Estimation of Clarinet Reed Parameters by Inverse Modelling

Chatziioannou, V., & Van Walstijn, M. (2012). Estimation of Clarinet Reed Parameters by Inverse Modelling.Acta Acustica united with Acustica, 98(4), 629 – 639. https://doi.org/10.3813/AAA.918543

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Estimation of clarinet reed parameters by inverse modelling

Vasileios Chatziioannou

Institute of Musical Acoustics

University of Music and performing Arts, Vienna

Anton von Webern Platz 1, 1030 Vienna, Austria

chatziioannou@mdw.ac.at

Maarten van WalstijnSonic Arts Research Centre

Queen’s University Belfast

BT7 1NN Belfast, Northern Ireland

m.vanwalstijn@qub.ac.uk

November 5, 2012

Abstract

Analysis of the acoustical functioning of musicalinstruments invariably involves the estimation ofmodel parameters. The broad aim of this paperis to develop methods for estimation of clarinetreed parameters that are representative of actualplaying conditions. This presents various chal-lenges because of the difficulties of measuring thedirectly relevant variables without interfering withthe control of the instrument. An inverse mod-elling approach is therefore proposed, in which theequations governing the sound generation mecha-nism of the clarinet are employed in an optimi-sation procedure to determine the reed parame-ters from the mouthpiece pressure and volume flowsignals. The underlying physical model capturesmost of the reed dynamics and is simple enoughto be used in an inversion process. The opti-misation procedure is first tested by applying itto numerically synthesised signals, and then ap-plied to mouthpiece signals acquired during notesblown by a human player. The proposed inversemodelling approach raises the possibility of re-vealing information about the way in which theembouchure-related reed parameters are controlledby the player, and also facilitates physics-based re-synthesis of clarinet sounds.

1 Introduction

The physics of single-reed woodwind instrumentshas been studied throughout the past century. In1929 Bouasse presented a series of observations onthe functioning of woodwind instruments [1]. Inlater work, a mathematical framework for study-ing the oscillations of such instruments was devel-oped [2, 3, 4, 5, 6], which provided the necessarybasis for the formulation of full, predictive physi-cal models (see, e.g. [7, 8]). Some uncertaintiesremain though, particularly regarding the complexfluid dynamics in the reed channel, and how thisinteracts with the reed motion [9, 10].

In most studies, the reed is modelled as a single-mass harmonic oscillator, which can be justified byconsidering that its dimensions are small comparedto the wavelengths inside the instrument. Sev-eral researchers have provided estimations of thelumped parameters of such a simplified mechani-cal reed model. In 1969 Nederveen measured thecompliance of the reed by observing its deforma-tion in static experiments [6]. Worman [11] deter-mined the effective stiffness, damping, and massper unit area of an isolated reed experimentallyand his results were adopted by many subsequentauthors [12, 13, 14]. Another measurement of thereed stiffness was provided by Gilbert [15] and his

1

results were in agreement with the compliance mea-sured by Nederveen. It is worthwhile noting that,apart from Worman, the above researchers useddry reeds, thus the estimations may not be rep-resentative of playing conditions. Moreover, theinfluence of the player’s lip is often neglected, andthe interaction with the mouthpiece lay was gen-erally treated in a simplified way, by imposing a‘hard’ beating condition.

The first published attempt to measure reed stiff-ness under actual playing conditions was made byBoutillon and Gibiat [16]. They derived the stiff-ness of a saxophone reed by establishing a balancebetween the reactive powers of the reed and theair-column. The obtained range of values was ofthe same order as that found by Nederveen andGilbert. Gazengel [17] developed a model in whichthe reed-lay interaction is modelled by varying thelumped parameters within the reed cycle. Thelumped parameter functions were deduced theoret-ically by considering a distributed interaction withthe mouthpiece lay, using simplified geometries forboth the reed and the mouthpiece. More recently ithas been shown how the lumped reed model param-eters can be estimated from a distributed modelwhile considering a more realistic geometry of thesystem, also incorporating the lip influence [18]. Inparticular it has been demonstrated how the effec-tive stiffness per unit area and the effective movingsurface of the reed can be estimated as functions ofthe pressure difference across the reed. The notionof a variable stiffness in order to capture the non-linear interaction of the reed with the mouthpiecelay was also suggested in an experimental study byDalmont et al. [19].

In the present study a method is presented to es-timate lumped reed model parameters from mouth-piece oscillations. The reed model is formulated us-ing a constant stiffness, and the reed-lay interactionis instead modelled by means of a conditional re-pelling force, in a way similar to modelling of pianohammer-string interaction [20]. The objective be-hind this is to formulate a more simple reed modelwith relatively few parameters that neverthelesscaptures most of the dynamics of the reed-player-mouthpiece system, so that it can sufficiently adaptto experimental data. This model is then employedin an inverse modelling procedure which, given thepressure and flow signals in the mouthpiece, esti-mates the lumped reed parameters. When usingsignals measured during human player control of

the instrument, the estimated parameters partlyreflect the actions of the player. This raises thepossibility of physics-based re-synthesis of clarinetsounds [21, 22], and also opens up a new way ofinvestigating how players control the instrumentin practice. In addition, the results of the in-verse modelling approach can give new perspectivesand insights into the validity and accuracy of thelumped reed model.

Studying the functioning and control of wind in-struments using an inverse modelling approach hasbeen attempted before, usually starting from a sin-gle pressure signal, which imposes fairly strong lim-itations on the inversion process. For example, in-vestigations into the inversion of a trumpet modelhave shown that additional assumptions and con-straints have to be applied in order to avoid prob-lems of uniqueness and non-invertibility [23, 24].Regarding the clarinet, several attempts have beenmade to identify the sound generation loop of a(simplified) clarinet using an artificial blowing set-up [25, 26], but the employed sensing methods havenot yet been demonstrated to be sufficiently accu-rate for subsequent parameter estimation. In orderto avoid or reduce some of the problems encoun-tered in these studies, this paper addresses the de-termination of clarinet reed parameters from twoin-duct signals, namely the pressure and volumeflow in the mouthpiece. For estimation from exper-imental data, the flow signal is difficult to measuredirectly, but can be inferred from pressure signalsacquired in the instrument bore.

The paper is structured as follows: Starting froma brief discussion on recent distributed modellingresults, the lumped formulation of the reed is re-considered under the inverse modelling scope insection 2, and a full time-domain model of a sim-plified clarinet without toneholes is presented. Theinversion of the model is discussed in section 3, in-cluding an analysis of the performance for numeri-cally generated pressure and flow signals. Then insection 4, the results of applying the inversion tosignals obtained under real playing conditions arepresented and discussed. Finally, the conclusionsare summarised in section 5.

2

uf

0

ymur

p

pm

y

Figure 1: Schematic illustration of a clarinet mouth-piece. The mouthpiece tip displacement is denotedwith y, the equilibrium opening with ym, the flow inthe reed channel with uf , and the reed induced flowwith ur, while p and pm represent the mouthpiece pres-sure and the mouth pressure, respectively.

2 Time-domain modelling

2.1 Distributed reed modelling re-

sults

Figure 1 shows a schematic depiction of a clarinetmouthpiece and the associated parameters. Themechanical response of the reed-mouthpiece sys-tem can be studied via numerical simulations witha distributed reed model. This allows bringing inthe geometrical detail of the reed-mouthpiece sys-tem, and facilitates investigating the interactionbetween the reed and the mouthpiece lay. In sev-eral earlier studies, the reed has been modelled as aone-dimensional clamped bar [27, 28]; later refine-ments to this approach were made regarding theformulation of the lip-reed and the reed-lay inter-action [29]. More recently, a two-dimensional dis-tributed model of the reed has been developed [30]which also captures the torsional behaviour of thereed; details of the numerical formulation of thismodel can be found in [31, Chapter 3]. One of themain results that can be obtained with these simu-lations is the quasi-static mechanical behaviour ofthe reed. The solid grey curve in Figure 2 shows atypical example of plotting the pressure differenceagainst the tip displacement from quasi-static nu-merical experiments with the 2-D reed model pre-sented in [30]. The plot demonstrates that the reedbehaves largely as a linear spring in the range oftip displacements in which there is little interactionbetween the reed and the lay. For larger displace-ments, the lay exerts additional contact forces onthe reed, which ‘bend’ the curve upwards.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

2000

4000

6000

8000

reed displacement [mm]

pres

sure

diff

eren

ce [N

/m2 ]

2D−distributed (quasi−static)collision model

Figure 2: An example of (quasi-static) pressure dif-ference versus reed tip displacement as computed withthe distributed model (grey solid curve). The dashedcurve indicates the lumped model approximation withequation (1). The reed tip touches the mouthpiece layat y = 0.4mm.

2.2 A lumped reed model

In previous publications, the relationship betweenpressure difference and reed tip displacement wasused to deduce the stiffness per unit area as a (non-linear) function of the pressure difference [18, 30].In the current study, the reed stiffness is kept con-stant, and as in hammer-string modelling [20], thereed-lay interaction is modelled with a conditionalcontact force based on a power-law. For quasi-static reed motion, the reed tip motion is then gov-erned by

k y + kc (⌊y − yc⌋)α = ∆p, (1)

where ∆p = pm−p is the pressure difference acrossthe reed, k is the effective reed stiffness per unitarea, kc and α are power-law constants, and ycrepresents the displacement value above which thecontact force becomes active, i.e.

⌊y − yc⌋ ={

y − yc if y > yc0, otherwise.

(2)

A similar ‘collision model’ approach can be foundin recent methods for synthesis-oriented time-domain modelling of reed woodwinds [32, 33]. Thedashed line in Figure 2 shows the approximationof equation (1) to the distributed model result forα = 2. The fit is not accurate for near-closure dis-placements, but in practice (when the reed behavesdynamically), the clarinet configurations we stud-ied do not support the production of mouthpiecepressure signals larger than about 4400 N/m2 (even

3

for blowing pressures that close the reed). Hencethis discrepancy hardly influences the simulationresults, and consequently has little bearing on theinversion process. Therefore α = 2, which gener-ally gives an excellent fit for the range of y-valuesjust above yc, was used in all computations.A lumped reed model that also incorporates

some reed dynamics may then be formulated byadding mass and damping to (1):

md2y

dt2+mg

dy

dt+ k y + kc (⌊y − yc⌋)α = ∆p, (3)

where m is the effective reed mass and g is thedamping per unit area; these reed parameters canin principle also be determined from distributedmodel simulation results [18].

2.3 The mouthpiece flow

As indicated in Figure 1, the flow into the mouth-piece consists of two components, namely the flowuf through reed channel and the volume flux ur

induced by the reed motion:

u = ur + uf . (4)

The flow through the reed channel is usually as-sumed to be governed by Bernoulli’s law, consid-ering the reed opening surface as a rectangle withsides λ and h [9, 19, 34]:

uf = λ h

√

2∆p

ρ, (5)

where h = ym−y is the reed opening, λ is the effec-tive width of the reed and ρ the air density. Thisformulation is based on the simplifying assumptionthat both the side openings and the vena contractaeffect result in a simple scaling of the flow throughthe reed channel. A more complex calculation ofuf is possible [34], on the basis of analytical for-mulations of the vena contracta effect [35], but itsvalidity in dynamic regimes is difficult to establish.The reed-induced flow ur is commonly calculated

as [6, 7]

ur = Sdy

dt, (6)

where S is the effective reed surface. Althoughfrom distributed simulations S can be said to bevarying with reed position, it has been shown thatany variations in it do not significantly affect themodel output [18, 31]. Therefore S is treated hereas a constant.

p0

u0

11 12 15

31.6 51 165.6 57

p

u

Figure 3: Schematic bore profile of the simplified clar-inet bore (dimensions in mm).

2.4 Coupling to the bore

In order to simulate clarinet tones, the responseof the bore must be modelled. Instead of using arealistic clarinet bore, the simplified configurationdepicted in Figure 3 was employed, as it allows amuch more reliable and accurate response predic-tion by theory than a bore with many toneholesand a flared bell. As will be seen in section 4, thisis useful in providing an intermediate validationof the sensing approach when applied to measuredsignals.Considering only plane waves, the response of

the bore can be calculated via convolution of theforward-travelling pressure wave p+ at the mouth-piece entry with the bore reflection function [14]:

p− = rf ∗ p+, (7)

where p− is the returning wave. The relationshipwith the mouthpiece flow is

Z0 u = p+ − p−, (8)

where Z0 is the characteristic impedance at themouthpiece entry. Digital waveguide modelling[36, 37] was employed to pre-compute the bore re-flection function rf . This requires first construct-ing an axially symmetric representation of the in-strument bore and mouthpiece. The main bore ofthe instrument is modelled as a straight cylinderand - following [37] - the mouthpiece is modelledas a cylindrical plus a conical section (see Figure 3for dimensions). In order to determine the dimen-sions, a 3-D model of the interior shape of an actualmouthpiece was generated (see Figure 4) by 3-Dscanning of a silicone cast, using the mouthpieceas a mould. The dimensions of the cylindrical partof the model are taken to have the same volume asthe complex-shaped interior shape of the first part

4

Figure 4: Model of the interior of a clarinet mouth-piece.

of the actual mouthpiece, whereas the dimensionsof the conical part match the corresponding sectionof the actual mouthpiece.A method to numerically solve the complete cou-

pled system is given in Appendix A. Exciting thesystem with a constant blowing pressure pm andusing the lumped model parameters shown on thefirst column of Table 1 yields the pressure and flowsignals in the mouthpiece depicted in Figure 5.These signals, which were generated using a 100kHz sample rate, are used in the next section asthe target signals of an inverse modelling method.

3 Inverse modelling

Starting from the assumption that the pressure andflow signals in the mouthpiece are known, the aimis now to reverse the modelling direction, i.e. toestimate the physical model parameters from thesignals. At first glance, it may look possible to de-sign an optimisation method that finds the modelparameters that minimise an objective function ina least square error sense. While such a methodis indeed used in this study, initial attempts im-mediately revealed that the search space for thisproblem contains many local minima and spurioussolutions, and that for the optimisation to be suc-cessful, a rather good initial guess is required. Thestrategy taken here is therefore to split the pro-cedure into two optimisation stages, which will bereferred to as step 1 and step 2 (see Figure 6). Thefirst step is based on simplifying the equations thatgovern the reed motion, which allows formulatinga closed-form expression that relates the pressureand flow signals in the mouthpiece. This step pro-duces a first estimate of a reduced set of param-eters, which then form a part of a parameter setthat can be used as a suitable initial guess for thesecond optimisation step, which really aims to find

0 0.05 0.1 0.15−2000

−1000

0

1000

2000

time [s]

pres

sure

[N/m

2 ]

0 0.05 0.1 0.150

1

2x 10

−4

time [s]

flow

[m3 /s

]

Figure 5: Pressure (top) and flow (bottom) signals inthe mouthpiece, calculated using the parameters in thefirst column of Table 1.

a close match between the synthesis signals and thetarget signals. Convergence to spurious solutionsthat do not lie in the physical range of the modelparameters is avoided this way.

3.1 First optimisation step

In the first step only a rough estimation of a pa-rameter set is aimed for, which if used for resynthe-sis produces signals that are to some extent simi-lar to the target signals. The highest priority inthis step is not accuracy, but robustness. Thiscan be achieved by simplifying the model equationssomewhat, thus optimising a smaller parameter set.Aiming solely at the steady-state part of the sig-nal, where dynamic effects are relatively small (see,for example, Figures 4 and 5 in [34]), a good wayto simplify the model is to neglect reed dampingand inertia. Another aspect that is temporarilydisregarded is the reed-lay interaction. With thosesimplifying assumptions the reed motion is propor-tional to the pressure difference:

y =∆p

k=

pm − p

k. (9)

Starting from (4), (5), and (6), and now consider-ing also negative pressure differences, the total flowinto the mouthpiece can be written as a function

5

pm m, g, λ, kc

from theory

step 1

step 2

k, S, ym

p, u

p, u

resynthesis

signals

p, u

target

signals

k, S, ym, pm,m, g, λ, kc

optimised parameters

from pressure signal

Figure 6: Two-step optimisation routine. The tar-get signals are fed into the first optimisation step toestimate a parameter set. This set along with theoret-ical values for the remaining model parameters is thestarting point for the second optimisation step. Theparameters estimated in the two steps can be used forresynthesis of the pressure and flow signals.

of pressure only:

u = σ λ ⌊ym − y⌋√

2 |pm − p|ρ

︸ ︷︷ ︸

uf

+Sdy

dt︸︷︷︸

ur

(9)= −σλ

k

√2

ρ|pm − p| 32 + σymλ

√2

ρ|pm − p| 12

− S

k

dp

dt

= σ

√2

ρ

(

c1 |pm − p|3

2 + c2 |pm − p|1

2

)

+ c3dp

dt,

= c1d1 + c2d2 + c3d3,

(10)

where σ is the sign of (pm − p) and

c1 = −λ/kc2 = ymλc3 = −S/k

⇒

k = −λ/c1ym = c2/λS = λ c3/c1

, (11)

and where we have the three signals

d1 = q |pm − p|3

2 , d2 = q |pm − p|1

2 , d3 =dp

dt, (12)

where q = σ√

2/ρ. A characteristic feature of thisapproach is that the flow ur in equation (10) bringsinto effect the derivative of the pressure with re-spect to time (dp/dt). This gives a simple toolfor distinguishing between the opening and closingphases of the reed motion, i.e. the two branches

0 50 100 150 200 250 300 350

10−2

10−1

100

iterations

F1

closingopening

0 100 200 300 400 500 60010

0

101

102

103

iterations

F2

Figure 7: Evolution of the objective functions F1 (top)and F2 (bottom).

that appear if we plot flow against pressure differ-ence (see Figure 8). Hence step 1 is run separatelyfor two parts of the data; once for the openingstate (dp/dt > 0) and once for the closing state(dp/dt < 0) of the reed motion; although usableresults can be estimated from either branch, theresults used for step 2 are obtained from the clos-ing branch estimates.

Equation (10) relates the pressure and the flowinside the mouthpiece, which are assumed known inthis context. For the mouth pressure pm a reason-ably good estimate can be obtained directly fromthe pressure signal vector p as pm = max |p|. Theair density ρ is also assumed to be known, so thetask is to find the three coefficients c1, c2, and c3.To this purpose, the square of an L2 norm is usedas an objective function, namely

F1 = ||(u− u)||22 , (13)

in order to minimise the difference between the tar-get flow signal vector u and the flow signal vectoru calculated with (10). By substituting (10) into(13) and taking the second-order partial derivativesof the objective function F1 with respect to ci (i =1,2,3), one obtains

∂2F1

∂c2i= 2

N∑

n=1

d2i (nT ), (14)

where n indexes the sample in the vector u andN isthe vector length. Hence the second derivatives ofF1 are guaranteed to be positive, thus the objective

6

0 500 1000 1500 2000 2500 3000 35000.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

−4

pressure difference [N/m2]

flow

[m3 /s

]

targetcalculated

Figure 8: Flow into the mouthpiece over pressure dif-ference, for the original model (grey) and as calcu-lated by equation (10) using the estimated parameters(black). Note that the closing and opening phase arecalculated separately, and plotted as separate curves(they cross at about ∆p = 150 N/m2).

function is concave, which enables a robust opti-misation with standard methods; the Nelder-Meadsimplex method [38] proved both efficient and suf-ficient in terms of robustness in this case. This pro-cedure yields an optimised set of three coefficients,but these are related to four physical parameters(λ, k, ym and S), so the actual physical parameterscannot be determined with (11) without further in-formation. To address this redundancy problem, λis simply set to the actual value of the geometricalreed width and its optimisation is only carried outduring the second step. The optimisation methodconverged after 175 iterations for the closing andafter 348 iterations for the opening branch (withthe termination criterion being a relative changein all the estimated parameters smaller than 1%).The evolution of the objective function is shown inthe upper plot of Figure 7.

One way to evaluate the obtained results of thisfirst estimation of the physical model parametersis to compare the target flow with the flow as cal-culated with (10) using the set of the estimatedparameters. These signals are plotted in Figure 8over the pressure difference across the reed. A sec-ond way of comparing is to feed the optimised pa-rameters (which are listed in the third column ofTable 1) to the time-domain model and comparethe resulting pressure and flow in the mouthpiece(the ‘resynthesised signals’) to the target signalssynthesised using the original model (see Figure 9).

0.18 0.182 0.184 0.186 0.188 0.19−2000

−1000

0

1000

2000

time [s]

pres

sure

[N/m

2 ]

0.18 0.182 0.184 0.186 0.188 0.190

1

2

3x 10

−4

time [s]

flow

[m3 /s

]

targetstep 1step 2

targetstep 1step 2

Figure 9: Pressure (top) and flow (bottom) signalsin the mouthpiece as synthesised using the estimatedparameters after the first and the second optimisationsteps, compared to the target signals (original model,grey solid).

In summary, step 1 is successful in that the esti-mated parameters enable the synthesis of sustainedoscillatory clarinet signals that, to a certain degree,resemble the target signals. This was found to bethe case for a wide range of signals generated usingdifferent bore configurations and excitation param-eters, and also when some noise was added [39].

3.2 Second optimisation step

For the second optimisation step, we use as objec-tive function the L2 norm of the difference betweenthe target and the resynthesised pressure signals atsteady state,

F2 = ||p− p||2 , (15)

and employ the Rosenbrock method [40] to lo-cate the corresponding optimum set of parameters.Comparing the pressure rather than the flow sig-nals, as in equation (13), ensures better conver-gence properties. The Rosenbrock algorithm isa direct search method, that can go through anM-dimensional search space. Starting with a setof M orthogonal directions, the algorithm movestowards those directions that reduce the value ofthe objective function (for minimisation problems)and then it changes the directions to a new or-thogonal set, more likely to yield better results.It has the advantage that by changing the set ofthe search directions, it can adapt to narrow ‘val-leys’ that may appear in the search-space. In addi-

7

tion, by expanding the motion towards successfuldirections and reducing that towards unsuccessfulones, it has the ability to avoid getting trappedwithin regions of local minima. The choice of theRosenbrock method stems from the fact that, eventhough it is computationally expensive, it is robustfor high dimensional problems, as long as a goodstarting point (that is given by the first optimi-sation step) is available, whereas the Nelder-Meadsimplex method turns out to fail when the dimen-sions of the problem increase [41].In our application, we run the clarinet simulation

after each parameter search within the Rosenbrockalgorithm, to synthesise the pressure signal in themouthpiece and compare it to the target one. Incontrast to the first optimisation step, it is now pos-sible to include in the model almost all the phys-ical model parameters, namely k, S, ym, pm, massm, damping g, effective width λ and collision coef-ficient kc; the only parameter that is not optimisedis yc, which is assumed not to deviate much fromthe value yc = 0.24 mm, as taken from the dis-tributed model result in Figure 2, and is set tothat value. The reason for this is that when yc isalso optimised, step 2 is far more likely to convergeto spurious solutions.Since the target signals are governed by the same

model that is used in the inversion, the parameterscan be recovered, i.e. the estimated parameters arenearly identical to the original model parametersupon convergence. For the target signals calcu-lated in section 2.4, using a 50 ms signal window,the algorithm converged (to a threshold relative er-ror set to 0.03%) after 538 iterations. The resultingevolution of the objective function is shown in thelower plot of Figure 7. Applying the algorithm tomore data sets showed that the rapid improvementin the first few iterations followed by a slower con-vergence period is typical for step 2. A comparisonbetween the target signals and the resynthesisedpressure and flow in the mouthpiece can been seenin Figure 9. The estimated parameters from bothoptimisation steps are listed in Table 1 .

4 Application to measured

signals

The next step is to investigate how the methodol-ogy works for measured signals. The main objec-tive within the scope of this paper is to test whether

Table 1: Theoretical (target) parameters and esti-mated parameters after the two optimisation steps.

parameter theoretical step 1 step 2k [Pa/m] 8.66e6 8.72e6 8.68e6S [m2] 7.62e-5 9.66e-5 7.65e-5ym [m] 4e-4 4.4e-4 3.99e-4pm [Pa] 1800 1683 1799λ [m] 0.013 — 0.0129m [kg/m2] 0.05 — 0.050g [1/s] 3000 — 2979kc [Pa/m

2] 8.23e10 — 8.35e10

the approach still works without problems in thatscenario, i.e. whether physically plausible parame-ters are estimated and whether the signals can bereasonably accurately resynthesised. The startingpoint in this investigation is a set of signals mea-sured at a specific point in the bore of a simplifiedclarinet (geometry shown in Figure 3) under play-ing conditions. The details of the employed tech-nique to measure these signals - the basics of whichare explained in [42] - are outside the scope of thispaper, and will be published in separate articles.In the experiments, the player generated a sus-

tained note of about 4 seconds. The signals cap-tured by three microphones embedded in the sidewall of the main cylindrical bore (spaced 2 cmapart) were then processed using an adaptive filter-ing approach in order to derive the pressure and theflow at the reference plane. This method involvesestimation of the parameters that model the trans-fer function between the microphones, and adaptsto time-varying playing conditions (including themean flow in the bore). All signals were acquiredand processed using a 100 kHz sample rate.

4.1 Signal Translation

Given the pressure (p0) and volume flow (u0) atthe reference plane (see Figure 3), the first stepin the processing chain is to translate these tothe corresponding pressure (p) and flow (u) at themouthpiece entry. This can be achieved with classi-cal transmission-line theory using ABCD matrices[43]. That is, in the frequency-domain the follow-ing calculation is performed

[PU

]

=

[A BC D

] [P0

U0

]

, (16)

8

0 500 1000 1500 2000−40

−20

0

20

40

frequency [Hz]

norm

alis

ed im

peda

nce

[dB

]

measuredtheory

Figure 10: Theoretical and experimental inputimpedance of the flanged pipe used for the experi-ments.

for a discrete set of frequencies, and FFTs are usedto transform between the time and the frequencydomain. The matrix elements are calculated fromthe geometry and the estimated air constants [43].Linear-phase low-pass FIR filtering [44] with a 7.25kHz cut-off is applied to both signals in order toremove high frequency errors that arise from thesingularities inherent to the multiple-microphonewave separation method.The translation procedure can be validated to

some extent by comparing the theoretical (nor-malised) input impedance of the instrument - alsocalculated using transmission-line theory - to thefrequency-domain ratio P/(Z0U) (see Figure 10).The measured data is contaminated by noise (thelevel of which cannot be reduced by averaging orusing longer signals since only short-time blockscan be assumed to have constant conditions), butthe global fit to theory is good. A flanged end wasused (see Figure 3), the termination impedance ofwhich is given in [45]. An unflanged end was triedas well, but this created relatively large discrep-ancies between the theoretical and experimentalimpedance curves, which is due to not having atermination impedance description of similar pre-diction accuracy available for an unflanged openpipe. Note that since the input impedance repre-sents the same information as the bore reflectionfunction rf , a good match here is crucial, as other-wise step 2 of the optimisation routine (that relieson the accuracy of rf) would fail.

4.2 Optimisation

The inversion algorithm is now applied to the ex-perimentally determined mouthpiece pressure and

0.22 0.222 0.224 0.226 0.228 0.23−2000

−1000

0

1000

2000

time [s]

pres

sure

[N/m

2 ]

targetresynthesised

0.22 0.222 0.224 0.226 0.228 0.23−1

0

1

2x 10

−4

time [s]

flow

[m3 /s

]

targetresynthesised

Figure 11: Pressure (top) and flow (bottom) signalresyntesised after step 2, compared to the measuredtarget signals (grey-solid). The flow signals do notcontain the mean flow component.

flow signals. In order to ensure sufficiently constantconditions (i.e. all parameters involved can be as-sumed constant over the window period), a shortwindow of 20 ms was used.

A couple of issues have to be dealt with beforeproceeding with the optimisation. Firstly, the mea-sured signals are zero-mean, while the time-domainclarinet model produces a flow signal that containsthe mean flow component. Hence a high-pass filter-ing routine is applied to the output of the model,so that the estimated and measured flow can bedirectly compared. Secondly, a problem that hasto be tackled before proceeding with step 2 is thatthe signals obtained from the measurements andthe ones resynthesised using the lumped model willnow not be in phase. This was not the case in sec-tion 3.2, since both signals were generated numeri-cally, using the same excitation model. Therefore asimple synchronisation procedure is now built intostep 2.

The resynthesis signals resulting from applyingthe complete optimisation routine are shown inFigure 11. As can be seen, the pressure signal isresynthesised accurately. With numerically synthe-sised target signals, the match in the flow signalswould then automatically also be good, since theforward and the inverse procedure use exactly thesame physical model. This is however not the casefor measured target signals. As can be seen fromthe lower plot of Figure 11, the flow match is glob-ally good, but small deviations in the waveform

9

remain, and are not diminishing with more step 2iterations. Explaining the causes of these small de-viations from the available data is difficult, but itis nevertheless useful to make an initial assessmentby considering the different stages in the processingchain at which errors may arise:

(A) errors introduced at the measurement stage,in particular due to non-perfect calibration,

(B) errors introduced in translating the mea-sured signals (p0, u0) to the mouthpiece signals(p, u), in particular errors caused by using asimplified geometry of the mouthpiece,

(C) errors due to the simplified reed dynamics,

(D) errors introduced by having simplified the fluiddynamics in the mouthpiece and the reedchannel to a quasi-static approximation.

The fairly good match in input impedance (see Fig-ure (10)) suggests that the errors under (A) and(B) are relatively small. Regarding (C), the oscil-latory motion per cycle in the measured flow signalmight indicate some form of reed resonance whichthe inversion procedure does not manage to fullyadapt to. Regarding (D), it is not yet understoodto what extent equation (5) is valid for dynamicregimes, during which the “vena contracta” factorat low ∆p regimes may not be constant [10]; fur-ther uncertainties exist regarding the effect of tur-bulence and how the lateral flow into the mouth-piece can affect the mouthpiece flow. To come tomore conclusive explanations, these issues wouldfirst have to be investigated separately (where pos-sible), which in turn would require experiments inwhich the conditions can be controlled more pre-cisely.All of the estimated parameters (see Table 2)

are physically feasible. In fact most parametersare not too far off the corresponding value foundin the literature (see theoretical values in Table 1),apart from the damping per unit area g, which inthis case was considerably smaller.

5 Conclusions

In this paper an inversion procedure has beenpresented which, given pressure and flow signalsin a clarinet mouthpiece, can estimate physicallymeaningful parameters. The procedure optimises

Table 2: Estimated parameters extracted from themeasured signals after the two optimisation steps.

parameter step 1 step 2k [Pa/m] 1.49e7 1.24e7S [m2] 2.1e-4 1.77e-4ym [m] 2.49e-4 2.55e-4pm [Pa] 1658 1671λ [m] — 0.012m [kg/m2] — 0.056g [1/s] — 453kc [Pa/m

2] — 3.7e11

the parameters of a simplified lumped model thataims to capture most of the dynamics of the reed-mouthpiece system. In this model, the (non-linear)interaction of the reed with the mouthpiece layis modelled using a repelling force when the reedstarts being in contact with the lay (and before itcloses completely). This allows the use of just twoconstant parameters, namely the effective stiffnessper unit area of the reed and the beating stiffnesscoefficient, in order to model the reed-lay interac-tion. The use of two constant parameters, ratherthen a generally varying parameter (e.g. k(∆p)) orthe use of ‘hard beating’ is one of several modellingchoices made in order to facilitate robust inversion.

The optimisation procedure consists of twosteps, where the first step - that relies on sev-eral simplifications - serves as a robust initial-guessprovider for the fine-tuning second step. For nu-merically generated target signals, the inversionprocedure recovers the model parameters, indicat-ing the correctness of the method, and extensivetesting with different signal sets has demonstratedits robustness. For measured target signals, thereis no reference for the estimated parameters, but asshown in section 4 the procedure can successfullyresynthesise the pressure and flow signals, and theoptimised parameters all lie in a physically feasiblerange. Any small discrepancies display themselvesmainly in the flow signal. When applied to tran-sient oscillations, the presented method does notgive consistent results; in particular the first opti-misation step appears to require a time-invariantpressure-flow relationship in a data window of suf-ficient length in order to systematically convergeto physically plausible results.

In summary, the results indicate that the pro-

10

posed inversion approach is a promising new way ofestimating reed parameters from player-controlledwoodwind oscillations. Improvements would bepossible if the parameter yc could also be opti-mised, and/or if a refinement can be made in thedescriptions of the reed motion and the fluid dy-namics in the mouthpiece. Both of these stepswould probably require the measurement of a sig-nal that represents the effective reed opening. Aninteresting future research direction may thereforebe to apply the method to oscillations generatedwith an artificial blowing machine, which wouldallow observing the relevant phenomena.

Acknowledgements

This work was supported by the Engineering andPhysical Sciences Research Council (UK), GrantNo EP/D074983/1. The authors are grateful toGiovanni de Sanctis (Queen’s University Belfast)for providing the measurement signals acquired un-der playing conditions, as well as for further fruitfuldiscussions and advice.

A Solving the coupled sys-

tem

The system of equations to be solved is describedby equations (3 - 8). This is done numerically us-ing finite difference approximations of any of thederivative terms in these equations, leading to thefollowing set of discrete-time equations:

y(n+ 1) = a1y(n) + a2y(n− 1) +

b1∆p(n) + a3 (⌊y(n)− yc⌋)α ,(A.1)h(n+ 1) = ym − y(n+ 1), (A.2)

ur(n+ 1) = Sy(n+ 1)− y(n)

∆T, (A.3)

p−(n+ 1) =

Nf∑

i=0

rf (i) p+(n− i), (A.4)

uf(n+ 1) = λ h(n+ 1)

√

2∆p(n + 1)

ρ, (A.5)

u(n+ 1) = ur(n+ 1) + uf(n+ 1), (A.6)

Z0u(n+ 1) = p+(n+ 1)− p−(n+ 1), (A.7)

p(n+ 1) = p+(n+ 1) + p−(n + 1), (A.8)

∆p(n+ 1) = pm(n+ 1)− p(n + 1), (A.9)

with equation (A.1) yielding the displacement atthe next time step (y(n + 1)) knowing the cur-rent values ∆p(n) and y(n) and the previous valuey(n−1), where time is discretised as t = n∆T . Thecoefficients a1, a2, a3 and b1 were calculated hereusing centered difference operators. The numericaldispersion and attenuation introduced in these ap-proximations is negligible in the frequency range ofinterest due to the use of a high (100 kHz) samplerate, and this also ensures that the relevant stabil-ity bounds are satisfied for any physically feasibleparameters. From the displacement at time n + 1it is possible to calculate the reed opening h andthe reed induced flow ur. However, for the com-putation of uf equation(A.5) needs to be writtenas

∆p = sign(∆p)ρ

2(λh)2u2f . (A.10)

Substituting

∆p = pm − p = pm − 2p− − Z0uf − Z0ur, (A.11)

we get

sign(∆p)u2f + 2

(λh)2Z0

ρ︸ ︷︷ ︸

Λ

uf

+2(λh)2

ρ(2p− + Z0ur − pm)

︸ ︷︷ ︸

Γ

= 0

⇒ sign(∆p)u2f + 2Λuf + Γ = 0,

(A.12)

with h, ur, pm and p− all evaluated at the nexttime step (t = (n + 1)∆t). There are two cases toconsider depending on the sign of ∆p.

• If sign(∆p) = 1:

uf =−2Λ±

√4Λ2 − 4Γ

2= −Λ ±

√Λ2 − Γ

(A.13)and since ∆p ≥ 0 ⇒ uf ≥ 0 then

√Λ2 − Γ ≥ Λ ⇒ Γ ≤ 0 (A.14)

• If sign(∆p) = −1:

uf =−2Λ±

√4Λ2 + 4Γ

−2= −Λ ±

√Λ2 + Γ

(A.15)and since ∆p < 0 ⇒ uf < 0 then

√Λ2 + Γ > Λ ⇒ Γ > 0. (A.16)

11

Wrapping up both cases, uf can be directly calcu-lated by first determining the sign of Γ and thencarrying out the following operation

if Γ(n + 1) ≤ 0 then

uf(n+ 1) = −Λ(n+ 1) +√

Λ(n+ 1)2 − Γ(n+ 1)

if Γ(n + 1) > 0 then

uf(n+ 1) = Λ(n+ 1)−√

Λ(n+ 1)2 + Γ(n + 1).(A.17)

Next the total flow trough the reed is calculatedwith A.6; the calculation of Γ(n+1) requires knowl-edge of p−(n+ 1), which is obtained by the convo-lution in equation (A.4), where Nf is the lengthof rf . Finally, the outgoing pressure wave is cal-culated using (A.7), and the result is then used toupdate the mouthpiece pressure p(n+1) with (A.8)and the pressure difference ∆p(n + 1) with (A.9).

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