On the Synthesis of Guitar Plucks
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A CTA A CUSTICA UNITED WITH A CUSTICA
Vol. 90 (2004) 928 – 944
On the Synthesis of Guitar Plucks
J. Woodhouse
Cambridge University Engineering Department, Trumpington St, Cambridge CB2 1PZ, UK.
jw12@eng.cam.ac.uk
Summary
Several different approaches are described to the problem of synthesising the pluck response of a guitar, starting
from information about string properties and the input admittance at the bridge of the guitar. Synthesis can be
carried out in the frequency domain, the time domain, or via modal superposition. Within these categories there
are further options. A range of methods is developed and implemented, and their performance on a particular test
case compared. The two most successful methods are found to be modal synthesis using the first-order method,
and frequency-domain synthesis. Of the two, frequency domain synthesis proves to be faster. A significant con-
clusion is that the coupled string/body modes of a normal classical guitar do not show “veering” behaviour except
at low frequencies, so that it is important to use a synthesis method which incorporates fully the effect of string
damping: methods based on first finding undamped modes give poor results.
PACS no. 43.40.Cw, 43.75.Gh
1. Introduction to the synthesis of plucked-string transients are examined,
and in a companion paper [1] the predictions are compared
On the face of it, vibrational analysis of the guitar is rather with measurements.
simple. When a guitar string is plucked the player cre- The vibration of an ideal stretched string is treated in ev-
ates certain initial conditions of displacement and velocity ery vibration textbook, and the specific behaviour of mu-
in the string and the guitar body, then releases the string sical strings has also been researched extensively (see for
so that they vibrate freely. Provided the motion is small example [2, 3, 4]). Similarly, many measurements have
enough for linear theory to apply, the resulting vibration been published of the vibration behaviour of guitar and vi-
and sound radiation is given simply by a superposition of olin bodies (see for example [2, 5]). It seems remarkable,
the transient responses of the various coupled string/body therefore, that there is no published work which explores
modes of the instrument. (“Body” is used here to include all the issues of putting the two together to give a fully
the effect of the air within the cavity and outside.) The detailed method to synthesise the coupled string/body vi-
player can, to a certain extent, control the amplitudes mak- bration from the separate knowledge of string and body
ing up the modal mixture, but the frequency, decay rate and behaviour. Parts of the problem have been tackled, cer-
radiation behaviour of each mode are governed by the in- tainly. A thorough and insightful treatment has been given
strument’s construction and stringing, and the player can by Gough [6] of the coupling of one string overtone to one
only exert a very minor influence on them for musical ef- mode of an instrument body. The specific coupled vibra-
fect. tion problem of the cello’s wolf note has been discussed by
Somewhere in the details of these transient decaying several authors (e.g. Schelleng [7]). In a very recent study,
sounds lies the information which determines perceptual Derveaux et al. [8] have presented numerical simulations
judgements: discrimination and quality rating between which include a string, an idealised guitar body and the
instruments, and the musical “palette” available to the surrounding air. There is also an extensive literature on the
player. It is not clear precisely which features of these tran- general problem of computational methods for vibration
sients have audible consequences: it is likely that in the analysis of complex structures, including methods based
context of musical performance, rather subtle effects may on assembling the behaviour of a complete system from
sometimes matter. In pursuit of the goal of relating musi- results for separate subsystems (see for example [9]).
cal qualities to the constructional details of an instrument However, none of these treatments entirely addresses
it is necessary to develop theoretical models of the system, the problem of synthesis of the transient response to a
and to be useful such models must be capable of achieving pluck over the full frequency range of interest for musi-
an accurate match to the observed physical behaviour of cal applications. That is the task of this paper, and the
an instrument. In this paper, various possible approaches attempt to solve it and to compare the results with mea-
surements will reveal some surprising subtleties. Many of
Received 1 October 2003, these are associated in one way or another with the correct
accepted 26 March 2004. treatment of vibration damping: after all, one of the major
928 c S. Hirzel Verlag EAAWoodhouse: On the synthesis of guitar plucks ACTA A CUSTICA UNITED WITH A CUSTICA
Vol. 90 (2004)
criteria for a successful synthesis method is that it should have length L, tension T , mass per unit length and bend-
give good predictions of decay rates. These vary strongly ing rigidity B . (For a homogeneous solid string of radius
with frequency, and reproducing this pattern by synthesis a and Young’s modulus E , B = Ea4 =4.) In the sim-
is a challenging problem. plest case, the string undergoes transverse vibration in a
It should be emphasised that the objective here is to single plane normal to the guitar’s soundboard, with trans-
develop synthesis methods which are physically accurate verse displacement y (x t) at position x and time t. At one
in the sense of matching in as much detail as possible end, x = 0 say, the string is assumed for the moment to
the measured behaviour of real guitars. The purpose of be rigidly anchored so that it has no displacement. At the
the models is primarily to address questions of interest to other end, x = L, it is attached to the body of the gui-
acoustical guitar makers and players, so that it is important tar so that string and body have identical displacements. If
to keep a clear link between physical parameters of the string vibration with both polarisations is included, trans-
instrument and the parameters of the model. This results verse displacement z (x t) will be assumed in the plane
in significantly different priorities to work which is con- parallel to the guitar’s soundboard.
cerned with designing synthesis algorithms for real-time For the body, the most direct characterisation of dy-
musical performance (see for example [10]). The empha- namic behaviour is via the input admittance at the point
sis there is inevitably on making such simplifications as on the bridge where the string makes contact, or more
are possible without sacrificing too much in sound quality, generally via the admittance matrix at this point. We may
in the interests of speed. The approach taken here is to de- choose to process this admittance to extract modal proper-
fer questions about auditory consequences until the model ties: each mode has an effective mass, effective stiffness,
has been demonstrated “complete” by laboratory measure- damping factor, and angle of movement at the bridge. It
ments. It will then be possible to proceed by subtraction, also has a radiation efficiency and pattern, which would
suppressing details of the model and assessing by listen- be relevant if the desire was to synthesise the sound field.
ing tests whether the change is significant to a listener or However, the emphasis here will be on predicting the tran-
player of the guitar. sient response measured on the structure. So far as this
It is worth noting the complexity of the synthesis task. paper is concerned, sound radiation is relevant only as
The radiated sound from a typical note on a classical gui- one mechanism contributing to the modal damping fac-
tar shows clear spectral peaks up to at least 5 kHz. This tors. Other effects of the air around the guitar body are
gives a useful target frequency range for synthesis. It cor- included implicitly: they influence the input admittance,
responds to about the 60th harmonic of the lowest note on and the “modes” into which this may be decomposed are
a normal guitar (82 Hz). Each of these isolated string “har- the coupled air-structure modes. It will be seen that this
monics” can appear in two polarisations. A typical guitar sub-problem is already quite challenging, without the ad-
body structure has of the order of 250 vibration modes in ditional layer of difficulties associated with radiation and
this range (plus a roughly comparable number of modes of room acoustics.
the internal air cavity [11]). Assembling these numbers, it The relation between input admittance (velocity per unit
is clear that an accurate synthesis method may have to ac- force) Y (! ) of the body at the string’s attachment point
count correctly for several hundred degrees of freedom. It and the modal properties is given by the standard formula
may come as a surprise to learn that very few vibration pre- (see for example Skudrzyk [12])
dictions of this degree of complexity have ever been con- X i!
firmed in detail by experiment. Although very large Finite- Y (! ) = m ( ! 2 + i!! ; ! 2 ) (1)
Element models are routinely used for vibration prediction k k k k k
of industrial structures of many kinds, where these have where ! is the angular frequency, and the k th mode of vi-
been checked against measurements they rarely give full bration of the body (including the surrounding air) in the
agreement in detail beyond the first few modes (although absence of the strings has effective mass mk , natural fre-
qualitative and statistical aspects of the predictions may quency !k and modal damping factor k (or corresponding
work to higher frequencies). Not for the first time, musical modal Q-factor Qk = 1=k ). The “effective mass” is re-
acoustics is “pushing the envelope” of vibration analysis. lated directly to the k th mode shape uk : if the mode shape
is normalised in the usual way with respect to the system
2. Overview of synthesis methods mass matrix [11], then
2.1. Characterising string and body
mk = 1=u2k (bridge) (2)
There is a wide variety of possible approaches to synthesis.
Before going into details, it is useful to review these meth- where (bridge) connotes the attachment point of the string.
ods in outline and note their potential strengths and weak- At least for the moment, the mode shapes, and hence
nesses. All methods start from the same basic information. the effective masses, will be assumed to be real: in other
The string is characterised by a small number of funda- words, proportional damping is assumed for the body vi-
mental physical properties: tension, mass per unit length, bration. Damping will also be assumed to be small, so that
bending stiffness and damping. Other relevant properties, k 1 or Qk 1. Very similar expressions to equa-
such as wave speed and characteristic impedance, follow tion (1) hold for the more general case of the 2 2 ad-
from these. Symbolically, the string will be assumed to mittance matrix at the bridge, which is needed when both
929A CTA A CUSTICA UNITED WITH A CUSTICA Woodhouse: On the synthesis of guitar plucks
Vol. 90 (2004)
string polarisations are included in the synthesis model. incide, the undamped modal calculation will always pro-
Details of this case are given in section 3.3. (Note that the duce two separated frequencies, with the two correspond-
expression (1) appears at first sight to be different in form ing coupled modes involving significant motion of both
to the expression for admittance given by Christensen and string and body, in phase for one mode and in antiphase
Vistisen [13, equation(6a)], which takes explicit account for the other. The Rayleigh calculation of modal damp-
of the structure and the enclosed air. However, if the de- ing factors will assign both modes equal damping at this
nominator of their expression is factorised and the whole “tuned” point. This effect, giving relatively high damping
expression then expanded in partial fractions, it takes the to both modes when playing a note very close to a strong
form (1) above.) body resonance, is often described by guitar players with
a phrase like “this note falls flat”. However, the intrinsic
2.2. Modal synthesis internal damping of the string mode will generally be far
lower than that of the body mode, and under these circum-
The first class of synthesis methods works by computing
stances Gough shows that the correct result may be very
the coupled string/body modes, together with appropriate
different from that just described. If the disparity of damp-
frequencies and damping factors, then using modal super-
ing between the string and the body is large compared
position to construct the transient response. This could be
to a measure of the strength of coupling between string
the response to an idealised pluck, a step function of force
and body, then when the string mode is tuned through the
applied at a single point of the string, or more generally it
frequency of the body mode, behaviour is seen similar to
could be the response to any given patterns of initial dis-
that illustrated in Figure 1. Figure 1(a) shows the variation
placement and velocity: for example, the low-pass filtering
of the two modal frequencies through the “tuning” range.
effect associated with the finite stiffness and radius of cur-
The frequencies calculated ignoring damping show “veer-
vature of the player’s fingertip could be included (see for
ing” behaviour, but the correct frequencies allowing for
example Benade [14, chapter 8]).
the effects of damping cross over. Figure 1(b) shows the
The simplest modal approach proceeds in four stages.
corresponding behaviour of the modal loss factors. When
First, a suitable set of generalised coordinates is chosen
calculated by the Rayleigh argument based on undamped
to describe the motion of the string and body, and the
modes, the two curves cross so that at the “tuned” condi-
mass and stiffness matrices for the coupled system are
tion the damping factors are equal. In the correct calcu-
worked out in terms of these generalised coordinates. For
lation, by contrast, the two damping factors remain very
the present purpose, the most natural choice of generalised
different throughout the range, so that it is always possi-
coordinates consists of (i) the mode amplitudes of the body
ble to identify one “string” mode with low damping and
in the absence of strings; (ii) a set of Fourier series coeffi-
one “body” mode with higher damping. The key to the
cients for the string displacement, in other words the am-
difference in behaviour is that the mode shapes become
plitudes of what would be the string modes if both ends
significantly complex in the correct calculation, invalidat-
of the string were rigidly fixed; and (iii) one or two “con-
ing the assumption underlying the Rayleigh method. The
straint modes” (see e.g. [9]), to allow the string to move at
detailed calculation leading to this figure will be described
the end attached to the body. The resulting mass and stiff-
in section 3.4.
ness matrices will be calculated in detail in section 3.1. In
To avoid the problem just illustrated, it is necessary to
the second stage of the pluck calculation, the modes and
tackle the rather murky question of how the damping of
natural frequencies of the coupled string/body system are
the string/body system should be modelled, so that real-
computed in the absence of damping by standard eigen-
value/eigenvector analysis. Third, a suitable Q-factor is
istic modal behaviour of the coupled, damped system can
be calculated. Unfortunately, there is no universal damping
calculated for each mode using a small-damping argument
model which has the same physical credibility as the stan-
based on Rayleigh’s principle [15, 16]. Finally, the tran-
dard models of elastic and inertial effects through the stiff-
sient response to an ideal pluck at a given position on the
ness and mass matrices. When an explicit damping model
string can be calculated using a standard modal superposi-
is needed, it is conventional to use the “viscous damping
tion result: again, details are given in section 3.1.
model” first introduced by Rayleigh [18]. If it is assumed
The method just outlined is precisely what is used in
that the rate of energy dissipation depends only on the in-
the vast majority of calculations of the vibration response
stantaneous values of the generalised velocities, then for
of complex structures. For example, almost all Finite-
small motions it is reasonable to approximate this dissipa-
Element calculations work this way (although they usu-
tion rate by a quadratic expression in the velocities, and
thus introduce a third matrix, the dissipation matrix C , so
ally use a cruder estimate of modal damping than that
resulting from the Rayleigh’s quotient method). Unfortu-
that the equations of free motion of the system take the
nately, as pointed out by Gough [6], this method does not
form
always give a good approximation to the correct answer.
The crucial issue concerns what is usually called “veer- M q + C q_ + K q = 0 (3)
ing” behaviour [17]. Imagine adjusting the tuning of the
string so that one particular string overtone passes close where q is the vector of generalised coordinates, and M
to the frequency of a body mode. At the point where the and K are the mass and stiffness matrices respectively.
string and body resonant frequencies were expected to co- Whether and when it is acceptable to use this viscous
930Woodhouse: On the synthesis of guitar plucks ACTA A CUSTICA UNITED WITH A CUSTICA
Vol. 90 (2004)
with M and K by the usual (real) mode shapes. There
are some mathematical conditions under which this is jus-
tified (see Caughey [21]), but there is rarely any reason
to expect these conditions to hold for a physical system.
Proportional damping is simply a mathematically conve-
nient assumption, which usually works well enough. In
the string/body coupling problem, Gough’s result shows
that we cannot make this assumption.
To retain a “modal” approach while solving equations
(3) with a non-proportional damping matrix C , the usual
method is to recast the equations into first-order form. This
can be done in more than one way, but the most direct is
to define a double-length vector
q
p = q_ (4)
and a 2N 2N matrix
0 I
A = ;M 1 C ;M 1 K
; ; (5)
where 0 denotes the N N zero matrix and I the N N
unit matrix, N being the number of degrees of freedom.
The equations (3) can then be rewritten
p_ = Ap: (6)
Now a modal solution p(t) = v k ek t leads to
Avk = k vk : (7)
Solving this eigenvalue/eigenvector problem yields the
(complex) mode vectors and associated complex natural
frequencies, and there are then standard formulae for fre-
Figure 1. Veering and non-veering behaviour in a simple model quency response functions and impulse response functions
of one string mode coupled to one body mode. The uncoupled in terms of modal sums (see for example Newland [22,
Q
body mode has a fixed frequency and a -factor of 100, while the chapter 8]).
Q
uncoupled string mode has a -factor of 3500 and a frequency In summary, there are two types of modal synthesis
varying over a small range. The variation of (a) the coupled mode which could be used for guitar plucks. One is based on
frequencies and (b) the modal damping factors is plotted against undamped modes, post-processed to give damping fac-
the frequency ratio of the two uncoupled modes. The dashed lines tors, while the other uses the first-order approach to give
show “veering” behaviour, when the calculation is based on un- damped modes directly (assuming a viscous model for
damped modes followed by Rayleigh’s principle to determine damping in the whole system). The first of these is sim-
modal damping, while the solid line uses the first-order method to pler and involves smaller matrices, but may give wrong
find damped modes directly and shows “non-veering” behaviour.
answers under some circumstances. The second is more
The detailed theory is given in section 3.4
accurate, but at the cost of doubling the size of the eigen-
problem to be solved. Both methods share certain advan-
tages over some other methods to be discussed shortly.
model of damping is a subject of some debate (see for ex- First, they work in terms of degrees of freedom which are
ample [19, 20]). It will be used provisionally in this study, very natural: loosely, “string mode amplitudes” and “body
but it should be kept in mind that experimental results may mode amplitudes”. This is very convenient for paramet-
require the issue to be revisited in the future. ric studies; for example, to explore the influence on sound
In the undamped problem, it is always possible to re- quality of particular body mode frequencies and damping
duce M and K to diagonal form by using the modal am- factors. A second advantage is that once the modes have
plitudes as new generalised coordinates. With three matri- been computed, the desired transient response is given by
ces M , C and K , it is not possible to diagonalise all three an explicit formula which introduces no further approx-
simultaneously by using modal coordinates. It is there- imations or numerical difficulties. If a calculation of the
fore common to introduce a further approximation at this radiated sound is wanted, the modal formulation is again
point, so-called “proportional damping”. Since informa- convenient: the modal sum expressing the transient re-
tion about C for real systems is usually rather sketchy, sponse simply has to be weighted by factors describing
it may be assumed that it is diagonalised simultaneously the modal radiation strengths (e.g. [23, 24]).
931A CTA A CUSTICA UNITED WITH A CUSTICA Woodhouse: On the synthesis of guitar plucks
Vol. 90 (2004)
A disadvantage of these methods is that they require the or of using the measured admittance directly, thus avoid-
body response to be expressed in terms of modal prop- ing the complication of modal fitting at high frequencies.
erties over the entire frequency range. This is simple and A potential disadvantage of the method is its reliance on
natural at low frequencies, but by frequencies of the or- inverse Fourier transformation at the final stage. Given the
der of 1 kHz in a typical classical guitar the modal overlap inevitable discrete frequency resolution and finite band-
factor of the body starts to become significant, and above width it is hard to guarantee an answer which is absolutely
that it is neither easy nor unambiguous to extract modal causal. However, it will be seen in section 4 that a suffi-
parameters from a measured admittance. ciently causal result can be achieved, and that makes this
method very attractive.
2.3. Frequency domain synthesis Finally, it should be noted that there is a completely
A completely different approach to the synthesis problem different approach to frequency-domain synthesis based
is to deal with the string/body coupling in the frequency on equation (3). By adding a vector of appropriate gen-
domain, and then use an inverse FFT to create the required eralised forces on the right-hand side of this equation and
time-varying transient response. This method is attractive then taking the Fourier transform, it is clear that the matrix
because the string and the body are coupled together at a of transfer functions can be derived directly by inverting
single point, and there is a very simple method which can the so-called “dynamic stiffness matrix”
be applied to any such problem: if two systems have in-
put admittances Y1 (! ) and Y2 (! ) and they are then rigidly D = ;!2M + i!C + K: (10)
connected at the points at which these admittances are de-
fined, then the coupled system has an admittance Y at that Using the same expressions for M , C and K as are needed
point satisfying for modal synthesis, this inversion can be carried out fre-
1 = 1 + 1: quency by frequency, then the relevant transfer function
Y Y1 Y2 (8) extracted from the matrix, and an inverse FFT performed
to give the transient response. This method has the virtue
This result expresses the fact that at the coupling point, of great conceptual simplicity, but in practice it is ex-
the two subsystems have equal velocities while the total tremely slow. The method is useful as a cross-check on
applied force is the sum of the forces applied to the two the coding of other methods, but is not a serious contender
separate subsystems. In the more general case in which for an efficient synthesis algorithm.
the coupling applies to more than one direction of motion,
then an equivalent result applies to the relevant admittance 2.4. Time domain synthesis
matrices: Finally, there is a class of possible synthesis methods
Y ; 1
= Y1 + Y2 :
;1 ; 1
(9) which work directly in the time domain. Two such meth-
ods can be found in the existing literature: to use a finite-
The body is naturally described in terms of its admittance difference approach to integrate directly the differential
(or admittance matrix), as already described. By calculat- equation governing string motion [25], or to adapt the syn-
ing the corresponding admittance at the end of a string, thesis method used in bowed-string studies, the so-called
which will be derived in section 3.2, these results can be “digital waveguide” approach (see for example [10, 26]).
applied immediately to give the impedance or impedance The finite-difference method starts from the differential
matrix of the strung guitar at the bridge. equations for the string and body motion. The various
To use this method to derive a pluck response, use can derivatives are approximated by difference formulae, and
be made of the reciprocal theorem of vibration response. the result is a formulation which lends itself to forward
We wish to find the body vibration which results from a integration by time marching. However, to apply such a
step function of force applied at a given position on the method to the model which will be used here, includ-
string. Reciprocally, we can consider applying the force at ing such effects as string bending stiffness, frequency-
the bridge, and calculate the resulting motion at the rele- dependent damping and accurate body modes up to at least
vant point on the string (and in the relevant direction there, 5 kHz, would require a level of detail in the implementa-
if string polarisations are to be taken into account). To tion which goes well beyond published work, and probably
solve this reciprocal problem, we simply have to multi- poses significant challenges of accuracy and speed, so this
ply two transfer functions together. The first is the coupled approach will not be explored further here.
admittance just calculated, which gives the velocity at the The digital waveguide approach is mathematically clo-
bridge when a force is applied there. The second is the sely related to the finite-difference method, but expressed
dimensionless transfer function between a given displace- in a more flexible and versatile form which does not rely
ment applied at one end of a string and the corresponding on underlying differential equations. The various physi-
displacement at the point where the pluck is to be applied. cal effects from the strings and the body are taken into
This second transfer function is also derived in section 3.2. account via “reflection functions”, essentially impulse re-
The advantages of this frequency-domain approach are sponses which encapsulate the various processes of dis-
that it is very simple, and that it gives the choice of ex- sipation, dispersion and reflection which occur as a wave
pressing the body admittance in terms of modes, as before, travels back and forth along the string. Any process which
932Woodhouse: On the synthesis of guitar plucks ACTA A CUSTICA UNITED WITH A CUSTICA
Vol. 90 (2004)
can be represented by a digital filter of some kind can be 3.1. Computing the coupled modes
incorporated. All the modal methods require the mass, stiffness and
Earlier bowed-string studies have provided almost all damping matrices for the coupled string/body system.
the ingredients necessary to attempt a synthesis of a gui- These are derived here for the case of a single polarisa-
tar pluck comparable to those possible by the other meth- tion of string vibration. The extension to include two po-
ods already mentioned. The basic behaviour of transverse larisations is dealt with in section 3.3. A convenient and
waves on an ideal string is handled directly by the dig- simple approach to the problem is to use Rayleigh’s prin-
ital waveguide method. The dispersive effect of bending ciple based on three sets of generalised coordinates: the
stiffness in the string can be handled using a reflection modes of the string with pinned ends, the modes of the un-
function reported by Woodhouse [27], and this has been strung body as experienced through the coupling point at
shown to work well to reproduce the pluck response of a the bridge, and one “constraint mode” which is the static
violin E string held by relatively rigid terminations [28]. response of the string when the end at the bridge is allowed
For a trial of the method, internal damping in the string to move [9]. Accordingly, write the string displacement in
can be handled using a “constant-Q” reflection function the form
[29]. Frequency-dependent string damping could in prin-
X
Ns
y(x) = a0 Lx + aj sin jx
ciple be incorporated by designing a suitable digital fil-
ter. Body resonances can be incorporated via a suitable
reflection function based on the impulse response of the
L
j =1
(11)
where the quantities a0 , aj are the (time-dependent) am-
body [30]. Using the modal properties of the body input
plitudes of the constraint mode and the j th pinned string
admittance, this can be efficiently implemented using re-
mode respectively, and Ns is the number of string modes
cursive IIR digital filters, one per mode (see for example
[31]). Bowed-string studies have not generally considered
used in the calculation. (Pinned boundary conditions for
the second polarisation of string motion, but there is no
the string in isolation seem appropriate, even in the pres-
difficulty in principle in incorporating this into the model.
ence of bending stiffness, because of the “crossed cylin-
In short, existing algorithms can be assembled to give a
der” geometry of a string over a fret or over the bridge
time-domain synthesis method for a guitar pluck which
saddle.) Similarly, denote the amplitude of motion at the
bridge of the k th body mode by bk . For the string and body
addresses the same physical phenomena which will be in-
motion to be equal at the point x = L,
cluded in the other methods.
However, the results of a test case were disappointing.
In section 4, the results will be shown of syntheses by var-
X
Nb
ious of the methods described above, using a standard set a0 = bk (12)
of parameters: 6 seconds of synthesis, a sampling rate of k=1
22050 Hz, and enough string and body modes to cover the
range up to 5 kHz. At this sampling rate, the digital waveg- where Nb is the number of body modes used in the calcu-
uide synthesis as described above proved to have poor ac- lation.
curacy, and more seriously it frequently resulted in unsta- The total potential energy of the system is then
ble, growing behaviour when trying to cope with the rather
1 XNb
1 Z L @y 2
low inherent damping of guitar strings. This instability had
its origin in problems with the “stiff string” reflection func-
V = 2 sk bk + 2 T
2
@x dx
0
tion, with appropriate parameter values for classical gui-
k=1
Z L @ 2y 2
tar strings. None of the other methods described earlier + 12 B @x2 dx (13)
have the possibility of instability in this sense. These faults 0
can be corrected, mainly by using a higher sampling rate.
where the three terms arise respectively from the stored en-
Equally, different digital filters could no doubt be designed
ergy in the body resonances, the energy in the string due to
to improve matters, as has been done in work aimed at
tension effects, and the energy in the string due to bending
stiffness effects. The terms sk denote the effective stiff-
musical synthesis (e.g. [10]). However, in the implemen-
tation used here and in comparison with other methods
nesses of the body modes, defined by
presented, this method does not offer an accurate and ef-
ficient synthesis method for guitar plucks for “laboratory” sk = mk !k2 : (14)
purposes, and it will not be considered further here.
Substituting equation (11), eliminating a0 using (12) and
3. Implementation details evaluating the integrals leads to
In this section a number of technical aspects of the syn-
X X " #2
V = 12 sk b2k + 21 TL
Nb Nn
thesis algorithms are dealt with, in order that explicit im- bk
plementations of various candidate methods can be made. k=1 k=1
2 X 1 B4 X
+ 12 T
The results of these are then compared with each other in Ns Ns
section 4, and with experimental measurements in a com-
2L j=1 j a
2 2
j + 2 2L3 j=1 j aj
4 2
(15)
panion paper [1].
933A CTA A CUSTICA UNITED WITH A CUSTICA Woodhouse: On the synthesis of guitar plucks
Vol. 90 (2004)
from which the stiffness matrix may be immediately de- As is usual with a “constraint mode” formulation, the stiff-
duced. In terms of the coordinate vector ness matrix has no string/body coupling terms: these occur
q=
a a a : : : b b : : : t (16)
only in the mass matrix (in submatrix M12 ) [9].
1 2 3 1 2
The next requirement is to specify the associated damp-
it is given by ing behaviour. The simplest possible model will be taken.
K 0
Each body mode and each string mode (in the absence
K= 0 K22
11 of body coupling) will be assigned an individual modal
Q-factor. In practice, the body damping factors are deter-
mined as part of the process of modal fitting to measured
where
admittance behaviour, while the string damping factors
K11 = diag T
2
+ B4 4T2 + 16B4 will follow a systematic trend with frequency of the kind
2L 2L3 2L 2L3 measured by Valette [3]. Notice, though, that an assump-
: : : j 2T j 4 B4 : : :
2 2 tion underlies this apparently obvious damping model:
L + 2L3 by specifying only “modal damping factors”, proportional
damping is implicitly being assumed for the string and
and the body considered in isolation. In the light of the earlier
2 s1 + T=L T=L T=L 3 comments about the dubious justification for any assump-
66 T=L s2 + T=L 77 tion of proportional damping this may seem a little odd.
6
K22 = 66 .
.. . 77 : (17) However, it seems to be the only realistic assumption at
64 T=L
..
sk + T=L 75
7 the present time, given the difficulty of obtaining reliable
experimental data on non-proportionally damped systems.
.. .. This is an issue to which it may be necessary to return in
. .
the future: for example, the “body modes” used here are
In a similar way, the total kinetic energy of the system is in reality coupled air/body modes, and it is possible that
;!2 T~ where this coupling may lead to significantly complex modes,
X ZL and also to complex “modal masses” in the formulation
T~ = 12 mk b2k + 21 y2 dx
Nb used here.
0 Two different methods for incorporating damping into
k=1
X "X
Nn #2 X
modal synthesis of the coupled string–body system were
1 Nb
1 L
= 2 mk b2k + 2 3 bk + 12 L
Ns
a2j
described in section 2.2: via post-processing of the un-
2 damped modes using Rayleigh’s principle, and via a vis-
k=1
X 0 Ns
k=1
Nn ! X
1 j =1 cous damping matrix C . For the Rayleigh method, the ex-
; 1 2L bk @ (;1) aj A
j pression for the total potential energy (13) of the string and
2 k=1 j =1
j (18) body is modified by replacing the tension, bending rigid-
ity and effective stiffnesses of the body modes by complex
values embodying the desired Q-factors:
from which the mass matrix may be written down:
M M T ;! T (1 + i=QT )
M = M11t M12
12 22 B ;! B (1 + i=QB )
where sk ;! sk (1 + i=Qk ):
(20)
M11 = diag L=2 L=2 : : : L=2 : : : The damping factors associated with the tension and the
2 L= L= L= 3 bending rigidity can vary with the frequency of the mode
66 ;L=2 ;L=2 ;L=2 77 in question if desired. The Rayleigh quotient incorporat-
M12 = 66
6 ... .. 77 ing these complex constants but evaluated using the un-
j +1 L (;1)j +1 L 7
. damped mode vector as a trial function gives an approx-
64 (;1)j+1 L
j (;1) j j
75 imation to the complex frequency of the damped system
.. .. [16], and hence the Q-factor of the coupled string/body
. . mode in question:
and
2 m1 + L L L 3 1 =mf!n2 g utn K un 0
QnWoodhouse: On the synthesis of guitar plucks ACTA A CUSTICA UNITED WITH A CUSTICA
Vol. 90 (2004)
with is the corresponding motion at the plucking point x from
(n)
B4 4T2 + 16B4 equation (11), where ak denotes the value of ak appro-
K11 = diag 2T
2
0
+
LQT 2L3QB 2LQT 2L3QB priate to the nth mode.
A similar modal sum can be used if damped modes
: : : j2LQ
2 T 2
+ 2jLB
4 4
3Q
::: and frequencies have been calculated using the first-order
T B form:
and X
2N
2s + T
1 T T
3 g(t) = n n en t (25)
66 Q LQT TLQT
1
s2
LQT
+ T
LQT
77 n=1
6 .. Q2 LQT 77
K22 = 666
where n is as above, and
0
.
..
. 77(22)
:
X
64 LQT T sk
Qk + LQT T 75 n=
N
Xnk where X = Vright
1
;
M ;1
. .. .. k=Ns +1
.
For the viscous damping model, the dissipation matrix and Vright
;1
is the 2N N matrix which is the right-hand
takes the simple diagonal form half of the inverse of the matrix whose columns are the
eigenvectors of the matrix A: see [22, equation (8.110) et
C = C011 C0
seq.].
22
3.2. Frequency response functions of the string
where
p The frequency-domain synthesis methods require as input
C11 = 2T diag Q1 Q2 : : : Qj : : : the drive-point impedance at the end of the string, and also
the transfer function linking motion at the end to motion
s1 s2 sj
ps m ps m ps m at the required plucking point. For the case of an ideal, un-
C22 = diag Q1 1 Q2 2 : : : Qk k : : : (23) damped string there are analytic formulae for both. How-
1 2 k ever, the effects of bending stiffness and, more impor-
and Qsj is the Q-factor of the j th string mode. tantly, string damping are essential to an accurate synthe-
sis. To include these effects, one simple approach is to ex-
This completes the information needed to calculate the
pand the analytic expressions in partial fractions, interpret
coupled modes, either undamped or damped. The calcu-
lation of the response to an ideal pluck at position x is
the terms as corresponding to the modes of the string, then
adjust the complex pole frequencies to allow for stiffness
then simple. In the first case, suppose that the undamped
modes un and natural frequencies !n have been calcu-
and damping. The residues associated with these poles are
lated from M and K , and the appropriate modal Q factors
kept unchanged – this amounts to the same approximation
Qn determined from equation (21). Each mode should be as assuming that the string in isolation has proportional
damping.
Suppose the string is fixed at position x = 0, and that a
normalised so that
utn M un = 1: harmonic displacement we i!t is imposed at x = L. To sat-
isfy the fixed boundary condition, the string displacement
A standard formula for impulse response can now be used. must take the form
y = a sin !x=c
Note that the velocity response of the body to an impulse
applied to the string will be the same as the acceleration
response of the body to a step function force applied at the p
where a is a constant and c = T= is the wave speed
same point. This acceleration response is then given by on the string. Imposing the other boundary condition fixes
the value of a, so that the response is
X
N ;
g(t) = n n cos !nt e ; !n t=2Qn (24) sin !x=c :
n=1 y = w sin !L=c (26)
where
X Now if the end motion is caused by a force f e i!t applied
n = (n)
bk to the string, force balance requires
k
@y
cos !L=c
is the modal motion at the bridge from equation (12),
(n)
f = T @x = Tw!
c sin !L=c
where bk denotes the value of bk appropriate to the nth x=L
mode, and
X
so the required end impedance is
= n x=L +
Ns ;
aj sin jx=L
(n) f = T cot !L = ;iZ cot !L
n
j =1 Z = i!w ic c 0
c (27)
935A CTA A CUSTICA UNITED WITH A CUSTICA Woodhouse: On the synthesis of guitar plucks
Vol. 90 (2004)
where Z0 is the characteristic impedance of the string. parallel to the bridge saddle, and Y12 is the cross admit-
The impedance Z has poles at !L=c = n , n = tance. As before, these can be expressed in terms of body
0 1 2 : : :. It is readily shown that all poles have the modes. However, we now need an additional property of
same residue ;iT=L, so each mode: the angle k of the body motion at the bridge
X measured from the normal direction. Then
Z = ; iLT 1 X
1
! ; nc=L : (28)
Y11 (!) = i! cos2 k
n=;1
k
mk (!k ; i!!k k ; !2 ) 2
X
Y22 (!) = m (!2 i;! sin
The corresponding expression with (small) modal dam- 2
k
i
!! k ; !2 )
ping included is
" X
k k k k
Z = ; iLT !1 + 1
1
! ; !j (1 + isj =2)
and
X i! cos k sin k
Y12 (!) = Y21 (!) = mk (!k2 ; i!!k k ; !2 ) :
j =1
# k
(32)
1
+ ! + ! (1 ; i =2)
j sj This is all that is needed for frequency-domain synthe-
" X # sis via equation (8), except to note that the string be-
i T
;L ! + 1 1
2 ! ; i! j sj
!2 ; i!!j sj ; !j2
haviour is rotationally symmetric so that the correspond-
ing impedance matrix is simply the impedance Z from
(29)
j =1
equation (29) multiplied by a 2 2 unit matrix.
where sj = 1=Qsj is the loss factor of the j th string For modal synthesis, it is necessary to introduce a sec-
mode. The modal frequency can be adjusted to allow for ond constraint mode, identical in form to the first but in the
small bending stiffness in the string by setting plane parallel to the soundboard. Then express the string
i c
B j 2 motion in that plane analogous to equation (11), as
!j L 1 + 2T L X
Ns
z (x) = a0 Lx + aj sin jx
(30)
L
0 0
(33)
a result which follows from Rayleigh’s principle applied j =1
to the string terms of equation (13). A similar procedure where the quantities a00 , a0j are the equivalents of a0 , aj
can be applied to equation (26) to provided a damped ap- for this plane. Now from continuity of displacement at the
proximation to the transfer function from the end of the
X X
bridge,
string to the plucking point: Nb Nb
a0 = bk cos k a0 =
0
bk sin k
y x + c X(;1)j 2! sin jx=L :
(34)
1
k=1 k=1
w L L j=1 !2 ; i!!j sj ; !j2 (31) in place of equation (12). The calculation of kinetic and
P
potential energies then follows very similar lines, except
that the term bk ]2 is replaced by
Note that equations (29) and (31) both give the correct
static response: when ! = 0 the string deflects in a straight hX i2 hX i2
bk cos k + bk sin k
line, the same shape used earlier as the “constraint mode”,
and the impedance is then spring-like with a stiffness T=L. XX
Note in passing that equations (29) and (31) could also
= bk bl cos( k ; l ): (35)
k l
be derived by a different approach, which is commonly
used in earthquake engineering (e.g. [32]). Starting from In terms of an extended vector of generalised coordinates
the expression (11), the applied end displacement can only
q=
a a a : : : a a a : : : b b : : : t
0 0 0
(36)
drive directly the motion described by the constraint mode. 1 2 3 1 2 3 1 2
That motion has associated with it an inertial reaction
force (a “d’Alembert force”) distributed along the string, 2K 0 0 3
the result for the stiffness matrix is
K = 4 0 K11 0 5
which can be resolved into components driving the other 11
modes of the string, leading to expressions for the desired
transfer functions.
0 0 K33
where K11 is as before and
3.3. String polarisations
The modal and frequency-domain synthesis methods can 2K33 = T cos( )T cos( k )T (37) 3
s1 + L 1; 2 1;
be readily extended to include two polarisations of string 66 cos( )T s2 + TL L 77
66 L.. 77
1; 2
motion. At the bridge, motion normal and parallel to the L
soundboard has to be taken into account, so in place of
the input admittance Y there is a matrix. It is convenient
66 . ..
. 77 :
to define Y11 to be the previous admittance in the normal
64 cos( L k )T
1;
sk + L
T 75
direction, so that Y22 is the admittance in the direction
. .. . ..
936Woodhouse: On the synthesis of guitar plucks ACTA A CUSTICA UNITED WITH A CUSTICA
Vol. 90 (2004)
The mass matrix is given by where K11
0
is as before and
2s + T cos(1 ;2 )T
cos(1 ;k )T
3
2M 0 M 3
1
66 cos(
Q 1 LQT
)T
1; 2 s2
LQT
+ LQT T
LQT
77
M = 4 0 M11 M23 5
11 13
6 .. T LQ Q2 77
M13t M23t M33 K33 = 666
0
.
..
. 77
64 cos(LQ
)T
1; 2
T
sk
Qk + LQT T 75
. ..
where M11 is as before,
.. .
2 cos L and
1 cos L 2
2C 0 0 3
66 ; cos2 L 1
; 2
cos L 2
66 C = 4 0 C11 0 5 :
11
(40)
M13 = 66
..
. 0 0 C22
64 (;1)j+1 cosj L (;1)j+1 cosj L
1 2
.. It is worth noting what the proportional damping assump-
3
. tion means in this particular context: if the body damping
cos j L is proportional, then in a given mode the connection point
; cos2j L 777 at the bridge moves along a straight line (at the angle k ) as
.. 77 assumed here. If the body damping were not proportional,
(;1)j+1 cosjj L 775
. this would allow the possibility of a phase difference be-
tween the components of body motion, so that the contact
.. point would travel around an ellipse.
2 sin L .
1 sin L 2
3.4. The criterion for ‘non-veering’
66 ; sin2L
1
; 2
sin L 2
66 From the matrices derived in sections 3.1 and 3.3, it is sim-
M23 = 66
..
. ple to recover Gough’s criterion for ‘non-veering’ men-
64 (;1)j+1 sinj L (;1)j+1 sinj L
1 2 tioned earlier. At a frequency where modal overlap is low,
it may be reasonable to approximate the behaviour when a
..
3
. string overtone falls close to a body resonance by keeping
sin j L just the two generalised coordinates concerned. The sys-
; sin2j L 777 tem matrices then take the general form
.. 77 M= m 1 m3 k1 0 c1 0
(;1) j +1
.
sin j L
775 m3 m2 K = 0 k2 C = 0 c2 (41)
j
. .. where the individual symbols are defined by comparison
with equations (17), (19) and (23). The complex natural
frequencies are the roots ! of the equation
and n o
det ; !2 M + i!C + K = 0
2M33 = L cos( )L cos( k )L 3 (38)
66 cos(m1 + 3)L m2 +3 L
1; 2 1; which reduces to
3
77 ;!2 + i!c =m ; !2
66 3.. 77
1; 2
66 .
3
..
. 77 : ;!12 + i!c1=m1 ; !2 = !42 (42)
2 2
64 cos( 3 k )L 75
2
1;
mk + L3
. . where
.. ..
!j2 = kj =mj j = 1 2
If the assumption of proportional damping for the string and
and body separately is retained, then the matrices K 0 and
C may be obtained by simple extension of the earlier re- 2 = m23 =(m1 m2 ):
sults:
This can be turned into a quadratic equation in ! 2 if the
2K 0 0 3 0
damping terms within the two bracketed expressions are
K = 4 0 K11 0 5
11 approximated by
0 0
(39)
0 0 K33 0 i!c1 =m1 i!12 =Q1 and i!c2=m2 i!22 =Q2
937A CTA A CUSTICA UNITED WITH A CUSTICA Woodhouse: On the synthesis of guitar plucks
Vol. 90 (2004)
in terms of the Q-factors Q1 , Q2 of the string and body
-3
x 10
2
modes respectively. At the tuned condition
1
!12 = !22
0
the roots for = !=!1 are then given by
h
Non-veering criterion
-1
2 + i=Q1 + i=Q2
2
p 2 i -2
4 (1 + i=Q1 )(1 + i=Q2 ) ; (1=Q1 ; 1=Q2)2
h i1
-3
2(1 ; ) :
;
2
(43) -4
The two roots are separated by an approximately real -5
quantity if the expression inside the square root has a pos-
-6
itive real part. This would mean that the two modes have 100 200 500 1000 2000
similar Q factors but different frequencies, i.e. that veer-
Frequency (Hz)
ing has occurred. Conversely, if the expression inside the
Figure 2. The “non-veering” criterion (45), for each overtone of
square root has a negative real part then the two modes
have similar frequencies but different Q factors, as seen in
every note on every string of a test guitar. Positive values indicate
non-veering behaviour.
Figure 1 for the non-veering case. Assuming small damp-
ing so that the two expressions (1+i=Qj ) inside the square
root can be ignored, the condition for non-veering is thus ted quantity is always positive so that “non-veering” is the
approximately rule. Synthesis using the undamped coupled modes and the
42 < (1=Q1 ; 1=Q2)2 : (44)
Rayleigh estimate of damping is likely to give seriously
misleading answers at all higher frequencies. Of course,
The non-veering case illustrated in Figure 1 has Q1 = the analysis described in this subsection will not apply
3500, Q2 = 100, and = 0:0035, slightly below the strictly once the modal overlap becomes significant, but
threshold value of 0:0049 given by equation (44). one might expect the qualitative conclusion to hold never-
Substituting the actual values from equation (19), the theless. Detailed simulation comparisons will be shown in
condition becomes section 4.3.
8 1 1 2 A similar calculation can be carried out allowing for
2 j 2 (1=3 + mk =L) < Qk ; Qsj
both polarisations of string motion: in other words, a pair
(45)
of string modes coupled to one body mode. Equations (41)
are replaced by 3 3 matrices extracted from equations
or more interestingly (37), (38) and (40). However, it is not necessary to work
j 2 > 2 (1=3 + m =L8)(1=Q ; 1=Q )2 :
through the details of this case to see what the answer will
(46) be. The string behaviour is rotationally symmetric: there is
k k sj
no significance to the particular two polarisation directions
For a given string, only the lower modes are likely to cou- chosen, and any two orthogonal directions would give the
ple strongly to body modes. Higher values of j are pro- same answer. However, the body mode behaviour is not
gressively more likely to satisfy this non-veering criterion. rotationally symmetric: the chosen mode involves motion
To see what this means in practice requires numerical at the bridge at the appropriate angle k , and in the per-
data. As will be described in section 4 and more fully pendicular direction the body has no motion. The coupled
in the companion paper [1], a plausible set of parame- string/body modes must respect the plane of symmetry as-
ters has been determined for a particular guitar and its set sociated with this body motion. The result is that string
of strings. Using these, condition (45) can be illustrated motion in the plane parallel to the body motion will couple
graphically. Each of the six strings is considered in turn, to the body exactly as analysed above, while the perpen-
and for each string every note from the open string to the dicular polarisation will be unaffected by body coupling.
12th fret is considered. For each of these notes, all string This means that at frequencies where modal overlap in
overtones below 2 kHz have been calculated, and for each the body is low, so that this approximation should hold,
of these the nearest body mode found, and the criterion one string polarisation is uncoupled to the body motion
(45) evaluated for this combination of string mode and and therefore unable to radiate sound. The player will in
body mode. Figure 2 shows the result: a symbol is plotted general excite a mixture of both polarisations, and they
for every overtone at every fret on every string, showing will vibrate with two different decay factors, but the ra-
the value of the right-hand side minus the left-hand side diated sound will show little trace of the “double decay”
of this inequality. From the figure it is clear that “veering” which occurs in other systems with similar behaviour, such
behaviour (negative values) is only predicted for this gui- as coupled piano strings [33]. The slow-decaying mode
tar below about 300 Hz. Above that, the value of the plot- can only radiate via distant modes which have a differ-
938Woodhouse: On the synthesis of guitar plucks ACTA A CUSTICA UNITED WITH A CUSTICA
Vol. 90 (2004)
ent angle k . (This simple result would change if non- -10
proportional damping of the body mode was allowed, so
that the body motion was no longer along a single line.) -20
At higher frequencies where modal overlap in the body be-
-30
Admittance (dB re 1 m s- 1 N- 1)
comes significant the behaviour may be different: a given
string overtone will couple significantly to more than one -40
body mode, with different values of k , and it is likely that
both polarisations would couple to body motion and thus -50
be able to radiate sound to some extent. To assess what
-60
really happens at these higher frequencies is one of the
major reasons for carrying out full syntheses using both -70
string polarisations. Results will be shown in section 4.5.
-80
3.5. Body coupling via the fingerboard
So far, it has been assumed that the string couples to the -90
100 200 500 1000 2000 5000
body only at the bridge. However, in reality there will ob- Frequency (Hz)
viously be some coupling through the nut or fret. Pub-
lished measurements of the admittance of a typical guitar Y
Figure 3. Calibrated admittances 11 (solid line), 12 (dashed Y
at the frets [34, 35] suggest that this would make only a
Y
line) and 22 (dash-dot line) determined from a test guitar by
measurement followed by modal fitting, as described in the text
small difference to the predictions, so it will not be taken and more fully in [1].
into account in the remainder of this paper. However, it
would be straightforward to extend the methods devel-
oped here to allow for a non-rigid nut or fret. For the For comparisons between methods in this section, a stan-
modal methods, additional constraint modes would be in- dardised test case will be used. This relates to the low-
cluded which would be the mirror image of those used so est note of a normally-tuned guitar, the open E string
far: moving at the fret but fixed at the bridge. To extend (82.41 Hz). The string length is 650 mm, and the string
the frequency-domain methods requires a little more care, properties correspond to a D’Addario “Pro Arte compos-
since the coupling would no longer be at a single point. ite, hard tension” string: tension 71.6 N, mass per unit
However, the matrix version of the coupling formula (9) length 0.0062 kg/m, string Q factor assumed to be 3500 for
can be extended to multiple-point coupling, by assembling all modes, and bending stiffness 5:7 10;5 Nm2 . Modal-
matrices which contain as many dimensions as there are based methods allow for 65 string modes, giving a maxi-
degrees of freedom constrained by the coupling: they will
be 2 2 matrices if one string polarisation is allowed, and
mum frequency of 5211 Hz (including the effect of bend-
4 4 if two are allowed. Note that these matrices would ing stiffness). The value for bending stiffness comes from
measurements reported in the companion paper [1]. The
involve not only the admittances at the bridge and fret, but constant-Q damping model is not realistic, but the numer-
also the transfer admittances between bridge and fret. ical value used here is of the correct order of magnitude
The most likely context in which string/body coupling
for the mid-frequency overtones of the string.
through the frets might be significant is for the higher frets
The determination of the body vibration model is also
which lie over the soundboard rather than on the neck. It
explained in the companion paper [1]. It uses 240 modes,
is a common complaint of guitar players that some notes
up to a maximum frequency of 5190 Hz. Up to 1500 Hz
in these positions “fall flat”, presumably implying a decay
the modal parameters of mass, frequency, damping and an-
rate more rapid than is desirable. Guitar makers often in-
gle were explicitly fitted to the measured admittance ma-
clude additional bracing under this area of the soundboard
trix of a test guitar. The higher modes were determined by
to reduce the problem. Another context in which coupling
a “statistical fit” to the measurements, which uses a ran-
at the fret might be important would be in analysing the be-
dom number generator to give frequencies with the correct
haviour of solid-body electric guitars. Although the body
density and spacing statistics, as well as damping factors,
vibration does not directly produce the sound, the decay
modal masses and angles with approximately the correct
statistical distribution. The resulting admittances Y11 , Y12
times of the strings are still of great concern to players,
and Y22 are plotted in Figure 3.
and these decay times are governed by mechanical rather
than electrical effects. One would guess that in this case
For each method tested, a pluck at a point 20 mm from
the energy loss through the frets might be greater than that
the bridge has been synthesised. For methods involving
through the bridge.
both polarisations of string motion, a plucking angle of
45 to the normal to the soundboard is assumed. The out-
4. Comparisons between methods put variable is the acceleration of the body at the bridge,
in the normal direction. This is a convenient quantity for
4.1. The test case comparison with experiments, to be discussed in the com-
The modal and frequency-domain synthesis methods de- panion paper. It is also the simplest quantity which ap-
scribed in section 2 can now be implemented and tested. proximately mirrors the spectral characteristics of the radi-
939A CTA A CUSTICA UNITED WITH A CUSTICA Woodhouse: On the synthesis of guitar plucks
Vol. 90 (2004)
120
80
100
60
80
Body acceleration (m s- 2)
Body acceleration (dB)
40
60
20
40
0
20
-20
0
-40
-20
-60 (a)
-40
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 200 400 600 800
Time (s) Frequency (Hz)
Figure 4. Comparison of the first few cycles of synthesised
plucks using the frequency domain method (upper curve) and 80
the first-order modal method (lower curve). Body acceleration in
response to a pluck amplitude of 1 N is plotted. The upper curve 60
is offset for clarity.
Body acceleration (dB) 40
ated sound (but keep in mind that modal radiation efficien- 20
cies are not included here). Plucks were synthesised with
0
a sampling rate of 22050 Hz, high enough that the Nyquist
frequency is well above the highest frequency included in -20
the model. For efficiency of the frequency-domain meth-
ods, the length of the synthesised pluck was chosen to have -40
217 samples to expedite FFT calculation. This results in a
-60 (b)
synthesis time of 5.94 s, a time chosen to be long enough
3000 3200 3400 3600 3800
to minimise “leakage” problems in the methods which use Frequency (Hz)
an inverse FFT to obtain the transient response.
Figure 5. Comparison of the Fourier spectrum of synthesised
4.2. Accuracy and speed plucks using the frequency domain method (upper curve), the
first-order modal method (middle curve) and undamped modes
All the synthesis methods were coded in Matlab, using plus Rayleigh’s principle (lower curve). The curves are offset for
vectorised constructions wherever possible to maximise clarity: the upper curve shows the true values. Plots (a) and (b)
speed. The first question to ask is whether all the meth- show the same spectra over different frequency ranges.
ods will run at all given the complexity of the problem.
The answer to this is affirmative: in particular, Matlab’s
eigenvalue/eigenvector routine was able to cope with both plitude before the pluck, because of the finite length and
styles of modal synthesis. For the first-order method with resolution of the inverse FFT, but this seems acceptably
both string polarisations, the matrix A has dimension 700 small. The top two curves of Figure 5a show a frequency-
and is not sparse, so it was not obvious a priori that the domain comparison of the two plucks at low frequencies.
calculation would be accurate. However, no difficulty was Again, excellent agreement between the two methods is
encountered. shown: the only visible deviations occur near the antires-
To check whether the computed answers are reliable, onances. This time, as one would expect, the frequency-
it is useful to compare results by the first-order modal domain approach gives the more accurate answers while
method (based on equation 25) with corresponding results the first-order modal approach shows minor errors due to
by the frequency domain approach (equation 8) using the the finite FFT used to generate the plot. This level of agree-
damped-string results (29) and (31). These two methods ment extends over the entire frequency range of the syn-
use entirely independent approaches and coding, but em- thesis: Figure 5b shows a typical comparison at higher fre-
body similar approximations. Figures 4 and 5 show that quencies. The conclusion is that both methods give an ac-
excellent agreement can be obtained between these two curate synthesis for the test case, whether one or two string
methods, when applied to the most challenging case with polarisations is required.
both string polarisations. Figure 4 shows a time-domain Timings for a selection of synthesis methods are given
comparison of the first few cycles following the pluck. The in Table I. The absolute times are of little significance be-
frequency-domain method shows motion at very low am- cause they depend on the machine and the detailed coding,
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