METAL STRING
Physical modelling of bowed strings a new
model and algorithm
Marco Palumbi
Centro Ricerche Musicali
Via Lamarmora, 18
00185 Roma, Italy
Lorenzo Seno
Centro Ricerche Musicali
Via Lamarmora, 18
00185 Roma, Italy
http://www.crmmusic.org
Abstract
This paper introduces a new simulating model of a bowed string,
whose algorithm is based upon a method quite different from the well
know waveguide approach the currently used model one in today
musical research.
Our approach is, at present, more computationally complex, but the
control of the various physical parameters and of their meaning is
very straightforward. Furthermore the algorithm makes no assumption on
the time invariance of the system, so that the variation of any
parameter does not introduce artifacts.
Our model implements both the viscous friction of the string with the
air, and the internal friction, and a discontinuous bow, which
includes also a parametric noise model (i.e., the noise is not simply
added to the sound). I.e. the energy dissipation due to internal
viscouslike behavior of the string.
The friction of the bow is represented by a discontinuous function,
which simulates the thermal behavior of the rosin by means of a
hysteresis mechanism, and models also the roughness of the bow by
means of parametric white noise.
The approach brings to an inherently timevariant system, so player
can freely change any parameter without artifacts. The timevarying
continuously controllable parameters are the tension/density ratio of
the string (i.e. the pitch of the note), the two friction
coefficients, the speed and the pressure of the bow. In today
implementation, the bowing point (b) is variable but discrete.
The computation algorithm here discussed is an intermix of a method
similar to the finite element one for what concerns the integration
over the space, and to the finite difference method for what concerns
the integration of the evolution of the system.
Our model can be used to play in extreme parametric conditions, beyond
the bowed strings performance tradition.
Although our model fully neglects twisting and longitudinal motions,
as well as the bridge admittance, it produces very likely sounds. One
can maybe infer that these characteristics are less important than one
may suspect.
Using a software based on our this model, the Italian composer
Michelangelo Lupone wrote the tape part of the string quartet "Corda
di metallo" ("Metal string""), (whose first world
performance was held in Rome in 1997 by the Kronos Quartett  Rome
1997).
Introduction
Musical Research in the field of musical instruments
synthesis by Physical modeling is dominated by the tendency to use
delay lines (or waveguides) as a solving algorithm [4][5][6][10]. This
is mainly because of the low computational cost of delay lines, and
may be because it is a form of historic tribute to the first method
invented  the KarplusStrong algorithm [7][8]. Our approach is
instead an evolution of the finite difference method, as an attempt to
bypass its limitations. Finite difference method is equivalent to a
physical model of a discrete massspring system, which inherently
brings to partials with subharmonic ratios. Our string model is
continuous, thus not suffering this inconvenient.
We started in the autumn of 1996 our studies and researches about
physical modeling. Non only the methods, by also the goals of our work
are quite different from the prevailing one in this research field.
The results of our readings of the related literature were that the
prevailing approach was the waveguide. On the other hand, in the
literature on the physics of the violin (starting from Raman, Cremer,
et al.) was not clearly stated the importance of the various factors
in the final behavior of the bowed string instruments. At that time
was also available to us a powerful computing system, based on a
parallel architecture of DSP Texas C40.
Under these circumstances we believed that an approach not so
sensitive to the computational cost was suitable, favoring instead a
direct relationship between the parameters of the model and the
physics of the phenomena. We hoped that this approach would make
easier the learning of the importance of the various physical factors
in the sound of the bowed strings. For us, in the evaluations of the
sounds, the musical point of view was the most important one.
Because of our personal tendency, and because of our bonds to
contemporary composers, we were more interested in the search of new
interesting sounds hardly obtainable within the bounds of the
actual physical world  rather than in the imitation of true,
traditionally "correctly" played, instruments.
Our model (in the form of a computer software) was actually in
effect used by the composer Michelangelo Lupone to get suggestions
about new performing techniques and to compose the magnetic tape part
of the "Metal string" quartet (from whom the title of this
paper) for strings, tape and spatialiser (Kronos Quartet, Rome,
1997).
The model
The string
Without loss of generality, consider a string of unit length, whose
free PDE of the free string is:
with boundary conditions:
y(0,t) = y(1,t) = 0
Where:
T (Newton) tension of the string.
m (Kg/m) linear density of the string.
S (sec^{1}) coefficient due to the viscous friction with
air.
S_{i }(m^{2}/sec) coefficient due to the viscous
internal friction.
A few words about the presence (and the absence) of some terms. The
classical dispersive term is absent:
This term is due to the stiffness of the string. It is responsible
of the dependence of the propagation speed on the frequency. Because
of these different speeds, partials are in superharmonic frequency
relationship.
This has important effects on the timbre of the sounds, especially
with stiff strings e.g. like in the low section of the piano.
Strings used in bowed instruments (as in the violin family) have
low stiffness, but some researchers believe it is the reason of the
socalled "rounding effect" [1][2]. that can be easily
observed by means of experiments.
Other researchers [3] (Woodhouse, 1992) suggested that this effect
is due, for the most part, to, to the action of the bow, particularly
to the hysteresis in the friction behavior of the rosin.
We skipped that term mainly because our integration method
introduces at present some computational error whose effect is to
slightly move the partial frequencies away from each other the
inclusion of this term in the computation being a straightforward
task. On the other hand, strings used in the violin family have
normally very low stiffness.
A few words now about the last, righthand, mixed term generally
neglected in the literature. It represents the effect of the energy
losses due to internal, viscous friction, which offer resistance to
changes in the curvature.
This phenomenon is responsible of a main behavior of actual, free
running strings: the higher the frequency partial, the faster the
dumping. For instance, if you pluck a string, during the transient you
can hear many partials a rich sound. Instead in the release part
of the sound you can hear just the fundamental.
The pitch of the sound, in our model, is due to the parameter T/m
and/or to the boundary conditions (specifically, to the parameter L).
You can thus obtain a variation of the pitch either varying T/m or
varying the length L of the string two independent ways to obtain
the same effect.
Varying the tension corresponds to physically twisting the tuning
peg. Length variation is quite similar to the action of the lefthand
of the performer.
In both circumstances, the system (and the corresponding PDE) is
timevariant. To avoid artifacts, the integration method must not make
any implicit assumption on timeinvariance. The way we integrate the
evolution of the system a finite difference method  does not make
actually any assumption about the timeinvariance of T/m nor of any
other parameter like internal and external dumping. . You can thus
modify in any desired way these parameters without any artifact.
Varying the length (a spatial parameter) sets some subtle question:
you must correctly transfer the final to the initial condition (the
shape of the string) from one length to the next one. We have not yet
implemented this feature.
The stimulus
There are several ways to excite a string: f.i. plucking, hitting
or bowing.
At a first glance you can consider the excitation as a timevarying
function applied to a specific point x_{b} of the string in
the wise of a force, or a boundary condition for the speed or the
displacement y.
A more realistic approach requires the consideration of the
interactions between the excitation function and the string
.This is the case of the exciting behavior of the bow.. In our
model we consider as excitation functions the pressure and the speed
of the bow, and as interacting state variable the location of bowing
point x_{b}, the speed and the acceleration of the bowing
point.
Our bow is in some way quite different from that usually
implemented by means of continuous speed/force functions. During the
stick state, the bowing point xb _{. }is glued to the bow, and
we force the speed of that point to be equal to the bow speed. (i.e.
the relative speed is zero), by applying a suitable acceleration to
the point xb _{. }During the slip state, the bowing force F is
added to the forces balance in that point.
The bowing force depends on the relative speed, in the way shown in
the figure underneath.
As you can see, the "stick"ฎ "slip" transition
has a threshold that is twice the "slip"ฎ "stick"
transition (which happens for zero relative speed).
State diagram of the bowing point.
Acceleration (force / linear density) versus pointbow relative speed.
The effect of the bowing pressure is to linearly scale the force
(both threshold and continuous curve). One may explore other kinds of
dependency.
We don't take explicitly into consideration the temperature of the
point  a variable affecting the fluidity and thus the friction
behavior of the rosin (a natural polymer with a complex behavior).
The shown threshold emulates however the cooling effect of the
rosin during the stick time, during which the dissipation is zero. The
noise due to the rubbing of the bow over the string is modeled by
means of white noise added to the F curve, in such a way to preserve
the curve as a "maximum value" of the random value. This is
a true "noise model", and adds parametric noise to the
model, which imparts a "chaotic" behavior due to the random
interaction between bow and string
The sound
The sound is taken as the displacement of the spatial sampling
point nearer to the bridge. One can think of this point as a
"secant" approximation of the tangent to the string in the
bridge position (i.e. the first spatial derivative in the origin),
being the tangent the expression of the strain against the bridge.
The method of calculus
To obtain a continuous model of the shape of the string, we
decompose it into a series of sinusoids of spatial frequency, which
are multiples of that corresponding to twice the string length.
Thus we imagine the string as periodically and antisymmetrically
infinitely expanded in the space. The series is truncated to the first
N terms, which represents the maximum spatial frequency allowed to the
string. For the free evolution, this limits to N also the partials of
the sound. In order to derive the coefficients a_{n} of such
series (i.e., to setup a continuous model of the string shape) we
must know the position of the string in N distinct points  the N
spatial sampling points:
Given:
(x_{1},y_{1}),(x_{2},y_{2})
(x_{N}y_{N})
One can write:
Where M is a matrix whose elements are:
So that:
Thanks to the knowledge of an analytic form of the solution, we can
easily derive its differential properties in the unit interval 0L; in
particular we can formally get the curvatures (i.e. second spatial
derivatives) in each of the sampling points.
:Leaving out the boring algebra, the calculation of the curvatures
can be performed through the product of a constant matrix (which is a
function of the sampling points abscissas) with by the vector of
positions y.
Knowing the curvatures means breaking the equation of the string
into N, second order, independent, motion equations of the sampled
points., in which the left term (acceleration) results from the
curvature, the velocities (which appear in the term of viscous
friction with air) and from the time variation of the curvature itself
(term of internal viscous friction). Our today software computes the
motion of each point by means of a finite difference method, with an
oversampling factor of 4, to reduce approximation errors.
Results
The model has been widely tested and used with N=16 and a four
times oversampling factor. In these conditions, it runs near to real
time on a Pentium 266 system. Current researches are made on better
methods of integration, in order to reduce the oversampling factor,
reaching thus the real time on today's commercial systems. Research
directions are also toward a global reduction of the complexity, to
improve the number of harmonics generated, and to reach some degree of
polyphony. The bowing point can be made continuous by a technique
similar to polyphase filtering, thus requiring memory but no further
computational complexity. In the same way can be also implemented
supplementary bonds along the string, f.i. slight fingerings like
those used to produce harmonics.
As to the excitations, we implemented simple models of pinch and
percussion that brought very satisfactory results from the standpoint
of imitation. As to the bowing, we made simulations with the specific
goal to imitate sound emissions of actual bowed instruments and they
gave encouraging results in the case of little variations of the
pitch, particularly as to the bow attack transients and the evolution
of longlasting sounds. Furthermore, the sounds generated with our
model approximate very well, from an acoustic standpoint, the actual
ones, especially in the variation of pressure, velocity and position
also in nonstandard conditions of execution.
The model has been used with "impossible articulations"
(see [9]) not only in the sense that sounds with parameter variations
timings that can't be performed by humans have been produced, but also
in the extended meaning that parameters, which are physically
unreachable, like f.i. internal friction, have been varied during
sound emission.
Conclusions
The evaluation of the various direction of research strongly
depends on ones goals. If we assume the imitation of actual
instruments as a primary goal of the physical model simulation, we
probably know how to interpret the results we obtained. But if our
goal is instead to provide a new synthesis instrument, is both better
controllable and more stimulating when compared to other methods, then
the evaluation criteria become more complex and, in our opinion,
finally depend strongly on musical considerations.
In this latter case the imitation of actual instruments is relevant
but only in an indirect way. It is just a guideline to verify the
correctness of the hypothesis and the methods adopted, but is no
longer, by its own, an interesting goal.
From the point of view of contemporary music, the modeling and
implementation of features and behaving that are physically
impractical may result more important and interesting even if
unverifiable.
If we assume this point of view, the key of the evolution of our
research lies in the hands of musicians and composers rather than in
those of the researchers. Thus its clear that such a research can
be led but through a strong interaction between researchers and
musicians.
Acknowledgements
Thanks to Michelangelo Lupone for his precious suggestions and for
encouragement, and for composing and playing our instrument.
Thanks to Marco Giordano who reviewed the paper.
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