving surface of the reed.The mouthpiece pressurep can be decomposed into a wave

going into(p+) and out(p−) of the bore, which are related to thetotal volume flowu = ur + uf by

Z0u = p+ − p−, (5)

whereZ0 is the characteristic impedance at the mouthpiece entry.Combining equations (3) and (5) yields the non-linear equation foruf

sign(uf )ρ

2(λh)2u2

f + Z0uf + (2p− − pm + Z0ur) = 0, (6)

wherepm is the blowing pressure andρ the air density.The above lumped element is coupled to a digital bore model,

constructed using wave variables [12], to create a feedback loopthat completes the digital representation of the instrument. A the-oretical approximation of the parameters of this physical modelenables the synthesis of the pressure and flow signals in the mouth-piece. These numerically synthesised signals can be used as aninput to the presented optimisation routine, estimating a new setof physical model parameters that can be directly compared to thetheoretical ones used during the simulation. Furthermore, using

the same physical model to resynthesise these signals gives us theopportunity to directly compare the input signals with the onesobtained with the estimated parameters. Since they have beenboth numerically generated by the same model, any differencesbetween these signals will have been introduced during the opti-misation process.

This lies in contrast to the optimisation based on input signalsfrom naturally performed sounds, since in that case errors can alsobe attributed to (1) the inability of the model to capture all thephysical details of the experiment and (2) measurement errors.

3. FIRST OPTIMISATION STEP

The first step towards the parameter optimisation is based on thesimplyfying assumption that the the reed displacementy is propor-tional to the pressure difference∆p across it [13]:

y = C∆p = C(pm − p), (7)

whereC is the compliance of the reed [14] andp the pressureinside the mouthpiece. The reed openingh can be related toy as

h = ym − y, (8)

with ym the closing position of the reed.Under this assumption the effects of inertia forces due to the

mass of the reed and frictional forces due to internal damping areneglected. It can be argued that even though these forces mightdominate the transient behaviour of the system, their effect almostvanishes at steady state (see Figures 4 and 5 in [8]), and it is thesteady state of the input signal that is going to be used for optimi-sation purposes, thus allowing the above assumption to be made.

The total flow into the mouthpiece as a function of the reeddisplacementy is

u = uf + ur

= (−λy + λym)

s2(pm − p)

ρ− CSr

dp

dt(9)

= c1

r2

ρ(pm − p)3/2 + c2

r2

ρ(pm − p)1/2 + c3

dp

dt,

with

8<:

c1 = −Cλc2 = ymλc3 = −CSr

Since the effective stiffnessKa is the reciprocal of the complianceC we can estimate physical model parameters from the arbitraryparametersc1, c2 andc3 using the following relationships:

8<:

Ka = −λ/c1

ym = c2/λSr = λc3/c1

,

Note that during this processh is not restricted to positive val-ues. This can be enforced by adding a constraint of the formc2 + c1∆p > 0, or it can be dealt with during the fine-tuningstage, at the second optimisation step.

The main feature of this technique stems from the fact that in-cluding the reed induced flowur in equation (9) brings into effectthe derivative of the pressure with respect to time (dp/dt). Since

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0 500 1000 1500 2000 2500 3000 35007

7.5

8

8.5

9

9.5

10

10.5

11

11.5

12x 10

6

∆ P [N/m2]

Kα [N

/m3 ]

Figure 2:Effective stiffness per unit area as predicted by the me-chanical properties of the reed [9] (dotted-black) and as estimatedfrom the numerically synthesised signals (grey).

the reed is assumed to move in phasewith the pressure differenceacross it, the model can now distinguish between the opening andclosing phases of the reed motion. As such, the two branches thatappear if we plot flow over pressure difference can be treated sepa-rately. This branch separation allows an optimisation process to beapplied twice; once for the opening state of the reed (dp/dt < 0)and once for the closing state (dp/dt > 0). Taking as our ob-jective function the mean square error between the original flowsignal and the estimated flow as calculated from equation (9), andusing the Nelder-Mead optimisation algorithm [15, 16], we canget a first estimate forKa, Sr, ym andpm [13]. These parameterscan be then fed into the lumped model to resynthesise the signalsin the mouthpiece. In the case ofKa, and since it is expected tobehave as a function of pressure difference, the estimated value istreated as the (constant) value ofKa for a low pressure difference(when the reed behaves linearly, for there is no interaction with thelay), whereas for higher values of∆p it rises to around 1.5 timesits value, as predicted by theory (see Figure 2).

One way to evaluate the obtained results of this first estimationof the physical model parameters is to compare the flow signalthat was used as an input with the estimated flow that is synthe-sised using the set of the estimated parameters. These are plottedin Figure 3 over the pressure difference across the reed. It canbe deduced that for pressure difference values within the range[500, 3000]N/m2 the estimation is good enough to be used as astarting point for a second optimisation method. It is also possibleto feed the new parameter set to the lumped model and comparethe signals in the mouthpiece that have been used as input and theresynthesised signals. A zoomed-in version of these signals is plot-ted in Figure 4. Finally the values of the estimated parameters andthe parameters used to create the input signals are listed in table 1,including the parameters estimated from the second optimisationmethod, as explained in the next section.

4. SECOND OPTIMISATION STEP

Having established a method to get a first estimate of the physicalmodel parameters allows us to proceed with a second optimisationmethod that completes the routine presented in the current paper.The parameters obtained so far enable the synthesis of oscillatory

0 500 1000 1500 2000 2500 3000 3500 4000−0.5

0

0.5

1

1.5

2

2.5x 10

−4

∆ P [N/m2]

u tota

l [m

3 ]

Figure 3: Flow into the mouthpiece over pressure difference, forthe original model (black) and as calculated using the estimatedparameters (grey).

signals; they lie within a range so that the simulation of sustainedclarinet notes is possible. As seen in Section 3 it is possible torecreate a note using these parameters, avoiding any regions whereblowing thresholds are not reached[17]. It remains to fine-tune theestimated parameters so that we get a better match for the originaland the resynthesised signals.

Using as our objective function the mean square error of thepressure signals at the steady state, we employ the Rosenbrockmethod [18, 19] to locate the optimum set of parameters. TheRosenbrock algorithm is a direct search method, that can go throughann-dimensional search space. Starting with a set ofn orthogo-nal directions, the algorithm moves towards those directions thatreduce the value of the objective function (for minimisation prob-lems) and then it changes the directions to a new orthogonal set,more likely to yield better results. It has the advantage that bychanging the set of the search directions, it can adapt to narrow“valleys” that can appear in the search-space. In addition, by ex-panding the motion towards successful directions and reducingthat towards unsuccessful ones, it has the ability to avoid gettingtrapped within regions of local minima.

In our application, we run the clarinet simulation after eachparameter search within the Rosenbrock algorithm, to synthesisethe pressure signal in the mouthpiece and compare it to the originalone. In contrast to the first optimisation step, it is now possibleto include in the model all the physical parameters that governthe oscillations of the system, namelyKa, Sr, ym, pm, massm,

Table 1:Theoretical vs. estimated parameters.

theoretical estimated (I) estimated (II) unitsKa 8.66 · 106 8.97 · 106 8.67 · 106 N/m3

Sr 7.61 · 105 8.42 · 105 8.33 · 105 m2

ym 4 · 10−4 3.2 · 10−4 3.68 · 10−4 mpm 1800 1919 1825 Paλ 0.013 — 0.0142 mm 0.05 — 0.044 Kg/m2

g 3000 — 3805 1/sec

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0.085 0.09 0.095 0.1 0.105−2000

−1500

−1000

−500

0

500

1000

1500

2000

time [sec]

pres

sure

[N/m

2 ]

Figure 4:Pressure signals in the mouthpiece for the original model(dashed-black) and as synthesised using the estimated parametersafter the first optimisation step (grey).

dampingg and effective widthλ. In addition, forKa a secondparameter is introduced, corresponding to its maximum value athigh∆p.

The resynthesised pressure signal should, at every iteration,lie closer to the pressure signal that was used as an input. Thusby starting with two signals that lie reasonably close to each other,something achieved in the previous section, it is possible to reacha suitable set of parameters that produces almost identical results.Again, working only with numerically generated signals, and sincewe are using the same model to create both the input and the resyn-thesised signal, an almost perfect match is required to indicate theefficiency of the method. A comparison between the input andthe resulting pressure and flow in the mouthpiece can been seen inFigure 5. This two-stage optimisation routine can also be appliedto signals measured under real playing conditions, as long as thepressure and flow signals are known.

0.096 0.098 0.1 0.102 0.104 0.106−2000

−1000

0

1000

2000

time [sec]

pres

sure

[N/m

2 ]

0.096 0.098 0.1 0.102 0.104 0.106−1

0

1

2

x 10−4

time [sec]

u tota

l [m3 ]

Figure 5: Pressure signals in the mouthpiece for the original(dashed-black) and the resynthesised (grey) sound (top) and thecorresponding flow signals (bottom).

s1 s0 s2

reference plane

p0

u0

pu

mouthpiece

Figure 6:Schematic depiction of the experimental setup.

5. SIGNAL MEASUREMENT

The experimental data is obtained from experiments with blow-ing a simplified clarinet, the schematic bore profile of which isshown in Figure 6. Following [12], the mouthpiece is modelledas a cylindrical plus a conical section, where the first is a heavilysimplified axially symmetric representation of the entry of the realmouthpiece; this approach is somewhat justified by the fact thatthe dimensions of the real mouthpiece geometry are small com-pared to the wavelength. Another reason to use such a simplemodel is that the fluid dynamics in this area are generally muchmore complicated than in the remaining part of the bore, involvingcomplex phenomena such as jet formation and its attachment andde-attachment from the side wall. For dynamic cases (i.e. when thereed moves), this behaviour is not yet understood well [8, 20, 21],and in the light of such modelling uncertainties, the best approachseems to use a simple model. The dimensions of the conical sec-tion as well as the step in the radius can be measured directly;the length of the cylindrical section was determined from the mea-sured volume of the section (≈ 3 ml).

In the experiments, the player generates a sustained note ofabout 4 seconds. The signals captured by the three microphonesembedded in the side wall of the main cylindrical bore are thenprocessed using adaptive delay-loop filtering in order to derive thepressure and flow at the reference plane. This method involvesestimation of the parameters that model the transfer function be-tween the microphones, adapting to the playing conditions (thereader is referred to [7] for a more detailed description).

Once the pressure (p0) and volume flow (u0) at the refer-ence plane are measured, classical transmission-line theory usingABCD matrices [22] is applied in order to derive the correspond-ing pressure (p) and flow (u) at the mouthpiece entry. Zero-phaselowpass filtering with a 7.25 kHz cut-off is applied to both sig-nals in order to remove high frequency errors that arise from thesingularities inherent to the three-microphone adaptive delay-loopfiltering method.

6. OPTIMISATION ROUTINE FOR THE MEASUREDSIGNALS

Having obtained the signals of pressure and flow in the mouth-piece we can directly proceed to the first step of our optimisationroutine. Equation (9) is used to form the objective function forthe Nelder-Mead method. Working on a “slice” of the measuredsignal that closely resembles the steady state of a sustained note,it is still possible to distinguish between the opening and closingstates of the reed motion, by calculatingdp/dt from the pressuresignal. As in Section 3, optimising separately for each branch andaveraging the obtained results gives a first estimate for the physicalmodel parameters, the validity of which is suggested by the com-parison of the measured flow and the calculated flow, as depictedin Figure 7.

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0 500 1000 1500 2000 2500 3000 3500 4000−2

0

2

4

6

8

10

12

14x 10

−5

∆ P [N/m2]

u tota

l [m3 ]

Figure 7: Flow into the mouthpiece over pressure difference, forthe measured data (dotted-black) and as calculated using the esti-mated parameters after the first optimisation step (grey).

SinceKa is known not to be constant, it is possible to get abetter estimation by feeding the rest of the estimated parametersto equation (9) and solving forKa as a function of∆p (sinceu isknown from the measurements). The pressure and flow signals inthe mouthpiece, as resynthesised using the parameters estimatedform this first-step optimisation, are compare to the original onesin Figure 8. The main cause of the deviation of the flow signal ofthis figure, as compared to the one in Figure 7, is the inclusion ofarbitrary mass and damping parameters during its resynthesis.

In order to transfer the above results to the second stage ofour optimisation routine, we have to adapt our model to the di-mensions of the experimental setup, used to obtain the measure-ments, as described in Section 5. The bore impedance (Z0) asseen from the reference plane was calculated from theory [22]. Afrequency-domain comparison betweenP0(ω) andZ0(ω)U0(ω)showed a good match between theory and measurement, validat-ing the derivation of a suitable reflection function fromZ0(ω) foruse in the time-domain clarinet model.

0.006 0.008 0.01 0.012 0.014 0.016 0.018−2000

−1000

0

1000

2000

time [sec]

pres

sure

[N/m

2 ]

0.006 0.008 0.01 0.012 0.014 0.016 0.018

−1

0

1

x 10−4

time [sec]

u tota

l [m3 ]

Figure 8: Pressure signals in the mouthpiece for the original(dashed-black) and the resynthesised (grey) sound (top), and cor-responding flow signals (bottom), after the first optimisation step.

Another problem that has to be tackled before proceeding withthe Rosenbrock algorithm is that the measured pressure signal andthe resynthesised signal from the lumped model might not be inphase. This was not a problem in Section 4, as the two signalswere generated using the same model. Now it is not guaranteed,constituting the synchronisation of the two signals necessary. Thiscan be achieved by shifting the numerically synthesised signal forthe required amount of samples, until it lies in phase with the orig-inal (measured) signal. (Note that such a synchronisation will berepeated several times during the optimisation routine, to ensurethat the compared signals lie in phase.) The results of the wholeoptimisation routine, applied to a small, steady part of the mea-sured signal, are shown in Figure 9.

7. DISCUSSION

The pressure signal in the mouthpiece can be resynthesised to matchthe measured one. For the flow, however, even though the resyn-thesised signal lies closer to the original one after the second op-timisation stage, a perfect match was not obtained. The use of adifferent objective function for the Rosenbrock algorithm, that in-cluded both the pressure and flow signals in the mouthpiece, didnot improve the estimation process. This indicates that the focusshould be shifted towards improving the model rather than the op-timisation routine.

For the measured signals there are non-linearities and uncer-tainties in the fluid dynamics that are not incorporated in the physi-cal model. These may stem from (1) a yet unpredictable behaviourof the “vena contracta” factor at low∆p regimes [23], and (2) theeffects of turbulence. On the other hand, applying the optimisationroutine on numerically generated data succeeded in resynthesisingboth pressure and flow signals. Since in that case both the inputand estimated signals are generated using the same model, the re-lation between pressure and flow in the mouthpiece remains thesame.

Future studies will focus on attributing the difference of theflow signals to appropriate physical phenomena, so that the modelcan regenerate both signals in the mouthpiece. It has to be pointedout here that the goal of our study remains to estimate parameters

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time [sec]

pres

sure

[N/m

2 ]

0.006 0.008 0.01 0.012 0.014 0.016 0.018

−1

0

1

x 10−4

time [sec]

u tota

l [m3 ]

Figure 9: Pressure signals in the mouthpiece for the original(dashed-black) and the resynthesised (grey) sound (top), and flowsignals (bottom), after the second optimisation step.

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that have a physical meaning. Thus, even though black-box tech-niques could parameterise the reed non-linearity, most of the re-sulting parameters would not have a direct physical interpretation.Hence such an approach would be less in line with our objectives.

8. CONCLUSIONS

A two-stage optimisation routine, that uses the Nelder-Mead andRosenbrock algorithms, can estimate physical model parameters,suitable for clarinet sound resynthesis. Starting from signals mea-sured under real playing conditions, the pressure signal can be re-generated using the estimated parameters as input to the model.For the flow in the mouthpiece, though, our model has to be re-fined, in order to improve the resynthesis accuracy.

9. ACKNOWLEDGEMENTS

The authors would like to thank Giovanni de Sanctis (Queen’s Uni-versity Belfast), for providing the apparatus to measure the internalbore pressure signals.

10. REFERENCES

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