Modeling and simulation of musical instruments
Antoine Chaigne
Unit of Mechanical Engineering (UME), ENSTA ParisTech, Palaiseau, France
Summary
The sound of musical instruments is the result of multiple and complex oscillations of their constitutive parts, including the surrounding air and internal cavities. At a large scale these oscillations
are governed by standard models, such as the well-known wave, string or plate equations. However,
the sounds obtained by solving these equations usually are of poor quality and do not mimic existing instruments convincingly. One dicult challenge is to rene the models by taking physically
and perceptually relevant phenomena into account. Depending on the family of instruments under
investigation, these renements (small scale) might be related to structural parameters (geometry,
amplitude of vibration, material properties, heterogeneity) and/or to coupling between their constitutive parts (uid-structure interaction, radiation). Each renement usually requires specic numerical
solving methods that need to be tested in terms of stability, accuracy and convergence. Developing
models and simulations of musical instruments has many benets. First, one can isolate phenomena
that are intrinsically linked in the reality, and examine their eects from both a physical and perceptual point of view. Listening to solutions also indicates the gap to ll in our understanding of
the instruments. Finally, simulations allow modications of parameters far beyond the possibilities of
the real world, and can thus be considered as essential complements of experimentation. This paper
presents an overview of some selected recent results obtained in the modeling of musical instruments.
Each result illustrates either basic physical phenomena or the numerical strategies to simulate them:
elasto-acoustic coupling in stringed instruments and timpani, uid-structure interaction and moving
mesh in vocal folds, nonlinear vibrations and chaos in gongs and cymbals, nonlinear strings and
coupling to bridge in pianos.
PACS no. 43.75.-z, 43.40.-r, 43.58.Ta
1.
Introduction
Music can be made with a large variety of sound
sources. A simple piece of wood or metal is sucient
for musicians to develop their creativity and give rise
to emotion through sound. The control of sound with
a clear intention is the major element that makes a
distinction between music and noise. Thus, almost
any vibrating structure or air column can pretend
Figure 1. General functional scheme of a musical instrument.
becoming a musical instrument. Over the centuries,
a number of elementary structures became more and
more sophisticated due to the objective of producing
achieved in case of free oscillations, as the time of
more sounds, with a larger dynamic range, allowing
interaction between the exciter and the instrument is
easier
small compared to the period of the sound. However,
playability
and
facilitating
ajustments
by
makers.
it becomes more problematic, both theoretically and
In this paper, we are interested in musical instru-
experimentally, for self-sustained oscillations, where
ments
This
the coupling between the exciter and the resonating
means rst that, in our models, we try to isolate as
part of the instrument is permanent. The exciter can
much as possible the action of the player from the
either be a device (bow) driven by the player, or
oscillations of the instrument. This can be reasonably
even a part of his (her) own body (vocal folds, lips,
viewed
as
mechanical
sound
sources.
air pressure from the lungs, air jet through the lips).
Figure 1 shows the basic function scheme of most
(c) European Acoustics Association
musical instruments.
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Chaigne: Modeling musical instruments
In what follows, the excitation at the input of
another structural dynamics context, such as bridge
musical instruments will be represented by imposed
or ship oscillations. Other requirements in instrument
mechanical
or
modeling are due to both the audible range and large
pressure. These quantities might vary with time and
spectral domain of most instruments. Low-pitched
can be distributed in space. From a physical point of
instruments, like timpani, show partials with notice-
view, musical instruments are described in terms of
able energy up to 1 kHz. Classical guitar tones show
geometry, materials and air-structure coupling. We
energy up to 3 to 5 kHz, depending on the pitch and
wish to understand the inuence of structural and
loudness of the played notes. Most piano and violin
uid parameters on the quality of sound, playability,
tones shows signicant energy up to 10 kHz, and the
radiation eciency and directivity. We ignore room
spectrum of cembalo tones can extend beyond 20 kHz.
eects
quantities
and
all
such
phenomena
as
force,
that
aect
velocity
the
sound
during its propagation through space, although we
Summarizing briey, musical instruments are de-
are aware of the fact that these factors might aect
signed to generate attractive sounds for human lis-
the judgment of the listeners.
teners and, as a consequence, their physical properties
must be considered under the light of auditory percep-
Modeling is one strategy among others to study a
tion. However, the concept of musical sound quality
musical instrument. It consists in describing both the
is rather delicate to dene accurately. Most instru-
geometry and behavior of its constitutive parts, the
ment makers know that appropriate choice of ma-
coupling between them and the excitation process. In
terials, assembly, ne tuning and other adjustments
general, the obtained set of equations is so complex
are essential in this context. Our goal is to investi-
that only a numerical resolution is conceivable. It also
gate through modeling the links between these ad-
requires in most cases to develop dedicated methods
justments and the sounds produced. Other blurring
and schemes. In this paper, focus are put on the
aspects, such as musical tradition, economy and art
use of time-domain methods. Such methods appear
market also take a signicant part in the judgment of
to be often unavoidable, because of the presence
players and listeners: these considerations will not be
of nonlinearities in the model. Another attractive
discussed here.
feature of time-domain modeling is due to the generation of solutions (waveforms) that can directly be
heard through loudspeakers, thus allowing auditory
2.
sults
evaluation of the model. Listening to mechanical
and acoustical quantities such as force, pressure,
velocity
or
acceleration
is
an
interesting
way
to
Examples of recently obtained re-
2.1.
Coupling of structures with air and cavities
understand the physics of musical instruments using
our ecient auditory system. The prime expected
Almost all percussive and string instruments are ex-
result of modeling is to gain a better understanding
amples of vibrating structures coupled with air and
of the physics of the instruments. Models are also
cavities. We examine below the representative cases
used to predict the eects of structural and material
of timpani and guitars, showing the inuence of these
modications. Even strong and fanciful modications
coupling on the produced sounds. The case of vocal
of geometry and material properties can be simu-
folds illustrate the coupling of structures with air ow,
lated, thus broadening the timbral space of a family
giving rise to self-sustained oscillations.
of instruments, in continuity with the real world.
In the next section, selected examples of recently
2.1.1.
obtained comprehensive results, through modeling
Timpani
The main vibrating element in timpani is the circular
and simulations, on coupled and nonlinear structures
membrane (or head). Because it is light and exible,
in musical instruments are presented and discussed.
its coupling with air cannot be neglected. To be convinced of this, one can speak (or sing) in front of the
The main challenging diculty of physical model-
head and touch it gently while speaking: the oscilla-
ing of musical instruments lies in the required degree
tions of the membrane are clearly perceived. We face
of accuracy needed, in view of the sensitivity of the
here a general problem of uid-structure interaction
human ear. A few hertz variation of the modal proper-
where the structural resonator (the head) is coupled
ties of a guitar soundboard, for example, consecutive
on one side to the external air, and, on the other side,
to material properties (temperature, moisture) or to
to the air enclosed in the cavity (see Figure 2).
slight modications of the structure (position and
The
sounds
of
timpani
are
characterised
by
a
height of the ribs, thickness) modies the coupling
strong impact followed by damped oscillations. Strik-
with the strings, and, in turn, both spectral and
ing the head near the center yields a rapidly decaying
temporal envelope so that a signicant proportion of
pressure, so that no clear pitch can be attributed to
listeners can hear the dierences. Oppositely, similar
the sound. In most case, the player hits the head near
variations
the edge, which induces less radiation damping and,
do
not
have
so
much
consequences
in
FORUM ACUSTICUM 2011
27. June - 1. July, Aalborg
Chaigne: Modeling musical instruments
10
146 222
0
a
u(t)
−10
pe(0-)
(S)
y
0
•
pi(0+)
Magnitude (dB)
x
w(x,y,t)
149
z
g(x,y,xo,yo)
292 358
359
219
−20
425
−30
−40
319
314
135
−50
(Si)
289
249
200
489
552
493 555
400 481
621
557
−60
−70
0
100
200
300
400
500
Frequency (Hz)
600
700
10
Figure 2. Sketch of a kettledrum. The mallet with dis-
u
strikes the head of radius
head of displacement
pe − pi .
w
a
on a spot
g.
0
The
is subjected to the pressure jump
The bowl is assumed to be rigid.
in turn, a longer tone. Spectral analysis of timpani
sounds (see Figure 3) show that the eigenfrequencies
dier from those predicted for a membrane in vacuo
Magnitude (dB)
placement
151
149
222 292
−10
353
−20
308352
−30
285
416
387
477
537 592
−40
signicantly. This is a consequence of air-membrane
−50
coupling. In addition, one can see that the main
−60
652
626
138
245
peaks usually are splitted, a consequence of heterogeneous tension eld in the membrane [3]. Timpani
−70
0
100
200
mallets are generally soft, so that the force impulse
300
400
500
Frequency (Hz)
600
700
is broad. As a consequence, only a limited number of
Figure 3. Comparison of real and simulated timpani spec-
low-frequency modes are excited. Internal damping in
tra.
the head material (mylar) and radiation increase with
frequency, which contributes to reduce the spectral
unforeseen result: the existence of a guided pressure
amplitude of the peaks in the upper frequency range.
wave on the membrane.
Frequency-domain methods exist for solving airmembrane coupling problems, and some of them were
used in the case of timpani to predict eigenfrequencies
and decay times of the main partials using a simplied geometry for the bowl [4]. Time-domain modeling account for this as well, and, in addition, for the
transient, peak splitting, coupled complex modes and
for the real geometry of the instrument (see Figure
3). The price to pay lies in the thorough preliminary
mathematical analysis of the global problem and of
its numerical approximation (see Section 3).
In
time-domain
methods,
all
mechanical
and
acoustical variables are computed step by step from
the origin of time corresponding to the impact of
the mallet against the head. The input data are the
mallet's mass and initial velocity, the compression
Figure 4. Simulation of elastic waves on timpani head.
Each picture represents the displacement of the head at
time intervals of 2ms. Time goes from left to right and
from top to bottom.
law of the felt and the width of the surface in
contact with the head. This allows to examine the
propagation of the elastic waves on the membrane
(see
Figure
4),
and
of
the
acoustic
waves
both
2.1.2.
Guitar
in the external space and inside the kettle. One
The extension of timpani model to guitar requires
can see these waves on the video at the address:
two major modications: rst, the membrane oper-
videotheque.inria.fr/videotheque/doc/411.
Looking at the video shows, among other things, an
ator must be replaced by a plate operator to account for the soundboard. A Kirchho-Love model
FORUM ACUSTICUM 2011
27. June - 1. July, Aalborg
Chaigne: Modeling musical instruments
can be used to a rst approximation, although the
2.1.3.
presence of ribs reduces the frequency range of va-
Vocal folds
Recent modeling of vocal folds coupled to air ow was
lidity of this theory. The second dierence is due to
motivated by the objective to examine the ow sep-
the presence of a soundhole that couples the inter-
aration from the volds. The separation points are in-
val cavity to the surrounding air [5]. In most previous
uenced by many factors: interaction of the main jet
studies, the cavity is treated as a separated resonator
with turbulent and vortical structures in the uid,
coupled to the plate. This approximation is valid in
ow interuption during glottal closure, and formation
the low-frequency range (below the rst top-plate res-
of the new jet when reopening. The importance of
onance) as seen in Figure 5. However, with increas-
the location of these points are conrmed by many
ing frequency, the guitar box must be considered as a
previous workd devoted to voice production and, in
semi-open waveguide that communicates with the ex-
particular, in computational models of phonation.
terior eld. A numerical formulation that consider the
structure-air coupling as a whole is then necessary.
MAGNITUDE (dB)
0
−20
−40
−60
0
0.1
0.2
0.3
0.4
0.5
FREQUENCY (kHz)
0.6
0.7
0.8
0
Figure 6. Comparison between simulated (top) and mea-
MAGNITUDE (dB)
sured (bottom) velocity elds in glottis.
−20
A 2D numerical model of vocal folds coupled to air
ow was developed and is presented briey in Section 3. In parallel, a self-oscillating four times scaled
−40
model of the system was build to validate the simulations [29]. For simplicity, only one fold was allowed
to move. A PIV (Particle Image Velocimetry) system was used to visualize the ow downstream of the
−60
0
0.1
0.2
0.3
0.4
0.5
FREQUENCY (kHz)
0.6
0.7
glottis and to measure the velocity eld (see Figure
0.8
6). The system was equipped with pressure transduc-
Figure 5. Admittances at the bridge in a simulated guitar
ers and accelerometers to record the time history of
showing the inuence of air-soundboard-cavity coupling.
the signicant dynamical variables.The results show
(Top) Guitar in vacuo, (Bottom) Guitar coupled to the
a good match between simulations and measurements
air and cavity. Plate thickness 2.9 mm.
in terms of maximum jet velocity and general ow
structure.
The position of the separation points is expressed
with the ow separation ratio
The
video
available
at
the
address
videotheque.inria.fr/videotheque/doc/514
Amin
F SR = A/Amin where
A of the
is the minimum of the cross-section
glottis. Figure 7 shows good agreement between mea-
shows the results obtained with a numerical time-
sured and simulated FSR for the left separation point
domain
a
(on the left side of the ow). It can be seen that
ctitious domain method is used (see Section 3) in
FSR varies signicantly during one oscillation cycle
order to account for the geometrical shape of the
(f= 13.2 Hz) and that it is dierent on both sides of
instrument.
the glottis.
model
of
the
guitar.
As
for
timpani,
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Chaigne: Modeling musical instruments
Excitation
Excitation
0.5
0
Acceleration
Accélération
du gong
−0.5
0
10
20
40
50
60
70
80
50
60
70
80
70
80
0.5
0
−0.5
−1
0
10
periodic
Régime
motion
périodique
20
30
40
Régime
quasiperiodic
motion
Régime
chaotic
chaotique
motion
quasi-périodique
Bifurcations
1500
Frequency
Fréquence
(Hz) − ∆(Hz)
f = 1.35 Hz
30
response of the gong
1
1000
500
Figure 7. Comparison between measured (top) and predicted (bottom) left ow separation ratio.
2.2.
0
Nonlinear vibrations
0
10
20
30
40
50
Temps time
(s) − ∆(s)
t = 0.372 s
60
Figure 8. Basic experiments of forced sinusoidal excitation
The presence of nonlinearities is essential in the be-
of a gong.
havior of bowed strings and wind instruments, since
their function is to transform continuous (or slowly
varying) energy input into rapid self-sustained oscilla-
of the vibration is comparable to the thickness of
tions [1]. For instruments subject to free oscillations,
the structure [26]. Since impulse excitation is hard
linear models might appear to be sucient in most
to analyse due to the number of input components,
cases, provided that the magnitude of the motion re-
the basic experiments used for investigating cymbal
mains small. However, a number of recent studies has
and gongs vibrations are based on a forced sinusoidal
shown that the linear approximation does not hold for
excitation as shown in Figure 8. The instrument is ex-
strings, especially for pianos, even in the medium dy-
cited at a frequency close to one of its lowest modes
namic range. For metallic percussive instruments, like
with a slowing increasing amplitude. The vibration
gongs and cymbals, the nonlinear regime is inherent
is recorded pointwise using either a laser vibrome-
and models of such sources must account for it.
ter or an accelerometer. Under low-amplitude forcing,
the response shows rst periodic signal whose distor-
2.2.1.
Gongs and cymbals
sion increases with amplitude. The Fourier spectrum
When metallic percussive instruments, such as gongs
shows harmonics of the excitation frequency. For a
and cymbals, are struck softly, the resulting sounds
given amplitude threshold, the spectrum shows sud-
generally yield a clear pitch. The spectrum shows a
denly additional peaks. The frequencies of these peaks
limited number of peaks corresponding to the excited
(also called combination resonances) correspond to
modes. With hard strokes on thin structures (crash
existing linear modes of the structure (we can decide,
fm
fn ) and are
fm ± fn = fk
cymbals) the perception of pitch disappears. Spectral
for example, to designate them
analysis shows a broadband continuous spectrum.
governed by simple laws of the form
It has been shown that the sounds obtained exhibit
where
the properties of deterministic chaotic signals: great
Theoretical proof of these combinations are found in
sensitivity
examining the coupling between nonlinearly coupled
to
initial
conditions
and
exponential
fk
and
is one harmonic frequency of the excitation.
separation of neighbouring points in phase space with
oscillators.
time [17].
With increasing amplitude, the system reaches a second threshold where the vibration leaves its quasi-
This nonlinear behavior is the result of geometri-
periodic state and becomes chaotic. It has been shown
cal nonlinearities. It has been proved that these non-
recently that this regime can be analyzed with the the-
linearities are noticeable as soon as the amplitude
ory of wave turbulence. Wave turbulence is often used
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Chaigne: Modeling musical instruments
Figure 10. String-soundboard coupling in pianos. The resulting force acting on the soundboard is the sum of the
Figure 9. Measured piano tones spectra (note C2 - fundamental 65 Hz)) showing nonlinear eects. Comparison
between the
piano
case (top) and the
fortissimo
case (bot-
force
Ft
due to the transverse string motion, and the pro-
FL
jection
of its longitudinal motion consecutive to the
curvature of the soundboard.
tom) shows rst a larger excited bandwith in the latter
case, which is essentially due to the nonlinear behavior of
transverse motion of the string (see Figure 10).
the hammer felt. Secondly, one can see additional peaks
in the
fortissimo
case that are not predicted by the linear
string model.
Another widely recognized feature of piano strings
lies in their intrinsic stiness, which is responsible for
inharmonicity [10]. This phenomenon also has been
in uid dynamics for dynamical and statistical analysis of a set of nonlinearly interacting dispersive waves,
when the nonlinear terms of the dynamics are small
compared to the linear ones. It turns out that these
assumptions are fullled in the case of thin structures,
proved to be perceived by listeners [11]. Thus, an appropriate model must account for string stiness in
addition to geometric nonlinearity. In this context, a
nonlinear Timoshenko string model is presented in the
next section.
such as those governed by the Von Kármán equations
[25]. One interest of wave turbulence theory is that
3.
it can predict the spectrum of the vibration. Clearly,
and performances
experiments are in agreement with theoretical predictions that shows cascade of energy transfer from large
to small-scale structures in the vibrating solid. Recent
simulations based on a time-domain discretization of
these equations are convincing and are, to our knowledge, the rst examples of simulation of gongs and
cymbal-like structures [30].
2.2.2.
Summary of numerical methods
3.1.
Fictitious domain methods
For the 3D simulations of timpani and guitar, we had
to face two major diculties. First, due to the size
of the problem, it is preferable to use a regular mesh
for the pressure eld. Secondly, the complex shape
of the structure has to be taken into account. The
ctitious domain method is appropriate for this. In-
Piano
The piano is probably today one of the most sophisticated instrument, and its modeling remains challenging. The rst simulations of linear piano strings excited by a hammer were made nearly twenty years
ago [6]. In between, a number of papers have shown
evidence of spectral energy in piano tones due to nonlinear vibrations of the string [7]. This nonlinearity
is mainly due to the variation of string tension with
amplitude. As a consequence, the transverse motion
of the string is coupled to its longitudinal component,
and the additional, or so-called phantom, partials
in the spectrum are the result of both quadratic and
cubic nonlinearities (see Figure 9).
A recent work shows that these additional partials
have audible eects [8]. This incited us to develop a
nonlinear model of piano string coupled to a soundboard [9]. Due to the curvature of the soundboard,
stead of computing internal pressure
and
pe
pi
in the cavity
λ is intro[p] = pe −pi
in the outer eld, a new unknown
duced which represents the pressure jump
across the structure (bowl or box). This has the advantage of eliminating spurious diraction that would
result from the selection of dierent pressure meshes
inside and outside the instrument, with the necessity
of adapting both meshes in the vicinity of the bowl
surface. Instead of that, we obtain a uniform cubic
mesh for the complete pressure eld that ignores the
geometry of the instrument.
In addition, the mesh of new variable
λ
is inde-
pendent of the pressure mesh. It is now dened on a
surface, which reduces complexity and improves the
accuracy (see Figure 11).
3.2.
Nonlinear Timoshenko string
(string
The main requirements for the modeling of piano
tension) is converted into a transverse component
strings are: geometric nonlinearity, broadband (au-
which is added to the force component due to the
dible range), stiness and coupling to soundboard.
the
longitudinal
component
of
the
force
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Chaigne: Modeling musical instruments
Figure 11. Meshes used for timpani modeling within the
framework of the ctitious domain method. (a) Cubic
mesh for the acoustic pressure and velocity ; (b) Finite
Elements P1 for the pressure jump across the head ; (c)
Renement of the P1 mesh for the membrane displace-
Figure 12. Comparison between measured and simulated
piano waveforms. String velocity of string
E3
- fundamen-
tal 165 Hz - recorded at a distance of 6 cm from the agrae.
ment.
a fully implicit scheme is used that guarantees the
As shown in [31], this imposes to use implicit discretization schemes in order to guarantee the stability. A good strategy for discretizing nonlinear coupled
systems is to construct an energy-preserving scheme
from which the stability condition can be derived. An
additional advantage of such an approach is that is
yields the coupling conditions naturally. For stringsoundboard coupling, for example, we derive a condition of force applied to the soundboard and, recipro-
stability. A high sampling rate is necessary to limit
the numerical dispersion to an acceptable proportion
(typically 1 % within the audible range). This allows
a good reproduction of the intrinsic dispersive properties of the sti string and, in particular, the reproduction of the precursors, clearly visible for the bass
strings.
3.3.
Gongs and cymbals
cally, a velocity condition for the string end attached
The generic model for metallic percussion instruments
to it. A prestressed Timoshenko model was used for
of circular geometry is a thin shallow spherical shell
the strings in order to fulll both conditions of sti-
with additional terms accounting for geometric non-
ness and broadband validity. An exact formulation for
linearity. It is written in nondimensional form (remov-
the geometric nonlinearity was used, since it has been
ing the damping terms for clarity) as follows [34]:
 4
 ∇ w + εq ∇2 F + wtt = εc L(w, F ) + εq p,
4
 ∇4 F − a ∇2 w = − 1 L(w, w),
Rh3
2
shown that approximate nonlinear models may lead
to signicant errors as the amplitude of string vibrations increases [9]. With the energy method, the exact
formulation yields a scheme whose complexity is comparable to approximate nonlinear equations. Remov-
where
ing (for simplicity) the additional internal damping
stress function,
terms and one transverse component of the string, we
p
get:
sure, due to the action of a mallet). The coecient
εq = 12(1 − ν 2 )h/R is a quadratic perturbation term,
2 4
4
whereas εc = 12(1 − ν )h /a is a cubic perturbation
[

(
)

∂
(ρA
∂
u)
−
∂
AGk ′ ∂x u − ϕ + EA ∂x u
t
t
x




]

∂x u



= 0,
− (EA − T0 ) √


(∂x u)2 + (1 + ∂x v)2






[
∂
(ρA
∂
v)
−
∂
EA ∂x v
t
t
x



]

1 + ∂x v



− (EA − T0 ) √
= 0,


(∂x u)2 + (1 + ∂x v)2







(
)

∂t (ρI ∂t ϕ) − ∂x [EI ∂x ϕ] − AGk ′ ∂x u − ϕ = 0,
w
is the transverse displacement,
L
F
the Airy
a bilinear quadratic operator and
the loading term (surface density force, or pres-
term, with
h the thickness of the shell, a the radius of
its projection on a plane perpendicular to its axis, and
R
its radius of curvature. Even for a small curvature
(high radius) the quadratic term is rapidly dominant,
so that only quadratic nonlinearities are observable in
practice.Depending on the instrument, various boundary conditions can be written: clamped at the center
and free at the edge for cymbals, clamped or simply
supported at the edge for gongs, in order to account
(at least partially) for the presence of the rim.
u is the transverse displacement of the string, v
its longitudinal displacement, ϕ the Timoshenko angle, ρ the density, A the cross-section, E the Young's
modulus, T0 the tension at rest, G the shear mod′
ulus, I the stiness inertial coecient and k the
As for piano strings, a good strategy for putting
Timoshenko parameter. For the discrete formulation,
they can approximate the boundaries very accurately.
where
these equations into a discrete form is to use energypreserving schemes. Time-space discretization can be
obtained with nite dierences. With a circular geometry, polar coordinates might be interesting since
FORUM ACUSTICUM 2011
27. June - 1. July, Aalborg
Chaigne: Modeling musical instruments
Ωt )
allows to compute all variables (including
at the
next time step. The numerical formulation is obtained
with a Finite Element Method using a weak form of
the ALE Navier-Stokes equation. It is implemented
with the help of the open-source library
4.
Figure 13. Computational domain of vocal folds and deformable mesh.
Melina
[32].
Conclusion and open questions
Within the last two decades, signicant advances
have been made in musical instruments modeling.
Dierent tracks and strategies were used in various
communities
(signal
processing,
psychoacoustics,
However, some diculties can appear near the cen-
theoretical mechanics and acoustics, applied math-
ter except if one excludes a small circle near the
ematics
center from the computational domain. Implicit -
exchanges
between
nite dierences schemes should be used to guarantee
processing
community
the stability of the algorithm, and an additional vari-
real-time computation [12] and automatic extraction
able (adjustment) parameter helps in optimizing the
of
scheme in terms of numerical dispersion. It has been
simulated musical sounds are used to establish link
shown that such a model was able to account for the
between physical and perceptual data [15], [16]. In
main characteristics of gongs and cymbals: pitch glide,
this paper, we attempted to summarize the main
hardening-softening behavior, bifurcations, combina-
results obtained in time and space modeling of some
tion of resonances, chaos and wave turbulence [2].
selected instruments, with particular emphasis to
3.4.
and
numerical
parameters
analysis)
them
[18].
[20],
gives
In
with
[19].
great
stimulating
The
signal
importance
psychoacoustics,
real
to
and
air-structure coupling and nonlinear vibrations. Both
Vocal folds
the interest of time-domain modeling and necessity
The purpose is to model self-sustained oscillations
of broadband modeling were underlined, in order
of human vocal folds consecutive to airow. The
to account for transients. A number of numerical
airow is modeled by nonstationary incompressible
methods were presented, allowing the modeling of
(Mach number = 0.1) Navier-Stokes equations in a
complex
2D computational domain
Ωt ,
which is deformed in
geometries,
geometrical
heterogeneous
nonlinearities.
materials
Time-space
and
modeling
time due to vocal fold vibration. The prime objective
allow the display of wave propagation in air and
is to account for the ow separation in glottis. A
structures in simulated instruments, thus shedding
nite element model is developed that allows to
new light on elastic and acoustic phenomena.
investigate upstream and downstream pressure and
Despite
velocity elds (see Figure 13).
undoubted
progress,
numerical
models
today are only used in a limited number of practical
to
cases for the design of instruments. This probably
the presence of a deformable mesh since the folds
arises from the fact that the simulated tones are still
move with time. To circumvent this, we reformulate
perceived as too crude by makers who have gained
the problem using an Arbitrary Lagrangian-Eulerian
over he years a particularly sharp auditory expertise
(ALE) approach, which yields [28]:
on
 A
D
u + [(u − w).∇] u + ∇p − νδu = 0
Dt

divu = 0
models can be used with benet for the understand-
The
main
diculty
of
the
problem
is
due
in
Ωt
in
Ωt
real
instruments'
tones.
However,
incomplete
ing of basic physical phenomena. Also the eects
of structural modications can be simulated prior
to making, which contributes to reduce the part of
empiricism.
where
u
is the ow velocity,
kinematic viscosity.
p
the pressure and
ν
the
w
denotes the domain velocity
DA
Dt is the ALEderivative which can be discretized even for time-
disciplinary
dependent computational domains. For the structural
denition of appropriate constitutive equations for
part, the elastic folds are modeled as rigid bodies sup-
complex materials such as felt, wood, composites,
ported by springs and dampers. This model accounts
and polymers used in instrument making, and/or the
for two basic modes of motion: vertical shigft and
eects of thermal and fabrication processes are linked
rocking. The full problem is solved as follows: Start-
with central questions of the mechanics of materials
(velocity of the mesh points) and
t
Ωt ,
The acoustics of musical instruments is a crossdomain.
A
number
of
challenging
research topics now emerge in various elds. The
the resulting
and structures. Also the problem of modeling dissi-
force and momentum acting on the folds are calcu-
pation processes in the audio range remain essential:
lated from the uid pressure and velocity elds. This
in most cases, an accurate description of the losses
ing at time
with known domain
FORUM ACUSTICUM 2011
27. June - 1. July, Aalborg
Chaigne: Modeling musical instruments
would lead to an unacceptable proportion of the
[4] R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson,
complete model of a given instrument, and one has to
R. I. Mills, and T. D. Rossing: Eects of air loading
content ourselves with approximations, mean values
and general tendencies. This might be unsatisfactory
in terms of perception, since, as one knows, a wrong
estimation of damping for a unique frequency (or
for a limited frequency band) is sucient to yield
an unadequate simulated tone. Accurate description
of constitutive equations including dissipation also
is a key point for the elaboration of equivalent (or
substitution) materials.
on timpani membrane vibrations , J. Acoust. Soc. Am.
76 (1984) 1336-1345.
[5] G. Derveaux, A. Chaigne, E. Becache, P. Joly: Timedomain simulation of a guitar. I: Model and method,
J. Acoust. Soc. Am.
114 (2003) 3368-3383.
[6] A. Chaigne, A. Askenfelt: Numerical simulations of piano strings. I: A physical model for a struck string using nite dierence methods, J. Acoust. Soc. Am.
95
(1994) 1112-1118.
[7] H. A. Conklin, Jr.: Generation of partials due to nonlinear mixing in a stringed instrument, J. Acoust. Soc.
Within
the
framework
of
a
given
model,
one
Am.
105 (1999) 536-545.
can think of applying optimization procedures for
[8] B. Bank and H.M. Lehtonen: Perception of longitudi-
instrument making. Such techniques were already
nal components in piano string vibrations, J. Acoust.
used in the case of violins where one objective is
Soc. Am.
128 (2010) EL 117-123.
to know how to carve the soundboard for obtaining
[9] J. Chabassier, A. Chaigne: Modeling and numerical
a desired set of modes [23], [13], [22]. It can be
simulation of a nonlinear system of piano strings cou-
anticipated that elaborate models serve as starting
pled to a soundboard, Proc. 2010 Int. Congress Acous-
points of control strategies for a larger number of
elastic and geometric parameters. The questions of
robustness and reproductibility are also important
in the context of computer-aided-lutherie: to what
extent is the sound of a instrument modied if we
have to change only one part (a violin bridge, for
example) ? Can we compensate modications of one
component (the wood structure of a soundboard) by
modifying other components ?
Tribology applied to musical instruments may also
lead to interesting research tracks. Interesting results
were obtained on rosin in the context of bow-string
contact [24]. Other problems such as the modeling of
string-nail contact at the end of piano strings might
be useful. Progress in robotics can also lead to sub-
tics, paper 590, 1-8.
[10] H. Fletcher, Normal Vibration Frequencies of a Sti
Piano String, J. Acoust. Soc. Am.
36 (1964) 203-209.
[11] H. Järveläinen, V. Välimäki and M. Karjalainen:
Audibility of the timbral eects of inharmonicity in
stringed instrument tones, Acoustics Research Letters
Online
2 (2001) 79-84.
[12] V. Välimäki, J. Pakarinen, C. Erkut and M. Karjalainen: Discrete-time modelling of musical instruments, Rep. Prog. Phys.
69 (2006) 1–78.
[13] Y. Yu, I.G. Jang, I. K.Kim, B. M. Kwak: Nodal line
optimization and its application to violin top plate design, J. Sound Vib.
329 (2010) 4785–4796.
[14] M. French: Structural modication of stringed instruments, Mech. Syst. and Signal Proc.
21 (2007) 98-107.
[15] S. McAdams, A. Chaigne and V. Roussarie: The psychomechanics of simulated sound sources. I: Material
stitutes for piano keys and hammer, whose design did
properties of impacted bars, J. Acoust. Soc. Am.,
not change signicantly for one century [21]. Finally,
(2004) 1306-1320.
there is still a need for perceptive studies on musical
[16] S.
McAdams,
V.
Roussarie,
A.
Chaigne
115
and
B.
sounds, especially in relation with structural changes
Giordano: The psychomechanics of simulated sound
of instruments and/or for evaluating perceptual dis-
sources: Perception of damping characteristics in im-
tances between real instruments and models.
Acknowledgement
The author wishes to thank V. Doutaut and P. Sinigaglia (ITEMM) for the lending of an upright Pleyel
piano, and René Caussé (IRCAM) for putting a Steinway grand piano at our disposal for measurements.
pacted plates, J. Acoust. Soc. Am.,
128
(2010) 1401-
1413.
[17] C. Touzé, A. Chaigne: Lyapunov exponents from experimental time series: application to cymbal vibrations, Acustica united with Acta Acustica
86
(2000),
557-567.
[18] J. Bensa, O. Gipouloux and R. Kronland-Martinet:
Parameter tting for piano sound synthesis by physical
modeling, J. Acoust. Soc. Am.
118 (2005), 495-504.
[19] P. Guillemain, J. Kergomard, S. Bilbao, M. Campbell
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