International audienceA new method for measuring the nonlinear characteristic of the clarinet exciter, which binds the air flow entering into the clarinet with the pressure drop (∆p) across the reed, is described. It uses a clarinet mouthpiece equipped with a reed and an artificial lip whose position ψ is controlled by a micrometer screw. The mouthpiece is connected to a bottle in which a moderate vacuum is generated at the beginning of the experiment. After a short time lapse, the opening of the reed occurs. The ther-modynamics of the volume in isochoric conditions enables the calculation of the volume velocity entering the mouthpiece from the pressure measurement. 13 reeds with 10 different embouchures are measured. The measurements enabled the estimation of the equivalent aeraulic section S(∆p, ψ). We propose a model of S as a convex function of ∆p and ψ, defined as the sum of two 1D stiffening springs plus a porosity constant. The mean standard error of the model is 0.2%

Arancón, Alberto,

Dalmont, Jean-Pierre

Gazengel, Bruno

TAILLARD, Pierre-André

English

Taillard, Pierre-André

Muñoz Arancón, Alberto

Hal-Diderot

Analysis of Nonlinear Characteristics of the Clarinet Exciter Obtained via a New Measurement Method

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Analysis of Nonlinear Characteristics of the Clarinet
Exciter Obtained via a New Measurement Method
Pierre-André Taillard, Jean-Pierre Dalmont, Bruno Gazengel, Alberto
Arancón
To cite this version:
Pierre-André Taillard, Jean-Pierre Dalmont, Bruno Gazengel, Alberto Arancón. Analysis of Nonlinear
Characteristics of the Clarinet Exciter Obtained via a New Measurement Method. International
Symposium on Musical Acoustics, Jun 2017, Montréal, Canada. ￿hal-02466949￿
Analysis of Nonlinear Characteristics of the Clarinet Exciter Obtained via a New
Measurement Method
Pierre-Andre´ Taillard,1† Jean-Pierre Dalmont,2 Bruno Gazengel,2 Alberto Mun˜oz Aranco´n2
1FHNW Schola Cantorum Basiliensis, CP 257, 4009 Basel, Switzerland
2Laboratoire d’Acoustique de l’Universite´ du Maine (UMR CNRS 6613), 72085 Le Mans, France
†taillard@hispeed.ch
ABSTRACT
A new method for measuring the nonlinear characteristic of the
clarinet exciter, which binds the air flow entering into the clar-
inet with the pressure drop (∆p) across the reed, is described.
It uses a clarinet mouthpiece equipped with a reed and an arti-
ficial lip whose position ψ is controlled by a micrometer screw.
The mouthpiece is connected to a bottle in which a moderate
vacuum is generated at the beginning of the experiment. After
a short time lapse, the opening of the reed occurs. The ther-
modynamics of the volume in isochoric conditions enables the
calculation of the volume velocity entering the mouthpiece
from the pressure measurement. 13 reeds with 10 different
embouchures are measured. The measurements enabled the
estimation of the equivalent aeraulic section S(∆p, ψ). We
propose a model of S as a convex function of ∆p and ψ, de-
fined as the sum of two 1D stiffening springs plus a porosity
constant. The mean standard error of the model is 0.2%.
1. INTRODUCTION
Since the invention of the instrument, about 300 years ago,
the clarinettists complain about the difficulty to find musically
suitable reeds (reeds with a “good vibration”). Many scientific
studies are devoted to this topic, for instance [1, 2, 3, 4]. Some
authors measure the nonlinear characteristics of the clarinet
exciter (reed+mouthpiece+lip), for instance [5]. This paper
describes a new, precise measurement method of these charac-
teristics. We are convinced that the main musical differences
between clarinet reeds are related to the quasistatic aeraulic
behavior. The “vibration” of the reed is probably not the major
point, but the ability of the reed i) not to interfere negatively
with the sound production of the clarinet, ii) to enable an ef-
ficient sound control by the lip and iii) to allow an operating
blowing pressure which is comfortable for the clarinettist.
This paper is organized as follow: in Sec. 2 the setup of the
measurement method is described and its thermodynamical
behavior is modeled in Sec. 3. Sec. 4 illustrates an application
of the method to clarinet reeds and proposes a quasistatic
model of the aeraulic section as a function of the pressure drop
and the lip pressure.
2. DESCRIPTION OF THE SETUP
The measurement is divided in 2 phases: 1) calibration of the
diaphragm, 2) measurement of the nonlinear characteristics of
the reeds. The underlaying thermodynamic problem must be
solved for each phase.
b)d)
a)
c)
b)d)
a)
c)
g)
h)
f) k)
e)
i) j)
1) 2)
Figure 1. Schematic setup of the problems 1) and 2). a) di-
aphragm, b) piezoelectric pressure sensor (bottle), c) thin tube
(about 1 mm inner diameter), d) cock valve, e) adaptation bar-
rel with absorbing foam (damping the acoustic oscillations), f)
piezoelectric pressure sensor (mouthpiece), g) clarinet mouth-
piece, h) artificial lip (silicon 10 mm thick), i) steel beam
(diameter 3 mm) glued to the artificial lip, j) micrometer screw
(controlling the position ψ of the artificial lip), k) clarinet reed
2.1. Problem 1): discharge through a diaphragm
A diaphragm is connected to a hermetically closed, rigid vol-
ume (bottle, 3.178 liters, well isolated thermally and opaque to
the light). A moderate vacuum (about 15-20 kPa) is generated
in the bottle trough a cock valve at the beginning of the experi-
ment (while the diaphragm is closed). After about 1 minute
(allowing the observation of the heat exchange with the bottle),
the diaphragm is quickly opened and the discharge starts (the
air comes back into the bottle). The pressure in the bottle is
measured with a piezoelectric sensor (Endevco). See Fig. 1.
The purpose is to determine the effective aeraulic section
of the diaphragm Sdia from the pressure measurement (via the
computation of temperature and flow rate).
Proceedings of the 2017 International Symposium on Musical Acoustics, 18–22 June, Montreal, Canada Peer Reviewed Paper
13
0.5 1.0 1.5 2.0
t [s]
-15
-10
-5
P [kPa]
1)
2)
Figure 2. Typical measurements of ∆P during the discharge
(zoom) for Problems 1) and 2): 1) diaphragm of 1.5mm di-
ameter. 2) reed #J03 with embouchure ψ = 7 [arbitrary
units]
2.2. Problem 2): discharge through a varying aeraulic
section
The setup of Problem 1) is completed by a clarinet mouth-
piece and a reed. An artificial lip (silicon) compresses the reed
against the lay of the mouthpiece. Its position (normal to the
table of the reed and denoted as ψ) is controlled by a microm-
eter screw. The reed is not moistened before measurement, in
order to avoid a bias in the measurement due to drying.
The purpose is to determine the effective aeraulic section
of the channel (slit between reed and mouthpiece) in qua-
sistatic conditions from the pressure measurement inside the
mouthpiece p and the aeraulic section of the diaphragm Sdia
calibrated with Problem 1). The measurement of the pressure
in the bottle serves only as a control for the computations.
This way the delicate problem of pairing between sensors can
be avoided and the precision and the reproducibility of the
measurements is increased.
Typical discharge measurements for problems 1) and 2)
are depicted on Fig. 2. The total duration of one experiment
is about 90 s.
3. THERMODYNAMIC MODEL
3.1. Laws of thermodynamics
This subsection recalls some laws of thermodynamics, using
mainly the traditional notations.
3.1.1. Constants and parameters
Ideal gaz constants : R = 8.314J/mol/K
γ = 1.4 (for diatomic gazes), Density of air : ρ,
Pressure : P Atmospheric pressure : P0
Pressure drop : ∆P = P − P0
Volume : V Flow rate : U Aeraulic section : S
R×number of moles of gaz : N = nR
Absolute temperature : T Ambient temperature : T0
Heat capacity at constant volume : Cv = 1/(γ − 1)N
Internal energy : E Work : W Heat : Q
Sample rate, time step : fs = 1/ts (typically fs = 5000 Hz)
Parameters with uppercase generally refer to values inside
the bottle. In lowercase the same parameters refers to the value
inside the mouthpiece.
3.1.2. Laws
Summary of thermodynamical laws: (1) Ideal gas law, (2) First
principle of thermodynamics, (3) Newton’s law of cooling
(thermostat) and (4) Bernoulli’s law:
P V = N T (1)
dE = dW + dQ (2)
dTth/dt = −r(T − T0) (3)
U = sign(∆P )S
√
2|∆P |/ρ (4)
Remarks:
Newton’s law of cooling: r is a positive constant, which has
to be determined experimentally.
Bernoulli’s law: valid for incompressible fluids and large ducts,
comparatively to the aeraulic section S, for Reynolds numbers
Re ' U/(ν√piS) > 2000 (ν = 15.6× 10−6 for air at 25◦C).
For compressible fluids the equation is approximately valid
for the conditions at the output of the jet.
3.2. Isochoric model
The heat variation due to the thermostatic effect of the bottle
is:
dQ = Cv dTth (5)
The work of small air volume dV leaving the jet outgoing
the diaphragm with a temperature Tjet and a pressure P is
dW = dV P = dN Tjet. On the other side, the variation of
energy is dE = 1/(γ− 1)((N + dN)(T + dT )− (dN Tjet +
N T )). Applying Eqs. 1, 2, 3 and 5, we obtain the equation
of our thermodynamic model in isochoric conditions (i.e. at
constant volume):
dN =
dP V +N r (T − T0)dt
γ Tjet
(6)
Considering an adiabatic expansion in the jet, the temperature
of the air leaving the jet is Tjet = T0 (P/P0)
γ−1
γ .
3.3. Discrete time scheme for Problem 1
During an experiment, the pressure in the bottle P [t] is mea-
sured with time steps ts : Pm = P [mts]. We have to deduce
from the equations above the temperature Tm and the number
of moles (×R) of air in the bottle, Nm.
Initialization
At time step m = 0 (before generating the vacuum), the
air in the bottle is at ambient temperature T0 and atmospheric
pressure P0. Applying Eq. 1, the initial quantity of air in the
bottle is: N0 = P0V/T0.
Proceedings of the 2017 International Symposium on Musical Acoustics, 18–22 June, Montreal, Canada Peer Reviewed Paper
14
Iterations form > 0
dN =
V (Pm − Pm−1) +Nm−1 r ts (Tm−1 − T0)
γ Tjet
Nm = Nm−1 + dN
Tm = PmV/Nm
Um = dN Tm/(Pm ts) (7)
3.4. Validation with known diaphragms
0 5000 10000 15000 20000
ΔP [Pa]0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
d [mm]
Figure 3. Color lines: aeraulic diameter of the tested di-
aphragms computed with the isochoric scheme Eq. 7. Each
diaphragm is measured 3 times with different initial conditions.
Thin dark lines: nominal diameter of the diaphragms.
A validation of the model was performed with a series of
chamfered diaphragms of nominal diameters 1, 1.5, 2, 2.5, 3
and 3.5mm. Eq. 4 allows the computation of the equivalent aer-
aulic section S from U , ρ (in the bottle) and ∆p, from which
we deduce the equivalent aeraulic diameter of the diaphragms.
Fig. 3 depicts the results. The following values of r were
determined by optimization (among 3 tests with different ini-
tial depressions for each diaphragm): 0.25, 0.32, 0.39, 0.46,
0.54, 0.61, for the diameters 1 to 3.5mm. This accounts approx-
imately for the greater heat exchange with the bottle when the
flow rate through the diaphragm is high. Before the discharge
r = 0.20 (measured value in static conditions).
The aeraulic section of the diaphragm Sdia used in Problem
2) is calibrated with this method (in our case 2.9mm). The
adaptation barrel with absorbing foam (damping the acoustic
oscillations) belongs formally to the diaphragm (like every
aeraulic resistance downstream the mouthpiece sensor).
3.5. Discrete time scheme for Problem 2
This problem is subdivided in 2 subproblems:
A) compute the net mass flow dn entering into the mouthpiece
B) compute the net mass flow dN entering into the bottle
The total mass flow entering trough the channel is dNch =
dn+dN , from which Uch = T0 (p/P0)
γ−1
γ dNch/(p dt) and
its corresponding aeraulic section Sch is deduced with Eq. 4.
Subproblem A) The mouthpiece is treated with Eq. 7 in
which the variables of the mouthpiece replace those of the bot-
tle (P → p, U → u, and so on). An adiabatic approximation
can be used for this case : dQ = 0, thus r = 0.
Subproblem B) the combination of Eq. 6 with Bernoulli’s
law Eq. 4 enables the calculation of the pressure in the bottle
Pm (at discrete time m) from the corresponding measurement
of the pressure in the mouthpiece, denoted pm.
Initialization Like Problem 1), additionally: p0 = P0.
Iterations form > 0
δp = pm−1 − Pm−1, ρ = (M Nm−1)/V
dN = sign(δp)Pm−1 Sdia
√
|2δp/ρ|) ts/Tm−1
dP = (Nm−1 r ts (T0 − Tm−1) + dN γ Tjet)/V
Nm = Nm−1 + dN
Pm = Pm−1 + dP
Tm = PmV/Nm
Um = dN Tm/(Pm ts) (8)
with M = 0.028965/R = 0.00348388 for the air.
4. MEASUREMENTS OF CLARINET REEDS
0 2 4 6 8 10 12
Δp [kPa]
2
4
6
8
10
S [mm2]
J03
0 2 4 6 8 10 12
Δp [kPa]
2
4
6
8
10
S [mm2]
J12
Figure 4. Aeraulic section S(∆p, ψ) determined with Eq. 8
for the reeds J03 and J12, ψ = 1 . . . 9 (color lines). Below the
transition to turbulent flow, S is approximated from the optic
sensors inside the mouthpiece.
A series of 14 clarinet reeds was measured with the de-
scribed method (Rigotti and Rico, strength 3 and 3 1/2). 10
Proceedings of the 2017 International Symposium on Musical Acoustics, 18–22 June, Montreal, Canada Peer Reviewed Paper
15
0 2 4 6 8 10 12
2
4
6
8
Δp kPa]
ψ
a
rb
it
ra
ry
u
n
it
s
]
J03
0 2 4 6 8 10 12
2
4
6
8
Δp kPa]
ψ
a
rb
it
ra
ry
u
n
it
s
]
J12
Figure 5. Contour plot of the second derivative of the aeraulic
section ∂
2S
∂∆p2 for the reeds J03 (Rigotti 3) and J12 (Rico 3 1/2),
ψ = 1 . . . 9, according to the proposed model Eq. 9 (arbitrary
color scale)
different positions ψ of the artificial lip were tested (in steps by
0.2 mm). Unfortunately our vacuum cleaner was not powerful
enough for creating the vacuum in the bottle necessary to close
some strong reeds with a loose embouchure. Complete mea-
surements are available only for 13 reeds and 8 embouchures.
4.1. Measurements of the aeraulic section S
Fig. 4 depicts the results of the measurement of the aeraulic
section S(∆p, ψ) for 2 reeds. Fig. 5 illustrates ∂
2S
∂∆p2 for the
same reeds. An interesting feature is present for all reeds: we
observe 2 different slopes along whom the second derivative
is almost constant. The same observation can be done with
∂2S
∂∆ψ2 . Along these slopes, ∆p and ψ are partially decoupled.
The observed behavior can be approximated as a sum of 2
1D stiffening springs and a porosity constant:
S(∆p, ψ) ' S1(∆p+ k1ψ) + S2(∆p+ k2ψ) + kp (9)
The following values were determined by optimization for
our setup : k1 = 1374.2 and k2 = 670.3 (same values for
all reeds). The porosity constant kp accounts for a residual
flow which do not vanish for high values of ∆p and must be
determined individually for each reed.
This model implies that S(∆p, ψ) is a convex function
of ∆p and of ψ. In other words, the second derivatives of S
are non-negative. The stiffening springs can be implemented
with non-negative, reproducing kernels (for instance gaussian
kernel) as bandlimited functions. We used a kernel allowing a
real-time implementation of the measured exciter for synthesis
purpose (but this is beyond the scope of this paper).
The mean standard error of the model is around 0.02mm2.
This represents a height of 1.5 µm for a rectangular chan-
nel, 13mm wide, or 0.2% of the maximal measured aeraulic
section.
5. CONCLUSIONS AND PERSPECTIVES
The precision of the method enables a detailed study of the
nonlinear characteristics of clarinet reeds. A great variety
of quasistatic behaviors could be measured, confirming the
every-day experience of the clarinetists with their reeds and
the importance of the non-linear contact between reed and
mouthpiece. A comparison with 3D simulations about the
dichotomic sensitivity of the reed to the lip pressure shows
that k1 accounts mainly for air coming from the middle of the
channel while k2 concerns mainly the air entering from the
sides of the channel.
This short paper could only demonstrate the main features
of the method, but could not investigate the musical conse-
quences of the measurements for reed makers and clarinettists,
explaining relationships with geometric and flexural measure-
ments, nor demonstrate the ability of the model for a real-time
simulation of the measured reeds.
Aknowledgments We thank the engineering school HE-
Arc (Neuchaˆtel, Switzerland) for the facilities granted.
REFERENCES
[1] D. Casadonte, “The Clarinet Reed: An Introduction to
its Biology, Chemistry, and Physics,” Ph.D. dissertation,
Ohio State University, 1995.
[2] M. O. Van Walstijn, “Discrete-time modelling of brass
and reed woodwind instruments with application to mu-
sical sound synthesis,” Ph.D. dissertation, University of
Edinburgh, 2002.
[3] T. Guimezanes, “Etude expe´rimentale et nume´rique de
l’anche de clarinette,” Ph.D. dissertation, Universite´ du
Maine, Le Mans, France, 2007.
[4] P.-A. Taillard, F. Laloe¨, M. Gross, J.-P. Dalmont, and
J. Kergomard, “Statistical estimation of mechanical pa-
rameters of clarinet reeds using experimental and numeri-
cal approaches,” Acta Acustica united with Acustica, vol.
100, no. 3, pp. 555–573, 2014.
[5] J. Dalmont, J. Gilbert, and S. Ollivier, “Nonlinear charac-
teristics of single-reed instruments: Quasistatic volume
flow and reed opening measurements,” The Journal of the
Acoustical Society of America, vol. 114, p. 2253, 2003.
Proceedings of the 2017 International Symposium on Musical Acoustics, 18–22 June, Montreal, Canada Peer Reviewed Paper
16


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Analysis of Nonlinear Characteristics of the Clarinet Exciter Obtained via a New Measurement Method

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