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Guitar Virtual Instrument using Physical Modelling with Collision Simulation

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We have created a guitar virtual instrument by simulating string vibration using a finite difference method to solve a modified one-dimensional wave equation with damping and stiffness, along with a collision system that allows the simulated guitar to perform a variety of articulations. Convolution with impulse response is also used to enhance the realism of the sound. The core model design, implementation approach and the optimization techniques are presented in this paper.
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Guitar Virtual Instrument using Physical Modelling with
Collision Simulation
Ka-wing Ho*
Yiu Ling*
Chuck-jee Chau
Department of Computer Science and Engineering
The Chinese University of Hong Kong
Shatin, Hong Kong
kwho
@link.cuhk.edu.hk
yling
@link.cuhk.edu.hk
chuckjee
@cse.cuhk.edu.hk
* These authors contributed equally to this work.
ABSTRACT
We have created a guitar virtual instrument by simulating
string vibration using a finite difference method to solve a
modified one-dimensional wave equation with damping
and stiffness, along with a collision system that allows the
simulated guitar to perform a variety of articulations.
Convolution with impulse response is also used to enhance
the realism of the sound. The core model design,
implementation approach and the optimization techniques
are presented in this paper.
1. INTRODUCTION
Guitar is a very popular musical instrument that can
produce tones with rich dynamics because of the variety of
ways to interact with the strings. A small difference in
finger motion can make a big difference in tones.
Sample-based synthesis is commonly used for existing
commercial guitar virtual instruments. However, there
may be difficulties in smoothly transitioning between
articulations and providing fine-grained controls due to the
number of samples required. Although multiple physical
modelling schemes had been proposed [1], they cannot
fully simulate the large range of articulations that can be
performed on guitars. Therefore, we would like to develop
a physical modelling scheme with a newly added collision
system to emulate the way guitars are played so that better
flexibility can be achieved for music production.
We have prepared a website1 with demo videos and
articulation animations to demonstrate our prototype
program for practical uses.
2. STRING VIBRATION MODEL
2.1 Wave Equation
To model the vibration of a guitar string, we use a partial
differential equation (PDE). The equation is modified from
the 1D wave equation [2]:
(1)
The 1D wave equation allows us to simulate the vibration
of an ideal string where the motion is strictly confined to a
2D plane. This has the advantages of being less complex
and more lightweight than simulating strings in 3D space.
Based on the presentation by Shuppius [3], we get
equation (2) which accounts for stiffness as well as
frequency dependent and independent damping factors:



(2)
To allow the tension of the string to be changed over
time, we introduce a new variable tension factor T(t):



(3)





  
In equation (3), y is a function of x and t, representing the
displacement of the string segments along the y-direction
at position x at time t, see Fig. 1. By changing T and μ, we
can set the pitch of the string, and by changing the damping
and stiffness factors, we can modify the timbre.
2.2 Boundary and Initial Conditions
To allow our model to simulate strings fixed by the bridge
and nut of the guitar, we define the boundary condition as:
 (4)
In our model, x = 0 and x = L are assumed to be the
position of the nut and bridge of the guitar respectively,
where L is the length of the string.
As we will discuss in Section 3, collision is used for
applying excitation to the string during the simulation, thus
we can set the initial condition to be at the resting position:
(5)
2.3 Finite Difference Method
It is difficult to analytically solve a complicated PDE such
as the one we use. Collision and external force simulation
also add additional complexity. Therefore, we choose to
use a numerical approximation with a finite difference
scheme based on that presented by Shuppius [3] instead.
Copyright: © 2020 Ho et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution License 3.0
Unported, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original author and source are credited.
1 https://weonix.github.io/Physical-Guitar-ICMC-Demo/
We discretize the model by dividing the space domain
along the string into nodes with distance Δx between
adjacent ones. The time domain is also divided into
discrete steps with interval Δt in between. Then we can
define the temporal and spatial resolution as such:
  (6)
  (7)
We can then obtain our finite difference scheme by
substituting each partial differential with a finite difference
approximation. The formula we obtain consists of terms of
y(x ± nΔx, t ± mΔx), where m and n are non-negative
integers. After moving the term y(x, t + Δt) to one side, we
get a formula that computes y(x, t + Δt) using the y(x ±
nΔx, t - mΔx) terms where m ≤ 0. Hence, we get a stepwise
algorithm that computes the new state of the string at each
time step using the information of previous steps.
Figure 1. Coordinate system and node setup of the string
model. (Not to scale)
2.4 Stability Condition
The stability of a finite difference scheme is determined by
the CourantFriedrichsLewy condition [2]:

  (8)
In our model, u is the string wave speed, defined by:
(9)
A finite difference model can remain stable only if the
Courant number C is less than or equal to a fixed value
Cmax which depends on the nature of the model. Through
our experimentation, we have found that the Cmax of our
model is around 1. To ensure the model stability regardless
of varying tension, we must also set an upper limit to the
tension T as Tmax. Thus, for a given setting, the maximum
spatial resolution can be determined as such:
 
 (10)
3. COLLISION
We also add collision simulation into our stepwise
simulation algorithm. To simulate collision, we define a
number of colliders in the scene. A collider can represent
objects such as a guitarist’s finger that may interact with
the string, or a fret wire on the fretboard.
We perform collision by computing a vertical collision
boundary based on the collider placements at each time
step. Then if the y-axis displacement of a node exceeds its
collision boundary, it will be forcefully set to the boundary,
so that the string’s motion will be obstructed by the
colliders appropriately.
To improve the performance, we can also label the
colliders as “static” or “dynamic” and do not recompute
the collision boundary of static colliders at each simulation
cycle unless they are moved explicitly.
3.1 Soft Collision
To better model colliders that are not completely rigid (e.g.
a guitarist’s finger and palm), two parameters: softness and
elasticity are used to approximate this property.
To simulate softness, when a string node exceeds its
collision boundaries, instead of setting its displacement
value to be exactly at the collision boundary, it is set to be
a ratio (i.e. the softness value) between the displacement
and the boundary, so that the string can partially overlap
with the collider before being pushed out over time.
Elasticity determines the energy loss that the string
experiences when it overlaps with colliders. At each
simulation step, if a string node is overlapping with a
collider, the new displacement of the node will be moved
to a position closer to its original displacement, essentially
slowing down the node movement.
4. ARTICULATION
Using our collision model, we have a lot more flexibility
in manipulating the string over the traditional method of
simply setting initial conditions (e.g. a triangular shape).
We make use of fixed colliders placed along the string to
represent the fretboard. We can then use other colliders to
press onto the string against the frets and to pull on the
string before releasing, to simulate string fretting and
plucking motions as in Fig. 2 and 3.
Figure 2. The fretboard arrangement of the string model
at the resting position. (Not to scale)
Figure 3. Fretting and plucking colliders acting on a
string to perform a plucking mo. (Not to scale)
Because of the limitation of simulating string vibration
on a 2D plane, when we place the fret colliders along the
string, the string may hit the fretboard and produce
undesirable sounds. To alleviate this, we placed the
colliders at a relatively steep slope, so that the unused frets
are less likely to collide with the vibrating string.
Using our method, incidental noise caused by the
interactions between the string and colliders, like the noise
of a fretting finger being released, will be generated as a
natural result, which enhances the realism of our model.
By using different combinations of collider motions and
parameter changes, we can implement a variety of
articulations used by real guitar performance, such as palm
mute, tapping, slapping, harmonics, sliding, hammer-on
and pull-off. Effects like fretted and fretless guitars or
using fingers and plectrums can also be simulated. Some
techniques used to reproduce these effects are listed below:
Bending and Vibrato: The pitch of the note that is
being played can be controlled by modulating the tension
of the vibrating string, resulting in a pitch-changing effect.
Palm Mute: We place a soft collider representing the
palm pressed against the string near the bridge to absorb
some of the string vibration and produce a muted sound.
Dead Note: The fretting collider only presses softly
against the frets while the frequency-independent damping
factor of the string is increased so that the notes played
become short and muted.
Slapping and Popping: We reduce the slope of the fret
colliders so that the string would be more likely to collide
with the fret wires and fingerboard to produce a collision
noise typical to the slapping and popping articulations.
Natural and Artificial Harmonics: Before we play a
note, we place an additional soft and non-elastic collider
that overlaps the string at a specific harmonic ratio of the
string length, which results in a unique harmonic tone.
Sliding: We can simply slide the fretting collider along
the string as a note is being played, so the length of the
vibrating string and the pitch of the note will be changed.
Finger and Plectrum: To simulate the difference
between plucking with a finger and plectrum, we use a
relatively soft collider to represent the finger, and increase
the softness of the collider as we release the finger to
emulate the string sliding off the soft flesh, which produces
a softer and duller tone than the one with a hard plectrum.
5. AUDIO GENERATION
In order to generate audio from string vibration, we first
take a sample of the displacement of the string at a specific
position(s) after every simulation step (Δt). Then we can
make use of the current and previous displacements to
compute the velocity of the string at the sampling point(s),
which will be used as the amplitude for audio output.
To simulate other elements that affect the guitar tone,
such as acoustic guitar bodies, or properties of pickups and
microphones, we take inspiration from a project by
Harriman [4] and choose to use convolution with impulse
response samples. This allows us to emulate a large range
of guitar models and environments without much increase
in the model complexity.
5.1 Magnetic Pickup
To simulate a magnetic pickup used for electric guitars, we
can set the sampling position as the pickup position, which
can be adjusted to produce different tones like in real
electric guitars, and then we can apply convolution with an
impulse response sample obtained from the real pickup to
simulate the tone of this pickup. Multiples pickups can also
be used at once by adding up the audio signals obtained
from multiple sampling positions.
5.2 Acoustic Body
For acoustic guitars, most of their sound comes from the
vibration of the strings being transferred through the
bridge of the guitar into the body, which would reverberate
and amplify the sound. Therefore, we can sample near the
bridge, and apply a convolution with an impulse response
sample which carries the information of the body
reverberation and microphone or piezo pickup signal
transformation to simulate the acoustic body and use of
microphones or piezo pickups.
6. VIRTUAL INSTRUMENT
IMPLEMENTATION
To produce a practical virtual instrument prototype, we
develop our program with JUCE in C++, which is a library
that handles some parts of audio software implementation
like audio processing, MIDI, VST interface and GUI.
Figure 4. A screenshot of the prototype program.
To simulate a full guitar, we have multiple copies of the
string model running at once (six for a typical guitar), each
tuned to a different note. When a MIDI note is played, an
appropriate string is chosen to play the note while
minimizing the hand movement, to mimic the behavior of
a guitarist. We also provide a strumming mode so the
fingering when playing multiple strings at once (or with
small time offsets) can be controlled more easily.
For performing and controlling different articulations,
we define a MIDI map for the ease of MIDI programming
and live performance. We also provide a set of key-
switches to switch between different articulations modes.
To give more fine-grained controls over the guitar
articulations, the note-on and note-off velocities of the
key-switches and normal notes also affect the way a note
is played. For example, the note-on and note-off velocities
under the default articulation mode determine the plucking
intensity and release speed respectively.
7. PERFORMANCE
In order to provide stable simulation and good audio
quality, the simulation must run at least 44.1k cycles per
second with 100 string nodes. Further increase of the
temporal and spatial resolution would still lead to audible
quality differences. With six strings and even multiple
instances of the program, the program may not be practical
for an average consumer-level computer to run smoothly.
Therefore, optimization is necessary.
We make use of many techniques to improve the
performance such as multithreading for different strings
and redundant code reduction. With these optimization
techniques, the program has been significantly accelerated.
7.1 Single Instruction Multiple Data (SIMD)
Another important optimization technique we use is SIMD,
which is a technique to operate multiple data as vectors in
a single instruction so that the whole process can be
accelerated. It means that we can utilize SIMD to compute
multiple string nodes at the same time.
However, it cannot be achieved easily in the actual case
because of the essence of the finite difference method.
Originally, we store the nodes sequentially in an array and
then group the nodes into different vectors in that order.
However, each node needs to access its adjacent nodes
during computation, some of which would be in the same
vector and some would not, such inconsistent accessing
cannot be done efficiently with SIMD operations.
Therefore, we use an alternative string node ordering
method as shown in Fig. 5. With this method, every node
can simply access the same element location in the
adjacent vectors. An extra vector needs to be constructed
for the first vector to access its previous vector in every
simulation cycle. It is similar for the last vector. However,
the number of extra vectors is fixed. Hence, this cost is not
significant when the spatial resolution is high enough.
Figure 5. A list of vectors with size four, using the
alternative ordering method. Here the numbers in each
vector vi represent the index of the actual string nodes.
The general rule for this ordering method is defined by:

 (11)

(12)
where f is the function for mapping the index i (starting at
zero) in the array to the index (starting at zero) of the actual
node, g is the function for doing the inversed mapping, Size
is the vector size, and N is the number of vectors.
7.2 Performance After Optimization
A performance test was conducted with an AMD Ryzen 5
1600 six cores 12 threads CPU running at 3.6 GHz, with
16 GB DDR4 RAM running at 3000 Mhz. Four versions
including the program without optimization, with
optimization (except for SIMD), with SSE and with AVX2
are tested. For the optimized versions, we also tested their
performance with single threading and multithreading.
During the test, the simulation cycle was ran one million
times at 120 spatial resolution with six strings, and the total
computation time was measured. We ran the test three
times to obtain the average time rounded to the nearest
millisecond. As shown in Fig. 6. the performance
improves significantly after optimization and can be used
as a practical real-time music production program.
Figure 6. The result of the performance test, showing the
performance difference before and after different kinds
of optimization.
8. CONCLUSIONS
We have developed a guitar simulation with a physical
string model, with an additional collision system that can
simulate common guitar articulations. Convolution with
impulse response samples is utilized to improve the
realism of the simulated sound. Moreover, with
optimization, the prototype program of our model can run
reasonably well on desktop computers.
There is still room for improvement, such as the presence
of longitudinal wave and sympathetic vibration, but we
consider our simulation sufficient for producing natural
and convincing simulated sound for music production.
9. REFERENCES
[1] C. McKay, “A survey of physical modelling
techniques for synthesizing the classical guitar,” 2003.
[2] H. P. Langtangen and S. Linge, Finite Difference
Computing with PDEs A Modern Software Approach.
Cham: Springer International Publishing, 2018.
[3] M. Shuppius, “Physical modelling of guitar strings,”
presented at Audio Developer Conference 2017,
London. [Online]. Available:
https://www.youtube.com/watch?v=sxt5rxF_PdI.
[Accessed: 16-Dec-2019].
[4] J. Harriman, “Filtering Techniques for Piezoelectric
Transducers.” [Online]. Available:
https://ccrma.stanford.edu/~jiffer8/420/project.html.
[Accessed: 16-Dec-2019].
ResearchGate has not been able to resolve any citations for this publication.
A survey of physical modelling techniques for synthesizing the classical guitar
  • C Mckay
C. McKay, "A survey of physical modelling techniques for synthesizing the classical guitar," 2003.
Physical modelling of guitar strings
  • M Shuppius
M. Shuppius, "Physical modelling of guitar strings," presented at Audio Developer Conference 2017, London. [Online]. Available: https://www.youtube.com/watch?v=sxt5rxF_PdI. [Accessed: 16-Dec-2019].
Filtering Techniques for Piezoelectric Transducers
  • J Harriman
J. Harriman, "Filtering Techniques for Piezoelectric Transducers." [Online]. Available: https://ccrma.stanford.edu/~jiffer8/420/project.html. [Accessed: 16-Dec-2019].