ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005) 312 ndash 325

Physics-Based Methods for Modeling NonlinearVibrating Strings

Jyri Pakarinen Vesa Valimaki Matti KarjalainenHelsinki University of Technology Laboratory of Acoustics and Audio Signal Processing PO Box 3000 FI-02015 HUT Espoo Finland jyripakarinenhutfi

SummaryNonlinearity in the vibration of a string is responsible for interesting acoustical features in many plucked stringinstruments resulting in a characteristic and easily recognizable tone For this reason synthesis models have tobe capable of modeling this nonlinear behavior when high quality results are desired This study presents twonovel physical modeling algorithms for simulating the tension modulation nonlinearity in vibrating strings in aspatially distributed manner The first method uses fractional delay filters within a digital waveguide structureallowing the length of the string to be modulated during run time The second method uses a nonlinear finitedifference approach where the string state is approximated between sampling instants using similar fractionaldelay elements thus allowing run-time modulation of the temporal sampling location The magnitude of thetension modulation is evaluated from the elongation of the string at every time step in both cases Simulationresults of the two models are presented and compared Real-time sound synthesis of the kantele a traditionalFinnish plucked-string instrument with strong effect of tension modulation has been implemented using thenonlinear digital waveguide algorithm

PACS no 4375Gh 4375Wx

1 Introduction

Interest towards physical modeling for sound synthesispurposes has been increasing during the last few yearsThe advantages of the physical models over traditionalsound synthesis methods reside in physically meaningfulmodel parameters which allow a natural control of the syn-thesis engine There has also been a trend towards com-paring and unifying the existing physical modeling meth-ods with a focus on generating more flexible and efficientsound synthesis models [1] [2] This paper discusses andcompares two physical models for simulating the behav-ior of the nonlinear string a spatially distributed nonlineardigital waveguide string recently developed by the authors[3] and a new spatially distributed nonlinear finite differ-ence string model created as a part of a Masterrsquos Thesiswork at Helsinki University of Technology [4]

Since the purpose of physics-based modeling is to sim-ulate the physical phenomena in the system of interest itis of paramount importance to be familiar with the behav-ior of the real-world case before modeling can take placeWe will start by studying some physical properties of vi-brating strings in section 2 the main focus being on thosewhich contribute the most to the resulting sound

Section 3 will discuss digital waveguide modeling ofstrings and present a spatially distributed nonlinear digi-

Received 30 April 2004accepted 11 October 2004

tal waveguide string algorithm A synthesis model of thekantele a Finnish plucked-string instrument is presentedusing the nonlinear digital waveguide model in section 4We will tackle finite difference modeling of strings in sec-tion 5 and introduce a spatially distributed nonlinear finitedifference string algorithm in section 6 Simulation resultsand comparisons to measured data are presented in section7 and conclusions are briefly drawn in section 8

2 Basics of string mechanics

21 Linear string

Let us consider a homogeneous string which is com-pletely flexible linear and lossless (ie the stringrsquos totalenergy remains constant) Such a string is called an idealstring If we also consider the string moving only in onetransversal polarization (eg horizontal) the motion of anideal string can be characterized by the well-known one-dimensional (1-D) wave equation

yttt x cyxxt x (1)

Here yttt x denotes the second-order partial derivativeof the string displacement in the horizontal axis with re-spect to the time variable t yxxt x denotes the second-order partial derivative of the horizontal displacement withrespect to the longitudinal coordinate x and c denotes

312 c S Hirzel Verlag EAA

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

the transversal wave propagation velocity within the stringmedium The transversal wave velocity can be written as

c

rK

(2)

where K is the string tension and is the linear mass den-sity of the string

Obviously every real string vibration decays withtime This results mainly from three damping mecha-nisms [5] (1) air damping (2) internal damping in thestring medium and (3) mechanical energy transfer throughstring terminations If the losses are simply divided intofrequency-dependent and frequency independent termsthe lossy 1-D wave equation can be written as [6] [7]

Kyxxt x yttt x dytt x dyxxxt x (3)

Here d and d are coefficients that simulate the fre-quency-independent and frequency-dependent dampingrespectively Real strings also experience stiffness whichleads to dispersion of the wave velocities and inharmonic-ities in the resulting sound Since this paper focuses on thesound synthesis of strings with relatively high elasticityand small diameters the effects of stiffness are not dis-cussed here More in-depth study about this topic can befound for example in [8]

22 Nonlinear string

Although the linearity assumption usually simplifies cal-culations the vibration of real strings can be consideredlinear in the most coarse approximations only More realis-tic string models require abandoning the linearity assump-tion thus also introducing more complex formulations forstring behavior

Some interesting nonlinearities in string instruments in-clude the hammer-string contact in piano-like instrumentsand the bow-string interaction in bowed string instru-ments These phenomena fall out of the scope of this paperThe hammer-string nonlinearity is mainly due to the com-pression of the hammer felt and it is covered in many ear-lier studies [9] [10] [11] [12] and [13] to name few Thebow-string nonlinearity is mainly caused by the stick-slipcontact between the string and the bow Also this interac-tion is covered in many studies [14] [15] and [16] presentsome of them In the following we will study more thor-oughly the tension modulation nonlinearity

When a real string is displaced its length and there-fore also its tension is increased When the string returnscloser to its equilibrium state its length and tension are de-creased This mechanism where the tension is varied dueto transversal vibrations is called tension modulation andabbreviated TM

It is easy to see that the frequency of the TM is twice thefrequency of the transversal vibration since the transversalequilibrium produces minimum tension and both extremediplacements produce maximum tension In many musicalinstruments the string termination is rigid in the longitu-dinal direction so that TM cannot be effectively coupled

BridgeNut

K

Kz

Kx

Figure 1 Tension modulation exerts a vertical force Kz on thebridge If the bridge is able to move in the z-direction this vi-bration will be coupled with the string With a rigid bridge nocoupling will take place and the missing harmonics will not begenerated (after [19])

to the instrument body and therefore cannot be heard di-rectly In instruments where this is not the case partialscreated by TM can be found at twice the frequencies ofthe transversal modes and they are called phantom par-tials The generation of these partials is discussed in moredetail in [17] and in [18]

Due to decaying vibration the string displacement isclearly greater just after the pluck than at the end of the vi-bration Therefore also the amplitude envelope of TM hasthe same form as the transversal vibration itself As canbe seen in equation (2) the increase in tension results inthe increase of wave velocities which in turn leads to theincrease in frequency Thus the frequency of a pluckedstring glides from an initial value to the steady-state value(the frequency in the linear case) This is called initialpitch glide an effect which is most apparent in elasticstrings with large vibrational amplitudes and relatively lownominal string tension (such as electric guitar or kantelestrings)

Tension modulation is also responsible for another ac-oustic feature generation of missing harmonics Thismeans as its name implies that the harmonics whichshould be missing from the vibration can be found in thespectrum Generally the lack of certain harmonics in aplucked string instrument occurs due to the plucking lo-cation which heavily attenuates all harmonics that wouldhave a node at that point In elastic strings however themissing harmonics begin with a gradual increase near zeroamplitude until they reach their peak value and then de-cay off like all other harmonics In the following this phe-nomenon is discussed while avoiding to go too deep intothe mathematics A more in-depth study can be found froma paper by Legge and Fletcher [19]

Although the TM takes place in the longitudinal di-rection it can also excite the string in the transversal di-rections at the string termination point provided that thetermination is nonrigid in the transversal plane Figure 1clarifies this As can be seen in the illustration the vary-ing of the tension called tension modulation driving force(TMDF) [20] excites the bridge in the vertical directionIf the bridge has a nonzero mechanical admittance in thevertical direction TMDF will cause the bridge to move upand down

Let us now consider a case where a vibrating string car-ries two transversal modes n and m As stated above

313

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

the TM caused by mode n is periodic with a frequencycorresponding to mode n When this vibration is cou-pled through the nonrigid bridge with the transversal vi-bration of mode m the resulting vibration is the mode namplitude-modulated with mode m Thus this vibrationcan excite mode p only if p jnmj Clearly the samerule applies with m and n interchanged

Now it is clear that even if a string is plucked at x Lp the pth mode although supposed to be missing willreceive energy from other modes because of this nonlinearcoupling at the bridge It is important to note however thatthis phenomenon cannot excite all the modes eg if thestring is plucked near its center so that no even modes willbe present they will not be generated by this mechanismeither

Since this energy transfer from other modes is rathergradual than instantaneous the missing modes excited bythe TMDF will experience a gradual onset and behave likeother modes after reaching their peak values The risingrate of these missing harmonics can be shown to be pro-portional to the cube of the pluck amplitude [19]

In real musical instruments also another mechanism isresponsible for the generation of missing harmonics Thestring is often terminated behind the bridge and it under-goes a change in direction at the bridge location Figure 2illustrates this fact Now the TM can be directly coupledwith the vertical polarization due to angle at the bridgeThis means that the TMDF due to a transversal mode nwill have a frequency corresponding to mode n so thismechanism can excite only even modes thus rising theeven harmonics also in a middle-plucked string [19]

3 Digital waveguide approach

Digital waveguide (DWG) modeling is a term often en-countered when studying the synthesis of string instru-ments It is based on the fact that when an excitation signalis inserted into a string it is reflected at the boundaries andreturns to its initial position At its simplest form this canbe implemented as a single delay-loop with two consec-utive samples averaged as is done in the classic Karplus-Strong algorithm [21] An excellent introduction to DWGsused in modeling musical instruments can be found in[22]

The entire digital waveguide methodology is based onthe traveling-wave solution of the wave equation Thismeans that the solution to equation (1) can be seen as asuperposition of two waveforms traveling in opposite di-rections along the string This solution commonly knownas the traveling-wave solution or as the drsquoAlembertrsquos so-lution was first published by drsquoAlembert in 1747 It canbe presented in the mathematical language as [22]

yt x yrt xc ylt xc (4)

where yr and yl denote the wave components proceedingright and left respectively The traveling-wave solutionof the 1-D wave equation (1) can be converted into dig-ital form by sampling the wave components temporally at

BridgeNut

K

Kz

Figure 2 A more realistic bridge model The angle causes thetension to have a vertical component Kz This results in a TMDFwith frequency twice as high as the transversal frequency in thestring (after [19])

Delay of L-samples presenting y1

Delay of L-samples presenting yr

y nm( )-1

-1

Figure 3 A DWG model of an ideal string The wave reflection atthe fixed termination points is implemented with a sign changesince y yr yl The string excitation can be insertedeg by initializing the delay lines to nonzero values (after [23])

T and spatially at X intervals Formally this is done bychanging the variables in equation (4) [22]

x xm mX

t tn nT

If now the traveling waves are redefined as

yln ylnT

yrn yrnT

the discrete traveling-wave solution can be obtained

ytn xm

yrnm ylnm (5)

The term yrn m can be thought as yrn delayed bym samples Similarly the term yln can be thought asyln m delayed by m samples It is important to notethat equation (5) is not a mere approximation of equation(1) but yields exact results for bandlimited signals at thesampling instants within the limits of the numerical pre-cision of the samples [22] This kind of structure can beeasily implemented with two delay lines containing unitdelays and ynm can be obtained by summing the de-lay line values at correct locations The string state at thenext time step can be updated by simply shifting the sam-ples one step in the direction of the delay line A DWGmodel of an ideal string is shown in Figure 3

Correct tuning of the waveguide can be enabled byadding a fractional delay filter (ie a filter capable of pro-ducing also noninteger delay values) inside the DWG loop[24] Frequency-independent losses can easily be modeledin a DWG structure by inserting simple scaling coeffi-cients in the ideal DWG string structure [22] Frequency-

314

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

dependent losses can in turn be simulated using lowpassfilters (aka loop filters) inside the DWG loop

31 Time-varying digital waveguide string

When tension modulation is to be implemented in a stringmodel it clearly means that the fundamental frequency ofthe string must be modulated also (see equation 2) In aDWG string this corresponds to varying either the lengthof the delay lines or the temporal sampling instant Thissection discusses implementing the TM by varying thelength of the delay lines and is presented earlier in a re-cent publication by the authors [3] Since the delay linescan generally have integer-valued delays only directly al-tering them would lead to having the tension change in astepwise manner Obviously this behavior is not desiredand therefore fractional delay (FD) elements are used Foran in-depth study of FD elements see [24]

A first-order allpass filter was chosen for the FD ele-ment of our string model

Az a z

az (6)

where a is the filter coefficient which defines the length ofthe delay Notice that when a the allpass filter acts asa unit delay

The decision for using a first-order allpass filter withinthe string model was done partly because it is the simplestway to design an allpass filter approximating a given frac-tional delay [24] and partly because the first-order allpassfilter is the best choice for the fractional delay elementwhen delay values around unity are to be obtained [25]The phase response error caused by the allpass filter is notconsidered to pose a problem since its effect is negligiblein the audio frequency range assuming the sampling fre-quency to be reasonably high [26]

32 Distributed nonlinear DWG string

Previous works [27] [28] [29] use a single fractional de-lay element in a single-polarization string model or aDWG string terminated with a nonlinear double-spring[30] to simulate the nonlinear string This is done in or-der to reduce the computational complexity of the modelbut it has also some shortcomings Since the system is non-linear the FD elements cannot be lumped into one singleelement without giving up the idea of viewing the systemas a distributed model

In other words the whole string becomes a lumpedmodel and the termination point ldquobehindrdquo the FD elementbecomes the only location for gathering meaningful outputfrom the string Physically this would correspond to a sin-gle elastic element at the termination point of an otherwiserigid string A more realistic solution can be obtained if theelongation process is distributed along the delay line in asimilar way as in a real physical string where the elasticityis distributed along the string rather than lumped

The distributed nonlinearity can be implemented by ex-changing the delay lines of the DWG model of Figure 3

gz

-1

A z( )

v n( )s( )n

+-g

(a) (b)

Figure 4 Illustration of (a) a basic element and (b) how to getoutput data from a string consisting of these elements The di-rections of the wave components in (a) are opposite for adjacentelements so that in effect the unit delays and allpass filters areinterleaved for each delay line In (b) either the velocity or theslope of the string segment can be obtained if velocity is used asthe wave variable

F n( )

y n( )

HLHR

Figure 5 One-polarizational DWG string with time-varyinglength The string consists of the basic elements illustrated inFigure 4(a) HL and HR denote the loop filters simulating thefrequency-dependent losses The excitation to the string can beinserted as a force signal using an interaction element denotedby I The construction of the interaction element is illustrated inFigure 6

F n( )

F n( )

y n( )v n( )

2Z12Z2

Z2Z1

y n( )

Figure 6 The interaction element allows excitation signals to beinserted to the string during run-time The input signal F n canbe seen as a force signal and the output signal yn as a dis-placement signal The coefficients Z and Z represent the me-chanical impedances of the two string branches Implementationof the integrator block is depicted in Figure 7

with a structure consisting of allpass filters Then the ef-fective length of the delay lines can be changed by vary-ing the filter coefficients We will now introduce a de-lay block which contains a unit delay a first-order all-pass filter and two scaling coefficients for modeling thefrequency-independent losses This block is called a basicelement and it is illustrated in Figure 4(a) The unit delayin each basic element ensures that no delay-free loops are

315

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

formed when constructing models using these elementsFigure 4(b) shows how to obtain output data from a junc-tion between two basic elements A time-varying DWGstructure consisting of these elements is illustrated in Fig-ure 5

From the discussion in section 1 we can conclude that asuitable control signal for the FD elements can be derivedfrom the instantaneous elongation of the string In the fol-lowing since the longitudinal wave propagation velocityis considerably higher than the transversal wave velocitywe will assume that the longitudinal waves will propa-gate instantaneously through the string and the elongationcalculation and the FD parameter tuning can be done forthe whole string in one piece In practice the longitudinalwave velocity is typically only 5-20 times higher than thetransversal one but carrying out the FD parameter evalua-tions for multiple string segments would add a significantcomputational load likely without any audible advancesThe elongation of the string can be expressed as [19]

ldevt

Z lnom

q

yxt x

dx lnom (7)

where lnom is the nominal string length x is the spatial co-ordinate along the string and y is the displacement of thestring The first spatial derivative yx suggests the use ofslope waves in the elongation calculation and thus equa-tion (7) can be approximated for the digital waveguide as[28]

Ldevn

LnomXm

p srnm slnm Lnom (8)

where srnm and slnm are the slope waves at timeinstant n and position m propagating to the right and tothe left respectively Lnom is the rounded nominal stringlength To reduce the computational complexity equation(8) can be further simplified using a truncated Taylor seriesexpansion to [28]

Ldevn

LnomXm

srnm slnm

(9)

while still maintaining a sufficient accuracy The approx-imated delay variation of the total DWG can be obtainedfrom equation (9) as [28]

Ddevn

nXlnLnom

EA

K

Ldevl

Lnom (10)

where E is Youngrsquos modulus A is the cross-sectional areaof the string and K is the nominal tension correspond-ing to the string at rest The length of the string in samplesis denoted as Lnom lnomfscnom where fs is the tem-poral sampling frequency and cnom is the nominal wavepropagation speed

T

z-1

Figure 7 The integrator block is implemented by summing upconsecutive samples

Figure 8 Illustration of the kantele The string termination at var-ras is magnified for clarity A denotes the termination point forvertical vibration of the string while B denotes the terminationpoint for horizontal vibrationl stands for the distance betweenA and B

Since the system under consideration uses a distributedset of delay elements the desired delay for each basic ele-ment is

dpartial Ddev

Lnom

(11)

The coefficient a in equation (6) can now be expressed as[26 24]

a dpartial

dpartial (12)

where dpartial is the delay intended for a single allpass fil-ter Note that previous studies have used a different signfor a in equations (6) and (12) although the operation ofthe allpass filter remains the same

4 Synthesis model of the kantele usingnonlinear digital waveguides

In this section we demonstrate the nonlinear DWG for-mulation by constructing a two-polarizational synthesismodel of the kantele a Finnish folk music instrument

41 Acoustical analysis of the kantele

The kantele is a bridgeless plucked string instrument withusually five metal strings in its basic form (see Figure 8)The strings are terminated at one end by metal tuning pins

316

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

f nin( ) f nout( )

f ny( )

f nx( )

f nzy( )

f nz( )v nz( )

v ny( )

ZC

Zybridge

Zzbridge

C

Figure 9 A kantele string model The one-polarization stringmodel blocks are identical to what is illustrated in Figure 5 buthave different string lengths It is important to note that the cou-pling between the vibrational polarizations in a real physical sys-tem is more complicated but a simplified one-way coupling isused here for ease of simulation

which are screwed directly into the soundboard At the op-posite end all strings are wound once around a horizontalmetal bar called the varras and then knotted Because ofthe nonzero distance between the center of the varras andthe knot the vibrations in two polarization planes havedifferent effective lengths the varras is the terminationpoint for horizontal vibration while the knot acts as thetermination for the vertical vibration as illustrated in Fig-ure 8 This phenomenon causes the total vibration of thestring to have two fundamental components with slightlydifferent frequencies producing beating [29] A more de-tailed structure and acoustical analysis of the kantele canbe found in [29] A study of the history of kantele and anacoustically improved new design are presented in [31]

42 A novel kantele string model

The novel synthesis model of a single kantele string is con-structed using two single-polarization time-varying DWGmodels illustrated in Figure 5 and connecting them to-gether via a scaling coefficient for modeling the couplingbetween the two polarizations We restrict the couplingto being one-directional in order to avoid stability prob-lems which would otherwise rise due to the feedback loopformed from the interconnected strings as suggested byKarjalainen et al [32] Clearly the actual physical cou-pling is two-directional The elongation approximations ofthe strings and the resulting allpass filter coefficient valuesare evaluated separately for the two DWG models usingthe arithmetics described in section 32 The structure ofthe novel kantele string model is illustrated in Figure 9 Inthis model vyn and vzn represent the velocity signalscoming from the strings vibrating vertically and horizon-tally respectively

It is important to note that while vyn and vzn canbe obtained anywhere along the string in this case theyare evaluated at the termination points so that terminalimpedances can be used Zybridge and Zzbridge representthe vertical and horizontal terminal impedances respec-tively Zc stands for the coupling impedance from verti-cal to horizontal string vibration polarization and fyznrepresents the corresponding driving force The forces to

the termination caused by the two one-polarizational vi-brations are denoted by fyn and fzn The connec-tion from the elongation approximation block to the outputsimulates the direct coupling of the TM to the instrumentbody [20] A scaling coefficient denoted by C is usedto control the amount of this coupling The output of thewhole two-polarization string model is finally presentedas a force signal foutn excerted to the string terminationpoint It must also be noted that this model simulates onlya single kantele string and a model of the instrument bodymust also be added if realistic sound synthesis is desired

A real-time sound synthesis model of a kantele is con-structed using a block-based efficient audio-DSP-tool theBlockCompiler The algorithm used is efficient enough toprocess a five-string kantele model on an ordinary laptopcomputer at a 441 kHz sampling rate A detailed descrip-tion of the BlockCompiler is presented by Karjalainen[33]

5 Finite difference approach

In the previous section we discussed string synthesis viadiscretizing the drsquoAlembertrsquos solution to the 1-D waveequation Another approach is to discretize the wave equa-tion itself for example by substituting finite differenceterms for the derivatives in the wave equation (equation(1) in the case of an ideal string) This mode of opera-tion is commonly known as the finite difference method(FDM) and it was first used for sound synthesis purposesby Hiller and Ruiz in the early seventies [6] and [34] Fi-nite differences had already earlier been used in mathemat-ics for numerical solving of partial differential equationsA fine introduction to FDM in the synthesis of pluckedstring instruments can be found in [35] Below we followthe guidelines provided in [22] in deriving the FDM recur-rence equation

51 Ideal finite difference string

The partial derivatives in the 1-D wave equation (1) can bereplaced by finite differences 1

ytt x yt x ytT x

T(13)

and

yxt x yt x yt xX

X (14)

Using the finite difference approximation for the second-order derivatives in the wave equation (1) leads to

Kyt xX yt x yt xX

X

ytT x yt x ytT x

T (15)

1 It is important to note that the finite difference scheme used in equa-tions (13) and (14) was only chosen for simplicity and other schemescould be used as well For a discussion of using an implicit finite-difference scheme see section 8

317

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Solving (15) we get

ytT x KT

X

yt xX yt x yt xX

yt x ytT x (16)

Next we define the relationship between the spatial andtemporal sampling steps with [35]

r cT

X (17)

where the ldquoless than unityrdquo-restriction is called the VonNeumann stability condition Using this together with thedefinition of transversal wave velocity (equation 2) equa-tion (16) can be written as

ytT x ryt xX yt xX

(18)

ryt x ytT x

If we now do the discretization by denoting t tn nT and x xm mX as we did in section 3 we endup with the finite difference approximation [35]

ynm rynm ynm

(19)

rynm ynm

If we set r (19) becomes

ynm ynm ynm

ynm (20)

which is the finite difference equation of an ideal stringThe equality of equation (20) can be checked by substitut-ing the waveguide decomposition (equation 5) in the right-hand side of equation (20) [22]

Since the length of the string must again have integervalues correct tuning of the string becomes difficult It hasbeen shown [35] that choosing r in equation (17) re-sults in lowering the fundamental frequency of the stringTherefore the finite difference string can be tuned via theparameter r

Choosing r also gives raise to an unwanted nu-merical dispersion phenomenon called grid dispersion [7]where the wave velocity in the numerical implementationwill be less than the ideal physical wave velocity This ar-tificial dispersion affects primarily the upper harmonicswhere the frequencies will be underestimated If a typi-cal error of in the generated frequencies is allowedthe difference between the tuning coefficient r and unityshould not be greater than [35] If the constraints be-tween the correct tuning and grid dispersion do not yieldsatisfactory results the spatial density of the grid shouldbe increased This is known as spatial oversampling

52 Boundary conditions and string excitation

Since the spatial coordinate m of the string must liebetween and Lnom problems arise near the ends ofthe string when evaluating equation (20) because spatialpoints outside the string are needed The problem can

be solved by introducing boundary conditions that definehow to evaluate the string movement when m orm Lnom The simplest approach introduced alreadyin [6] would be to assume that the string terminations berigid so that yn yn Lnom This results in aphase-inverting termination which suits perfectly the caseof an ideal string For other types of string terminationseveral models have been introduced (see eg [6] [35]and [36]) Generally the nonrigid string terminations leadto frequency-dependent losses in the string model

For the FDM string excitation a useful method has beenproposed in [36] It is conceptually simple and allows forinteraction with the string during run-time There

ynm ynm

un (21)

and

ynm ynm

un (22)

are inserted into the string which causes a ldquoboxcarrdquo blockfunction to spread in both directions from the excitationpoint pair The wave component un is now used as theexcitation signal in a similar way as the exciting force sig-nal F n in section 32

53 Finite difference approximation of a lossy string

Frequency-independent losses can be modeled in an FDMstring by discretizing the velocity-dependent dampingterm in the lossy 1D wave equation (3) This results in twoadditional scaling coefficients in the recurrence equation[35]

ynm pynm ynm

qynm (23)

where the values of p and q determine the amount oflosses Generally p and q may depend on the spatial indexm but since homogeneous strings are considered here thisdependency is omitted Values

q p jpj (24)

ensure the stability of a linear finite difference string withfrequency-independent losses [37] Note that the sign dif-ference of p and q in [37] has already been taken care ofin equation (23) Modeling of frequency-dependent lossesby discretizing the lossy wave equation leads to an implicitrecurrence equation which can be evaluated if suitableapproximations are made [35]

6 Nonlinear finite difference string

Implementing tension modulation in a digital waveguidestring in section 3 was not an overly difficult task This wasdue to the fact that the implementation of a DWG string isessentially a feedback loop with delay and therefore mod-ulating the delay time of this loop corresponded to modu-lating the wave velocities In FDM strings however such

318

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

Figure 10 Illustration of the nonlinear FD algorithm on a spatio-temporal grid The vertical axis denotes the time while the hor-izontal axis denotes the spatial location on the string The illus-tration is shown only for a string segment of length N forclarity In each step of the algorithm most recently evaluated val-ues are presented as black dots while earlier values are presentedas white dots

an approach would not lead to satisfactory results sincethe physical quantities (eg displacement) themselves arepresent in the string model and not their wave decompo-sitions2

Instead we concluded that in order to correctly modelthe TM in a FDM string we first have to evaluate the recur-rence equation and use these three snapshots of the string(at time instants n n and n) in interpolating two newstring states at time instants n and n where Using these two string states we then eval-uate the recurrence equation in order to obtain the stringstate at time n It is important to note that the in-terpolation here is in effect stretching the time axis so thatthe wave propagation velocities are altered whereas in theDWG model the allpass filters perform the interpolation inthe spatial domain

This algorithm can also be seen as using two FDM sys-tems in implementing the nonlinear string The elongationof the string would be evaluated from one system and theresult the stretched string state would be updated to theother system Figure 10 illustrates this procedure on thespatio-temporal grid

In step 1 the two initial states have been assigned forthe string and the state at the next instant (in the linearcase) is obtained by the standard recurrence equation (20)The grid values which represent the state of the string atthe corresponding time instant are circled in step 1 In step2 sample values corresponding to the TM have been inter-polated from the string states in step 1 In step 3 equation(20) has been applied on the values evaluated in step 2 in

2 Such a system which deals with the physical quantities themselves iscalled a Kirchhoff model as opposed to a wave model which deals withthe wave components of the physical quantities

(c)

(b)

(a)

t

t

t t

t

t

n

n

n n

n

n

n+1

n+1

n+1 n+1

n+1

n+1y n+ m( 1 )

y n+ m( 1 )

y n m( )

y n m( )

y n+ m( 1 )

y n- m( 1 )

m

m

m m

m

m

d

d

d

a

a

a

-a

-a

-a

z-1

z-1

z-1

n-1

n-1

n-1 n-1

n-1

n-1

Figure 11 Illustration of the interpolation process due to thechange in the stringrsquos length The spatio-temporal grids on theleft and right represent the linear and interpolated string statesrespectively The fractional delay value caused by the interpola-tion is denoted by d The interpolation process in (a) is simplifiedin (b) and further in (c)

order to obtain the string state corresponding to the changein tension The two most recently obtained states are nowtaken as the ldquoinitial statesrdquo in step 4 and we can return tostep 1

As seen in Figure 10 the tension modulation corre-sponds here to interpolating the string state in the tempo-ral domain The elongation of the FDM string was evalu-ated similarly to what was done in equation (9) except thathere the slope of the string was obtained by taking the dif-ference of the displacements between two adjacent stringsegments rather than summing up the slope wave compo-nents In the following we will have a closer look at theinterpolation process

61 String state interpolation

We chose again to use first-order allpass filters in inter-polating the string state from the linear model (step 2 inFigure 10) Figure 11(a) illustrates how the interpolatedvalue of ynm is obtained from the linear values Thespatio-temporal grid on the left represents the string statein the linear case while the spatio-temporal grid on theright represents the string state after spatial interpolationThe structure between the two grids is the block diagramof a first-order allpass filter (equation 6) The coefficient afor the allpass filter was evaluated as presented earlier byequations (9)ndash(12)

319

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

In this figure we notice that the allpass filter uses thevalue of ynm delayed by one sample thus corre-sponding to ynm Clearly this can be obtained directlyfrom the grid on the left and the branch on the left contain-ing the unit delay can be reformed The result is shown inFigure 11(b) Here we also note that the interpolation sys-tem uses its own output at the previous time instant This isactually the same as using the value of ynm becauseit is the same as the output of the interpolation process onetime step ago (this might be best understood by noting thatthe bottom row of step 4 in Figure 10 is the same as thebottom row of step 1 at the next time instant) Thus Fig-ure 11(b) can be further simplified to Figure 11(c)

Having this said the recurrence equation for the time-varying finite difference string with frequency-indepen-dent damping can be written as

ynm pynm ynm

qynm (25)

where

ynm aynm ynm

aynm

ynm aynm ynm

aynm

ynm aynm ynm

ayn m

Here the coefficients p and q incorporate the frequency-independent losses and y and y refer to the linear and in-terpolated strings respectively Simplifying and rearrang-ing we end up with an equation containing only terms ofy and the subscript may therefore be omitted

ynm paynm paynm

pynm qaynm pynm

paynm qynm

paynm qayn m (26)

This equation is illustrated with a block diagram in Fig-ure 12 along with its abstraction A nonlinear FDM stringcan be constructed by connecting several of these blockstogether and using the string elongation in controlling theamount of interpolation We will refer to such a block asa time-varying finite difference time-domain (FDTD) ele-ment Illustration of the lossless time-varying FDTD ele-ment can be found in Figure 13 where p and q equal unityand have therefore been left out

62 String excitation and termination

For the interaction with the time-varying FDTD stringmodel we chose to use the ldquoboxcarrdquo excitation model dis-cussed in section 52 so that the excitation signal couldagain be interpreted as a force signal Figure 14 presentsan interaction block to be used with a time-varying FDTDstring We will call such a block the FDTD interaction el-

FDTD

-pa

z-1

z-1

z-1

y n+1m( )

y nm( )

y n-2m( )

y n-1m( )

-pa

pa

qa

-q

-qa

pa

p p

Figure 12 Illustration of the time-varying FDTD element to-gether with its abstraction A lossless time-varying FDTD ele-ment can be found in Figure 13

-a

z-1

z-1

z-1

y n-1m( )

y n-2m( )

y nm( )

y n+1m( )

-a

-a

a

a

a

Figure 13 Illustration of the lossless time-varying FDTD ele-ment

ement Using these DSP blocks we can construct a one-polarization nonlinear FDTD (NFDTD) string as illus-trated in Figure 15

We chose to use rigid terminations for our nonlin-ear finite difference string model since the modeling offrequency-dependent losses is not a key aspect of thisstudy Fixed terminations do not ruin the generation ofmissing harmonics in our model either since the TMDFcoupling is implemented in a different manner as ex-plained below

63 NFDTD string with generation of missing har-monics

In order to model the generation of missing harmonics ina NFDTD string we constructed a model where an addi-tional interaction element is placed between the last FDTDelement and the termination for feeding the TMDF to thestring Since the spatial distance between the last FDTD

320

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

y n-1m+1( )

y n-1m( )

F n( )

F n( )

y nm+1( )

y nm( )

y n+1m+1( )

y n+1m( )

Figure 14 Illustration of the FDTD interaction element togetherwith its abstraction The excitation algorithm is defined by equa-tions (21) and (22)

F n( )

FDTD FDTD FDTD FDTD FDTD

Allpass-coefficientapproximation

Elongationapproximation

Figure 15 One-polarizational NFDTD string The string consistsof the time-varying FDTD elements illustrated in Figure 12 Thezero-blocks at the terminations give zero as an output regardlessof the input values thus implying a rigid termination The excita-tion to the string can be inserted as a force signal using a FDTDinteraction element illustrated in Figure 14

F n( )

FDTD FDTD FDTD

TMDF

FDTD

Allpass-coefficientapproximation

Elongationapproximation

y n L( -1)nom

Figure 16 Illustration of the NFDTD string with a generationmechanism for missing harmonics A second interaction elementis added in order to feed the TMDF into the string The scal-ing coefficient TMDF controls the amplitude of the missing har-monics The string elongation is approximated from the displace-ments of each FDTD element

element and the rigid termination is one sample the ver-tical component of the TMDF can be seen to be equal tothe product of the displacement of the last FDTD elementand the tension Here we can replace the tension signalby the elongation signal and introduce a scaling coeffi-cient TMDF to control the amount of TMDF to be in-serted to the interaction element at the termination Thisis illustrated in Figure 16 The generation of missing har-

monics in a NFDTD model will be further discussed in thefollowing section

7 Simulation results

In this section we present the results obtained from the twononlinear string algorithms discussed in sections 3 and 5The synthesis results are compared by simulating the samephenomena namely the initial pitch glide and the genera-tion of missing harmonics using the two models Stabilityissues and computational cost of the synthesis models arealso discussed

71 Synthesis results

The synthesis results reveal that both the nonlinear DWGand NFDTD models are able to realistically model the ini-tial pitch glide phenomenon Figure 17 illustrates the fun-damental frequency behavior of a recorded kantele toneand the two synthesized tones Here the horizontal dottedline approximates the mean value of perceptual detectionthreshold of an initial pitch glide The psychoacoustic de-tection threshold in the frequency region of these tones isabout 54 Hz [38] This shows that the fundamental fre-quency glide is an audible phenomenon in plucked stringinstruments such as the kantele even at modest pluckingamplitudes and thus it must be included in a synthesismodel if realistic tones are desired

The nonlinear DWG model used in this figure has a totaldelay line length of 55125 samples and the allpass coef-ficient a is scaled using a constant value of 09 in orderto correctly simulate the behavior of the recorded sampleThe NFDTD string consists of 56 FDTD elements andthe fine-tuning parameter (aka Courant number equa-tion 17) has a value of r 13 The allpass coefficienta is scaled using a coefficient in the NFDTD case

The modeling of the generation of missing harmonicscan be implemented similarly in the distributed nonlinearDWG model as was suggested in [28] If the boxcar inte-gration of equation (10) is replaced with a leaky integratorhaving the transfer function

Iz gp ap

apz (27)

the generation of the missing harmonics can be controlledvia the integration parameter ap The variable gp definesthe gain of the integration

Figure 18 shows the amplitude envelopes of the firstthree harmonics of a synthesized tone with two differ-ent ap parameter values The string was plucked close tord of its length and as can be seen in the figure themissing harmonic in (a) has a gradual increase after thebeginning transient after which it experiences an expo-nential decay like all other modes

It is worthwhile to note that the generation of missingharmonics in the nonlinear DWG model results from theproperties of the integration of the elongation approxima-tion and is therefore not a physically justified process Ba-sically here the integration error in the leaky integrator is

321

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Time [s]

fnom

Fre

quen

cy[H

z]

Figure 17 Fundamental frequencies as a function of time fora moderately-plucked recorded kantele tone (solid line) a syn-thesized nonlinear DWG tone (dashed line) and a synthesizedNFDTD tone (dash-dotted line) The fnom stands for the nomi-nal fundamental frequency of the string and the horizontal dottedline denotes the approximated detection threshold of a pitch drift(fnomHz) which suggests that the fundamental frequencydrifts in all cases are audible

responsible for feeding energy to the missing harmonicsAlso unlike the real physical phenomenon the generationof missing harmonics in the nonlinear DWG case does notdepend on the rigidity of the terminations Neverthelessthis feature can be exploited in emulating the real stringbehavior when the integration parameters are properly ad-justed Details on tuning the leaky integrator parameterscan be found in [28]

Modeling the generation of missing harmonics in aNFDTD string is however not so simple Even if a leakyintegrator is used in the elongation calculation its param-eters do not have a desirable effect on the missing har-monics This does not seem too surprising when consider-ing the major differences of these two algorithms and it isthe reason that forced us to use an alternative mechanismfor creating the missing harmonics in the previous section(Figure 16)

Figure 19 represents the behavior of the first three har-monics of a tone synthesized by this model It can be seenthat the missing harmonics can be ldquoliftedrdquo by choosing aproper value for TMDF The stability of the system how-ever poses an upper limit for the TMDF coefficient sincethe TMDF mechanism continuously feeds energy to thestring According to our experience generating missingharmonics with amplitudes greater than what is shown inFigure 19 is difficult

72 Stability issues and computational comparison

We found the nonlinear DWG algorithm to remain sta-ble for nearly all parameter and excitation values Onlyhighly exaggerated nonlinearity scaling values togetherwith high excitation impulses resulted in stability prob-lems We thus conclude that the nonlinear DWG waveg-uide has no real stability problems when synthesis of nat-ural plucked-instrument sounds are desired

We studied the stability of the NFDTD algorithm us-ing the Von Neumann analysis [39] in the time-invariantcase ie parameter a of equation (26) was kept constantThe basic idea of this method is to calculate the spatialFourier spectrum of the system under discussion at twoconsecutive time steps An amplification function which

(a)

(b)

Figure 18 Generation of the missing harmonics in the nonlinearDWG model can be controlled via the leaky integrator parame-ters Here the string was plucked approximately at rd of itslength so every 3rd harmonic should be missing from the re-sulting spectrum In a) ap and the third harmonicclearly rises after the initial transient In b) ap 13 andthe third harmonic is more attenuated

(a)

(b)

Figure 19 Generation of missing harmonics in a NFDTD stringThe string was plucked again approximately at rd of itslength and the coupling of the TMDF to the transversal vibra-tion was controlled using a scaling coefficient TMDF In a) thescaling coefficient has a value of TMDF and the missingthird harmonic can be seen rising after the initial transient In b)TMDF and generation of missing harmonics does not takeplace

shows how the spatial spectrum evolves with time canthen be derived from the two spectra If the absolute valueof this amplification function remains below unity stabil-ity is guaranteed Formally the Von Neumann analysis forthe NFDTD algorithm goes as follows [4]

If the spatial inverse Fourier transform is defined as

ynm FfY n g neim (28)

322

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

the transversal wave propagation velocity within the stringmedium The transversal wave velocity can be written as

c

rK

(2)

where K is the string tension and is the linear mass den-sity of the string

Obviously every real string vibration decays withtime This results mainly from three damping mecha-nisms [5] (1) air damping (2) internal damping in thestring medium and (3) mechanical energy transfer throughstring terminations If the losses are simply divided intofrequency-dependent and frequency independent termsthe lossy 1-D wave equation can be written as [6] [7]

Kyxxt x yttt x dytt x dyxxxt x (3)

Here d and d are coefficients that simulate the fre-quency-independent and frequency-dependent dampingrespectively Real strings also experience stiffness whichleads to dispersion of the wave velocities and inharmonic-ities in the resulting sound Since this paper focuses on thesound synthesis of strings with relatively high elasticityand small diameters the effects of stiffness are not dis-cussed here More in-depth study about this topic can befound for example in [8]

22 Nonlinear string

Although the linearity assumption usually simplifies cal-culations the vibration of real strings can be consideredlinear in the most coarse approximations only More realis-tic string models require abandoning the linearity assump-tion thus also introducing more complex formulations forstring behavior

Some interesting nonlinearities in string instruments in-clude the hammer-string contact in piano-like instrumentsand the bow-string interaction in bowed string instru-ments These phenomena fall out of the scope of this paperThe hammer-string nonlinearity is mainly due to the com-pression of the hammer felt and it is covered in many ear-lier studies [9] [10] [11] [12] and [13] to name few Thebow-string nonlinearity is mainly caused by the stick-slipcontact between the string and the bow Also this interac-tion is covered in many studies [14] [15] and [16] presentsome of them In the following we will study more thor-oughly the tension modulation nonlinearity

When a real string is displaced its length and there-fore also its tension is increased When the string returnscloser to its equilibrium state its length and tension are de-creased This mechanism where the tension is varied dueto transversal vibrations is called tension modulation andabbreviated TM

It is easy to see that the frequency of the TM is twice thefrequency of the transversal vibration since the transversalequilibrium produces minimum tension and both extremediplacements produce maximum tension In many musicalinstruments the string termination is rigid in the longitu-dinal direction so that TM cannot be effectively coupled

BridgeNut

K

Kz

Kx

Figure 1 Tension modulation exerts a vertical force Kz on thebridge If the bridge is able to move in the z-direction this vi-bration will be coupled with the string With a rigid bridge nocoupling will take place and the missing harmonics will not begenerated (after [19])

to the instrument body and therefore cannot be heard di-rectly In instruments where this is not the case partialscreated by TM can be found at twice the frequencies ofthe transversal modes and they are called phantom par-tials The generation of these partials is discussed in moredetail in [17] and in [18]

Due to decaying vibration the string displacement isclearly greater just after the pluck than at the end of the vi-bration Therefore also the amplitude envelope of TM hasthe same form as the transversal vibration itself As canbe seen in equation (2) the increase in tension results inthe increase of wave velocities which in turn leads to theincrease in frequency Thus the frequency of a pluckedstring glides from an initial value to the steady-state value(the frequency in the linear case) This is called initialpitch glide an effect which is most apparent in elasticstrings with large vibrational amplitudes and relatively lownominal string tension (such as electric guitar or kantelestrings)

Tension modulation is also responsible for another ac-oustic feature generation of missing harmonics Thismeans as its name implies that the harmonics whichshould be missing from the vibration can be found in thespectrum Generally the lack of certain harmonics in aplucked string instrument occurs due to the plucking lo-cation which heavily attenuates all harmonics that wouldhave a node at that point In elastic strings however themissing harmonics begin with a gradual increase near zeroamplitude until they reach their peak value and then de-cay off like all other harmonics In the following this phe-nomenon is discussed while avoiding to go too deep intothe mathematics A more in-depth study can be found froma paper by Legge and Fletcher [19]

Although the TM takes place in the longitudinal di-rection it can also excite the string in the transversal di-rections at the string termination point provided that thetermination is nonrigid in the transversal plane Figure 1clarifies this As can be seen in the illustration the vary-ing of the tension called tension modulation driving force(TMDF) [20] excites the bridge in the vertical directionIf the bridge has a nonzero mechanical admittance in thevertical direction TMDF will cause the bridge to move upand down

Let us now consider a case where a vibrating string car-ries two transversal modes n and m As stated above

313

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

the TM caused by mode n is periodic with a frequencycorresponding to mode n When this vibration is cou-pled through the nonrigid bridge with the transversal vi-bration of mode m the resulting vibration is the mode namplitude-modulated with mode m Thus this vibrationcan excite mode p only if p jnmj Clearly the samerule applies with m and n interchanged

Now it is clear that even if a string is plucked at x Lp the pth mode although supposed to be missing willreceive energy from other modes because of this nonlinearcoupling at the bridge It is important to note however thatthis phenomenon cannot excite all the modes eg if thestring is plucked near its center so that no even modes willbe present they will not be generated by this mechanismeither

Since this energy transfer from other modes is rathergradual than instantaneous the missing modes excited bythe TMDF will experience a gradual onset and behave likeother modes after reaching their peak values The risingrate of these missing harmonics can be shown to be pro-portional to the cube of the pluck amplitude [19]

In real musical instruments also another mechanism isresponsible for the generation of missing harmonics Thestring is often terminated behind the bridge and it under-goes a change in direction at the bridge location Figure 2illustrates this fact Now the TM can be directly coupledwith the vertical polarization due to angle at the bridgeThis means that the TMDF due to a transversal mode nwill have a frequency corresponding to mode n so thismechanism can excite only even modes thus rising theeven harmonics also in a middle-plucked string [19]

3 Digital waveguide approach

Digital waveguide (DWG) modeling is a term often en-countered when studying the synthesis of string instru-ments It is based on the fact that when an excitation signalis inserted into a string it is reflected at the boundaries andreturns to its initial position At its simplest form this canbe implemented as a single delay-loop with two consec-utive samples averaged as is done in the classic Karplus-Strong algorithm [21] An excellent introduction to DWGsused in modeling musical instruments can be found in[22]

The entire digital waveguide methodology is based onthe traveling-wave solution of the wave equation Thismeans that the solution to equation (1) can be seen as asuperposition of two waveforms traveling in opposite di-rections along the string This solution commonly knownas the traveling-wave solution or as the drsquoAlembertrsquos so-lution was first published by drsquoAlembert in 1747 It canbe presented in the mathematical language as [22]

yt x yrt xc ylt xc (4)

where yr and yl denote the wave components proceedingright and left respectively The traveling-wave solutionof the 1-D wave equation (1) can be converted into dig-ital form by sampling the wave components temporally at

BridgeNut

K

Kz

Figure 2 A more realistic bridge model The angle causes thetension to have a vertical component Kz This results in a TMDFwith frequency twice as high as the transversal frequency in thestring (after [19])

Delay of L-samples presenting y1

Delay of L-samples presenting yr

y nm( )-1

-1

Figure 3 A DWG model of an ideal string The wave reflection atthe fixed termination points is implemented with a sign changesince y yr yl The string excitation can be insertedeg by initializing the delay lines to nonzero values (after [23])

T and spatially at X intervals Formally this is done bychanging the variables in equation (4) [22]

x xm mX

t tn nT

If now the traveling waves are redefined as

yln ylnT

yrn yrnT

the discrete traveling-wave solution can be obtained

ytn xm

yrnm ylnm (5)

The term yrn m can be thought as yrn delayed bym samples Similarly the term yln can be thought asyln m delayed by m samples It is important to notethat equation (5) is not a mere approximation of equation(1) but yields exact results for bandlimited signals at thesampling instants within the limits of the numerical pre-cision of the samples [22] This kind of structure can beeasily implemented with two delay lines containing unitdelays and ynm can be obtained by summing the de-lay line values at correct locations The string state at thenext time step can be updated by simply shifting the sam-ples one step in the direction of the delay line A DWGmodel of an ideal string is shown in Figure 3

Correct tuning of the waveguide can be enabled byadding a fractional delay filter (ie a filter capable of pro-ducing also noninteger delay values) inside the DWG loop[24] Frequency-independent losses can easily be modeledin a DWG structure by inserting simple scaling coeffi-cients in the ideal DWG string structure [22] Frequency-

314

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

dependent losses can in turn be simulated using lowpassfilters (aka loop filters) inside the DWG loop

31 Time-varying digital waveguide string

When tension modulation is to be implemented in a stringmodel it clearly means that the fundamental frequency ofthe string must be modulated also (see equation 2) In aDWG string this corresponds to varying either the lengthof the delay lines or the temporal sampling instant Thissection discusses implementing the TM by varying thelength of the delay lines and is presented earlier in a re-cent publication by the authors [3] Since the delay linescan generally have integer-valued delays only directly al-tering them would lead to having the tension change in astepwise manner Obviously this behavior is not desiredand therefore fractional delay (FD) elements are used Foran in-depth study of FD elements see [24]

A first-order allpass filter was chosen for the FD ele-ment of our string model

Az a z

az (6)

where a is the filter coefficient which defines the length ofthe delay Notice that when a the allpass filter acts asa unit delay

The decision for using a first-order allpass filter withinthe string model was done partly because it is the simplestway to design an allpass filter approximating a given frac-tional delay [24] and partly because the first-order allpassfilter is the best choice for the fractional delay elementwhen delay values around unity are to be obtained [25]The phase response error caused by the allpass filter is notconsidered to pose a problem since its effect is negligiblein the audio frequency range assuming the sampling fre-quency to be reasonably high [26]

32 Distributed nonlinear DWG string

Previous works [27] [28] [29] use a single fractional de-lay element in a single-polarization string model or aDWG string terminated with a nonlinear double-spring[30] to simulate the nonlinear string This is done in or-der to reduce the computational complexity of the modelbut it has also some shortcomings Since the system is non-linear the FD elements cannot be lumped into one singleelement without giving up the idea of viewing the systemas a distributed model

In other words the whole string becomes a lumpedmodel and the termination point ldquobehindrdquo the FD elementbecomes the only location for gathering meaningful outputfrom the string Physically this would correspond to a sin-gle elastic element at the termination point of an otherwiserigid string A more realistic solution can be obtained if theelongation process is distributed along the delay line in asimilar way as in a real physical string where the elasticityis distributed along the string rather than lumped

The distributed nonlinearity can be implemented by ex-changing the delay lines of the DWG model of Figure 3

gz

-1

A z( )

v n( )s( )n

+-g

(a) (b)

Figure 4 Illustration of (a) a basic element and (b) how to getoutput data from a string consisting of these elements The di-rections of the wave components in (a) are opposite for adjacentelements so that in effect the unit delays and allpass filters areinterleaved for each delay line In (b) either the velocity or theslope of the string segment can be obtained if velocity is used asthe wave variable

F n( )

y n( )

HLHR

Figure 5 One-polarizational DWG string with time-varyinglength The string consists of the basic elements illustrated inFigure 4(a) HL and HR denote the loop filters simulating thefrequency-dependent losses The excitation to the string can beinserted as a force signal using an interaction element denotedby I The construction of the interaction element is illustrated inFigure 6

F n( )

F n( )

y n( )v n( )

2Z12Z2

Z2Z1

y n( )

Figure 6 The interaction element allows excitation signals to beinserted to the string during run-time The input signal F n canbe seen as a force signal and the output signal yn as a dis-placement signal The coefficients Z and Z represent the me-chanical impedances of the two string branches Implementationof the integrator block is depicted in Figure 7

with a structure consisting of allpass filters Then the ef-fective length of the delay lines can be changed by vary-ing the filter coefficients We will now introduce a de-lay block which contains a unit delay a first-order all-pass filter and two scaling coefficients for modeling thefrequency-independent losses This block is called a basicelement and it is illustrated in Figure 4(a) The unit delayin each basic element ensures that no delay-free loops are

315

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

formed when constructing models using these elementsFigure 4(b) shows how to obtain output data from a junc-tion between two basic elements A time-varying DWGstructure consisting of these elements is illustrated in Fig-ure 5

From the discussion in section 1 we can conclude that asuitable control signal for the FD elements can be derivedfrom the instantaneous elongation of the string In the fol-lowing since the longitudinal wave propagation velocityis considerably higher than the transversal wave velocitywe will assume that the longitudinal waves will propa-gate instantaneously through the string and the elongationcalculation and the FD parameter tuning can be done forthe whole string in one piece In practice the longitudinalwave velocity is typically only 5-20 times higher than thetransversal one but carrying out the FD parameter evalua-tions for multiple string segments would add a significantcomputational load likely without any audible advancesThe elongation of the string can be expressed as [19]

ldevt

Z lnom

q

yxt x

dx lnom (7)

where lnom is the nominal string length x is the spatial co-ordinate along the string and y is the displacement of thestring The first spatial derivative yx suggests the use ofslope waves in the elongation calculation and thus equa-tion (7) can be approximated for the digital waveguide as[28]

Ldevn

LnomXm

p srnm slnm Lnom (8)

where srnm and slnm are the slope waves at timeinstant n and position m propagating to the right and tothe left respectively Lnom is the rounded nominal stringlength To reduce the computational complexity equation(8) can be further simplified using a truncated Taylor seriesexpansion to [28]

Ldevn

LnomXm

srnm slnm

(9)

while still maintaining a sufficient accuracy The approx-imated delay variation of the total DWG can be obtainedfrom equation (9) as [28]

Ddevn

nXlnLnom

EA

K

Ldevl

Lnom (10)

where E is Youngrsquos modulus A is the cross-sectional areaof the string and K is the nominal tension correspond-ing to the string at rest The length of the string in samplesis denoted as Lnom lnomfscnom where fs is the tem-poral sampling frequency and cnom is the nominal wavepropagation speed

T

z-1

Figure 7 The integrator block is implemented by summing upconsecutive samples

Figure 8 Illustration of the kantele The string termination at var-ras is magnified for clarity A denotes the termination point forvertical vibration of the string while B denotes the terminationpoint for horizontal vibrationl stands for the distance betweenA and B

Since the system under consideration uses a distributedset of delay elements the desired delay for each basic ele-ment is

dpartial Ddev

Lnom

(11)

The coefficient a in equation (6) can now be expressed as[26 24]

a dpartial

dpartial (12)

where dpartial is the delay intended for a single allpass fil-ter Note that previous studies have used a different signfor a in equations (6) and (12) although the operation ofthe allpass filter remains the same

4 Synthesis model of the kantele usingnonlinear digital waveguides

In this section we demonstrate the nonlinear DWG for-mulation by constructing a two-polarizational synthesismodel of the kantele a Finnish folk music instrument

41 Acoustical analysis of the kantele

The kantele is a bridgeless plucked string instrument withusually five metal strings in its basic form (see Figure 8)The strings are terminated at one end by metal tuning pins

316

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

f nin( ) f nout( )

f ny( )

f nx( )

f nzy( )

f nz( )v nz( )

v ny( )

ZC

Zybridge

Zzbridge

C

Figure 9 A kantele string model The one-polarization stringmodel blocks are identical to what is illustrated in Figure 5 buthave different string lengths It is important to note that the cou-pling between the vibrational polarizations in a real physical sys-tem is more complicated but a simplified one-way coupling isused here for ease of simulation

which are screwed directly into the soundboard At the op-posite end all strings are wound once around a horizontalmetal bar called the varras and then knotted Because ofthe nonzero distance between the center of the varras andthe knot the vibrations in two polarization planes havedifferent effective lengths the varras is the terminationpoint for horizontal vibration while the knot acts as thetermination for the vertical vibration as illustrated in Fig-ure 8 This phenomenon causes the total vibration of thestring to have two fundamental components with slightlydifferent frequencies producing beating [29] A more de-tailed structure and acoustical analysis of the kantele canbe found in [29] A study of the history of kantele and anacoustically improved new design are presented in [31]

42 A novel kantele string model

The novel synthesis model of a single kantele string is con-structed using two single-polarization time-varying DWGmodels illustrated in Figure 5 and connecting them to-gether via a scaling coefficient for modeling the couplingbetween the two polarizations We restrict the couplingto being one-directional in order to avoid stability prob-lems which would otherwise rise due to the feedback loopformed from the interconnected strings as suggested byKarjalainen et al [32] Clearly the actual physical cou-pling is two-directional The elongation approximations ofthe strings and the resulting allpass filter coefficient valuesare evaluated separately for the two DWG models usingthe arithmetics described in section 32 The structure ofthe novel kantele string model is illustrated in Figure 9 Inthis model vyn and vzn represent the velocity signalscoming from the strings vibrating vertically and horizon-tally respectively

It is important to note that while vyn and vzn canbe obtained anywhere along the string in this case theyare evaluated at the termination points so that terminalimpedances can be used Zybridge and Zzbridge representthe vertical and horizontal terminal impedances respec-tively Zc stands for the coupling impedance from verti-cal to horizontal string vibration polarization and fyznrepresents the corresponding driving force The forces to

the termination caused by the two one-polarizational vi-brations are denoted by fyn and fzn The connec-tion from the elongation approximation block to the outputsimulates the direct coupling of the TM to the instrumentbody [20] A scaling coefficient denoted by C is usedto control the amount of this coupling The output of thewhole two-polarization string model is finally presentedas a force signal foutn excerted to the string terminationpoint It must also be noted that this model simulates onlya single kantele string and a model of the instrument bodymust also be added if realistic sound synthesis is desired

A real-time sound synthesis model of a kantele is con-structed using a block-based efficient audio-DSP-tool theBlockCompiler The algorithm used is efficient enough toprocess a five-string kantele model on an ordinary laptopcomputer at a 441 kHz sampling rate A detailed descrip-tion of the BlockCompiler is presented by Karjalainen[33]

5 Finite difference approach

In the previous section we discussed string synthesis viadiscretizing the drsquoAlembertrsquos solution to the 1-D waveequation Another approach is to discretize the wave equa-tion itself for example by substituting finite differenceterms for the derivatives in the wave equation (equation(1) in the case of an ideal string) This mode of opera-tion is commonly known as the finite difference method(FDM) and it was first used for sound synthesis purposesby Hiller and Ruiz in the early seventies [6] and [34] Fi-nite differences had already earlier been used in mathemat-ics for numerical solving of partial differential equationsA fine introduction to FDM in the synthesis of pluckedstring instruments can be found in [35] Below we followthe guidelines provided in [22] in deriving the FDM recur-rence equation

51 Ideal finite difference string

The partial derivatives in the 1-D wave equation (1) can bereplaced by finite differences 1

ytt x yt x ytT x

T(13)

and

yxt x yt x yt xX

X (14)

Using the finite difference approximation for the second-order derivatives in the wave equation (1) leads to

Kyt xX yt x yt xX

X

ytT x yt x ytT x

T (15)

1 It is important to note that the finite difference scheme used in equa-tions (13) and (14) was only chosen for simplicity and other schemescould be used as well For a discussion of using an implicit finite-difference scheme see section 8

317

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Solving (15) we get

ytT x KT

X

yt xX yt x yt xX

yt x ytT x (16)

Next we define the relationship between the spatial andtemporal sampling steps with [35]

r cT

X (17)

where the ldquoless than unityrdquo-restriction is called the VonNeumann stability condition Using this together with thedefinition of transversal wave velocity (equation 2) equa-tion (16) can be written as

ytT x ryt xX yt xX

(18)

ryt x ytT x

If we now do the discretization by denoting t tn nT and x xm mX as we did in section 3 we endup with the finite difference approximation [35]

ynm rynm ynm

(19)

rynm ynm

If we set r (19) becomes

ynm ynm ynm

ynm (20)

which is the finite difference equation of an ideal stringThe equality of equation (20) can be checked by substitut-ing the waveguide decomposition (equation 5) in the right-hand side of equation (20) [22]

Since the length of the string must again have integervalues correct tuning of the string becomes difficult It hasbeen shown [35] that choosing r in equation (17) re-sults in lowering the fundamental frequency of the stringTherefore the finite difference string can be tuned via theparameter r

Choosing r also gives raise to an unwanted nu-merical dispersion phenomenon called grid dispersion [7]where the wave velocity in the numerical implementationwill be less than the ideal physical wave velocity This ar-tificial dispersion affects primarily the upper harmonicswhere the frequencies will be underestimated If a typi-cal error of in the generated frequencies is allowedthe difference between the tuning coefficient r and unityshould not be greater than [35] If the constraints be-tween the correct tuning and grid dispersion do not yieldsatisfactory results the spatial density of the grid shouldbe increased This is known as spatial oversampling

52 Boundary conditions and string excitation

Since the spatial coordinate m of the string must liebetween and Lnom problems arise near the ends ofthe string when evaluating equation (20) because spatialpoints outside the string are needed The problem can

be solved by introducing boundary conditions that definehow to evaluate the string movement when m orm Lnom The simplest approach introduced alreadyin [6] would be to assume that the string terminations berigid so that yn yn Lnom This results in aphase-inverting termination which suits perfectly the caseof an ideal string For other types of string terminationseveral models have been introduced (see eg [6] [35]and [36]) Generally the nonrigid string terminations leadto frequency-dependent losses in the string model

For the FDM string excitation a useful method has beenproposed in [36] It is conceptually simple and allows forinteraction with the string during run-time There

ynm ynm

un (21)

and

ynm ynm

un (22)

are inserted into the string which causes a ldquoboxcarrdquo blockfunction to spread in both directions from the excitationpoint pair The wave component un is now used as theexcitation signal in a similar way as the exciting force sig-nal F n in section 32

53 Finite difference approximation of a lossy string

Frequency-independent losses can be modeled in an FDMstring by discretizing the velocity-dependent dampingterm in the lossy 1D wave equation (3) This results in twoadditional scaling coefficients in the recurrence equation[35]

ynm pynm ynm

qynm (23)

where the values of p and q determine the amount oflosses Generally p and q may depend on the spatial indexm but since homogeneous strings are considered here thisdependency is omitted Values

q p jpj (24)

ensure the stability of a linear finite difference string withfrequency-independent losses [37] Note that the sign dif-ference of p and q in [37] has already been taken care ofin equation (23) Modeling of frequency-dependent lossesby discretizing the lossy wave equation leads to an implicitrecurrence equation which can be evaluated if suitableapproximations are made [35]

6 Nonlinear finite difference string

Implementing tension modulation in a digital waveguidestring in section 3 was not an overly difficult task This wasdue to the fact that the implementation of a DWG string isessentially a feedback loop with delay and therefore mod-ulating the delay time of this loop corresponded to modu-lating the wave velocities In FDM strings however such

318

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

Figure 10 Illustration of the nonlinear FD algorithm on a spatio-temporal grid The vertical axis denotes the time while the hor-izontal axis denotes the spatial location on the string The illus-tration is shown only for a string segment of length N forclarity In each step of the algorithm most recently evaluated val-ues are presented as black dots while earlier values are presentedas white dots

an approach would not lead to satisfactory results sincethe physical quantities (eg displacement) themselves arepresent in the string model and not their wave decompo-sitions2

Instead we concluded that in order to correctly modelthe TM in a FDM string we first have to evaluate the recur-rence equation and use these three snapshots of the string(at time instants n n and n) in interpolating two newstring states at time instants n and n where Using these two string states we then eval-uate the recurrence equation in order to obtain the stringstate at time n It is important to note that the in-terpolation here is in effect stretching the time axis so thatthe wave propagation velocities are altered whereas in theDWG model the allpass filters perform the interpolation inthe spatial domain

This algorithm can also be seen as using two FDM sys-tems in implementing the nonlinear string The elongationof the string would be evaluated from one system and theresult the stretched string state would be updated to theother system Figure 10 illustrates this procedure on thespatio-temporal grid

In step 1 the two initial states have been assigned forthe string and the state at the next instant (in the linearcase) is obtained by the standard recurrence equation (20)The grid values which represent the state of the string atthe corresponding time instant are circled in step 1 In step2 sample values corresponding to the TM have been inter-polated from the string states in step 1 In step 3 equation(20) has been applied on the values evaluated in step 2 in

2 Such a system which deals with the physical quantities themselves iscalled a Kirchhoff model as opposed to a wave model which deals withthe wave components of the physical quantities

(c)

(b)

(a)

t

t

t t

t

t

n

n

n n

n

n

n+1

n+1

n+1 n+1

n+1

n+1y n+ m( 1 )

y n+ m( 1 )

y n m( )

y n m( )

y n+ m( 1 )

y n- m( 1 )

m

m

m m

m

m

d

d

d

a

a

a

-a

-a

-a

z-1

z-1

z-1

n-1

n-1

n-1 n-1

n-1

n-1

Figure 11 Illustration of the interpolation process due to thechange in the stringrsquos length The spatio-temporal grids on theleft and right represent the linear and interpolated string statesrespectively The fractional delay value caused by the interpola-tion is denoted by d The interpolation process in (a) is simplifiedin (b) and further in (c)

order to obtain the string state corresponding to the changein tension The two most recently obtained states are nowtaken as the ldquoinitial statesrdquo in step 4 and we can return tostep 1

As seen in Figure 10 the tension modulation corre-sponds here to interpolating the string state in the tempo-ral domain The elongation of the FDM string was evalu-ated similarly to what was done in equation (9) except thathere the slope of the string was obtained by taking the dif-ference of the displacements between two adjacent stringsegments rather than summing up the slope wave compo-nents In the following we will have a closer look at theinterpolation process

61 String state interpolation

We chose again to use first-order allpass filters in inter-polating the string state from the linear model (step 2 inFigure 10) Figure 11(a) illustrates how the interpolatedvalue of ynm is obtained from the linear values Thespatio-temporal grid on the left represents the string statein the linear case while the spatio-temporal grid on theright represents the string state after spatial interpolationThe structure between the two grids is the block diagramof a first-order allpass filter (equation 6) The coefficient afor the allpass filter was evaluated as presented earlier byequations (9)ndash(12)

319

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

In this figure we notice that the allpass filter uses thevalue of ynm delayed by one sample thus corre-sponding to ynm Clearly this can be obtained directlyfrom the grid on the left and the branch on the left contain-ing the unit delay can be reformed The result is shown inFigure 11(b) Here we also note that the interpolation sys-tem uses its own output at the previous time instant This isactually the same as using the value of ynm becauseit is the same as the output of the interpolation process onetime step ago (this might be best understood by noting thatthe bottom row of step 4 in Figure 10 is the same as thebottom row of step 1 at the next time instant) Thus Fig-ure 11(b) can be further simplified to Figure 11(c)

Having this said the recurrence equation for the time-varying finite difference string with frequency-indepen-dent damping can be written as

ynm pynm ynm

qynm (25)

where

ynm aynm ynm

aynm

ynm aynm ynm

aynm

ynm aynm ynm

ayn m

Here the coefficients p and q incorporate the frequency-independent losses and y and y refer to the linear and in-terpolated strings respectively Simplifying and rearrang-ing we end up with an equation containing only terms ofy and the subscript may therefore be omitted

ynm paynm paynm

pynm qaynm pynm

paynm qynm

paynm qayn m (26)

This equation is illustrated with a block diagram in Fig-ure 12 along with its abstraction A nonlinear FDM stringcan be constructed by connecting several of these blockstogether and using the string elongation in controlling theamount of interpolation We will refer to such a block asa time-varying finite difference time-domain (FDTD) ele-ment Illustration of the lossless time-varying FDTD ele-ment can be found in Figure 13 where p and q equal unityand have therefore been left out

62 String excitation and termination

For the interaction with the time-varying FDTD stringmodel we chose to use the ldquoboxcarrdquo excitation model dis-cussed in section 52 so that the excitation signal couldagain be interpreted as a force signal Figure 14 presentsan interaction block to be used with a time-varying FDTDstring We will call such a block the FDTD interaction el-

FDTD

-pa

z-1

z-1

z-1

y n+1m( )

y nm( )

y n-2m( )

y n-1m( )

-pa

pa

qa

-q

-qa

pa

p p

Figure 12 Illustration of the time-varying FDTD element to-gether with its abstraction A lossless time-varying FDTD ele-ment can be found in Figure 13

-a

z-1

z-1

z-1

y n-1m( )

y n-2m( )

y nm( )

y n+1m( )

-a

-a

a

a

a

Figure 13 Illustration of the lossless time-varying FDTD ele-ment

ement Using these DSP blocks we can construct a one-polarization nonlinear FDTD (NFDTD) string as illus-trated in Figure 15

We chose to use rigid terminations for our nonlin-ear finite difference string model since the modeling offrequency-dependent losses is not a key aspect of thisstudy Fixed terminations do not ruin the generation ofmissing harmonics in our model either since the TMDFcoupling is implemented in a different manner as ex-plained below

63 NFDTD string with generation of missing har-monics

In order to model the generation of missing harmonics ina NFDTD string we constructed a model where an addi-tional interaction element is placed between the last FDTDelement and the termination for feeding the TMDF to thestring Since the spatial distance between the last FDTD

320

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

y n-1m+1( )

y n-1m( )

F n( )

F n( )

y nm+1( )

y nm( )

y n+1m+1( )

y n+1m( )

Figure 14 Illustration of the FDTD interaction element togetherwith its abstraction The excitation algorithm is defined by equa-tions (21) and (22)

F n( )

FDTD FDTD FDTD FDTD FDTD

Allpass-coefficientapproximation

Elongationapproximation

Figure 15 One-polarizational NFDTD string The string consistsof the time-varying FDTD elements illustrated in Figure 12 Thezero-blocks at the terminations give zero as an output regardlessof the input values thus implying a rigid termination The excita-tion to the string can be inserted as a force signal using a FDTDinteraction element illustrated in Figure 14

F n( )

FDTD FDTD FDTD

TMDF

FDTD

Allpass-coefficientapproximation

Elongationapproximation

y n L( -1)nom

Figure 16 Illustration of the NFDTD string with a generationmechanism for missing harmonics A second interaction elementis added in order to feed the TMDF into the string The scal-ing coefficient TMDF controls the amplitude of the missing har-monics The string elongation is approximated from the displace-ments of each FDTD element

element and the rigid termination is one sample the ver-tical component of the TMDF can be seen to be equal tothe product of the displacement of the last FDTD elementand the tension Here we can replace the tension signalby the elongation signal and introduce a scaling coeffi-cient TMDF to control the amount of TMDF to be in-serted to the interaction element at the termination Thisis illustrated in Figure 16 The generation of missing har-

monics in a NFDTD model will be further discussed in thefollowing section

7 Simulation results

In this section we present the results obtained from the twononlinear string algorithms discussed in sections 3 and 5The synthesis results are compared by simulating the samephenomena namely the initial pitch glide and the genera-tion of missing harmonics using the two models Stabilityissues and computational cost of the synthesis models arealso discussed

71 Synthesis results

The synthesis results reveal that both the nonlinear DWGand NFDTD models are able to realistically model the ini-tial pitch glide phenomenon Figure 17 illustrates the fun-damental frequency behavior of a recorded kantele toneand the two synthesized tones Here the horizontal dottedline approximates the mean value of perceptual detectionthreshold of an initial pitch glide The psychoacoustic de-tection threshold in the frequency region of these tones isabout 54 Hz [38] This shows that the fundamental fre-quency glide is an audible phenomenon in plucked stringinstruments such as the kantele even at modest pluckingamplitudes and thus it must be included in a synthesismodel if realistic tones are desired

The nonlinear DWG model used in this figure has a totaldelay line length of 55125 samples and the allpass coef-ficient a is scaled using a constant value of 09 in orderto correctly simulate the behavior of the recorded sampleThe NFDTD string consists of 56 FDTD elements andthe fine-tuning parameter (aka Courant number equa-tion 17) has a value of r 13 The allpass coefficienta is scaled using a coefficient in the NFDTD case

The modeling of the generation of missing harmonicscan be implemented similarly in the distributed nonlinearDWG model as was suggested in [28] If the boxcar inte-gration of equation (10) is replaced with a leaky integratorhaving the transfer function

Iz gp ap

apz (27)

the generation of the missing harmonics can be controlledvia the integration parameter ap The variable gp definesthe gain of the integration

Figure 18 shows the amplitude envelopes of the firstthree harmonics of a synthesized tone with two differ-ent ap parameter values The string was plucked close tord of its length and as can be seen in the figure themissing harmonic in (a) has a gradual increase after thebeginning transient after which it experiences an expo-nential decay like all other modes

It is worthwhile to note that the generation of missingharmonics in the nonlinear DWG model results from theproperties of the integration of the elongation approxima-tion and is therefore not a physically justified process Ba-sically here the integration error in the leaky integrator is

321

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Time [s]

fnom

Fre

quen

cy[H

z]

Figure 17 Fundamental frequencies as a function of time fora moderately-plucked recorded kantele tone (solid line) a syn-thesized nonlinear DWG tone (dashed line) and a synthesizedNFDTD tone (dash-dotted line) The fnom stands for the nomi-nal fundamental frequency of the string and the horizontal dottedline denotes the approximated detection threshold of a pitch drift(fnomHz) which suggests that the fundamental frequencydrifts in all cases are audible

responsible for feeding energy to the missing harmonicsAlso unlike the real physical phenomenon the generationof missing harmonics in the nonlinear DWG case does notdepend on the rigidity of the terminations Neverthelessthis feature can be exploited in emulating the real stringbehavior when the integration parameters are properly ad-justed Details on tuning the leaky integrator parameterscan be found in [28]

Modeling the generation of missing harmonics in aNFDTD string is however not so simple Even if a leakyintegrator is used in the elongation calculation its param-eters do not have a desirable effect on the missing har-monics This does not seem too surprising when consider-ing the major differences of these two algorithms and it isthe reason that forced us to use an alternative mechanismfor creating the missing harmonics in the previous section(Figure 16)

Figure 19 represents the behavior of the first three har-monics of a tone synthesized by this model It can be seenthat the missing harmonics can be ldquoliftedrdquo by choosing aproper value for TMDF The stability of the system how-ever poses an upper limit for the TMDF coefficient sincethe TMDF mechanism continuously feeds energy to thestring According to our experience generating missingharmonics with amplitudes greater than what is shown inFigure 19 is difficult

72 Stability issues and computational comparison

We found the nonlinear DWG algorithm to remain sta-ble for nearly all parameter and excitation values Onlyhighly exaggerated nonlinearity scaling values togetherwith high excitation impulses resulted in stability prob-lems We thus conclude that the nonlinear DWG waveg-uide has no real stability problems when synthesis of nat-ural plucked-instrument sounds are desired

We studied the stability of the NFDTD algorithm us-ing the Von Neumann analysis [39] in the time-invariantcase ie parameter a of equation (26) was kept constantThe basic idea of this method is to calculate the spatialFourier spectrum of the system under discussion at twoconsecutive time steps An amplification function which

(a)

(b)

Figure 18 Generation of the missing harmonics in the nonlinearDWG model can be controlled via the leaky integrator parame-ters Here the string was plucked approximately at rd of itslength so every 3rd harmonic should be missing from the re-sulting spectrum In a) ap and the third harmonicclearly rises after the initial transient In b) ap 13 andthe third harmonic is more attenuated

(a)

(b)

Figure 19 Generation of missing harmonics in a NFDTD stringThe string was plucked again approximately at rd of itslength and the coupling of the TMDF to the transversal vibra-tion was controlled using a scaling coefficient TMDF In a) thescaling coefficient has a value of TMDF and the missingthird harmonic can be seen rising after the initial transient In b)TMDF and generation of missing harmonics does not takeplace

shows how the spatial spectrum evolves with time canthen be derived from the two spectra If the absolute valueof this amplification function remains below unity stabil-ity is guaranteed Formally the Von Neumann analysis forthe NFDTD algorithm goes as follows [4]

If the spatial inverse Fourier transform is defined as

ynm FfY n g neim (28)

322

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

the TM caused by mode n is periodic with a frequencycorresponding to mode n When this vibration is cou-pled through the nonrigid bridge with the transversal vi-bration of mode m the resulting vibration is the mode namplitude-modulated with mode m Thus this vibrationcan excite mode p only if p jnmj Clearly the samerule applies with m and n interchanged

Now it is clear that even if a string is plucked at x Lp the pth mode although supposed to be missing willreceive energy from other modes because of this nonlinearcoupling at the bridge It is important to note however thatthis phenomenon cannot excite all the modes eg if thestring is plucked near its center so that no even modes willbe present they will not be generated by this mechanismeither

Since this energy transfer from other modes is rathergradual than instantaneous the missing modes excited bythe TMDF will experience a gradual onset and behave likeother modes after reaching their peak values The risingrate of these missing harmonics can be shown to be pro-portional to the cube of the pluck amplitude [19]

In real musical instruments also another mechanism isresponsible for the generation of missing harmonics Thestring is often terminated behind the bridge and it under-goes a change in direction at the bridge location Figure 2illustrates this fact Now the TM can be directly coupledwith the vertical polarization due to angle at the bridgeThis means that the TMDF due to a transversal mode nwill have a frequency corresponding to mode n so thismechanism can excite only even modes thus rising theeven harmonics also in a middle-plucked string [19]

3 Digital waveguide approach

Digital waveguide (DWG) modeling is a term often en-countered when studying the synthesis of string instru-ments It is based on the fact that when an excitation signalis inserted into a string it is reflected at the boundaries andreturns to its initial position At its simplest form this canbe implemented as a single delay-loop with two consec-utive samples averaged as is done in the classic Karplus-Strong algorithm [21] An excellent introduction to DWGsused in modeling musical instruments can be found in[22]

The entire digital waveguide methodology is based onthe traveling-wave solution of the wave equation Thismeans that the solution to equation (1) can be seen as asuperposition of two waveforms traveling in opposite di-rections along the string This solution commonly knownas the traveling-wave solution or as the drsquoAlembertrsquos so-lution was first published by drsquoAlembert in 1747 It canbe presented in the mathematical language as [22]

yt x yrt xc ylt xc (4)

where yr and yl denote the wave components proceedingright and left respectively The traveling-wave solutionof the 1-D wave equation (1) can be converted into dig-ital form by sampling the wave components temporally at

BridgeNut

K

Kz

Figure 2 A more realistic bridge model The angle causes thetension to have a vertical component Kz This results in a TMDFwith frequency twice as high as the transversal frequency in thestring (after [19])

Delay of L-samples presenting y1

Delay of L-samples presenting yr

y nm( )-1

-1

Figure 3 A DWG model of an ideal string The wave reflection atthe fixed termination points is implemented with a sign changesince y yr yl The string excitation can be insertedeg by initializing the delay lines to nonzero values (after [23])

T and spatially at X intervals Formally this is done bychanging the variables in equation (4) [22]

x xm mX

t tn nT

If now the traveling waves are redefined as

yln ylnT

yrn yrnT

the discrete traveling-wave solution can be obtained

ytn xm

yrnm ylnm (5)

The term yrn m can be thought as yrn delayed bym samples Similarly the term yln can be thought asyln m delayed by m samples It is important to notethat equation (5) is not a mere approximation of equation(1) but yields exact results for bandlimited signals at thesampling instants within the limits of the numerical pre-cision of the samples [22] This kind of structure can beeasily implemented with two delay lines containing unitdelays and ynm can be obtained by summing the de-lay line values at correct locations The string state at thenext time step can be updated by simply shifting the sam-ples one step in the direction of the delay line A DWGmodel of an ideal string is shown in Figure 3

Correct tuning of the waveguide can be enabled byadding a fractional delay filter (ie a filter capable of pro-ducing also noninteger delay values) inside the DWG loop[24] Frequency-independent losses can easily be modeledin a DWG structure by inserting simple scaling coeffi-cients in the ideal DWG string structure [22] Frequency-

314

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

dependent losses can in turn be simulated using lowpassfilters (aka loop filters) inside the DWG loop

31 Time-varying digital waveguide string

When tension modulation is to be implemented in a stringmodel it clearly means that the fundamental frequency ofthe string must be modulated also (see equation 2) In aDWG string this corresponds to varying either the lengthof the delay lines or the temporal sampling instant Thissection discusses implementing the TM by varying thelength of the delay lines and is presented earlier in a re-cent publication by the authors [3] Since the delay linescan generally have integer-valued delays only directly al-tering them would lead to having the tension change in astepwise manner Obviously this behavior is not desiredand therefore fractional delay (FD) elements are used Foran in-depth study of FD elements see [24]

A first-order allpass filter was chosen for the FD ele-ment of our string model

Az a z

az (6)

where a is the filter coefficient which defines the length ofthe delay Notice that when a the allpass filter acts asa unit delay

The decision for using a first-order allpass filter withinthe string model was done partly because it is the simplestway to design an allpass filter approximating a given frac-tional delay [24] and partly because the first-order allpassfilter is the best choice for the fractional delay elementwhen delay values around unity are to be obtained [25]The phase response error caused by the allpass filter is notconsidered to pose a problem since its effect is negligiblein the audio frequency range assuming the sampling fre-quency to be reasonably high [26]

32 Distributed nonlinear DWG string

Previous works [27] [28] [29] use a single fractional de-lay element in a single-polarization string model or aDWG string terminated with a nonlinear double-spring[30] to simulate the nonlinear string This is done in or-der to reduce the computational complexity of the modelbut it has also some shortcomings Since the system is non-linear the FD elements cannot be lumped into one singleelement without giving up the idea of viewing the systemas a distributed model

In other words the whole string becomes a lumpedmodel and the termination point ldquobehindrdquo the FD elementbecomes the only location for gathering meaningful outputfrom the string Physically this would correspond to a sin-gle elastic element at the termination point of an otherwiserigid string A more realistic solution can be obtained if theelongation process is distributed along the delay line in asimilar way as in a real physical string where the elasticityis distributed along the string rather than lumped

The distributed nonlinearity can be implemented by ex-changing the delay lines of the DWG model of Figure 3

gz

-1

A z( )

v n( )s( )n

+-g

(a) (b)

Figure 4 Illustration of (a) a basic element and (b) how to getoutput data from a string consisting of these elements The di-rections of the wave components in (a) are opposite for adjacentelements so that in effect the unit delays and allpass filters areinterleaved for each delay line In (b) either the velocity or theslope of the string segment can be obtained if velocity is used asthe wave variable

F n( )

y n( )

HLHR

Figure 5 One-polarizational DWG string with time-varyinglength The string consists of the basic elements illustrated inFigure 4(a) HL and HR denote the loop filters simulating thefrequency-dependent losses The excitation to the string can beinserted as a force signal using an interaction element denotedby I The construction of the interaction element is illustrated inFigure 6

F n( )

F n( )

y n( )v n( )

2Z12Z2

Z2Z1

y n( )

Figure 6 The interaction element allows excitation signals to beinserted to the string during run-time The input signal F n canbe seen as a force signal and the output signal yn as a dis-placement signal The coefficients Z and Z represent the me-chanical impedances of the two string branches Implementationof the integrator block is depicted in Figure 7

with a structure consisting of allpass filters Then the ef-fective length of the delay lines can be changed by vary-ing the filter coefficients We will now introduce a de-lay block which contains a unit delay a first-order all-pass filter and two scaling coefficients for modeling thefrequency-independent losses This block is called a basicelement and it is illustrated in Figure 4(a) The unit delayin each basic element ensures that no delay-free loops are

315

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

formed when constructing models using these elementsFigure 4(b) shows how to obtain output data from a junc-tion between two basic elements A time-varying DWGstructure consisting of these elements is illustrated in Fig-ure 5

From the discussion in section 1 we can conclude that asuitable control signal for the FD elements can be derivedfrom the instantaneous elongation of the string In the fol-lowing since the longitudinal wave propagation velocityis considerably higher than the transversal wave velocitywe will assume that the longitudinal waves will propa-gate instantaneously through the string and the elongationcalculation and the FD parameter tuning can be done forthe whole string in one piece In practice the longitudinalwave velocity is typically only 5-20 times higher than thetransversal one but carrying out the FD parameter evalua-tions for multiple string segments would add a significantcomputational load likely without any audible advancesThe elongation of the string can be expressed as [19]

ldevt

Z lnom

q

yxt x

dx lnom (7)

where lnom is the nominal string length x is the spatial co-ordinate along the string and y is the displacement of thestring The first spatial derivative yx suggests the use ofslope waves in the elongation calculation and thus equa-tion (7) can be approximated for the digital waveguide as[28]

Ldevn

LnomXm

p srnm slnm Lnom (8)

where srnm and slnm are the slope waves at timeinstant n and position m propagating to the right and tothe left respectively Lnom is the rounded nominal stringlength To reduce the computational complexity equation(8) can be further simplified using a truncated Taylor seriesexpansion to [28]

Ldevn

LnomXm

srnm slnm

(9)

while still maintaining a sufficient accuracy The approx-imated delay variation of the total DWG can be obtainedfrom equation (9) as [28]

Ddevn

nXlnLnom

EA

K

Ldevl

Lnom (10)

where E is Youngrsquos modulus A is the cross-sectional areaof the string and K is the nominal tension correspond-ing to the string at rest The length of the string in samplesis denoted as Lnom lnomfscnom where fs is the tem-poral sampling frequency and cnom is the nominal wavepropagation speed

T

z-1

Figure 7 The integrator block is implemented by summing upconsecutive samples

Figure 8 Illustration of the kantele The string termination at var-ras is magnified for clarity A denotes the termination point forvertical vibration of the string while B denotes the terminationpoint for horizontal vibrationl stands for the distance betweenA and B

Since the system under consideration uses a distributedset of delay elements the desired delay for each basic ele-ment is

dpartial Ddev

Lnom

(11)

The coefficient a in equation (6) can now be expressed as[26 24]

a dpartial

dpartial (12)

where dpartial is the delay intended for a single allpass fil-ter Note that previous studies have used a different signfor a in equations (6) and (12) although the operation ofthe allpass filter remains the same

4 Synthesis model of the kantele usingnonlinear digital waveguides

In this section we demonstrate the nonlinear DWG for-mulation by constructing a two-polarizational synthesismodel of the kantele a Finnish folk music instrument

41 Acoustical analysis of the kantele

The kantele is a bridgeless plucked string instrument withusually five metal strings in its basic form (see Figure 8)The strings are terminated at one end by metal tuning pins

316

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

f nin( ) f nout( )

f ny( )

f nx( )

f nzy( )

f nz( )v nz( )

v ny( )

ZC

Zybridge

Zzbridge

C

Figure 9 A kantele string model The one-polarization stringmodel blocks are identical to what is illustrated in Figure 5 buthave different string lengths It is important to note that the cou-pling between the vibrational polarizations in a real physical sys-tem is more complicated but a simplified one-way coupling isused here for ease of simulation

which are screwed directly into the soundboard At the op-posite end all strings are wound once around a horizontalmetal bar called the varras and then knotted Because ofthe nonzero distance between the center of the varras andthe knot the vibrations in two polarization planes havedifferent effective lengths the varras is the terminationpoint for horizontal vibration while the knot acts as thetermination for the vertical vibration as illustrated in Fig-ure 8 This phenomenon causes the total vibration of thestring to have two fundamental components with slightlydifferent frequencies producing beating [29] A more de-tailed structure and acoustical analysis of the kantele canbe found in [29] A study of the history of kantele and anacoustically improved new design are presented in [31]

42 A novel kantele string model

The novel synthesis model of a single kantele string is con-structed using two single-polarization time-varying DWGmodels illustrated in Figure 5 and connecting them to-gether via a scaling coefficient for modeling the couplingbetween the two polarizations We restrict the couplingto being one-directional in order to avoid stability prob-lems which would otherwise rise due to the feedback loopformed from the interconnected strings as suggested byKarjalainen et al [32] Clearly the actual physical cou-pling is two-directional The elongation approximations ofthe strings and the resulting allpass filter coefficient valuesare evaluated separately for the two DWG models usingthe arithmetics described in section 32 The structure ofthe novel kantele string model is illustrated in Figure 9 Inthis model vyn and vzn represent the velocity signalscoming from the strings vibrating vertically and horizon-tally respectively

It is important to note that while vyn and vzn canbe obtained anywhere along the string in this case theyare evaluated at the termination points so that terminalimpedances can be used Zybridge and Zzbridge representthe vertical and horizontal terminal impedances respec-tively Zc stands for the coupling impedance from verti-cal to horizontal string vibration polarization and fyznrepresents the corresponding driving force The forces to

the termination caused by the two one-polarizational vi-brations are denoted by fyn and fzn The connec-tion from the elongation approximation block to the outputsimulates the direct coupling of the TM to the instrumentbody [20] A scaling coefficient denoted by C is usedto control the amount of this coupling The output of thewhole two-polarization string model is finally presentedas a force signal foutn excerted to the string terminationpoint It must also be noted that this model simulates onlya single kantele string and a model of the instrument bodymust also be added if realistic sound synthesis is desired

A real-time sound synthesis model of a kantele is con-structed using a block-based efficient audio-DSP-tool theBlockCompiler The algorithm used is efficient enough toprocess a five-string kantele model on an ordinary laptopcomputer at a 441 kHz sampling rate A detailed descrip-tion of the BlockCompiler is presented by Karjalainen[33]

5 Finite difference approach

In the previous section we discussed string synthesis viadiscretizing the drsquoAlembertrsquos solution to the 1-D waveequation Another approach is to discretize the wave equa-tion itself for example by substituting finite differenceterms for the derivatives in the wave equation (equation(1) in the case of an ideal string) This mode of opera-tion is commonly known as the finite difference method(FDM) and it was first used for sound synthesis purposesby Hiller and Ruiz in the early seventies [6] and [34] Fi-nite differences had already earlier been used in mathemat-ics for numerical solving of partial differential equationsA fine introduction to FDM in the synthesis of pluckedstring instruments can be found in [35] Below we followthe guidelines provided in [22] in deriving the FDM recur-rence equation

51 Ideal finite difference string

The partial derivatives in the 1-D wave equation (1) can bereplaced by finite differences 1

ytt x yt x ytT x

T(13)

and

yxt x yt x yt xX

X (14)

Using the finite difference approximation for the second-order derivatives in the wave equation (1) leads to

Kyt xX yt x yt xX

X

ytT x yt x ytT x

T (15)

1 It is important to note that the finite difference scheme used in equa-tions (13) and (14) was only chosen for simplicity and other schemescould be used as well For a discussion of using an implicit finite-difference scheme see section 8

317

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Solving (15) we get

ytT x KT

X

yt xX yt x yt xX

yt x ytT x (16)

Next we define the relationship between the spatial andtemporal sampling steps with [35]

r cT

X (17)

where the ldquoless than unityrdquo-restriction is called the VonNeumann stability condition Using this together with thedefinition of transversal wave velocity (equation 2) equa-tion (16) can be written as

ytT x ryt xX yt xX

(18)

ryt x ytT x

If we now do the discretization by denoting t tn nT and x xm mX as we did in section 3 we endup with the finite difference approximation [35]

ynm rynm ynm

(19)

rynm ynm

If we set r (19) becomes

ynm ynm ynm

ynm (20)

which is the finite difference equation of an ideal stringThe equality of equation (20) can be checked by substitut-ing the waveguide decomposition (equation 5) in the right-hand side of equation (20) [22]

Since the length of the string must again have integervalues correct tuning of the string becomes difficult It hasbeen shown [35] that choosing r in equation (17) re-sults in lowering the fundamental frequency of the stringTherefore the finite difference string can be tuned via theparameter r

Choosing r also gives raise to an unwanted nu-merical dispersion phenomenon called grid dispersion [7]where the wave velocity in the numerical implementationwill be less than the ideal physical wave velocity This ar-tificial dispersion affects primarily the upper harmonicswhere the frequencies will be underestimated If a typi-cal error of in the generated frequencies is allowedthe difference between the tuning coefficient r and unityshould not be greater than [35] If the constraints be-tween the correct tuning and grid dispersion do not yieldsatisfactory results the spatial density of the grid shouldbe increased This is known as spatial oversampling

52 Boundary conditions and string excitation

Since the spatial coordinate m of the string must liebetween and Lnom problems arise near the ends ofthe string when evaluating equation (20) because spatialpoints outside the string are needed The problem can

be solved by introducing boundary conditions that definehow to evaluate the string movement when m orm Lnom The simplest approach introduced alreadyin [6] would be to assume that the string terminations berigid so that yn yn Lnom This results in aphase-inverting termination which suits perfectly the caseof an ideal string For other types of string terminationseveral models have been introduced (see eg [6] [35]and [36]) Generally the nonrigid string terminations leadto frequency-dependent losses in the string model

For the FDM string excitation a useful method has beenproposed in [36] It is conceptually simple and allows forinteraction with the string during run-time There

ynm ynm

un (21)

and

ynm ynm

un (22)

are inserted into the string which causes a ldquoboxcarrdquo blockfunction to spread in both directions from the excitationpoint pair The wave component un is now used as theexcitation signal in a similar way as the exciting force sig-nal F n in section 32

53 Finite difference approximation of a lossy string

Frequency-independent losses can be modeled in an FDMstring by discretizing the velocity-dependent dampingterm in the lossy 1D wave equation (3) This results in twoadditional scaling coefficients in the recurrence equation[35]

ynm pynm ynm

qynm (23)

where the values of p and q determine the amount oflosses Generally p and q may depend on the spatial indexm but since homogeneous strings are considered here thisdependency is omitted Values

q p jpj (24)

ensure the stability of a linear finite difference string withfrequency-independent losses [37] Note that the sign dif-ference of p and q in [37] has already been taken care ofin equation (23) Modeling of frequency-dependent lossesby discretizing the lossy wave equation leads to an implicitrecurrence equation which can be evaluated if suitableapproximations are made [35]

6 Nonlinear finite difference string

Implementing tension modulation in a digital waveguidestring in section 3 was not an overly difficult task This wasdue to the fact that the implementation of a DWG string isessentially a feedback loop with delay and therefore mod-ulating the delay time of this loop corresponded to modu-lating the wave velocities In FDM strings however such

318

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

Figure 10 Illustration of the nonlinear FD algorithm on a spatio-temporal grid The vertical axis denotes the time while the hor-izontal axis denotes the spatial location on the string The illus-tration is shown only for a string segment of length N forclarity In each step of the algorithm most recently evaluated val-ues are presented as black dots while earlier values are presentedas white dots

an approach would not lead to satisfactory results sincethe physical quantities (eg displacement) themselves arepresent in the string model and not their wave decompo-sitions2

Instead we concluded that in order to correctly modelthe TM in a FDM string we first have to evaluate the recur-rence equation and use these three snapshots of the string(at time instants n n and n) in interpolating two newstring states at time instants n and n where Using these two string states we then eval-uate the recurrence equation in order to obtain the stringstate at time n It is important to note that the in-terpolation here is in effect stretching the time axis so thatthe wave propagation velocities are altered whereas in theDWG model the allpass filters perform the interpolation inthe spatial domain

This algorithm can also be seen as using two FDM sys-tems in implementing the nonlinear string The elongationof the string would be evaluated from one system and theresult the stretched string state would be updated to theother system Figure 10 illustrates this procedure on thespatio-temporal grid

In step 1 the two initial states have been assigned forthe string and the state at the next instant (in the linearcase) is obtained by the standard recurrence equation (20)The grid values which represent the state of the string atthe corresponding time instant are circled in step 1 In step2 sample values corresponding to the TM have been inter-polated from the string states in step 1 In step 3 equation(20) has been applied on the values evaluated in step 2 in

2 Such a system which deals with the physical quantities themselves iscalled a Kirchhoff model as opposed to a wave model which deals withthe wave components of the physical quantities

(c)

(b)

(a)

t

t

t t

t

t

n

n

n n

n

n

n+1

n+1

n+1 n+1

n+1

n+1y n+ m( 1 )

y n+ m( 1 )

y n m( )

y n m( )

y n+ m( 1 )

y n- m( 1 )

m

m

m m

m

m

d

d

d

a

a

a

-a

-a

-a

z-1

z-1

z-1

n-1

n-1

n-1 n-1

n-1

n-1

Figure 11 Illustration of the interpolation process due to thechange in the stringrsquos length The spatio-temporal grids on theleft and right represent the linear and interpolated string statesrespectively The fractional delay value caused by the interpola-tion is denoted by d The interpolation process in (a) is simplifiedin (b) and further in (c)

order to obtain the string state corresponding to the changein tension The two most recently obtained states are nowtaken as the ldquoinitial statesrdquo in step 4 and we can return tostep 1

As seen in Figure 10 the tension modulation corre-sponds here to interpolating the string state in the tempo-ral domain The elongation of the FDM string was evalu-ated similarly to what was done in equation (9) except thathere the slope of the string was obtained by taking the dif-ference of the displacements between two adjacent stringsegments rather than summing up the slope wave compo-nents In the following we will have a closer look at theinterpolation process

61 String state interpolation

We chose again to use first-order allpass filters in inter-polating the string state from the linear model (step 2 inFigure 10) Figure 11(a) illustrates how the interpolatedvalue of ynm is obtained from the linear values Thespatio-temporal grid on the left represents the string statein the linear case while the spatio-temporal grid on theright represents the string state after spatial interpolationThe structure between the two grids is the block diagramof a first-order allpass filter (equation 6) The coefficient afor the allpass filter was evaluated as presented earlier byequations (9)ndash(12)

319

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

In this figure we notice that the allpass filter uses thevalue of ynm delayed by one sample thus corre-sponding to ynm Clearly this can be obtained directlyfrom the grid on the left and the branch on the left contain-ing the unit delay can be reformed The result is shown inFigure 11(b) Here we also note that the interpolation sys-tem uses its own output at the previous time instant This isactually the same as using the value of ynm becauseit is the same as the output of the interpolation process onetime step ago (this might be best understood by noting thatthe bottom row of step 4 in Figure 10 is the same as thebottom row of step 1 at the next time instant) Thus Fig-ure 11(b) can be further simplified to Figure 11(c)

Having this said the recurrence equation for the time-varying finite difference string with frequency-indepen-dent damping can be written as

ynm pynm ynm

qynm (25)

where

ynm aynm ynm

aynm

ynm aynm ynm

aynm

ynm aynm ynm

ayn m

Here the coefficients p and q incorporate the frequency-independent losses and y and y refer to the linear and in-terpolated strings respectively Simplifying and rearrang-ing we end up with an equation containing only terms ofy and the subscript may therefore be omitted

ynm paynm paynm

pynm qaynm pynm

paynm qynm

paynm qayn m (26)

This equation is illustrated with a block diagram in Fig-ure 12 along with its abstraction A nonlinear FDM stringcan be constructed by connecting several of these blockstogether and using the string elongation in controlling theamount of interpolation We will refer to such a block asa time-varying finite difference time-domain (FDTD) ele-ment Illustration of the lossless time-varying FDTD ele-ment can be found in Figure 13 where p and q equal unityand have therefore been left out

62 String excitation and termination

For the interaction with the time-varying FDTD stringmodel we chose to use the ldquoboxcarrdquo excitation model dis-cussed in section 52 so that the excitation signal couldagain be interpreted as a force signal Figure 14 presentsan interaction block to be used with a time-varying FDTDstring We will call such a block the FDTD interaction el-

FDTD

-pa

z-1

z-1

z-1

y n+1m( )

y nm( )

y n-2m( )

y n-1m( )

-pa

pa

qa

-q

-qa

pa

p p

Figure 12 Illustration of the time-varying FDTD element to-gether with its abstraction A lossless time-varying FDTD ele-ment can be found in Figure 13

-a

z-1

z-1

z-1

y n-1m( )

y n-2m( )

y nm( )

y n+1m( )

-a

-a

a

a

a

Figure 13 Illustration of the lossless time-varying FDTD ele-ment

ement Using these DSP blocks we can construct a one-polarization nonlinear FDTD (NFDTD) string as illus-trated in Figure 15

We chose to use rigid terminations for our nonlin-ear finite difference string model since the modeling offrequency-dependent losses is not a key aspect of thisstudy Fixed terminations do not ruin the generation ofmissing harmonics in our model either since the TMDFcoupling is implemented in a different manner as ex-plained below

63 NFDTD string with generation of missing har-monics

In order to model the generation of missing harmonics ina NFDTD string we constructed a model where an addi-tional interaction element is placed between the last FDTDelement and the termination for feeding the TMDF to thestring Since the spatial distance between the last FDTD

320

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

y n-1m+1( )

y n-1m( )

F n( )

F n( )

y nm+1( )

y nm( )

y n+1m+1( )

y n+1m( )

Figure 14 Illustration of the FDTD interaction element togetherwith its abstraction The excitation algorithm is defined by equa-tions (21) and (22)

F n( )

FDTD FDTD FDTD FDTD FDTD

Allpass-coefficientapproximation

Elongationapproximation

Figure 15 One-polarizational NFDTD string The string consistsof the time-varying FDTD elements illustrated in Figure 12 Thezero-blocks at the terminations give zero as an output regardlessof the input values thus implying a rigid termination The excita-tion to the string can be inserted as a force signal using a FDTDinteraction element illustrated in Figure 14

F n( )

FDTD FDTD FDTD

TMDF

FDTD

Allpass-coefficientapproximation

Elongationapproximation

y n L( -1)nom

Figure 16 Illustration of the NFDTD string with a generationmechanism for missing harmonics A second interaction elementis added in order to feed the TMDF into the string The scal-ing coefficient TMDF controls the amplitude of the missing har-monics The string elongation is approximated from the displace-ments of each FDTD element

element and the rigid termination is one sample the ver-tical component of the TMDF can be seen to be equal tothe product of the displacement of the last FDTD elementand the tension Here we can replace the tension signalby the elongation signal and introduce a scaling coeffi-cient TMDF to control the amount of TMDF to be in-serted to the interaction element at the termination Thisis illustrated in Figure 16 The generation of missing har-

monics in a NFDTD model will be further discussed in thefollowing section

7 Simulation results

In this section we present the results obtained from the twononlinear string algorithms discussed in sections 3 and 5The synthesis results are compared by simulating the samephenomena namely the initial pitch glide and the genera-tion of missing harmonics using the two models Stabilityissues and computational cost of the synthesis models arealso discussed

71 Synthesis results

The synthesis results reveal that both the nonlinear DWGand NFDTD models are able to realistically model the ini-tial pitch glide phenomenon Figure 17 illustrates the fun-damental frequency behavior of a recorded kantele toneand the two synthesized tones Here the horizontal dottedline approximates the mean value of perceptual detectionthreshold of an initial pitch glide The psychoacoustic de-tection threshold in the frequency region of these tones isabout 54 Hz [38] This shows that the fundamental fre-quency glide is an audible phenomenon in plucked stringinstruments such as the kantele even at modest pluckingamplitudes and thus it must be included in a synthesismodel if realistic tones are desired

The nonlinear DWG model used in this figure has a totaldelay line length of 55125 samples and the allpass coef-ficient a is scaled using a constant value of 09 in orderto correctly simulate the behavior of the recorded sampleThe NFDTD string consists of 56 FDTD elements andthe fine-tuning parameter (aka Courant number equa-tion 17) has a value of r 13 The allpass coefficienta is scaled using a coefficient in the NFDTD case

The modeling of the generation of missing harmonicscan be implemented similarly in the distributed nonlinearDWG model as was suggested in [28] If the boxcar inte-gration of equation (10) is replaced with a leaky integratorhaving the transfer function

Iz gp ap

apz (27)

the generation of the missing harmonics can be controlledvia the integration parameter ap The variable gp definesthe gain of the integration

Figure 18 shows the amplitude envelopes of the firstthree harmonics of a synthesized tone with two differ-ent ap parameter values The string was plucked close tord of its length and as can be seen in the figure themissing harmonic in (a) has a gradual increase after thebeginning transient after which it experiences an expo-nential decay like all other modes

It is worthwhile to note that the generation of missingharmonics in the nonlinear DWG model results from theproperties of the integration of the elongation approxima-tion and is therefore not a physically justified process Ba-sically here the integration error in the leaky integrator is

321

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Time [s]

fnom

Fre

quen

cy[H

z]

Figure 17 Fundamental frequencies as a function of time fora moderately-plucked recorded kantele tone (solid line) a syn-thesized nonlinear DWG tone (dashed line) and a synthesizedNFDTD tone (dash-dotted line) The fnom stands for the nomi-nal fundamental frequency of the string and the horizontal dottedline denotes the approximated detection threshold of a pitch drift(fnomHz) which suggests that the fundamental frequencydrifts in all cases are audible

responsible for feeding energy to the missing harmonicsAlso unlike the real physical phenomenon the generationof missing harmonics in the nonlinear DWG case does notdepend on the rigidity of the terminations Neverthelessthis feature can be exploited in emulating the real stringbehavior when the integration parameters are properly ad-justed Details on tuning the leaky integrator parameterscan be found in [28]

Modeling the generation of missing harmonics in aNFDTD string is however not so simple Even if a leakyintegrator is used in the elongation calculation its param-eters do not have a desirable effect on the missing har-monics This does not seem too surprising when consider-ing the major differences of these two algorithms and it isthe reason that forced us to use an alternative mechanismfor creating the missing harmonics in the previous section(Figure 16)

Figure 19 represents the behavior of the first three har-monics of a tone synthesized by this model It can be seenthat the missing harmonics can be ldquoliftedrdquo by choosing aproper value for TMDF The stability of the system how-ever poses an upper limit for the TMDF coefficient sincethe TMDF mechanism continuously feeds energy to thestring According to our experience generating missingharmonics with amplitudes greater than what is shown inFigure 19 is difficult

72 Stability issues and computational comparison

We found the nonlinear DWG algorithm to remain sta-ble for nearly all parameter and excitation values Onlyhighly exaggerated nonlinearity scaling values togetherwith high excitation impulses resulted in stability prob-lems We thus conclude that the nonlinear DWG waveg-uide has no real stability problems when synthesis of nat-ural plucked-instrument sounds are desired

We studied the stability of the NFDTD algorithm us-ing the Von Neumann analysis [39] in the time-invariantcase ie parameter a of equation (26) was kept constantThe basic idea of this method is to calculate the spatialFourier spectrum of the system under discussion at twoconsecutive time steps An amplification function which

(a)

(b)

Figure 18 Generation of the missing harmonics in the nonlinearDWG model can be controlled via the leaky integrator parame-ters Here the string was plucked approximately at rd of itslength so every 3rd harmonic should be missing from the re-sulting spectrum In a) ap and the third harmonicclearly rises after the initial transient In b) ap 13 andthe third harmonic is more attenuated

(a)

(b)

Figure 19 Generation of missing harmonics in a NFDTD stringThe string was plucked again approximately at rd of itslength and the coupling of the TMDF to the transversal vibra-tion was controlled using a scaling coefficient TMDF In a) thescaling coefficient has a value of TMDF and the missingthird harmonic can be seen rising after the initial transient In b)TMDF and generation of missing harmonics does not takeplace

shows how the spatial spectrum evolves with time canthen be derived from the two spectra If the absolute valueof this amplification function remains below unity stabil-ity is guaranteed Formally the Von Neumann analysis forthe NFDTD algorithm goes as follows [4]

If the spatial inverse Fourier transform is defined as

ynm FfY n g neim (28)

322

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

dependent losses can in turn be simulated using lowpassfilters (aka loop filters) inside the DWG loop

31 Time-varying digital waveguide string

When tension modulation is to be implemented in a stringmodel it clearly means that the fundamental frequency ofthe string must be modulated also (see equation 2) In aDWG string this corresponds to varying either the lengthof the delay lines or the temporal sampling instant Thissection discusses implementing the TM by varying thelength of the delay lines and is presented earlier in a re-cent publication by the authors [3] Since the delay linescan generally have integer-valued delays only directly al-tering them would lead to having the tension change in astepwise manner Obviously this behavior is not desiredand therefore fractional delay (FD) elements are used Foran in-depth study of FD elements see [24]

A first-order allpass filter was chosen for the FD ele-ment of our string model

Az a z

az (6)

where a is the filter coefficient which defines the length ofthe delay Notice that when a the allpass filter acts asa unit delay

The decision for using a first-order allpass filter withinthe string model was done partly because it is the simplestway to design an allpass filter approximating a given frac-tional delay [24] and partly because the first-order allpassfilter is the best choice for the fractional delay elementwhen delay values around unity are to be obtained [25]The phase response error caused by the allpass filter is notconsidered to pose a problem since its effect is negligiblein the audio frequency range assuming the sampling fre-quency to be reasonably high [26]

32 Distributed nonlinear DWG string

Previous works [27] [28] [29] use a single fractional de-lay element in a single-polarization string model or aDWG string terminated with a nonlinear double-spring[30] to simulate the nonlinear string This is done in or-der to reduce the computational complexity of the modelbut it has also some shortcomings Since the system is non-linear the FD elements cannot be lumped into one singleelement without giving up the idea of viewing the systemas a distributed model

In other words the whole string becomes a lumpedmodel and the termination point ldquobehindrdquo the FD elementbecomes the only location for gathering meaningful outputfrom the string Physically this would correspond to a sin-gle elastic element at the termination point of an otherwiserigid string A more realistic solution can be obtained if theelongation process is distributed along the delay line in asimilar way as in a real physical string where the elasticityis distributed along the string rather than lumped

The distributed nonlinearity can be implemented by ex-changing the delay lines of the DWG model of Figure 3

gz

-1

A z( )

v n( )s( )n

+-g

(a) (b)

Figure 4 Illustration of (a) a basic element and (b) how to getoutput data from a string consisting of these elements The di-rections of the wave components in (a) are opposite for adjacentelements so that in effect the unit delays and allpass filters areinterleaved for each delay line In (b) either the velocity or theslope of the string segment can be obtained if velocity is used asthe wave variable

F n( )

y n( )

HLHR

Figure 5 One-polarizational DWG string with time-varyinglength The string consists of the basic elements illustrated inFigure 4(a) HL and HR denote the loop filters simulating thefrequency-dependent losses The excitation to the string can beinserted as a force signal using an interaction element denotedby I The construction of the interaction element is illustrated inFigure 6

F n( )

F n( )

y n( )v n( )

2Z12Z2

Z2Z1

y n( )

Figure 6 The interaction element allows excitation signals to beinserted to the string during run-time The input signal F n canbe seen as a force signal and the output signal yn as a dis-placement signal The coefficients Z and Z represent the me-chanical impedances of the two string branches Implementationof the integrator block is depicted in Figure 7

with a structure consisting of allpass filters Then the ef-fective length of the delay lines can be changed by vary-ing the filter coefficients We will now introduce a de-lay block which contains a unit delay a first-order all-pass filter and two scaling coefficients for modeling thefrequency-independent losses This block is called a basicelement and it is illustrated in Figure 4(a) The unit delayin each basic element ensures that no delay-free loops are

315

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

formed when constructing models using these elementsFigure 4(b) shows how to obtain output data from a junc-tion between two basic elements A time-varying DWGstructure consisting of these elements is illustrated in Fig-ure 5

From the discussion in section 1 we can conclude that asuitable control signal for the FD elements can be derivedfrom the instantaneous elongation of the string In the fol-lowing since the longitudinal wave propagation velocityis considerably higher than the transversal wave velocitywe will assume that the longitudinal waves will propa-gate instantaneously through the string and the elongationcalculation and the FD parameter tuning can be done forthe whole string in one piece In practice the longitudinalwave velocity is typically only 5-20 times higher than thetransversal one but carrying out the FD parameter evalua-tions for multiple string segments would add a significantcomputational load likely without any audible advancesThe elongation of the string can be expressed as [19]

ldevt

Z lnom

q

yxt x

dx lnom (7)

where lnom is the nominal string length x is the spatial co-ordinate along the string and y is the displacement of thestring The first spatial derivative yx suggests the use ofslope waves in the elongation calculation and thus equa-tion (7) can be approximated for the digital waveguide as[28]

Ldevn

LnomXm

p srnm slnm Lnom (8)

where srnm and slnm are the slope waves at timeinstant n and position m propagating to the right and tothe left respectively Lnom is the rounded nominal stringlength To reduce the computational complexity equation(8) can be further simplified using a truncated Taylor seriesexpansion to [28]

Ldevn

LnomXm

srnm slnm

(9)

while still maintaining a sufficient accuracy The approx-imated delay variation of the total DWG can be obtainedfrom equation (9) as [28]

Ddevn

nXlnLnom

EA

K

Ldevl

Lnom (10)

where E is Youngrsquos modulus A is the cross-sectional areaof the string and K is the nominal tension correspond-ing to the string at rest The length of the string in samplesis denoted as Lnom lnomfscnom where fs is the tem-poral sampling frequency and cnom is the nominal wavepropagation speed

T

z-1

Figure 7 The integrator block is implemented by summing upconsecutive samples

Figure 8 Illustration of the kantele The string termination at var-ras is magnified for clarity A denotes the termination point forvertical vibration of the string while B denotes the terminationpoint for horizontal vibrationl stands for the distance betweenA and B

Since the system under consideration uses a distributedset of delay elements the desired delay for each basic ele-ment is

dpartial Ddev

Lnom

(11)

The coefficient a in equation (6) can now be expressed as[26 24]

a dpartial

dpartial (12)

where dpartial is the delay intended for a single allpass fil-ter Note that previous studies have used a different signfor a in equations (6) and (12) although the operation ofthe allpass filter remains the same

4 Synthesis model of the kantele usingnonlinear digital waveguides

In this section we demonstrate the nonlinear DWG for-mulation by constructing a two-polarizational synthesismodel of the kantele a Finnish folk music instrument

41 Acoustical analysis of the kantele

The kantele is a bridgeless plucked string instrument withusually five metal strings in its basic form (see Figure 8)The strings are terminated at one end by metal tuning pins

316

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

f nin( ) f nout( )

f ny( )

f nx( )

f nzy( )

f nz( )v nz( )

v ny( )

ZC

Zybridge

Zzbridge

C

Figure 9 A kantele string model The one-polarization stringmodel blocks are identical to what is illustrated in Figure 5 buthave different string lengths It is important to note that the cou-pling between the vibrational polarizations in a real physical sys-tem is more complicated but a simplified one-way coupling isused here for ease of simulation

which are screwed directly into the soundboard At the op-posite end all strings are wound once around a horizontalmetal bar called the varras and then knotted Because ofthe nonzero distance between the center of the varras andthe knot the vibrations in two polarization planes havedifferent effective lengths the varras is the terminationpoint for horizontal vibration while the knot acts as thetermination for the vertical vibration as illustrated in Fig-ure 8 This phenomenon causes the total vibration of thestring to have two fundamental components with slightlydifferent frequencies producing beating [29] A more de-tailed structure and acoustical analysis of the kantele canbe found in [29] A study of the history of kantele and anacoustically improved new design are presented in [31]

42 A novel kantele string model

The novel synthesis model of a single kantele string is con-structed using two single-polarization time-varying DWGmodels illustrated in Figure 5 and connecting them to-gether via a scaling coefficient for modeling the couplingbetween the two polarizations We restrict the couplingto being one-directional in order to avoid stability prob-lems which would otherwise rise due to the feedback loopformed from the interconnected strings as suggested byKarjalainen et al [32] Clearly the actual physical cou-pling is two-directional The elongation approximations ofthe strings and the resulting allpass filter coefficient valuesare evaluated separately for the two DWG models usingthe arithmetics described in section 32 The structure ofthe novel kantele string model is illustrated in Figure 9 Inthis model vyn and vzn represent the velocity signalscoming from the strings vibrating vertically and horizon-tally respectively

It is important to note that while vyn and vzn canbe obtained anywhere along the string in this case theyare evaluated at the termination points so that terminalimpedances can be used Zybridge and Zzbridge representthe vertical and horizontal terminal impedances respec-tively Zc stands for the coupling impedance from verti-cal to horizontal string vibration polarization and fyznrepresents the corresponding driving force The forces to

the termination caused by the two one-polarizational vi-brations are denoted by fyn and fzn The connec-tion from the elongation approximation block to the outputsimulates the direct coupling of the TM to the instrumentbody [20] A scaling coefficient denoted by C is usedto control the amount of this coupling The output of thewhole two-polarization string model is finally presentedas a force signal foutn excerted to the string terminationpoint It must also be noted that this model simulates onlya single kantele string and a model of the instrument bodymust also be added if realistic sound synthesis is desired

A real-time sound synthesis model of a kantele is con-structed using a block-based efficient audio-DSP-tool theBlockCompiler The algorithm used is efficient enough toprocess a five-string kantele model on an ordinary laptopcomputer at a 441 kHz sampling rate A detailed descrip-tion of the BlockCompiler is presented by Karjalainen[33]

5 Finite difference approach

In the previous section we discussed string synthesis viadiscretizing the drsquoAlembertrsquos solution to the 1-D waveequation Another approach is to discretize the wave equa-tion itself for example by substituting finite differenceterms for the derivatives in the wave equation (equation(1) in the case of an ideal string) This mode of opera-tion is commonly known as the finite difference method(FDM) and it was first used for sound synthesis purposesby Hiller and Ruiz in the early seventies [6] and [34] Fi-nite differences had already earlier been used in mathemat-ics for numerical solving of partial differential equationsA fine introduction to FDM in the synthesis of pluckedstring instruments can be found in [35] Below we followthe guidelines provided in [22] in deriving the FDM recur-rence equation

51 Ideal finite difference string

The partial derivatives in the 1-D wave equation (1) can bereplaced by finite differences 1

ytt x yt x ytT x

T(13)

and

yxt x yt x yt xX

X (14)

Using the finite difference approximation for the second-order derivatives in the wave equation (1) leads to

Kyt xX yt x yt xX

X

ytT x yt x ytT x

T (15)

1 It is important to note that the finite difference scheme used in equa-tions (13) and (14) was only chosen for simplicity and other schemescould be used as well For a discussion of using an implicit finite-difference scheme see section 8

317

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Solving (15) we get

ytT x KT

X

yt xX yt x yt xX

yt x ytT x (16)

Next we define the relationship between the spatial andtemporal sampling steps with [35]

r cT

X (17)

where the ldquoless than unityrdquo-restriction is called the VonNeumann stability condition Using this together with thedefinition of transversal wave velocity (equation 2) equa-tion (16) can be written as

ytT x ryt xX yt xX

(18)

ryt x ytT x

If we now do the discretization by denoting t tn nT and x xm mX as we did in section 3 we endup with the finite difference approximation [35]

ynm rynm ynm

(19)

rynm ynm

If we set r (19) becomes

ynm ynm ynm

ynm (20)

which is the finite difference equation of an ideal stringThe equality of equation (20) can be checked by substitut-ing the waveguide decomposition (equation 5) in the right-hand side of equation (20) [22]

Since the length of the string must again have integervalues correct tuning of the string becomes difficult It hasbeen shown [35] that choosing r in equation (17) re-sults in lowering the fundamental frequency of the stringTherefore the finite difference string can be tuned via theparameter r

Choosing r also gives raise to an unwanted nu-merical dispersion phenomenon called grid dispersion [7]where the wave velocity in the numerical implementationwill be less than the ideal physical wave velocity This ar-tificial dispersion affects primarily the upper harmonicswhere the frequencies will be underestimated If a typi-cal error of in the generated frequencies is allowedthe difference between the tuning coefficient r and unityshould not be greater than [35] If the constraints be-tween the correct tuning and grid dispersion do not yieldsatisfactory results the spatial density of the grid shouldbe increased This is known as spatial oversampling

52 Boundary conditions and string excitation

Since the spatial coordinate m of the string must liebetween and Lnom problems arise near the ends ofthe string when evaluating equation (20) because spatialpoints outside the string are needed The problem can

be solved by introducing boundary conditions that definehow to evaluate the string movement when m orm Lnom The simplest approach introduced alreadyin [6] would be to assume that the string terminations berigid so that yn yn Lnom This results in aphase-inverting termination which suits perfectly the caseof an ideal string For other types of string terminationseveral models have been introduced (see eg [6] [35]and [36]) Generally the nonrigid string terminations leadto frequency-dependent losses in the string model

For the FDM string excitation a useful method has beenproposed in [36] It is conceptually simple and allows forinteraction with the string during run-time There

ynm ynm

un (21)

and

ynm ynm

un (22)

are inserted into the string which causes a ldquoboxcarrdquo blockfunction to spread in both directions from the excitationpoint pair The wave component un is now used as theexcitation signal in a similar way as the exciting force sig-nal F n in section 32

53 Finite difference approximation of a lossy string

Frequency-independent losses can be modeled in an FDMstring by discretizing the velocity-dependent dampingterm in the lossy 1D wave equation (3) This results in twoadditional scaling coefficients in the recurrence equation[35]

ynm pynm ynm

qynm (23)

where the values of p and q determine the amount oflosses Generally p and q may depend on the spatial indexm but since homogeneous strings are considered here thisdependency is omitted Values

q p jpj (24)

ensure the stability of a linear finite difference string withfrequency-independent losses [37] Note that the sign dif-ference of p and q in [37] has already been taken care ofin equation (23) Modeling of frequency-dependent lossesby discretizing the lossy wave equation leads to an implicitrecurrence equation which can be evaluated if suitableapproximations are made [35]

6 Nonlinear finite difference string

Implementing tension modulation in a digital waveguidestring in section 3 was not an overly difficult task This wasdue to the fact that the implementation of a DWG string isessentially a feedback loop with delay and therefore mod-ulating the delay time of this loop corresponded to modu-lating the wave velocities In FDM strings however such

318

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

Figure 10 Illustration of the nonlinear FD algorithm on a spatio-temporal grid The vertical axis denotes the time while the hor-izontal axis denotes the spatial location on the string The illus-tration is shown only for a string segment of length N forclarity In each step of the algorithm most recently evaluated val-ues are presented as black dots while earlier values are presentedas white dots

an approach would not lead to satisfactory results sincethe physical quantities (eg displacement) themselves arepresent in the string model and not their wave decompo-sitions2

Instead we concluded that in order to correctly modelthe TM in a FDM string we first have to evaluate the recur-rence equation and use these three snapshots of the string(at time instants n n and n) in interpolating two newstring states at time instants n and n where Using these two string states we then eval-uate the recurrence equation in order to obtain the stringstate at time n It is important to note that the in-terpolation here is in effect stretching the time axis so thatthe wave propagation velocities are altered whereas in theDWG model the allpass filters perform the interpolation inthe spatial domain

This algorithm can also be seen as using two FDM sys-tems in implementing the nonlinear string The elongationof the string would be evaluated from one system and theresult the stretched string state would be updated to theother system Figure 10 illustrates this procedure on thespatio-temporal grid

In step 1 the two initial states have been assigned forthe string and the state at the next instant (in the linearcase) is obtained by the standard recurrence equation (20)The grid values which represent the state of the string atthe corresponding time instant are circled in step 1 In step2 sample values corresponding to the TM have been inter-polated from the string states in step 1 In step 3 equation(20) has been applied on the values evaluated in step 2 in

2 Such a system which deals with the physical quantities themselves iscalled a Kirchhoff model as opposed to a wave model which deals withthe wave components of the physical quantities

(c)

(b)

(a)

t

t

t t

t

t

n

n

n n

n

n

n+1

n+1

n+1 n+1

n+1

n+1y n+ m( 1 )

y n+ m( 1 )

y n m( )

y n m( )

y n+ m( 1 )

y n- m( 1 )

m

m

m m

m

m

d

d

d

a

a

a

-a

-a

-a

z-1

z-1

z-1

n-1

n-1

n-1 n-1

n-1

n-1

Figure 11 Illustration of the interpolation process due to thechange in the stringrsquos length The spatio-temporal grids on theleft and right represent the linear and interpolated string statesrespectively The fractional delay value caused by the interpola-tion is denoted by d The interpolation process in (a) is simplifiedin (b) and further in (c)

order to obtain the string state corresponding to the changein tension The two most recently obtained states are nowtaken as the ldquoinitial statesrdquo in step 4 and we can return tostep 1

As seen in Figure 10 the tension modulation corre-sponds here to interpolating the string state in the tempo-ral domain The elongation of the FDM string was evalu-ated similarly to what was done in equation (9) except thathere the slope of the string was obtained by taking the dif-ference of the displacements between two adjacent stringsegments rather than summing up the slope wave compo-nents In the following we will have a closer look at theinterpolation process

61 String state interpolation

We chose again to use first-order allpass filters in inter-polating the string state from the linear model (step 2 inFigure 10) Figure 11(a) illustrates how the interpolatedvalue of ynm is obtained from the linear values Thespatio-temporal grid on the left represents the string statein the linear case while the spatio-temporal grid on theright represents the string state after spatial interpolationThe structure between the two grids is the block diagramof a first-order allpass filter (equation 6) The coefficient afor the allpass filter was evaluated as presented earlier byequations (9)ndash(12)

319

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

In this figure we notice that the allpass filter uses thevalue of ynm delayed by one sample thus corre-sponding to ynm Clearly this can be obtained directlyfrom the grid on the left and the branch on the left contain-ing the unit delay can be reformed The result is shown inFigure 11(b) Here we also note that the interpolation sys-tem uses its own output at the previous time instant This isactually the same as using the value of ynm becauseit is the same as the output of the interpolation process onetime step ago (this might be best understood by noting thatthe bottom row of step 4 in Figure 10 is the same as thebottom row of step 1 at the next time instant) Thus Fig-ure 11(b) can be further simplified to Figure 11(c)

Having this said the recurrence equation for the time-varying finite difference string with frequency-indepen-dent damping can be written as

ynm pynm ynm

qynm (25)

where

ynm aynm ynm

aynm

ynm aynm ynm

aynm

ynm aynm ynm

ayn m

Here the coefficients p and q incorporate the frequency-independent losses and y and y refer to the linear and in-terpolated strings respectively Simplifying and rearrang-ing we end up with an equation containing only terms ofy and the subscript may therefore be omitted

ynm paynm paynm

pynm qaynm pynm

paynm qynm

paynm qayn m (26)

This equation is illustrated with a block diagram in Fig-ure 12 along with its abstraction A nonlinear FDM stringcan be constructed by connecting several of these blockstogether and using the string elongation in controlling theamount of interpolation We will refer to such a block asa time-varying finite difference time-domain (FDTD) ele-ment Illustration of the lossless time-varying FDTD ele-ment can be found in Figure 13 where p and q equal unityand have therefore been left out

62 String excitation and termination

For the interaction with the time-varying FDTD stringmodel we chose to use the ldquoboxcarrdquo excitation model dis-cussed in section 52 so that the excitation signal couldagain be interpreted as a force signal Figure 14 presentsan interaction block to be used with a time-varying FDTDstring We will call such a block the FDTD interaction el-

FDTD

-pa

z-1

z-1

z-1

y n+1m( )

y nm( )

y n-2m( )

y n-1m( )

-pa

pa

qa

-q

-qa

pa

p p

Figure 12 Illustration of the time-varying FDTD element to-gether with its abstraction A lossless time-varying FDTD ele-ment can be found in Figure 13

-a

z-1

z-1

z-1

y n-1m( )

y n-2m( )

y nm( )

y n+1m( )

-a

-a

a

a

a

Figure 13 Illustration of the lossless time-varying FDTD ele-ment

ement Using these DSP blocks we can construct a one-polarization nonlinear FDTD (NFDTD) string as illus-trated in Figure 15

We chose to use rigid terminations for our nonlin-ear finite difference string model since the modeling offrequency-dependent losses is not a key aspect of thisstudy Fixed terminations do not ruin the generation ofmissing harmonics in our model either since the TMDFcoupling is implemented in a different manner as ex-plained below

63 NFDTD string with generation of missing har-monics

In order to model the generation of missing harmonics ina NFDTD string we constructed a model where an addi-tional interaction element is placed between the last FDTDelement and the termination for feeding the TMDF to thestring Since the spatial distance between the last FDTD

320

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

y n-1m+1( )

y n-1m( )

F n( )

F n( )

y nm+1( )

y nm( )

y n+1m+1( )

y n+1m( )

Figure 14 Illustration of the FDTD interaction element togetherwith its abstraction The excitation algorithm is defined by equa-tions (21) and (22)

F n( )

FDTD FDTD FDTD FDTD FDTD

Allpass-coefficientapproximation

Elongationapproximation

Figure 15 One-polarizational NFDTD string The string consistsof the time-varying FDTD elements illustrated in Figure 12 Thezero-blocks at the terminations give zero as an output regardlessof the input values thus implying a rigid termination The excita-tion to the string can be inserted as a force signal using a FDTDinteraction element illustrated in Figure 14

F n( )

FDTD FDTD FDTD

TMDF

FDTD

Allpass-coefficientapproximation

Elongationapproximation

y n L( -1)nom

Figure 16 Illustration of the NFDTD string with a generationmechanism for missing harmonics A second interaction elementis added in order to feed the TMDF into the string The scal-ing coefficient TMDF controls the amplitude of the missing har-monics The string elongation is approximated from the displace-ments of each FDTD element

element and the rigid termination is one sample the ver-tical component of the TMDF can be seen to be equal tothe product of the displacement of the last FDTD elementand the tension Here we can replace the tension signalby the elongation signal and introduce a scaling coeffi-cient TMDF to control the amount of TMDF to be in-serted to the interaction element at the termination Thisis illustrated in Figure 16 The generation of missing har-

monics in a NFDTD model will be further discussed in thefollowing section

7 Simulation results

In this section we present the results obtained from the twononlinear string algorithms discussed in sections 3 and 5The synthesis results are compared by simulating the samephenomena namely the initial pitch glide and the genera-tion of missing harmonics using the two models Stabilityissues and computational cost of the synthesis models arealso discussed

71 Synthesis results

The synthesis results reveal that both the nonlinear DWGand NFDTD models are able to realistically model the ini-tial pitch glide phenomenon Figure 17 illustrates the fun-damental frequency behavior of a recorded kantele toneand the two synthesized tones Here the horizontal dottedline approximates the mean value of perceptual detectionthreshold of an initial pitch glide The psychoacoustic de-tection threshold in the frequency region of these tones isabout 54 Hz [38] This shows that the fundamental fre-quency glide is an audible phenomenon in plucked stringinstruments such as the kantele even at modest pluckingamplitudes and thus it must be included in a synthesismodel if realistic tones are desired

The nonlinear DWG model used in this figure has a totaldelay line length of 55125 samples and the allpass coef-ficient a is scaled using a constant value of 09 in orderto correctly simulate the behavior of the recorded sampleThe NFDTD string consists of 56 FDTD elements andthe fine-tuning parameter (aka Courant number equa-tion 17) has a value of r 13 The allpass coefficienta is scaled using a coefficient in the NFDTD case

The modeling of the generation of missing harmonicscan be implemented similarly in the distributed nonlinearDWG model as was suggested in [28] If the boxcar inte-gration of equation (10) is replaced with a leaky integratorhaving the transfer function

Iz gp ap

apz (27)

the generation of the missing harmonics can be controlledvia the integration parameter ap The variable gp definesthe gain of the integration

Figure 18 shows the amplitude envelopes of the firstthree harmonics of a synthesized tone with two differ-ent ap parameter values The string was plucked close tord of its length and as can be seen in the figure themissing harmonic in (a) has a gradual increase after thebeginning transient after which it experiences an expo-nential decay like all other modes

It is worthwhile to note that the generation of missingharmonics in the nonlinear DWG model results from theproperties of the integration of the elongation approxima-tion and is therefore not a physically justified process Ba-sically here the integration error in the leaky integrator is

321

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Time [s]

fnom

Fre

quen

cy[H

z]

Figure 17 Fundamental frequencies as a function of time fora moderately-plucked recorded kantele tone (solid line) a syn-thesized nonlinear DWG tone (dashed line) and a synthesizedNFDTD tone (dash-dotted line) The fnom stands for the nomi-nal fundamental frequency of the string and the horizontal dottedline denotes the approximated detection threshold of a pitch drift(fnomHz) which suggests that the fundamental frequencydrifts in all cases are audible

responsible for feeding energy to the missing harmonicsAlso unlike the real physical phenomenon the generationof missing harmonics in the nonlinear DWG case does notdepend on the rigidity of the terminations Neverthelessthis feature can be exploited in emulating the real stringbehavior when the integration parameters are properly ad-justed Details on tuning the leaky integrator parameterscan be found in [28]

Modeling the generation of missing harmonics in aNFDTD string is however not so simple Even if a leakyintegrator is used in the elongation calculation its param-eters do not have a desirable effect on the missing har-monics This does not seem too surprising when consider-ing the major differences of these two algorithms and it isthe reason that forced us to use an alternative mechanismfor creating the missing harmonics in the previous section(Figure 16)

Figure 19 represents the behavior of the first three har-monics of a tone synthesized by this model It can be seenthat the missing harmonics can be ldquoliftedrdquo by choosing aproper value for TMDF The stability of the system how-ever poses an upper limit for the TMDF coefficient sincethe TMDF mechanism continuously feeds energy to thestring According to our experience generating missingharmonics with amplitudes greater than what is shown inFigure 19 is difficult

72 Stability issues and computational comparison

We found the nonlinear DWG algorithm to remain sta-ble for nearly all parameter and excitation values Onlyhighly exaggerated nonlinearity scaling values togetherwith high excitation impulses resulted in stability prob-lems We thus conclude that the nonlinear DWG waveg-uide has no real stability problems when synthesis of nat-ural plucked-instrument sounds are desired

We studied the stability of the NFDTD algorithm us-ing the Von Neumann analysis [39] in the time-invariantcase ie parameter a of equation (26) was kept constantThe basic idea of this method is to calculate the spatialFourier spectrum of the system under discussion at twoconsecutive time steps An amplification function which

(a)

(b)

Figure 18 Generation of the missing harmonics in the nonlinearDWG model can be controlled via the leaky integrator parame-ters Here the string was plucked approximately at rd of itslength so every 3rd harmonic should be missing from the re-sulting spectrum In a) ap and the third harmonicclearly rises after the initial transient In b) ap 13 andthe third harmonic is more attenuated

(a)

(b)

Figure 19 Generation of missing harmonics in a NFDTD stringThe string was plucked again approximately at rd of itslength and the coupling of the TMDF to the transversal vibra-tion was controlled using a scaling coefficient TMDF In a) thescaling coefficient has a value of TMDF and the missingthird harmonic can be seen rising after the initial transient In b)TMDF and generation of missing harmonics does not takeplace

shows how the spatial spectrum evolves with time canthen be derived from the two spectra If the absolute valueof this amplification function remains below unity stabil-ity is guaranteed Formally the Von Neumann analysis forthe NFDTD algorithm goes as follows [4]

If the spatial inverse Fourier transform is defined as

ynm FfY n g neim (28)

322

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

formed when constructing models using these elementsFigure 4(b) shows how to obtain output data from a junc-tion between two basic elements A time-varying DWGstructure consisting of these elements is illustrated in Fig-ure 5

From the discussion in section 1 we can conclude that asuitable control signal for the FD elements can be derivedfrom the instantaneous elongation of the string In the fol-lowing since the longitudinal wave propagation velocityis considerably higher than the transversal wave velocitywe will assume that the longitudinal waves will propa-gate instantaneously through the string and the elongationcalculation and the FD parameter tuning can be done forthe whole string in one piece In practice the longitudinalwave velocity is typically only 5-20 times higher than thetransversal one but carrying out the FD parameter evalua-tions for multiple string segments would add a significantcomputational load likely without any audible advancesThe elongation of the string can be expressed as [19]

ldevt

Z lnom

q

yxt x

dx lnom (7)

where lnom is the nominal string length x is the spatial co-ordinate along the string and y is the displacement of thestring The first spatial derivative yx suggests the use ofslope waves in the elongation calculation and thus equa-tion (7) can be approximated for the digital waveguide as[28]

Ldevn

LnomXm

p srnm slnm Lnom (8)

where srnm and slnm are the slope waves at timeinstant n and position m propagating to the right and tothe left respectively Lnom is the rounded nominal stringlength To reduce the computational complexity equation(8) can be further simplified using a truncated Taylor seriesexpansion to [28]

Ldevn

LnomXm

srnm slnm

(9)

while still maintaining a sufficient accuracy The approx-imated delay variation of the total DWG can be obtainedfrom equation (9) as [28]

Ddevn

nXlnLnom

EA

K

Ldevl

Lnom (10)

where E is Youngrsquos modulus A is the cross-sectional areaof the string and K is the nominal tension correspond-ing to the string at rest The length of the string in samplesis denoted as Lnom lnomfscnom where fs is the tem-poral sampling frequency and cnom is the nominal wavepropagation speed

T

z-1

Figure 7 The integrator block is implemented by summing upconsecutive samples

Figure 8 Illustration of the kantele The string termination at var-ras is magnified for clarity A denotes the termination point forvertical vibration of the string while B denotes the terminationpoint for horizontal vibrationl stands for the distance betweenA and B

Since the system under consideration uses a distributedset of delay elements the desired delay for each basic ele-ment is

dpartial Ddev

Lnom

(11)

The coefficient a in equation (6) can now be expressed as[26 24]

a dpartial

dpartial (12)

where dpartial is the delay intended for a single allpass fil-ter Note that previous studies have used a different signfor a in equations (6) and (12) although the operation ofthe allpass filter remains the same

4 Synthesis model of the kantele usingnonlinear digital waveguides

In this section we demonstrate the nonlinear DWG for-mulation by constructing a two-polarizational synthesismodel of the kantele a Finnish folk music instrument

41 Acoustical analysis of the kantele

The kantele is a bridgeless plucked string instrument withusually five metal strings in its basic form (see Figure 8)The strings are terminated at one end by metal tuning pins

316

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

f nin( ) f nout( )

f ny( )

f nx( )

f nzy( )

f nz( )v nz( )

v ny( )

ZC

Zybridge

Zzbridge

C

Figure 9 A kantele string model The one-polarization stringmodel blocks are identical to what is illustrated in Figure 5 buthave different string lengths It is important to note that the cou-pling between the vibrational polarizations in a real physical sys-tem is more complicated but a simplified one-way coupling isused here for ease of simulation

which are screwed directly into the soundboard At the op-posite end all strings are wound once around a horizontalmetal bar called the varras and then knotted Because ofthe nonzero distance between the center of the varras andthe knot the vibrations in two polarization planes havedifferent effective lengths the varras is the terminationpoint for horizontal vibration while the knot acts as thetermination for the vertical vibration as illustrated in Fig-ure 8 This phenomenon causes the total vibration of thestring to have two fundamental components with slightlydifferent frequencies producing beating [29] A more de-tailed structure and acoustical analysis of the kantele canbe found in [29] A study of the history of kantele and anacoustically improved new design are presented in [31]

42 A novel kantele string model

The novel synthesis model of a single kantele string is con-structed using two single-polarization time-varying DWGmodels illustrated in Figure 5 and connecting them to-gether via a scaling coefficient for modeling the couplingbetween the two polarizations We restrict the couplingto being one-directional in order to avoid stability prob-lems which would otherwise rise due to the feedback loopformed from the interconnected strings as suggested byKarjalainen et al [32] Clearly the actual physical cou-pling is two-directional The elongation approximations ofthe strings and the resulting allpass filter coefficient valuesare evaluated separately for the two DWG models usingthe arithmetics described in section 32 The structure ofthe novel kantele string model is illustrated in Figure 9 Inthis model vyn and vzn represent the velocity signalscoming from the strings vibrating vertically and horizon-tally respectively

It is important to note that while vyn and vzn canbe obtained anywhere along the string in this case theyare evaluated at the termination points so that terminalimpedances can be used Zybridge and Zzbridge representthe vertical and horizontal terminal impedances respec-tively Zc stands for the coupling impedance from verti-cal to horizontal string vibration polarization and fyznrepresents the corresponding driving force The forces to

the termination caused by the two one-polarizational vi-brations are denoted by fyn and fzn The connec-tion from the elongation approximation block to the outputsimulates the direct coupling of the TM to the instrumentbody [20] A scaling coefficient denoted by C is usedto control the amount of this coupling The output of thewhole two-polarization string model is finally presentedas a force signal foutn excerted to the string terminationpoint It must also be noted that this model simulates onlya single kantele string and a model of the instrument bodymust also be added if realistic sound synthesis is desired

A real-time sound synthesis model of a kantele is con-structed using a block-based efficient audio-DSP-tool theBlockCompiler The algorithm used is efficient enough toprocess a five-string kantele model on an ordinary laptopcomputer at a 441 kHz sampling rate A detailed descrip-tion of the BlockCompiler is presented by Karjalainen[33]

5 Finite difference approach

In the previous section we discussed string synthesis viadiscretizing the drsquoAlembertrsquos solution to the 1-D waveequation Another approach is to discretize the wave equa-tion itself for example by substituting finite differenceterms for the derivatives in the wave equation (equation(1) in the case of an ideal string) This mode of opera-tion is commonly known as the finite difference method(FDM) and it was first used for sound synthesis purposesby Hiller and Ruiz in the early seventies [6] and [34] Fi-nite differences had already earlier been used in mathemat-ics for numerical solving of partial differential equationsA fine introduction to FDM in the synthesis of pluckedstring instruments can be found in [35] Below we followthe guidelines provided in [22] in deriving the FDM recur-rence equation

51 Ideal finite difference string

The partial derivatives in the 1-D wave equation (1) can bereplaced by finite differences 1

ytt x yt x ytT x

T(13)

and

yxt x yt x yt xX

X (14)

Using the finite difference approximation for the second-order derivatives in the wave equation (1) leads to

Kyt xX yt x yt xX

X

ytT x yt x ytT x

T (15)

1 It is important to note that the finite difference scheme used in equa-tions (13) and (14) was only chosen for simplicity and other schemescould be used as well For a discussion of using an implicit finite-difference scheme see section 8

317

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Solving (15) we get

ytT x KT

X

yt xX yt x yt xX

yt x ytT x (16)

Next we define the relationship between the spatial andtemporal sampling steps with [35]

r cT

X (17)

where the ldquoless than unityrdquo-restriction is called the VonNeumann stability condition Using this together with thedefinition of transversal wave velocity (equation 2) equa-tion (16) can be written as

ytT x ryt xX yt xX

(18)

ryt x ytT x

If we now do the discretization by denoting t tn nT and x xm mX as we did in section 3 we endup with the finite difference approximation [35]

ynm rynm ynm

(19)

rynm ynm

If we set r (19) becomes

ynm ynm ynm

ynm (20)

which is the finite difference equation of an ideal stringThe equality of equation (20) can be checked by substitut-ing the waveguide decomposition (equation 5) in the right-hand side of equation (20) [22]

Since the length of the string must again have integervalues correct tuning of the string becomes difficult It hasbeen shown [35] that choosing r in equation (17) re-sults in lowering the fundamental frequency of the stringTherefore the finite difference string can be tuned via theparameter r

Choosing r also gives raise to an unwanted nu-merical dispersion phenomenon called grid dispersion [7]where the wave velocity in the numerical implementationwill be less than the ideal physical wave velocity This ar-tificial dispersion affects primarily the upper harmonicswhere the frequencies will be underestimated If a typi-cal error of in the generated frequencies is allowedthe difference between the tuning coefficient r and unityshould not be greater than [35] If the constraints be-tween the correct tuning and grid dispersion do not yieldsatisfactory results the spatial density of the grid shouldbe increased This is known as spatial oversampling

52 Boundary conditions and string excitation

Since the spatial coordinate m of the string must liebetween and Lnom problems arise near the ends ofthe string when evaluating equation (20) because spatialpoints outside the string are needed The problem can

be solved by introducing boundary conditions that definehow to evaluate the string movement when m orm Lnom The simplest approach introduced alreadyin [6] would be to assume that the string terminations berigid so that yn yn Lnom This results in aphase-inverting termination which suits perfectly the caseof an ideal string For other types of string terminationseveral models have been introduced (see eg [6] [35]and [36]) Generally the nonrigid string terminations leadto frequency-dependent losses in the string model

For the FDM string excitation a useful method has beenproposed in [36] It is conceptually simple and allows forinteraction with the string during run-time There

ynm ynm

un (21)

and

ynm ynm

un (22)

are inserted into the string which causes a ldquoboxcarrdquo blockfunction to spread in both directions from the excitationpoint pair The wave component un is now used as theexcitation signal in a similar way as the exciting force sig-nal F n in section 32

53 Finite difference approximation of a lossy string

Frequency-independent losses can be modeled in an FDMstring by discretizing the velocity-dependent dampingterm in the lossy 1D wave equation (3) This results in twoadditional scaling coefficients in the recurrence equation[35]

ynm pynm ynm

qynm (23)

where the values of p and q determine the amount oflosses Generally p and q may depend on the spatial indexm but since homogeneous strings are considered here thisdependency is omitted Values

q p jpj (24)

ensure the stability of a linear finite difference string withfrequency-independent losses [37] Note that the sign dif-ference of p and q in [37] has already been taken care ofin equation (23) Modeling of frequency-dependent lossesby discretizing the lossy wave equation leads to an implicitrecurrence equation which can be evaluated if suitableapproximations are made [35]

6 Nonlinear finite difference string

Implementing tension modulation in a digital waveguidestring in section 3 was not an overly difficult task This wasdue to the fact that the implementation of a DWG string isessentially a feedback loop with delay and therefore mod-ulating the delay time of this loop corresponded to modu-lating the wave velocities In FDM strings however such

318

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

Figure 10 Illustration of the nonlinear FD algorithm on a spatio-temporal grid The vertical axis denotes the time while the hor-izontal axis denotes the spatial location on the string The illus-tration is shown only for a string segment of length N forclarity In each step of the algorithm most recently evaluated val-ues are presented as black dots while earlier values are presentedas white dots

an approach would not lead to satisfactory results sincethe physical quantities (eg displacement) themselves arepresent in the string model and not their wave decompo-sitions2

Instead we concluded that in order to correctly modelthe TM in a FDM string we first have to evaluate the recur-rence equation and use these three snapshots of the string(at time instants n n and n) in interpolating two newstring states at time instants n and n where Using these two string states we then eval-uate the recurrence equation in order to obtain the stringstate at time n It is important to note that the in-terpolation here is in effect stretching the time axis so thatthe wave propagation velocities are altered whereas in theDWG model the allpass filters perform the interpolation inthe spatial domain

This algorithm can also be seen as using two FDM sys-tems in implementing the nonlinear string The elongationof the string would be evaluated from one system and theresult the stretched string state would be updated to theother system Figure 10 illustrates this procedure on thespatio-temporal grid

In step 1 the two initial states have been assigned forthe string and the state at the next instant (in the linearcase) is obtained by the standard recurrence equation (20)The grid values which represent the state of the string atthe corresponding time instant are circled in step 1 In step2 sample values corresponding to the TM have been inter-polated from the string states in step 1 In step 3 equation(20) has been applied on the values evaluated in step 2 in

2 Such a system which deals with the physical quantities themselves iscalled a Kirchhoff model as opposed to a wave model which deals withthe wave components of the physical quantities

(c)

(b)

(a)

t

t

t t

t

t

n

n

n n

n

n

n+1

n+1

n+1 n+1

n+1

n+1y n+ m( 1 )

y n+ m( 1 )

y n m( )

y n m( )

y n+ m( 1 )

y n- m( 1 )

m

m

m m

m

m

d

d

d

a

a

a

-a

-a

-a

z-1

z-1

z-1

n-1

n-1

n-1 n-1

n-1

n-1

Figure 11 Illustration of the interpolation process due to thechange in the stringrsquos length The spatio-temporal grids on theleft and right represent the linear and interpolated string statesrespectively The fractional delay value caused by the interpola-tion is denoted by d The interpolation process in (a) is simplifiedin (b) and further in (c)

order to obtain the string state corresponding to the changein tension The two most recently obtained states are nowtaken as the ldquoinitial statesrdquo in step 4 and we can return tostep 1

As seen in Figure 10 the tension modulation corre-sponds here to interpolating the string state in the tempo-ral domain The elongation of the FDM string was evalu-ated similarly to what was done in equation (9) except thathere the slope of the string was obtained by taking the dif-ference of the displacements between two adjacent stringsegments rather than summing up the slope wave compo-nents In the following we will have a closer look at theinterpolation process

61 String state interpolation

We chose again to use first-order allpass filters in inter-polating the string state from the linear model (step 2 inFigure 10) Figure 11(a) illustrates how the interpolatedvalue of ynm is obtained from the linear values Thespatio-temporal grid on the left represents the string statein the linear case while the spatio-temporal grid on theright represents the string state after spatial interpolationThe structure between the two grids is the block diagramof a first-order allpass filter (equation 6) The coefficient afor the allpass filter was evaluated as presented earlier byequations (9)ndash(12)

319

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

In this figure we notice that the allpass filter uses thevalue of ynm delayed by one sample thus corre-sponding to ynm Clearly this can be obtained directlyfrom the grid on the left and the branch on the left contain-ing the unit delay can be reformed The result is shown inFigure 11(b) Here we also note that the interpolation sys-tem uses its own output at the previous time instant This isactually the same as using the value of ynm becauseit is the same as the output of the interpolation process onetime step ago (this might be best understood by noting thatthe bottom row of step 4 in Figure 10 is the same as thebottom row of step 1 at the next time instant) Thus Fig-ure 11(b) can be further simplified to Figure 11(c)

Having this said the recurrence equation for the time-varying finite difference string with frequency-indepen-dent damping can be written as

ynm pynm ynm

qynm (25)

where

ynm aynm ynm

aynm

ynm aynm ynm

aynm

ynm aynm ynm

ayn m

Here the coefficients p and q incorporate the frequency-independent losses and y and y refer to the linear and in-terpolated strings respectively Simplifying and rearrang-ing we end up with an equation containing only terms ofy and the subscript may therefore be omitted

ynm paynm paynm

pynm qaynm pynm

paynm qynm

paynm qayn m (26)

This equation is illustrated with a block diagram in Fig-ure 12 along with its abstraction A nonlinear FDM stringcan be constructed by connecting several of these blockstogether and using the string elongation in controlling theamount of interpolation We will refer to such a block asa time-varying finite difference time-domain (FDTD) ele-ment Illustration of the lossless time-varying FDTD ele-ment can be found in Figure 13 where p and q equal unityand have therefore been left out

62 String excitation and termination

For the interaction with the time-varying FDTD stringmodel we chose to use the ldquoboxcarrdquo excitation model dis-cussed in section 52 so that the excitation signal couldagain be interpreted as a force signal Figure 14 presentsan interaction block to be used with a time-varying FDTDstring We will call such a block the FDTD interaction el-

FDTD

-pa

z-1

z-1

z-1

y n+1m( )

y nm( )

y n-2m( )

y n-1m( )

-pa

pa

qa

-q

-qa

pa

p p

Figure 12 Illustration of the time-varying FDTD element to-gether with its abstraction A lossless time-varying FDTD ele-ment can be found in Figure 13

-a

z-1

z-1

z-1

y n-1m( )

y n-2m( )

y nm( )

y n+1m( )

-a

-a

a

a

a

Figure 13 Illustration of the lossless time-varying FDTD ele-ment

ement Using these DSP blocks we can construct a one-polarization nonlinear FDTD (NFDTD) string as illus-trated in Figure 15

We chose to use rigid terminations for our nonlin-ear finite difference string model since the modeling offrequency-dependent losses is not a key aspect of thisstudy Fixed terminations do not ruin the generation ofmissing harmonics in our model either since the TMDFcoupling is implemented in a different manner as ex-plained below

63 NFDTD string with generation of missing har-monics

In order to model the generation of missing harmonics ina NFDTD string we constructed a model where an addi-tional interaction element is placed between the last FDTDelement and the termination for feeding the TMDF to thestring Since the spatial distance between the last FDTD

320

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

y n-1m+1( )

y n-1m( )

F n( )

F n( )

y nm+1( )

y nm( )

y n+1m+1( )

y n+1m( )

Figure 14 Illustration of the FDTD interaction element togetherwith its abstraction The excitation algorithm is defined by equa-tions (21) and (22)

F n( )

FDTD FDTD FDTD FDTD FDTD

Allpass-coefficientapproximation

Elongationapproximation

Figure 15 One-polarizational NFDTD string The string consistsof the time-varying FDTD elements illustrated in Figure 12 Thezero-blocks at the terminations give zero as an output regardlessof the input values thus implying a rigid termination The excita-tion to the string can be inserted as a force signal using a FDTDinteraction element illustrated in Figure 14

F n( )

FDTD FDTD FDTD

TMDF

FDTD

Allpass-coefficientapproximation

Elongationapproximation

y n L( -1)nom

Figure 16 Illustration of the NFDTD string with a generationmechanism for missing harmonics A second interaction elementis added in order to feed the TMDF into the string The scal-ing coefficient TMDF controls the amplitude of the missing har-monics The string elongation is approximated from the displace-ments of each FDTD element

element and the rigid termination is one sample the ver-tical component of the TMDF can be seen to be equal tothe product of the displacement of the last FDTD elementand the tension Here we can replace the tension signalby the elongation signal and introduce a scaling coeffi-cient TMDF to control the amount of TMDF to be in-serted to the interaction element at the termination Thisis illustrated in Figure 16 The generation of missing har-

monics in a NFDTD model will be further discussed in thefollowing section

7 Simulation results

In this section we present the results obtained from the twononlinear string algorithms discussed in sections 3 and 5The synthesis results are compared by simulating the samephenomena namely the initial pitch glide and the genera-tion of missing harmonics using the two models Stabilityissues and computational cost of the synthesis models arealso discussed

71 Synthesis results

The synthesis results reveal that both the nonlinear DWGand NFDTD models are able to realistically model the ini-tial pitch glide phenomenon Figure 17 illustrates the fun-damental frequency behavior of a recorded kantele toneand the two synthesized tones Here the horizontal dottedline approximates the mean value of perceptual detectionthreshold of an initial pitch glide The psychoacoustic de-tection threshold in the frequency region of these tones isabout 54 Hz [38] This shows that the fundamental fre-quency glide is an audible phenomenon in plucked stringinstruments such as the kantele even at modest pluckingamplitudes and thus it must be included in a synthesismodel if realistic tones are desired

The nonlinear DWG model used in this figure has a totaldelay line length of 55125 samples and the allpass coef-ficient a is scaled using a constant value of 09 in orderto correctly simulate the behavior of the recorded sampleThe NFDTD string consists of 56 FDTD elements andthe fine-tuning parameter (aka Courant number equa-tion 17) has a value of r 13 The allpass coefficienta is scaled using a coefficient in the NFDTD case

The modeling of the generation of missing harmonicscan be implemented similarly in the distributed nonlinearDWG model as was suggested in [28] If the boxcar inte-gration of equation (10) is replaced with a leaky integratorhaving the transfer function

Iz gp ap

apz (27)

the generation of the missing harmonics can be controlledvia the integration parameter ap The variable gp definesthe gain of the integration

Figure 18 shows the amplitude envelopes of the firstthree harmonics of a synthesized tone with two differ-ent ap parameter values The string was plucked close tord of its length and as can be seen in the figure themissing harmonic in (a) has a gradual increase after thebeginning transient after which it experiences an expo-nential decay like all other modes

It is worthwhile to note that the generation of missingharmonics in the nonlinear DWG model results from theproperties of the integration of the elongation approxima-tion and is therefore not a physically justified process Ba-sically here the integration error in the leaky integrator is

321

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Time [s]

fnom

Fre

quen

cy[H

z]

Figure 17 Fundamental frequencies as a function of time fora moderately-plucked recorded kantele tone (solid line) a syn-thesized nonlinear DWG tone (dashed line) and a synthesizedNFDTD tone (dash-dotted line) The fnom stands for the nomi-nal fundamental frequency of the string and the horizontal dottedline denotes the approximated detection threshold of a pitch drift(fnomHz) which suggests that the fundamental frequencydrifts in all cases are audible

responsible for feeding energy to the missing harmonicsAlso unlike the real physical phenomenon the generationof missing harmonics in the nonlinear DWG case does notdepend on the rigidity of the terminations Neverthelessthis feature can be exploited in emulating the real stringbehavior when the integration parameters are properly ad-justed Details on tuning the leaky integrator parameterscan be found in [28]

Modeling the generation of missing harmonics in aNFDTD string is however not so simple Even if a leakyintegrator is used in the elongation calculation its param-eters do not have a desirable effect on the missing har-monics This does not seem too surprising when consider-ing the major differences of these two algorithms and it isthe reason that forced us to use an alternative mechanismfor creating the missing harmonics in the previous section(Figure 16)

Figure 19 represents the behavior of the first three har-monics of a tone synthesized by this model It can be seenthat the missing harmonics can be ldquoliftedrdquo by choosing aproper value for TMDF The stability of the system how-ever poses an upper limit for the TMDF coefficient sincethe TMDF mechanism continuously feeds energy to thestring According to our experience generating missingharmonics with amplitudes greater than what is shown inFigure 19 is difficult

72 Stability issues and computational comparison

We found the nonlinear DWG algorithm to remain sta-ble for nearly all parameter and excitation values Onlyhighly exaggerated nonlinearity scaling values togetherwith high excitation impulses resulted in stability prob-lems We thus conclude that the nonlinear DWG waveg-uide has no real stability problems when synthesis of nat-ural plucked-instrument sounds are desired

We studied the stability of the NFDTD algorithm us-ing the Von Neumann analysis [39] in the time-invariantcase ie parameter a of equation (26) was kept constantThe basic idea of this method is to calculate the spatialFourier spectrum of the system under discussion at twoconsecutive time steps An amplification function which

(a)

(b)

Figure 18 Generation of the missing harmonics in the nonlinearDWG model can be controlled via the leaky integrator parame-ters Here the string was plucked approximately at rd of itslength so every 3rd harmonic should be missing from the re-sulting spectrum In a) ap and the third harmonicclearly rises after the initial transient In b) ap 13 andthe third harmonic is more attenuated

(a)

(b)

Figure 19 Generation of missing harmonics in a NFDTD stringThe string was plucked again approximately at rd of itslength and the coupling of the TMDF to the transversal vibra-tion was controlled using a scaling coefficient TMDF In a) thescaling coefficient has a value of TMDF and the missingthird harmonic can be seen rising after the initial transient In b)TMDF and generation of missing harmonics does not takeplace

shows how the spatial spectrum evolves with time canthen be derived from the two spectra If the absolute valueof this amplification function remains below unity stabil-ity is guaranteed Formally the Von Neumann analysis forthe NFDTD algorithm goes as follows [4]

If the spatial inverse Fourier transform is defined as

ynm FfY n g neim (28)

322

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

f nin( ) f nout( )

f ny( )

f nx( )

f nzy( )

f nz( )v nz( )

v ny( )

ZC

Zybridge

Zzbridge

C

Figure 9 A kantele string model The one-polarization stringmodel blocks are identical to what is illustrated in Figure 5 buthave different string lengths It is important to note that the cou-pling between the vibrational polarizations in a real physical sys-tem is more complicated but a simplified one-way coupling isused here for ease of simulation

which are screwed directly into the soundboard At the op-posite end all strings are wound once around a horizontalmetal bar called the varras and then knotted Because ofthe nonzero distance between the center of the varras andthe knot the vibrations in two polarization planes havedifferent effective lengths the varras is the terminationpoint for horizontal vibration while the knot acts as thetermination for the vertical vibration as illustrated in Fig-ure 8 This phenomenon causes the total vibration of thestring to have two fundamental components with slightlydifferent frequencies producing beating [29] A more de-tailed structure and acoustical analysis of the kantele canbe found in [29] A study of the history of kantele and anacoustically improved new design are presented in [31]

42 A novel kantele string model

The novel synthesis model of a single kantele string is con-structed using two single-polarization time-varying DWGmodels illustrated in Figure 5 and connecting them to-gether via a scaling coefficient for modeling the couplingbetween the two polarizations We restrict the couplingto being one-directional in order to avoid stability prob-lems which would otherwise rise due to the feedback loopformed from the interconnected strings as suggested byKarjalainen et al [32] Clearly the actual physical cou-pling is two-directional The elongation approximations ofthe strings and the resulting allpass filter coefficient valuesare evaluated separately for the two DWG models usingthe arithmetics described in section 32 The structure ofthe novel kantele string model is illustrated in Figure 9 Inthis model vyn and vzn represent the velocity signalscoming from the strings vibrating vertically and horizon-tally respectively

It is important to note that while vyn and vzn canbe obtained anywhere along the string in this case theyare evaluated at the termination points so that terminalimpedances can be used Zybridge and Zzbridge representthe vertical and horizontal terminal impedances respec-tively Zc stands for the coupling impedance from verti-cal to horizontal string vibration polarization and fyznrepresents the corresponding driving force The forces to

the termination caused by the two one-polarizational vi-brations are denoted by fyn and fzn The connec-tion from the elongation approximation block to the outputsimulates the direct coupling of the TM to the instrumentbody [20] A scaling coefficient denoted by C is usedto control the amount of this coupling The output of thewhole two-polarization string model is finally presentedas a force signal foutn excerted to the string terminationpoint It must also be noted that this model simulates onlya single kantele string and a model of the instrument bodymust also be added if realistic sound synthesis is desired

A real-time sound synthesis model of a kantele is con-structed using a block-based efficient audio-DSP-tool theBlockCompiler The algorithm used is efficient enough toprocess a five-string kantele model on an ordinary laptopcomputer at a 441 kHz sampling rate A detailed descrip-tion of the BlockCompiler is presented by Karjalainen[33]

5 Finite difference approach

In the previous section we discussed string synthesis viadiscretizing the drsquoAlembertrsquos solution to the 1-D waveequation Another approach is to discretize the wave equa-tion itself for example by substituting finite differenceterms for the derivatives in the wave equation (equation(1) in the case of an ideal string) This mode of opera-tion is commonly known as the finite difference method(FDM) and it was first used for sound synthesis purposesby Hiller and Ruiz in the early seventies [6] and [34] Fi-nite differences had already earlier been used in mathemat-ics for numerical solving of partial differential equationsA fine introduction to FDM in the synthesis of pluckedstring instruments can be found in [35] Below we followthe guidelines provided in [22] in deriving the FDM recur-rence equation

51 Ideal finite difference string

The partial derivatives in the 1-D wave equation (1) can bereplaced by finite differences 1

ytt x yt x ytT x

T(13)

and

yxt x yt x yt xX

X (14)

Using the finite difference approximation for the second-order derivatives in the wave equation (1) leads to

Kyt xX yt x yt xX

X

ytT x yt x ytT x

T (15)

1 It is important to note that the finite difference scheme used in equa-tions (13) and (14) was only chosen for simplicity and other schemescould be used as well For a discussion of using an implicit finite-difference scheme see section 8

317

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Solving (15) we get

ytT x KT

X

yt xX yt x yt xX

yt x ytT x (16)

Next we define the relationship between the spatial andtemporal sampling steps with [35]

r cT

X (17)

where the ldquoless than unityrdquo-restriction is called the VonNeumann stability condition Using this together with thedefinition of transversal wave velocity (equation 2) equa-tion (16) can be written as

ytT x ryt xX yt xX

(18)

ryt x ytT x

If we now do the discretization by denoting t tn nT and x xm mX as we did in section 3 we endup with the finite difference approximation [35]

ynm rynm ynm

(19)

rynm ynm

If we set r (19) becomes

ynm ynm ynm

ynm (20)

which is the finite difference equation of an ideal stringThe equality of equation (20) can be checked by substitut-ing the waveguide decomposition (equation 5) in the right-hand side of equation (20) [22]

Since the length of the string must again have integervalues correct tuning of the string becomes difficult It hasbeen shown [35] that choosing r in equation (17) re-sults in lowering the fundamental frequency of the stringTherefore the finite difference string can be tuned via theparameter r

Choosing r also gives raise to an unwanted nu-merical dispersion phenomenon called grid dispersion [7]where the wave velocity in the numerical implementationwill be less than the ideal physical wave velocity This ar-tificial dispersion affects primarily the upper harmonicswhere the frequencies will be underestimated If a typi-cal error of in the generated frequencies is allowedthe difference between the tuning coefficient r and unityshould not be greater than [35] If the constraints be-tween the correct tuning and grid dispersion do not yieldsatisfactory results the spatial density of the grid shouldbe increased This is known as spatial oversampling

52 Boundary conditions and string excitation

Since the spatial coordinate m of the string must liebetween and Lnom problems arise near the ends ofthe string when evaluating equation (20) because spatialpoints outside the string are needed The problem can

be solved by introducing boundary conditions that definehow to evaluate the string movement when m orm Lnom The simplest approach introduced alreadyin [6] would be to assume that the string terminations berigid so that yn yn Lnom This results in aphase-inverting termination which suits perfectly the caseof an ideal string For other types of string terminationseveral models have been introduced (see eg [6] [35]and [36]) Generally the nonrigid string terminations leadto frequency-dependent losses in the string model

For the FDM string excitation a useful method has beenproposed in [36] It is conceptually simple and allows forinteraction with the string during run-time There

ynm ynm

un (21)

and

ynm ynm

un (22)

are inserted into the string which causes a ldquoboxcarrdquo blockfunction to spread in both directions from the excitationpoint pair The wave component un is now used as theexcitation signal in a similar way as the exciting force sig-nal F n in section 32

53 Finite difference approximation of a lossy string

Frequency-independent losses can be modeled in an FDMstring by discretizing the velocity-dependent dampingterm in the lossy 1D wave equation (3) This results in twoadditional scaling coefficients in the recurrence equation[35]

ynm pynm ynm

qynm (23)

where the values of p and q determine the amount oflosses Generally p and q may depend on the spatial indexm but since homogeneous strings are considered here thisdependency is omitted Values

q p jpj (24)

ensure the stability of a linear finite difference string withfrequency-independent losses [37] Note that the sign dif-ference of p and q in [37] has already been taken care ofin equation (23) Modeling of frequency-dependent lossesby discretizing the lossy wave equation leads to an implicitrecurrence equation which can be evaluated if suitableapproximations are made [35]

6 Nonlinear finite difference string

Implementing tension modulation in a digital waveguidestring in section 3 was not an overly difficult task This wasdue to the fact that the implementation of a DWG string isessentially a feedback loop with delay and therefore mod-ulating the delay time of this loop corresponded to modu-lating the wave velocities In FDM strings however such

318

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

Figure 10 Illustration of the nonlinear FD algorithm on a spatio-temporal grid The vertical axis denotes the time while the hor-izontal axis denotes the spatial location on the string The illus-tration is shown only for a string segment of length N forclarity In each step of the algorithm most recently evaluated val-ues are presented as black dots while earlier values are presentedas white dots

an approach would not lead to satisfactory results sincethe physical quantities (eg displacement) themselves arepresent in the string model and not their wave decompo-sitions2

Instead we concluded that in order to correctly modelthe TM in a FDM string we first have to evaluate the recur-rence equation and use these three snapshots of the string(at time instants n n and n) in interpolating two newstring states at time instants n and n where Using these two string states we then eval-uate the recurrence equation in order to obtain the stringstate at time n It is important to note that the in-terpolation here is in effect stretching the time axis so thatthe wave propagation velocities are altered whereas in theDWG model the allpass filters perform the interpolation inthe spatial domain

This algorithm can also be seen as using two FDM sys-tems in implementing the nonlinear string The elongationof the string would be evaluated from one system and theresult the stretched string state would be updated to theother system Figure 10 illustrates this procedure on thespatio-temporal grid

In step 1 the two initial states have been assigned forthe string and the state at the next instant (in the linearcase) is obtained by the standard recurrence equation (20)The grid values which represent the state of the string atthe corresponding time instant are circled in step 1 In step2 sample values corresponding to the TM have been inter-polated from the string states in step 1 In step 3 equation(20) has been applied on the values evaluated in step 2 in

2 Such a system which deals with the physical quantities themselves iscalled a Kirchhoff model as opposed to a wave model which deals withthe wave components of the physical quantities

(c)

(b)

(a)

t

t

t t

t

t

n

n

n n

n

n

n+1

n+1

n+1 n+1

n+1

n+1y n+ m( 1 )

y n+ m( 1 )

y n m( )

y n m( )

y n+ m( 1 )

y n- m( 1 )

m

m

m m

m

m

d

d

d

a

a

a

-a

-a

-a

z-1

z-1

z-1

n-1

n-1

n-1 n-1

n-1

n-1

Figure 11 Illustration of the interpolation process due to thechange in the stringrsquos length The spatio-temporal grids on theleft and right represent the linear and interpolated string statesrespectively The fractional delay value caused by the interpola-tion is denoted by d The interpolation process in (a) is simplifiedin (b) and further in (c)

order to obtain the string state corresponding to the changein tension The two most recently obtained states are nowtaken as the ldquoinitial statesrdquo in step 4 and we can return tostep 1

As seen in Figure 10 the tension modulation corre-sponds here to interpolating the string state in the tempo-ral domain The elongation of the FDM string was evalu-ated similarly to what was done in equation (9) except thathere the slope of the string was obtained by taking the dif-ference of the displacements between two adjacent stringsegments rather than summing up the slope wave compo-nents In the following we will have a closer look at theinterpolation process

61 String state interpolation

We chose again to use first-order allpass filters in inter-polating the string state from the linear model (step 2 inFigure 10) Figure 11(a) illustrates how the interpolatedvalue of ynm is obtained from the linear values Thespatio-temporal grid on the left represents the string statein the linear case while the spatio-temporal grid on theright represents the string state after spatial interpolationThe structure between the two grids is the block diagramof a first-order allpass filter (equation 6) The coefficient afor the allpass filter was evaluated as presented earlier byequations (9)ndash(12)

319

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

In this figure we notice that the allpass filter uses thevalue of ynm delayed by one sample thus corre-sponding to ynm Clearly this can be obtained directlyfrom the grid on the left and the branch on the left contain-ing the unit delay can be reformed The result is shown inFigure 11(b) Here we also note that the interpolation sys-tem uses its own output at the previous time instant This isactually the same as using the value of ynm becauseit is the same as the output of the interpolation process onetime step ago (this might be best understood by noting thatthe bottom row of step 4 in Figure 10 is the same as thebottom row of step 1 at the next time instant) Thus Fig-ure 11(b) can be further simplified to Figure 11(c)

Having this said the recurrence equation for the time-varying finite difference string with frequency-indepen-dent damping can be written as

ynm pynm ynm

qynm (25)

where

ynm aynm ynm

aynm

ynm aynm ynm

aynm

ynm aynm ynm

ayn m

Here the coefficients p and q incorporate the frequency-independent losses and y and y refer to the linear and in-terpolated strings respectively Simplifying and rearrang-ing we end up with an equation containing only terms ofy and the subscript may therefore be omitted

ynm paynm paynm

pynm qaynm pynm

paynm qynm

paynm qayn m (26)

This equation is illustrated with a block diagram in Fig-ure 12 along with its abstraction A nonlinear FDM stringcan be constructed by connecting several of these blockstogether and using the string elongation in controlling theamount of interpolation We will refer to such a block asa time-varying finite difference time-domain (FDTD) ele-ment Illustration of the lossless time-varying FDTD ele-ment can be found in Figure 13 where p and q equal unityand have therefore been left out

62 String excitation and termination

For the interaction with the time-varying FDTD stringmodel we chose to use the ldquoboxcarrdquo excitation model dis-cussed in section 52 so that the excitation signal couldagain be interpreted as a force signal Figure 14 presentsan interaction block to be used with a time-varying FDTDstring We will call such a block the FDTD interaction el-

FDTD

-pa

z-1

z-1

z-1

y n+1m( )

y nm( )

y n-2m( )

y n-1m( )

-pa

pa

qa

-q

-qa

pa

p p

Figure 12 Illustration of the time-varying FDTD element to-gether with its abstraction A lossless time-varying FDTD ele-ment can be found in Figure 13

-a

z-1

z-1

z-1

y n-1m( )

y n-2m( )

y nm( )

y n+1m( )

-a

-a

a

a

a

Figure 13 Illustration of the lossless time-varying FDTD ele-ment

ement Using these DSP blocks we can construct a one-polarization nonlinear FDTD (NFDTD) string as illus-trated in Figure 15

We chose to use rigid terminations for our nonlin-ear finite difference string model since the modeling offrequency-dependent losses is not a key aspect of thisstudy Fixed terminations do not ruin the generation ofmissing harmonics in our model either since the TMDFcoupling is implemented in a different manner as ex-plained below

63 NFDTD string with generation of missing har-monics

In order to model the generation of missing harmonics ina NFDTD string we constructed a model where an addi-tional interaction element is placed between the last FDTDelement and the termination for feeding the TMDF to thestring Since the spatial distance between the last FDTD

320

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

y n-1m+1( )

y n-1m( )

F n( )

F n( )

y nm+1( )

y nm( )

y n+1m+1( )

y n+1m( )

Figure 14 Illustration of the FDTD interaction element togetherwith its abstraction The excitation algorithm is defined by equa-tions (21) and (22)

F n( )

FDTD FDTD FDTD FDTD FDTD

Allpass-coefficientapproximation

Elongationapproximation

Figure 15 One-polarizational NFDTD string The string consistsof the time-varying FDTD elements illustrated in Figure 12 Thezero-blocks at the terminations give zero as an output regardlessof the input values thus implying a rigid termination The excita-tion to the string can be inserted as a force signal using a FDTDinteraction element illustrated in Figure 14

F n( )

FDTD FDTD FDTD

TMDF

FDTD

Allpass-coefficientapproximation

Elongationapproximation

y n L( -1)nom

Figure 16 Illustration of the NFDTD string with a generationmechanism for missing harmonics A second interaction elementis added in order to feed the TMDF into the string The scal-ing coefficient TMDF controls the amplitude of the missing har-monics The string elongation is approximated from the displace-ments of each FDTD element

element and the rigid termination is one sample the ver-tical component of the TMDF can be seen to be equal tothe product of the displacement of the last FDTD elementand the tension Here we can replace the tension signalby the elongation signal and introduce a scaling coeffi-cient TMDF to control the amount of TMDF to be in-serted to the interaction element at the termination Thisis illustrated in Figure 16 The generation of missing har-

monics in a NFDTD model will be further discussed in thefollowing section

7 Simulation results

In this section we present the results obtained from the twononlinear string algorithms discussed in sections 3 and 5The synthesis results are compared by simulating the samephenomena namely the initial pitch glide and the genera-tion of missing harmonics using the two models Stabilityissues and computational cost of the synthesis models arealso discussed

71 Synthesis results

The synthesis results reveal that both the nonlinear DWGand NFDTD models are able to realistically model the ini-tial pitch glide phenomenon Figure 17 illustrates the fun-damental frequency behavior of a recorded kantele toneand the two synthesized tones Here the horizontal dottedline approximates the mean value of perceptual detectionthreshold of an initial pitch glide The psychoacoustic de-tection threshold in the frequency region of these tones isabout 54 Hz [38] This shows that the fundamental fre-quency glide is an audible phenomenon in plucked stringinstruments such as the kantele even at modest pluckingamplitudes and thus it must be included in a synthesismodel if realistic tones are desired

The nonlinear DWG model used in this figure has a totaldelay line length of 55125 samples and the allpass coef-ficient a is scaled using a constant value of 09 in orderto correctly simulate the behavior of the recorded sampleThe NFDTD string consists of 56 FDTD elements andthe fine-tuning parameter (aka Courant number equa-tion 17) has a value of r 13 The allpass coefficienta is scaled using a coefficient in the NFDTD case

The modeling of the generation of missing harmonicscan be implemented similarly in the distributed nonlinearDWG model as was suggested in [28] If the boxcar inte-gration of equation (10) is replaced with a leaky integratorhaving the transfer function

Iz gp ap

apz (27)

the generation of the missing harmonics can be controlledvia the integration parameter ap The variable gp definesthe gain of the integration

Figure 18 shows the amplitude envelopes of the firstthree harmonics of a synthesized tone with two differ-ent ap parameter values The string was plucked close tord of its length and as can be seen in the figure themissing harmonic in (a) has a gradual increase after thebeginning transient after which it experiences an expo-nential decay like all other modes

It is worthwhile to note that the generation of missingharmonics in the nonlinear DWG model results from theproperties of the integration of the elongation approxima-tion and is therefore not a physically justified process Ba-sically here the integration error in the leaky integrator is

321

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Time [s]

fnom

Fre

quen

cy[H

z]

Figure 17 Fundamental frequencies as a function of time fora moderately-plucked recorded kantele tone (solid line) a syn-thesized nonlinear DWG tone (dashed line) and a synthesizedNFDTD tone (dash-dotted line) The fnom stands for the nomi-nal fundamental frequency of the string and the horizontal dottedline denotes the approximated detection threshold of a pitch drift(fnomHz) which suggests that the fundamental frequencydrifts in all cases are audible

responsible for feeding energy to the missing harmonicsAlso unlike the real physical phenomenon the generationof missing harmonics in the nonlinear DWG case does notdepend on the rigidity of the terminations Neverthelessthis feature can be exploited in emulating the real stringbehavior when the integration parameters are properly ad-justed Details on tuning the leaky integrator parameterscan be found in [28]

Modeling the generation of missing harmonics in aNFDTD string is however not so simple Even if a leakyintegrator is used in the elongation calculation its param-eters do not have a desirable effect on the missing har-monics This does not seem too surprising when consider-ing the major differences of these two algorithms and it isthe reason that forced us to use an alternative mechanismfor creating the missing harmonics in the previous section(Figure 16)

Figure 19 represents the behavior of the first three har-monics of a tone synthesized by this model It can be seenthat the missing harmonics can be ldquoliftedrdquo by choosing aproper value for TMDF The stability of the system how-ever poses an upper limit for the TMDF coefficient sincethe TMDF mechanism continuously feeds energy to thestring According to our experience generating missingharmonics with amplitudes greater than what is shown inFigure 19 is difficult

72 Stability issues and computational comparison

We found the nonlinear DWG algorithm to remain sta-ble for nearly all parameter and excitation values Onlyhighly exaggerated nonlinearity scaling values togetherwith high excitation impulses resulted in stability prob-lems We thus conclude that the nonlinear DWG waveg-uide has no real stability problems when synthesis of nat-ural plucked-instrument sounds are desired

We studied the stability of the NFDTD algorithm us-ing the Von Neumann analysis [39] in the time-invariantcase ie parameter a of equation (26) was kept constantThe basic idea of this method is to calculate the spatialFourier spectrum of the system under discussion at twoconsecutive time steps An amplification function which

(a)

(b)

Figure 18 Generation of the missing harmonics in the nonlinearDWG model can be controlled via the leaky integrator parame-ters Here the string was plucked approximately at rd of itslength so every 3rd harmonic should be missing from the re-sulting spectrum In a) ap and the third harmonicclearly rises after the initial transient In b) ap 13 andthe third harmonic is more attenuated

(a)

(b)

Figure 19 Generation of missing harmonics in a NFDTD stringThe string was plucked again approximately at rd of itslength and the coupling of the TMDF to the transversal vibra-tion was controlled using a scaling coefficient TMDF In a) thescaling coefficient has a value of TMDF and the missingthird harmonic can be seen rising after the initial transient In b)TMDF and generation of missing harmonics does not takeplace

shows how the spatial spectrum evolves with time canthen be derived from the two spectra If the absolute valueof this amplification function remains below unity stabil-ity is guaranteed Formally the Von Neumann analysis forthe NFDTD algorithm goes as follows [4]

If the spatial inverse Fourier transform is defined as

ynm FfY n g neim (28)

322

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Solving (15) we get

ytT x KT

X

yt xX yt x yt xX

yt x ytT x (16)

Next we define the relationship between the spatial andtemporal sampling steps with [35]

r cT

X (17)

where the ldquoless than unityrdquo-restriction is called the VonNeumann stability condition Using this together with thedefinition of transversal wave velocity (equation 2) equa-tion (16) can be written as

ytT x ryt xX yt xX

(18)

ryt x ytT x

If we now do the discretization by denoting t tn nT and x xm mX as we did in section 3 we endup with the finite difference approximation [35]

ynm rynm ynm

(19)

rynm ynm

If we set r (19) becomes

ynm ynm ynm

ynm (20)

which is the finite difference equation of an ideal stringThe equality of equation (20) can be checked by substitut-ing the waveguide decomposition (equation 5) in the right-hand side of equation (20) [22]

Since the length of the string must again have integervalues correct tuning of the string becomes difficult It hasbeen shown [35] that choosing r in equation (17) re-sults in lowering the fundamental frequency of the stringTherefore the finite difference string can be tuned via theparameter r

Choosing r also gives raise to an unwanted nu-merical dispersion phenomenon called grid dispersion [7]where the wave velocity in the numerical implementationwill be less than the ideal physical wave velocity This ar-tificial dispersion affects primarily the upper harmonicswhere the frequencies will be underestimated If a typi-cal error of in the generated frequencies is allowedthe difference between the tuning coefficient r and unityshould not be greater than [35] If the constraints be-tween the correct tuning and grid dispersion do not yieldsatisfactory results the spatial density of the grid shouldbe increased This is known as spatial oversampling

52 Boundary conditions and string excitation

Since the spatial coordinate m of the string must liebetween and Lnom problems arise near the ends ofthe string when evaluating equation (20) because spatialpoints outside the string are needed The problem can

be solved by introducing boundary conditions that definehow to evaluate the string movement when m orm Lnom The simplest approach introduced alreadyin [6] would be to assume that the string terminations berigid so that yn yn Lnom This results in aphase-inverting termination which suits perfectly the caseof an ideal string For other types of string terminationseveral models have been introduced (see eg [6] [35]and [36]) Generally the nonrigid string terminations leadto frequency-dependent losses in the string model

For the FDM string excitation a useful method has beenproposed in [36] It is conceptually simple and allows forinteraction with the string during run-time There

ynm ynm

un (21)

and

ynm ynm

un (22)

are inserted into the string which causes a ldquoboxcarrdquo blockfunction to spread in both directions from the excitationpoint pair The wave component un is now used as theexcitation signal in a similar way as the exciting force sig-nal F n in section 32

53 Finite difference approximation of a lossy string

Frequency-independent losses can be modeled in an FDMstring by discretizing the velocity-dependent dampingterm in the lossy 1D wave equation (3) This results in twoadditional scaling coefficients in the recurrence equation[35]

ynm pynm ynm

qynm (23)

where the values of p and q determine the amount oflosses Generally p and q may depend on the spatial indexm but since homogeneous strings are considered here thisdependency is omitted Values

q p jpj (24)

ensure the stability of a linear finite difference string withfrequency-independent losses [37] Note that the sign dif-ference of p and q in [37] has already been taken care ofin equation (23) Modeling of frequency-dependent lossesby discretizing the lossy wave equation leads to an implicitrecurrence equation which can be evaluated if suitableapproximations are made [35]

6 Nonlinear finite difference string

Implementing tension modulation in a digital waveguidestring in section 3 was not an overly difficult task This wasdue to the fact that the implementation of a DWG string isessentially a feedback loop with delay and therefore mod-ulating the delay time of this loop corresponded to modu-lating the wave velocities In FDM strings however such

318

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

Figure 10 Illustration of the nonlinear FD algorithm on a spatio-temporal grid The vertical axis denotes the time while the hor-izontal axis denotes the spatial location on the string The illus-tration is shown only for a string segment of length N forclarity In each step of the algorithm most recently evaluated val-ues are presented as black dots while earlier values are presentedas white dots

an approach would not lead to satisfactory results sincethe physical quantities (eg displacement) themselves arepresent in the string model and not their wave decompo-sitions2

Instead we concluded that in order to correctly modelthe TM in a FDM string we first have to evaluate the recur-rence equation and use these three snapshots of the string(at time instants n n and n) in interpolating two newstring states at time instants n and n where Using these two string states we then eval-uate the recurrence equation in order to obtain the stringstate at time n It is important to note that the in-terpolation here is in effect stretching the time axis so thatthe wave propagation velocities are altered whereas in theDWG model the allpass filters perform the interpolation inthe spatial domain

This algorithm can also be seen as using two FDM sys-tems in implementing the nonlinear string The elongationof the string would be evaluated from one system and theresult the stretched string state would be updated to theother system Figure 10 illustrates this procedure on thespatio-temporal grid

In step 1 the two initial states have been assigned forthe string and the state at the next instant (in the linearcase) is obtained by the standard recurrence equation (20)The grid values which represent the state of the string atthe corresponding time instant are circled in step 1 In step2 sample values corresponding to the TM have been inter-polated from the string states in step 1 In step 3 equation(20) has been applied on the values evaluated in step 2 in

2 Such a system which deals with the physical quantities themselves iscalled a Kirchhoff model as opposed to a wave model which deals withthe wave components of the physical quantities

(c)

(b)

(a)

t

t

t t

t

t

n

n

n n

n

n

n+1

n+1

n+1 n+1

n+1

n+1y n+ m( 1 )

y n+ m( 1 )

y n m( )

y n m( )

y n+ m( 1 )

y n- m( 1 )

m

m

m m

m

m

d

d

d

a

a

a

-a

-a

-a

z-1

z-1

z-1

n-1

n-1

n-1 n-1

n-1

n-1

Figure 11 Illustration of the interpolation process due to thechange in the stringrsquos length The spatio-temporal grids on theleft and right represent the linear and interpolated string statesrespectively The fractional delay value caused by the interpola-tion is denoted by d The interpolation process in (a) is simplifiedin (b) and further in (c)

order to obtain the string state corresponding to the changein tension The two most recently obtained states are nowtaken as the ldquoinitial statesrdquo in step 4 and we can return tostep 1

As seen in Figure 10 the tension modulation corre-sponds here to interpolating the string state in the tempo-ral domain The elongation of the FDM string was evalu-ated similarly to what was done in equation (9) except thathere the slope of the string was obtained by taking the dif-ference of the displacements between two adjacent stringsegments rather than summing up the slope wave compo-nents In the following we will have a closer look at theinterpolation process

61 String state interpolation

We chose again to use first-order allpass filters in inter-polating the string state from the linear model (step 2 inFigure 10) Figure 11(a) illustrates how the interpolatedvalue of ynm is obtained from the linear values Thespatio-temporal grid on the left represents the string statein the linear case while the spatio-temporal grid on theright represents the string state after spatial interpolationThe structure between the two grids is the block diagramof a first-order allpass filter (equation 6) The coefficient afor the allpass filter was evaluated as presented earlier byequations (9)ndash(12)

319

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

In this figure we notice that the allpass filter uses thevalue of ynm delayed by one sample thus corre-sponding to ynm Clearly this can be obtained directlyfrom the grid on the left and the branch on the left contain-ing the unit delay can be reformed The result is shown inFigure 11(b) Here we also note that the interpolation sys-tem uses its own output at the previous time instant This isactually the same as using the value of ynm becauseit is the same as the output of the interpolation process onetime step ago (this might be best understood by noting thatthe bottom row of step 4 in Figure 10 is the same as thebottom row of step 1 at the next time instant) Thus Fig-ure 11(b) can be further simplified to Figure 11(c)

Having this said the recurrence equation for the time-varying finite difference string with frequency-indepen-dent damping can be written as

ynm pynm ynm

qynm (25)

where

ynm aynm ynm

aynm

ynm aynm ynm

aynm

ynm aynm ynm

ayn m

Here the coefficients p and q incorporate the frequency-independent losses and y and y refer to the linear and in-terpolated strings respectively Simplifying and rearrang-ing we end up with an equation containing only terms ofy and the subscript may therefore be omitted

ynm paynm paynm

pynm qaynm pynm

paynm qynm

paynm qayn m (26)

This equation is illustrated with a block diagram in Fig-ure 12 along with its abstraction A nonlinear FDM stringcan be constructed by connecting several of these blockstogether and using the string elongation in controlling theamount of interpolation We will refer to such a block asa time-varying finite difference time-domain (FDTD) ele-ment Illustration of the lossless time-varying FDTD ele-ment can be found in Figure 13 where p and q equal unityand have therefore been left out

62 String excitation and termination

For the interaction with the time-varying FDTD stringmodel we chose to use the ldquoboxcarrdquo excitation model dis-cussed in section 52 so that the excitation signal couldagain be interpreted as a force signal Figure 14 presentsan interaction block to be used with a time-varying FDTDstring We will call such a block the FDTD interaction el-

FDTD

-pa

z-1

z-1

z-1

y n+1m( )

y nm( )

y n-2m( )

y n-1m( )

-pa

pa

qa

-q

-qa

pa

p p

Figure 12 Illustration of the time-varying FDTD element to-gether with its abstraction A lossless time-varying FDTD ele-ment can be found in Figure 13

-a

z-1

z-1

z-1

y n-1m( )

y n-2m( )

y nm( )

y n+1m( )

-a

-a

a

a

a

Figure 13 Illustration of the lossless time-varying FDTD ele-ment

ement Using these DSP blocks we can construct a one-polarization nonlinear FDTD (NFDTD) string as illus-trated in Figure 15

We chose to use rigid terminations for our nonlin-ear finite difference string model since the modeling offrequency-dependent losses is not a key aspect of thisstudy Fixed terminations do not ruin the generation ofmissing harmonics in our model either since the TMDFcoupling is implemented in a different manner as ex-plained below

63 NFDTD string with generation of missing har-monics

In order to model the generation of missing harmonics ina NFDTD string we constructed a model where an addi-tional interaction element is placed between the last FDTDelement and the termination for feeding the TMDF to thestring Since the spatial distance between the last FDTD

320

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

y n-1m+1( )

y n-1m( )

F n( )

F n( )

y nm+1( )

y nm( )

y n+1m+1( )

y n+1m( )

Figure 14 Illustration of the FDTD interaction element togetherwith its abstraction The excitation algorithm is defined by equa-tions (21) and (22)

F n( )

FDTD FDTD FDTD FDTD FDTD

Allpass-coefficientapproximation

Elongationapproximation

Figure 15 One-polarizational NFDTD string The string consistsof the time-varying FDTD elements illustrated in Figure 12 Thezero-blocks at the terminations give zero as an output regardlessof the input values thus implying a rigid termination The excita-tion to the string can be inserted as a force signal using a FDTDinteraction element illustrated in Figure 14

F n( )

FDTD FDTD FDTD

TMDF

FDTD

Allpass-coefficientapproximation

Elongationapproximation

y n L( -1)nom

Figure 16 Illustration of the NFDTD string with a generationmechanism for missing harmonics A second interaction elementis added in order to feed the TMDF into the string The scal-ing coefficient TMDF controls the amplitude of the missing har-monics The string elongation is approximated from the displace-ments of each FDTD element

element and the rigid termination is one sample the ver-tical component of the TMDF can be seen to be equal tothe product of the displacement of the last FDTD elementand the tension Here we can replace the tension signalby the elongation signal and introduce a scaling coeffi-cient TMDF to control the amount of TMDF to be in-serted to the interaction element at the termination Thisis illustrated in Figure 16 The generation of missing har-

monics in a NFDTD model will be further discussed in thefollowing section

7 Simulation results

In this section we present the results obtained from the twononlinear string algorithms discussed in sections 3 and 5The synthesis results are compared by simulating the samephenomena namely the initial pitch glide and the genera-tion of missing harmonics using the two models Stabilityissues and computational cost of the synthesis models arealso discussed

71 Synthesis results

The synthesis results reveal that both the nonlinear DWGand NFDTD models are able to realistically model the ini-tial pitch glide phenomenon Figure 17 illustrates the fun-damental frequency behavior of a recorded kantele toneand the two synthesized tones Here the horizontal dottedline approximates the mean value of perceptual detectionthreshold of an initial pitch glide The psychoacoustic de-tection threshold in the frequency region of these tones isabout 54 Hz [38] This shows that the fundamental fre-quency glide is an audible phenomenon in plucked stringinstruments such as the kantele even at modest pluckingamplitudes and thus it must be included in a synthesismodel if realistic tones are desired

The nonlinear DWG model used in this figure has a totaldelay line length of 55125 samples and the allpass coef-ficient a is scaled using a constant value of 09 in orderto correctly simulate the behavior of the recorded sampleThe NFDTD string consists of 56 FDTD elements andthe fine-tuning parameter (aka Courant number equa-tion 17) has a value of r 13 The allpass coefficienta is scaled using a coefficient in the NFDTD case

The modeling of the generation of missing harmonicscan be implemented similarly in the distributed nonlinearDWG model as was suggested in [28] If the boxcar inte-gration of equation (10) is replaced with a leaky integratorhaving the transfer function

Iz gp ap

apz (27)

the generation of the missing harmonics can be controlledvia the integration parameter ap The variable gp definesthe gain of the integration

Figure 18 shows the amplitude envelopes of the firstthree harmonics of a synthesized tone with two differ-ent ap parameter values The string was plucked close tord of its length and as can be seen in the figure themissing harmonic in (a) has a gradual increase after thebeginning transient after which it experiences an expo-nential decay like all other modes

It is worthwhile to note that the generation of missingharmonics in the nonlinear DWG model results from theproperties of the integration of the elongation approxima-tion and is therefore not a physically justified process Ba-sically here the integration error in the leaky integrator is

321

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Time [s]

fnom

Fre

quen

cy[H

z]

Figure 17 Fundamental frequencies as a function of time fora moderately-plucked recorded kantele tone (solid line) a syn-thesized nonlinear DWG tone (dashed line) and a synthesizedNFDTD tone (dash-dotted line) The fnom stands for the nomi-nal fundamental frequency of the string and the horizontal dottedline denotes the approximated detection threshold of a pitch drift(fnomHz) which suggests that the fundamental frequencydrifts in all cases are audible

responsible for feeding energy to the missing harmonicsAlso unlike the real physical phenomenon the generationof missing harmonics in the nonlinear DWG case does notdepend on the rigidity of the terminations Neverthelessthis feature can be exploited in emulating the real stringbehavior when the integration parameters are properly ad-justed Details on tuning the leaky integrator parameterscan be found in [28]

Modeling the generation of missing harmonics in aNFDTD string is however not so simple Even if a leakyintegrator is used in the elongation calculation its param-eters do not have a desirable effect on the missing har-monics This does not seem too surprising when consider-ing the major differences of these two algorithms and it isthe reason that forced us to use an alternative mechanismfor creating the missing harmonics in the previous section(Figure 16)

Figure 19 represents the behavior of the first three har-monics of a tone synthesized by this model It can be seenthat the missing harmonics can be ldquoliftedrdquo by choosing aproper value for TMDF The stability of the system how-ever poses an upper limit for the TMDF coefficient sincethe TMDF mechanism continuously feeds energy to thestring According to our experience generating missingharmonics with amplitudes greater than what is shown inFigure 19 is difficult

72 Stability issues and computational comparison

We found the nonlinear DWG algorithm to remain sta-ble for nearly all parameter and excitation values Onlyhighly exaggerated nonlinearity scaling values togetherwith high excitation impulses resulted in stability prob-lems We thus conclude that the nonlinear DWG waveg-uide has no real stability problems when synthesis of nat-ural plucked-instrument sounds are desired

We studied the stability of the NFDTD algorithm us-ing the Von Neumann analysis [39] in the time-invariantcase ie parameter a of equation (26) was kept constantThe basic idea of this method is to calculate the spatialFourier spectrum of the system under discussion at twoconsecutive time steps An amplification function which

(a)

(b)

Figure 18 Generation of the missing harmonics in the nonlinearDWG model can be controlled via the leaky integrator parame-ters Here the string was plucked approximately at rd of itslength so every 3rd harmonic should be missing from the re-sulting spectrum In a) ap and the third harmonicclearly rises after the initial transient In b) ap 13 andthe third harmonic is more attenuated

(a)

(b)

Figure 19 Generation of missing harmonics in a NFDTD stringThe string was plucked again approximately at rd of itslength and the coupling of the TMDF to the transversal vibra-tion was controlled using a scaling coefficient TMDF In a) thescaling coefficient has a value of TMDF and the missingthird harmonic can be seen rising after the initial transient In b)TMDF and generation of missing harmonics does not takeplace

shows how the spatial spectrum evolves with time canthen be derived from the two spectra If the absolute valueof this amplification function remains below unity stabil-ity is guaranteed Formally the Von Neumann analysis forthe NFDTD algorithm goes as follows [4]

If the spatial inverse Fourier transform is defined as

ynm FfY n g neim (28)

322

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

Figure 10 Illustration of the nonlinear FD algorithm on a spatio-temporal grid The vertical axis denotes the time while the hor-izontal axis denotes the spatial location on the string The illus-tration is shown only for a string segment of length N forclarity In each step of the algorithm most recently evaluated val-ues are presented as black dots while earlier values are presentedas white dots

an approach would not lead to satisfactory results sincethe physical quantities (eg displacement) themselves arepresent in the string model and not their wave decompo-sitions2

Instead we concluded that in order to correctly modelthe TM in a FDM string we first have to evaluate the recur-rence equation and use these three snapshots of the string(at time instants n n and n) in interpolating two newstring states at time instants n and n where Using these two string states we then eval-uate the recurrence equation in order to obtain the stringstate at time n It is important to note that the in-terpolation here is in effect stretching the time axis so thatthe wave propagation velocities are altered whereas in theDWG model the allpass filters perform the interpolation inthe spatial domain

This algorithm can also be seen as using two FDM sys-tems in implementing the nonlinear string The elongationof the string would be evaluated from one system and theresult the stretched string state would be updated to theother system Figure 10 illustrates this procedure on thespatio-temporal grid

In step 1 the two initial states have been assigned forthe string and the state at the next instant (in the linearcase) is obtained by the standard recurrence equation (20)The grid values which represent the state of the string atthe corresponding time instant are circled in step 1 In step2 sample values corresponding to the TM have been inter-polated from the string states in step 1 In step 3 equation(20) has been applied on the values evaluated in step 2 in

2 Such a system which deals with the physical quantities themselves iscalled a Kirchhoff model as opposed to a wave model which deals withthe wave components of the physical quantities

(c)

(b)

(a)

t

t

t t

t

t

n

n

n n

n

n

n+1

n+1

n+1 n+1

n+1

n+1y n+ m( 1 )

y n+ m( 1 )

y n m( )

y n m( )

y n+ m( 1 )

y n- m( 1 )

m

m

m m

m

m

d

d

d

a

a

a

-a

-a

-a

z-1

z-1

z-1

n-1

n-1

n-1 n-1

n-1

n-1

Figure 11 Illustration of the interpolation process due to thechange in the stringrsquos length The spatio-temporal grids on theleft and right represent the linear and interpolated string statesrespectively The fractional delay value caused by the interpola-tion is denoted by d The interpolation process in (a) is simplifiedin (b) and further in (c)

order to obtain the string state corresponding to the changein tension The two most recently obtained states are nowtaken as the ldquoinitial statesrdquo in step 4 and we can return tostep 1

As seen in Figure 10 the tension modulation corre-sponds here to interpolating the string state in the tempo-ral domain The elongation of the FDM string was evalu-ated similarly to what was done in equation (9) except thathere the slope of the string was obtained by taking the dif-ference of the displacements between two adjacent stringsegments rather than summing up the slope wave compo-nents In the following we will have a closer look at theinterpolation process

61 String state interpolation

We chose again to use first-order allpass filters in inter-polating the string state from the linear model (step 2 inFigure 10) Figure 11(a) illustrates how the interpolatedvalue of ynm is obtained from the linear values Thespatio-temporal grid on the left represents the string statein the linear case while the spatio-temporal grid on theright represents the string state after spatial interpolationThe structure between the two grids is the block diagramof a first-order allpass filter (equation 6) The coefficient afor the allpass filter was evaluated as presented earlier byequations (9)ndash(12)

319

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

In this figure we notice that the allpass filter uses thevalue of ynm delayed by one sample thus corre-sponding to ynm Clearly this can be obtained directlyfrom the grid on the left and the branch on the left contain-ing the unit delay can be reformed The result is shown inFigure 11(b) Here we also note that the interpolation sys-tem uses its own output at the previous time instant This isactually the same as using the value of ynm becauseit is the same as the output of the interpolation process onetime step ago (this might be best understood by noting thatthe bottom row of step 4 in Figure 10 is the same as thebottom row of step 1 at the next time instant) Thus Fig-ure 11(b) can be further simplified to Figure 11(c)

Having this said the recurrence equation for the time-varying finite difference string with frequency-indepen-dent damping can be written as

ynm pynm ynm

qynm (25)

where

ynm aynm ynm

aynm

ynm aynm ynm

aynm

ynm aynm ynm

ayn m

Here the coefficients p and q incorporate the frequency-independent losses and y and y refer to the linear and in-terpolated strings respectively Simplifying and rearrang-ing we end up with an equation containing only terms ofy and the subscript may therefore be omitted

ynm paynm paynm

pynm qaynm pynm

paynm qynm

paynm qayn m (26)

This equation is illustrated with a block diagram in Fig-ure 12 along with its abstraction A nonlinear FDM stringcan be constructed by connecting several of these blockstogether and using the string elongation in controlling theamount of interpolation We will refer to such a block asa time-varying finite difference time-domain (FDTD) ele-ment Illustration of the lossless time-varying FDTD ele-ment can be found in Figure 13 where p and q equal unityand have therefore been left out

62 String excitation and termination

For the interaction with the time-varying FDTD stringmodel we chose to use the ldquoboxcarrdquo excitation model dis-cussed in section 52 so that the excitation signal couldagain be interpreted as a force signal Figure 14 presentsan interaction block to be used with a time-varying FDTDstring We will call such a block the FDTD interaction el-

FDTD

-pa

z-1

z-1

z-1

y n+1m( )

y nm( )

y n-2m( )

y n-1m( )

-pa

pa

qa

-q

-qa

pa

p p

Figure 12 Illustration of the time-varying FDTD element to-gether with its abstraction A lossless time-varying FDTD ele-ment can be found in Figure 13

-a

z-1

z-1

z-1

y n-1m( )

y n-2m( )

y nm( )

y n+1m( )

-a

-a

a

a

a

Figure 13 Illustration of the lossless time-varying FDTD ele-ment

ement Using these DSP blocks we can construct a one-polarization nonlinear FDTD (NFDTD) string as illus-trated in Figure 15

We chose to use rigid terminations for our nonlin-ear finite difference string model since the modeling offrequency-dependent losses is not a key aspect of thisstudy Fixed terminations do not ruin the generation ofmissing harmonics in our model either since the TMDFcoupling is implemented in a different manner as ex-plained below

63 NFDTD string with generation of missing har-monics

In order to model the generation of missing harmonics ina NFDTD string we constructed a model where an addi-tional interaction element is placed between the last FDTDelement and the termination for feeding the TMDF to thestring Since the spatial distance between the last FDTD

320

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

y n-1m+1( )

y n-1m( )

F n( )

F n( )

y nm+1( )

y nm( )

y n+1m+1( )

y n+1m( )

Figure 14 Illustration of the FDTD interaction element togetherwith its abstraction The excitation algorithm is defined by equa-tions (21) and (22)

F n( )

FDTD FDTD FDTD FDTD FDTD

Allpass-coefficientapproximation

Elongationapproximation

Figure 15 One-polarizational NFDTD string The string consistsof the time-varying FDTD elements illustrated in Figure 12 Thezero-blocks at the terminations give zero as an output regardlessof the input values thus implying a rigid termination The excita-tion to the string can be inserted as a force signal using a FDTDinteraction element illustrated in Figure 14

F n( )

FDTD FDTD FDTD

TMDF

FDTD

Allpass-coefficientapproximation

Elongationapproximation

y n L( -1)nom

Figure 16 Illustration of the NFDTD string with a generationmechanism for missing harmonics A second interaction elementis added in order to feed the TMDF into the string The scal-ing coefficient TMDF controls the amplitude of the missing har-monics The string elongation is approximated from the displace-ments of each FDTD element

element and the rigid termination is one sample the ver-tical component of the TMDF can be seen to be equal tothe product of the displacement of the last FDTD elementand the tension Here we can replace the tension signalby the elongation signal and introduce a scaling coeffi-cient TMDF to control the amount of TMDF to be in-serted to the interaction element at the termination Thisis illustrated in Figure 16 The generation of missing har-

monics in a NFDTD model will be further discussed in thefollowing section

7 Simulation results

In this section we present the results obtained from the twononlinear string algorithms discussed in sections 3 and 5The synthesis results are compared by simulating the samephenomena namely the initial pitch glide and the genera-tion of missing harmonics using the two models Stabilityissues and computational cost of the synthesis models arealso discussed

71 Synthesis results

The synthesis results reveal that both the nonlinear DWGand NFDTD models are able to realistically model the ini-tial pitch glide phenomenon Figure 17 illustrates the fun-damental frequency behavior of a recorded kantele toneand the two synthesized tones Here the horizontal dottedline approximates the mean value of perceptual detectionthreshold of an initial pitch glide The psychoacoustic de-tection threshold in the frequency region of these tones isabout 54 Hz [38] This shows that the fundamental fre-quency glide is an audible phenomenon in plucked stringinstruments such as the kantele even at modest pluckingamplitudes and thus it must be included in a synthesismodel if realistic tones are desired

The nonlinear DWG model used in this figure has a totaldelay line length of 55125 samples and the allpass coef-ficient a is scaled using a constant value of 09 in orderto correctly simulate the behavior of the recorded sampleThe NFDTD string consists of 56 FDTD elements andthe fine-tuning parameter (aka Courant number equa-tion 17) has a value of r 13 The allpass coefficienta is scaled using a coefficient in the NFDTD case

The modeling of the generation of missing harmonicscan be implemented similarly in the distributed nonlinearDWG model as was suggested in [28] If the boxcar inte-gration of equation (10) is replaced with a leaky integratorhaving the transfer function

Iz gp ap

apz (27)

the generation of the missing harmonics can be controlledvia the integration parameter ap The variable gp definesthe gain of the integration

Figure 18 shows the amplitude envelopes of the firstthree harmonics of a synthesized tone with two differ-ent ap parameter values The string was plucked close tord of its length and as can be seen in the figure themissing harmonic in (a) has a gradual increase after thebeginning transient after which it experiences an expo-nential decay like all other modes

It is worthwhile to note that the generation of missingharmonics in the nonlinear DWG model results from theproperties of the integration of the elongation approxima-tion and is therefore not a physically justified process Ba-sically here the integration error in the leaky integrator is

321

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Time [s]

fnom

Fre

quen

cy[H

z]

Figure 17 Fundamental frequencies as a function of time fora moderately-plucked recorded kantele tone (solid line) a syn-thesized nonlinear DWG tone (dashed line) and a synthesizedNFDTD tone (dash-dotted line) The fnom stands for the nomi-nal fundamental frequency of the string and the horizontal dottedline denotes the approximated detection threshold of a pitch drift(fnomHz) which suggests that the fundamental frequencydrifts in all cases are audible

responsible for feeding energy to the missing harmonicsAlso unlike the real physical phenomenon the generationof missing harmonics in the nonlinear DWG case does notdepend on the rigidity of the terminations Neverthelessthis feature can be exploited in emulating the real stringbehavior when the integration parameters are properly ad-justed Details on tuning the leaky integrator parameterscan be found in [28]

Modeling the generation of missing harmonics in aNFDTD string is however not so simple Even if a leakyintegrator is used in the elongation calculation its param-eters do not have a desirable effect on the missing har-monics This does not seem too surprising when consider-ing the major differences of these two algorithms and it isthe reason that forced us to use an alternative mechanismfor creating the missing harmonics in the previous section(Figure 16)

Figure 19 represents the behavior of the first three har-monics of a tone synthesized by this model It can be seenthat the missing harmonics can be ldquoliftedrdquo by choosing aproper value for TMDF The stability of the system how-ever poses an upper limit for the TMDF coefficient sincethe TMDF mechanism continuously feeds energy to thestring According to our experience generating missingharmonics with amplitudes greater than what is shown inFigure 19 is difficult

72 Stability issues and computational comparison

We found the nonlinear DWG algorithm to remain sta-ble for nearly all parameter and excitation values Onlyhighly exaggerated nonlinearity scaling values togetherwith high excitation impulses resulted in stability prob-lems We thus conclude that the nonlinear DWG waveg-uide has no real stability problems when synthesis of nat-ural plucked-instrument sounds are desired

We studied the stability of the NFDTD algorithm us-ing the Von Neumann analysis [39] in the time-invariantcase ie parameter a of equation (26) was kept constantThe basic idea of this method is to calculate the spatialFourier spectrum of the system under discussion at twoconsecutive time steps An amplification function which

(a)

(b)

Figure 18 Generation of the missing harmonics in the nonlinearDWG model can be controlled via the leaky integrator parame-ters Here the string was plucked approximately at rd of itslength so every 3rd harmonic should be missing from the re-sulting spectrum In a) ap and the third harmonicclearly rises after the initial transient In b) ap 13 andthe third harmonic is more attenuated

(a)

(b)

Figure 19 Generation of missing harmonics in a NFDTD stringThe string was plucked again approximately at rd of itslength and the coupling of the TMDF to the transversal vibra-tion was controlled using a scaling coefficient TMDF In a) thescaling coefficient has a value of TMDF and the missingthird harmonic can be seen rising after the initial transient In b)TMDF and generation of missing harmonics does not takeplace

shows how the spatial spectrum evolves with time canthen be derived from the two spectra If the absolute valueof this amplification function remains below unity stabil-ity is guaranteed Formally the Von Neumann analysis forthe NFDTD algorithm goes as follows [4]

If the spatial inverse Fourier transform is defined as

ynm FfY n g neim (28)

322

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

In this figure we notice that the allpass filter uses thevalue of ynm delayed by one sample thus corre-sponding to ynm Clearly this can be obtained directlyfrom the grid on the left and the branch on the left contain-ing the unit delay can be reformed The result is shown inFigure 11(b) Here we also note that the interpolation sys-tem uses its own output at the previous time instant This isactually the same as using the value of ynm becauseit is the same as the output of the interpolation process onetime step ago (this might be best understood by noting thatthe bottom row of step 4 in Figure 10 is the same as thebottom row of step 1 at the next time instant) Thus Fig-ure 11(b) can be further simplified to Figure 11(c)

Having this said the recurrence equation for the time-varying finite difference string with frequency-indepen-dent damping can be written as

ynm pynm ynm

qynm (25)

where

ynm aynm ynm

aynm

ynm aynm ynm

aynm

ynm aynm ynm

ayn m

Here the coefficients p and q incorporate the frequency-independent losses and y and y refer to the linear and in-terpolated strings respectively Simplifying and rearrang-ing we end up with an equation containing only terms ofy and the subscript may therefore be omitted

ynm paynm paynm

pynm qaynm pynm

paynm qynm

paynm qayn m (26)

This equation is illustrated with a block diagram in Fig-ure 12 along with its abstraction A nonlinear FDM stringcan be constructed by connecting several of these blockstogether and using the string elongation in controlling theamount of interpolation We will refer to such a block asa time-varying finite difference time-domain (FDTD) ele-ment Illustration of the lossless time-varying FDTD ele-ment can be found in Figure 13 where p and q equal unityand have therefore been left out

62 String excitation and termination

For the interaction with the time-varying FDTD stringmodel we chose to use the ldquoboxcarrdquo excitation model dis-cussed in section 52 so that the excitation signal couldagain be interpreted as a force signal Figure 14 presentsan interaction block to be used with a time-varying FDTDstring We will call such a block the FDTD interaction el-

FDTD

-pa

z-1

z-1

z-1

y n+1m( )

y nm( )

y n-2m( )

y n-1m( )

-pa

pa

qa

-q

-qa

pa

p p

Figure 12 Illustration of the time-varying FDTD element to-gether with its abstraction A lossless time-varying FDTD ele-ment can be found in Figure 13

-a

z-1

z-1

z-1

y n-1m( )

y n-2m( )

y nm( )

y n+1m( )

-a

-a

a

a

a

Figure 13 Illustration of the lossless time-varying FDTD ele-ment

ement Using these DSP blocks we can construct a one-polarization nonlinear FDTD (NFDTD) string as illus-trated in Figure 15

We chose to use rigid terminations for our nonlin-ear finite difference string model since the modeling offrequency-dependent losses is not a key aspect of thisstudy Fixed terminations do not ruin the generation ofmissing harmonics in our model either since the TMDFcoupling is implemented in a different manner as ex-plained below

63 NFDTD string with generation of missing har-monics

In order to model the generation of missing harmonics ina NFDTD string we constructed a model where an addi-tional interaction element is placed between the last FDTDelement and the termination for feeding the TMDF to thestring Since the spatial distance between the last FDTD

320

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

y n-1m+1( )

y n-1m( )

F n( )

F n( )

y nm+1( )

y nm( )

y n+1m+1( )

y n+1m( )

Figure 14 Illustration of the FDTD interaction element togetherwith its abstraction The excitation algorithm is defined by equa-tions (21) and (22)

F n( )

FDTD FDTD FDTD FDTD FDTD

Allpass-coefficientapproximation

Elongationapproximation

Figure 15 One-polarizational NFDTD string The string consistsof the time-varying FDTD elements illustrated in Figure 12 Thezero-blocks at the terminations give zero as an output regardlessof the input values thus implying a rigid termination The excita-tion to the string can be inserted as a force signal using a FDTDinteraction element illustrated in Figure 14

F n( )

FDTD FDTD FDTD

TMDF

FDTD

Allpass-coefficientapproximation

Elongationapproximation

y n L( -1)nom

Figure 16 Illustration of the NFDTD string with a generationmechanism for missing harmonics A second interaction elementis added in order to feed the TMDF into the string The scal-ing coefficient TMDF controls the amplitude of the missing har-monics The string elongation is approximated from the displace-ments of each FDTD element

element and the rigid termination is one sample the ver-tical component of the TMDF can be seen to be equal tothe product of the displacement of the last FDTD elementand the tension Here we can replace the tension signalby the elongation signal and introduce a scaling coeffi-cient TMDF to control the amount of TMDF to be in-serted to the interaction element at the termination Thisis illustrated in Figure 16 The generation of missing har-

monics in a NFDTD model will be further discussed in thefollowing section

7 Simulation results

In this section we present the results obtained from the twononlinear string algorithms discussed in sections 3 and 5The synthesis results are compared by simulating the samephenomena namely the initial pitch glide and the genera-tion of missing harmonics using the two models Stabilityissues and computational cost of the synthesis models arealso discussed

71 Synthesis results

The synthesis results reveal that both the nonlinear DWGand NFDTD models are able to realistically model the ini-tial pitch glide phenomenon Figure 17 illustrates the fun-damental frequency behavior of a recorded kantele toneand the two synthesized tones Here the horizontal dottedline approximates the mean value of perceptual detectionthreshold of an initial pitch glide The psychoacoustic de-tection threshold in the frequency region of these tones isabout 54 Hz [38] This shows that the fundamental fre-quency glide is an audible phenomenon in plucked stringinstruments such as the kantele even at modest pluckingamplitudes and thus it must be included in a synthesismodel if realistic tones are desired

The nonlinear DWG model used in this figure has a totaldelay line length of 55125 samples and the allpass coef-ficient a is scaled using a constant value of 09 in orderto correctly simulate the behavior of the recorded sampleThe NFDTD string consists of 56 FDTD elements andthe fine-tuning parameter (aka Courant number equa-tion 17) has a value of r 13 The allpass coefficienta is scaled using a coefficient in the NFDTD case

The modeling of the generation of missing harmonicscan be implemented similarly in the distributed nonlinearDWG model as was suggested in [28] If the boxcar inte-gration of equation (10) is replaced with a leaky integratorhaving the transfer function

Iz gp ap

apz (27)

the generation of the missing harmonics can be controlledvia the integration parameter ap The variable gp definesthe gain of the integration

Figure 18 shows the amplitude envelopes of the firstthree harmonics of a synthesized tone with two differ-ent ap parameter values The string was plucked close tord of its length and as can be seen in the figure themissing harmonic in (a) has a gradual increase after thebeginning transient after which it experiences an expo-nential decay like all other modes

It is worthwhile to note that the generation of missingharmonics in the nonlinear DWG model results from theproperties of the integration of the elongation approxima-tion and is therefore not a physically justified process Ba-sically here the integration error in the leaky integrator is

321

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Time [s]

fnom

Fre

quen

cy[H

z]

Figure 17 Fundamental frequencies as a function of time fora moderately-plucked recorded kantele tone (solid line) a syn-thesized nonlinear DWG tone (dashed line) and a synthesizedNFDTD tone (dash-dotted line) The fnom stands for the nomi-nal fundamental frequency of the string and the horizontal dottedline denotes the approximated detection threshold of a pitch drift(fnomHz) which suggests that the fundamental frequencydrifts in all cases are audible

responsible for feeding energy to the missing harmonicsAlso unlike the real physical phenomenon the generationof missing harmonics in the nonlinear DWG case does notdepend on the rigidity of the terminations Neverthelessthis feature can be exploited in emulating the real stringbehavior when the integration parameters are properly ad-justed Details on tuning the leaky integrator parameterscan be found in [28]

Modeling the generation of missing harmonics in aNFDTD string is however not so simple Even if a leakyintegrator is used in the elongation calculation its param-eters do not have a desirable effect on the missing har-monics This does not seem too surprising when consider-ing the major differences of these two algorithms and it isthe reason that forced us to use an alternative mechanismfor creating the missing harmonics in the previous section(Figure 16)

Figure 19 represents the behavior of the first three har-monics of a tone synthesized by this model It can be seenthat the missing harmonics can be ldquoliftedrdquo by choosing aproper value for TMDF The stability of the system how-ever poses an upper limit for the TMDF coefficient sincethe TMDF mechanism continuously feeds energy to thestring According to our experience generating missingharmonics with amplitudes greater than what is shown inFigure 19 is difficult

72 Stability issues and computational comparison

We found the nonlinear DWG algorithm to remain sta-ble for nearly all parameter and excitation values Onlyhighly exaggerated nonlinearity scaling values togetherwith high excitation impulses resulted in stability prob-lems We thus conclude that the nonlinear DWG waveg-uide has no real stability problems when synthesis of nat-ural plucked-instrument sounds are desired

We studied the stability of the NFDTD algorithm us-ing the Von Neumann analysis [39] in the time-invariantcase ie parameter a of equation (26) was kept constantThe basic idea of this method is to calculate the spatialFourier spectrum of the system under discussion at twoconsecutive time steps An amplification function which

(a)

(b)

Figure 18 Generation of the missing harmonics in the nonlinearDWG model can be controlled via the leaky integrator parame-ters Here the string was plucked approximately at rd of itslength so every 3rd harmonic should be missing from the re-sulting spectrum In a) ap and the third harmonicclearly rises after the initial transient In b) ap 13 andthe third harmonic is more attenuated

(a)

(b)

Figure 19 Generation of missing harmonics in a NFDTD stringThe string was plucked again approximately at rd of itslength and the coupling of the TMDF to the transversal vibra-tion was controlled using a scaling coefficient TMDF In a) thescaling coefficient has a value of TMDF and the missingthird harmonic can be seen rising after the initial transient In b)TMDF and generation of missing harmonics does not takeplace

shows how the spatial spectrum evolves with time canthen be derived from the two spectra If the absolute valueof this amplification function remains below unity stabil-ity is guaranteed Formally the Von Neumann analysis forthe NFDTD algorithm goes as follows [4]

If the spatial inverse Fourier transform is defined as

ynm FfY n g neim (28)

322

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

y n-1m+1( )

y n-1m( )

F n( )

F n( )

y nm+1( )

y nm( )

y n+1m+1( )

y n+1m( )

Figure 14 Illustration of the FDTD interaction element togetherwith its abstraction The excitation algorithm is defined by equa-tions (21) and (22)

F n( )

FDTD FDTD FDTD FDTD FDTD

Allpass-coefficientapproximation

Elongationapproximation

Figure 15 One-polarizational NFDTD string The string consistsof the time-varying FDTD elements illustrated in Figure 12 Thezero-blocks at the terminations give zero as an output regardlessof the input values thus implying a rigid termination The excita-tion to the string can be inserted as a force signal using a FDTDinteraction element illustrated in Figure 14

F n( )

FDTD FDTD FDTD

TMDF

FDTD

Allpass-coefficientapproximation

Elongationapproximation

y n L( -1)nom

Figure 16 Illustration of the NFDTD string with a generationmechanism for missing harmonics A second interaction elementis added in order to feed the TMDF into the string The scal-ing coefficient TMDF controls the amplitude of the missing har-monics The string elongation is approximated from the displace-ments of each FDTD element

element and the rigid termination is one sample the ver-tical component of the TMDF can be seen to be equal tothe product of the displacement of the last FDTD elementand the tension Here we can replace the tension signalby the elongation signal and introduce a scaling coeffi-cient TMDF to control the amount of TMDF to be in-serted to the interaction element at the termination Thisis illustrated in Figure 16 The generation of missing har-

monics in a NFDTD model will be further discussed in thefollowing section

7 Simulation results

In this section we present the results obtained from the twononlinear string algorithms discussed in sections 3 and 5The synthesis results are compared by simulating the samephenomena namely the initial pitch glide and the genera-tion of missing harmonics using the two models Stabilityissues and computational cost of the synthesis models arealso discussed

71 Synthesis results

The synthesis results reveal that both the nonlinear DWGand NFDTD models are able to realistically model the ini-tial pitch glide phenomenon Figure 17 illustrates the fun-damental frequency behavior of a recorded kantele toneand the two synthesized tones Here the horizontal dottedline approximates the mean value of perceptual detectionthreshold of an initial pitch glide The psychoacoustic de-tection threshold in the frequency region of these tones isabout 54 Hz [38] This shows that the fundamental fre-quency glide is an audible phenomenon in plucked stringinstruments such as the kantele even at modest pluckingamplitudes and thus it must be included in a synthesismodel if realistic tones are desired

The nonlinear DWG model used in this figure has a totaldelay line length of 55125 samples and the allpass coef-ficient a is scaled using a constant value of 09 in orderto correctly simulate the behavior of the recorded sampleThe NFDTD string consists of 56 FDTD elements andthe fine-tuning parameter (aka Courant number equa-tion 17) has a value of r 13 The allpass coefficienta is scaled using a coefficient in the NFDTD case

The modeling of the generation of missing harmonicscan be implemented similarly in the distributed nonlinearDWG model as was suggested in [28] If the boxcar inte-gration of equation (10) is replaced with a leaky integratorhaving the transfer function

Iz gp ap

apz (27)

the generation of the missing harmonics can be controlledvia the integration parameter ap The variable gp definesthe gain of the integration

Figure 18 shows the amplitude envelopes of the firstthree harmonics of a synthesized tone with two differ-ent ap parameter values The string was plucked close tord of its length and as can be seen in the figure themissing harmonic in (a) has a gradual increase after thebeginning transient after which it experiences an expo-nential decay like all other modes

It is worthwhile to note that the generation of missingharmonics in the nonlinear DWG model results from theproperties of the integration of the elongation approxima-tion and is therefore not a physically justified process Ba-sically here the integration error in the leaky integrator is

321

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Time [s]

fnom

Fre

quen

cy[H

z]

Figure 17 Fundamental frequencies as a function of time fora moderately-plucked recorded kantele tone (solid line) a syn-thesized nonlinear DWG tone (dashed line) and a synthesizedNFDTD tone (dash-dotted line) The fnom stands for the nomi-nal fundamental frequency of the string and the horizontal dottedline denotes the approximated detection threshold of a pitch drift(fnomHz) which suggests that the fundamental frequencydrifts in all cases are audible

responsible for feeding energy to the missing harmonicsAlso unlike the real physical phenomenon the generationof missing harmonics in the nonlinear DWG case does notdepend on the rigidity of the terminations Neverthelessthis feature can be exploited in emulating the real stringbehavior when the integration parameters are properly ad-justed Details on tuning the leaky integrator parameterscan be found in [28]

Modeling the generation of missing harmonics in aNFDTD string is however not so simple Even if a leakyintegrator is used in the elongation calculation its param-eters do not have a desirable effect on the missing har-monics This does not seem too surprising when consider-ing the major differences of these two algorithms and it isthe reason that forced us to use an alternative mechanismfor creating the missing harmonics in the previous section(Figure 16)

Figure 19 represents the behavior of the first three har-monics of a tone synthesized by this model It can be seenthat the missing harmonics can be ldquoliftedrdquo by choosing aproper value for TMDF The stability of the system how-ever poses an upper limit for the TMDF coefficient sincethe TMDF mechanism continuously feeds energy to thestring According to our experience generating missingharmonics with amplitudes greater than what is shown inFigure 19 is difficult

72 Stability issues and computational comparison

We found the nonlinear DWG algorithm to remain sta-ble for nearly all parameter and excitation values Onlyhighly exaggerated nonlinearity scaling values togetherwith high excitation impulses resulted in stability prob-lems We thus conclude that the nonlinear DWG waveg-uide has no real stability problems when synthesis of nat-ural plucked-instrument sounds are desired

We studied the stability of the NFDTD algorithm us-ing the Von Neumann analysis [39] in the time-invariantcase ie parameter a of equation (26) was kept constantThe basic idea of this method is to calculate the spatialFourier spectrum of the system under discussion at twoconsecutive time steps An amplification function which

(a)

(b)

Figure 18 Generation of the missing harmonics in the nonlinearDWG model can be controlled via the leaky integrator parame-ters Here the string was plucked approximately at rd of itslength so every 3rd harmonic should be missing from the re-sulting spectrum In a) ap and the third harmonicclearly rises after the initial transient In b) ap 13 andthe third harmonic is more attenuated

(a)

(b)

Figure 19 Generation of missing harmonics in a NFDTD stringThe string was plucked again approximately at rd of itslength and the coupling of the TMDF to the transversal vibra-tion was controlled using a scaling coefficient TMDF In a) thescaling coefficient has a value of TMDF and the missingthird harmonic can be seen rising after the initial transient In b)TMDF and generation of missing harmonics does not takeplace

shows how the spatial spectrum evolves with time canthen be derived from the two spectra If the absolute valueof this amplification function remains below unity stabil-ity is guaranteed Formally the Von Neumann analysis forthe NFDTD algorithm goes as follows [4]

If the spatial inverse Fourier transform is defined as

ynm FfY n g neim (28)

322

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

Time [s]

fnom

Fre

quen

cy[H

z]

Figure 17 Fundamental frequencies as a function of time fora moderately-plucked recorded kantele tone (solid line) a syn-thesized nonlinear DWG tone (dashed line) and a synthesizedNFDTD tone (dash-dotted line) The fnom stands for the nomi-nal fundamental frequency of the string and the horizontal dottedline denotes the approximated detection threshold of a pitch drift(fnomHz) which suggests that the fundamental frequencydrifts in all cases are audible

responsible for feeding energy to the missing harmonicsAlso unlike the real physical phenomenon the generationof missing harmonics in the nonlinear DWG case does notdepend on the rigidity of the terminations Neverthelessthis feature can be exploited in emulating the real stringbehavior when the integration parameters are properly ad-justed Details on tuning the leaky integrator parameterscan be found in [28]

Modeling the generation of missing harmonics in aNFDTD string is however not so simple Even if a leakyintegrator is used in the elongation calculation its param-eters do not have a desirable effect on the missing har-monics This does not seem too surprising when consider-ing the major differences of these two algorithms and it isthe reason that forced us to use an alternative mechanismfor creating the missing harmonics in the previous section(Figure 16)

Figure 19 represents the behavior of the first three har-monics of a tone synthesized by this model It can be seenthat the missing harmonics can be ldquoliftedrdquo by choosing aproper value for TMDF The stability of the system how-ever poses an upper limit for the TMDF coefficient sincethe TMDF mechanism continuously feeds energy to thestring According to our experience generating missingharmonics with amplitudes greater than what is shown inFigure 19 is difficult

72 Stability issues and computational comparison

We found the nonlinear DWG algorithm to remain sta-ble for nearly all parameter and excitation values Onlyhighly exaggerated nonlinearity scaling values togetherwith high excitation impulses resulted in stability prob-lems We thus conclude that the nonlinear DWG waveg-uide has no real stability problems when synthesis of nat-ural plucked-instrument sounds are desired

We studied the stability of the NFDTD algorithm us-ing the Von Neumann analysis [39] in the time-invariantcase ie parameter a of equation (26) was kept constantThe basic idea of this method is to calculate the spatialFourier spectrum of the system under discussion at twoconsecutive time steps An amplification function which

(a)

(b)

Figure 18 Generation of the missing harmonics in the nonlinearDWG model can be controlled via the leaky integrator parame-ters Here the string was plucked approximately at rd of itslength so every 3rd harmonic should be missing from the re-sulting spectrum In a) ap and the third harmonicclearly rises after the initial transient In b) ap 13 andthe third harmonic is more attenuated

(a)

(b)

Figure 19 Generation of missing harmonics in a NFDTD stringThe string was plucked again approximately at rd of itslength and the coupling of the TMDF to the transversal vibra-tion was controlled using a scaling coefficient TMDF In a) thescaling coefficient has a value of TMDF and the missingthird harmonic can be seen rising after the initial transient In b)TMDF and generation of missing harmonics does not takeplace

shows how the spatial spectrum evolves with time canthen be derived from the two spectra If the absolute valueof this amplification function remains below unity stabil-ity is guaranteed Formally the Von Neumann analysis forthe NFDTD algorithm goes as follows [4]

If the spatial inverse Fourier transform is defined as

ynm FfY n g neim (28)

322

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

where is the spatial frequency and i is the imaginaryunit the nonlinear finite-difference recurrence equation(26) can be written as

neim paneim paneim

pneim qaneim pneim

paneim qneim

paneim qaneim (29)

Dividing with neim and rearranging we have

paei paei

pei qa pei

paei q paei qa (30)

Using the Eulerrsquos equation leads to a simpler form

A B CD (31)

where

A pa cos

B qa p cos

C q pa cos

D qa (32)

In order to get the amplification function we would nowhave to solve the third-order equation (31) Unfortunatelythe solution of this equation is complicated and involvesdozens of terms If we want to consider the stability of thelossless NFDTD string we can substitute p q Thissimplifies the solution of equation (31) enough to enablenumerical stability analysis for the amplification functionThe absolute value of the amplification function is il-lustrated in Figure 20 as a function of the interpolationcoefficient a and the spatial frequency

It is important to note that this stability analysis is con-ducted on a lossless NFDTD string with constant inter-polation coefficient We can thus call this system time-invariant (normally the interpolation coefficient dependson the string elongation)

Figure 20 reveals that in the lossless case the time-invariant version of the NFDTD algorithm is unstable forall but very small a parameter values Making the algo-rithm time-variant results in an even more unstable systemIn a practical lossy string implementation however theNFDTD string remained stable for normal excitation am-plitudes (ie excitation amplitudes commonly used whenplaying real string instruments)

The computational complexities of the two algorithmsare different Since the models consist mainly of the basicstring blocks (basic elements in the DWG case and FDTDelements in the finite difference case) the differences inthe computation of the basic string blocks dominate thecomputational needs of the algorithms

The basic element (Figure 4) consists of four multipli-cations and two summations per time sample whereas theFDTD element (Figure 12) requires a total of nine multi-plications and eight summations for computing one time

a

Figure 20 Absolute value of amplification function of a NFDTDalgorithm The white color denotes areas where the amplificationfunction exceeds unity ie when the model becomes unstable

sample Although the interaction and termination blocksare much simpler in the finite difference case the typi-cally large number of the string elements turns the favorto the nonlinear DWG model If the computational cost ofthe string elongation approximation is taken into accountthe NFDTD algorithm can be seen to have twice the com-putational complexity of its digital waveguide counterpartFor a more thorough comparison of the two presented al-gorithms see [4]

8 Conclusions and future work

Two algorithms for modeling spatially distributed non-linear strings in a physically meaningful way were pre-sented a nonlinear digital waveguide algorithm and a non-linear finite difference algorithm The former uses first-order allpass filters distributed along a delay line for mod-ulating the total delay of the string loop while the latterone uses first-order allpass filters for interpolating betweentime samples in the linear recurrence equation Both tech-niques evaluate the control signals for the allpass filtersfrom the elongation of the string The amount of nonlin-earity among with other physical parameters can be ad-justed in both string models A physical model of a kantelestring was presented using the nonlinear digital waveguidestring algorithm

Realistic simulation of the inital pitch glide phenome-non can be performed with both algorithms but model-ing of the generation of missing harmonics can be realisti-cally obtained only using the nonlinear digital waveguidemodel due to stability problems of the nonlinear finite dif-ference algorithm Computational complexities of the twoalgorithms were also compared

As stated in section 51 the explicit finite differencescheme was chosen for simplicity Another option wouldbe to use an implicit scheme such as a scheme [40]where the temporal and spatial derivatives of the waveequation (equation 1) are averaged in space and time re-

323

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

ACTA ACUSTICA UNITED WITH ACUSTICA Pakarinen et al Modeling nonlinear vibrating stringsVol 91 (2005)

spectively Using such a scheme would lead to an uncon-ditionally stable finite difference algorithm and thus lib-erate us from the Von Neumann stability condition (equa-tion 17) The implicit form of this scheme would howevercall for a matrix formulation instead of a simple recurrenceequation and probably increase the computational load ofthe algorithm Construction of such an algorithm is left forfuture work

AcknowledgementThanks to Dr Cumhur Erkut and Dr Lutz Trautmann forsuggestions and discussions This work was supported bythe ALMA project (IST-2001-33059) the Academy ofFinland project SA 104934 and the Helsinki GraduateSchool of Electrical and Communications Engeneering

References

[1] M Karjalainen C Erkut Digital waveguides vs finitedifference schemes Equivalence and mixed modelingEURASIP Journal on Applied Signal Processing (June2004) 978ndash989 Special issue on Model-Based Sound Syn-thesis

[2] C Erkut M Karjalainen Finite difference method vs dig-ital waveguide method in string instrument modeling andsynthesis Proceedings of the International Symposiumon Musical Acoustics (ISMA 2002) Mexico City MexicoDecember 9-13 2002

[3] J Pakarinen M Karjalainen V Valimaki Modeling andreal-time synthesis of the kantele using distributed tensionmodulation Proc Stockholm Music Acoustics ConferenceStockholm Sweden August 6-9 2003 409ndash412

[4] J Pakarinen Spatially distributed computational modelingof a nonlinear vibrating string Diploma Thesis HelsinkiUniversity of Technology June 14 2004 Available on-lineat httpwwwacousticshutfipublications

[5] N H Fletcher T D Rossing The physics of musical in-struments Springer-Verlag New York USA 1988

[6] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part I Journal ofthe Audio Engineering Society 19 (June 1971) 462ndash470

[7] A Chaigne A Askenfelt Numerical simulations of pianostrings I A physical model for a struck string using finitedifference methods Journal of the Acoustical Society ofAmerica 95 (February 1994) 1112ndash1118

[8] M Podlesak A Lee Dispersion of waves in piano stringsJournal of the Acoustical Society of America 83 (1988)305ndash317

[9] D Hall Piano string excitation in the case of small ham-mer mass Journal of the Acoustical Society of America 79(1986) 141ndash147

[10] D Hall Piano string excitation II General solution for ahard narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 535ndash546

[11] D Hall Piano string excitation III General solution for asoft narrow hammer Journal of the Acoustical Society ofAmerica 81 (1987) 547ndash555

[12] H Suzuki Model analysis of a hammer-string interactionJournal of the Acoustical Society of America 82 (1987)1145ndash1151

[13] X Boutillon Model for piano hammers Experimental de-termination and digital simulation Journal of the Acousti-cal Society of America 83 (1988) 746ndash754

[14] M E McIntyre J Woodhouse On the fundamentals ofbowed string dynamics Acustica 43 (1979) 93ndash108

[15] J Woodhouse Idealised models of a bowed string Acus-tica 79 (1993) 233ndash250

[16] L Cremer The physics of the violin MIT Press Cam-bridge MA 1983

[17] H A Conklin Generation of partials due to nonlinear mix-ing in a stringed instrument Journal of the Acoustical So-ciety of America 105 (January 1999) 536ndash545

[18] B Bank L Sujbert Modeling the longitudinal vibration ofpiano strings Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 143ndash146

[19] K A Legge N H Fletcher Nonlinear generation of miss-ing modes on a vibrating string Journal of the AcousticalSociety of America 76 (July 1984) 5ndash12

[20] T Tolonen C Erkut V Valimaki M Karjalainen Simula-tion of plucked strings exhibiting tension modulation driv-ing force Proceedings of the International Computer MusicConference Beijing China October 22-28 1999 5ndash8

[21] K Karplus A Strong Digital synthesis of plucked-stringand drum timbres Computer Music Journal 7 (1983) 43ndash55

[22] J O Smith Principles of digital waveguide models of mu-sical instruments Applications of Digital Signal Processingto Audio and Acoustics (M Kahrs and K Brandenburgeds) (February 1998) 417ndash466

[23] J O Smith Physical modeling using digital waveguidesComputer Music Journal 16 (Winter 1992) 74ndash87

[24] T I Laakso V Valimaki M Karjalainen U K LaineSplitting the unit delay - tools for fractional delay filter de-sign IEEE Signal Processing Magazine 13 (1996) 30ndash60

[25] V Valimaki T I Laakso Principles of fractional delay fil-ters Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Istanbul Turkey5-9 June 2000 3870ndash3873

[26] V Valimaki Discrete-time modeling of acoustic tubes us-ing fractional delay filters Doctoral dissertation HelsinkiUniv of Technol Acoustics Lab Report Series Reportno 37 1995 Available on-line at httpwwwacous-ticshutfipublications

[27] V Valimaki T Tolonen M Karjalainen Plucked-stringsynthesis algorithms with tension modulation nonlinear-ity Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Phoenix Ari-zona March 15-19 1999 977ndash980

[28] T Tolonen V Valimaki M Karjalainen Modeling of ten-sion modulation nonlinearity in plucked strings IEEETransactions on Speech and Audio Processing 8 (May2000) 300ndash310

[29] C Erkut M Karjalainen P Huang V Valimaki Acous-tical analysis and model-based sound synthesis of the kan-tele Journal of the Acoustical Society of America 112 (Oc-tober 2002) 1681ndash1691

[30] J R Pierce S A Van Duyne A passive nonlinear digitalfilter design which facilitates physics-based sound synthe-sis of highly nonlinear musical instruments Journal of theAcoustical Society of America 101 (February 1997) 1120ndash1126

[31] J Polkki C Erkut H Penttinen M KarjalainenV Valimaki New designs for the kantele with improvedsound radiation Proc Stockholm Music Acoustics Confer-ence Stockholm Sweden August 6-9 2003 133ndash136

324

Pakarinen et al Modeling nonlinear vibrating strings ACTA ACUSTICA UNITED WITH ACUSTICA

Vol 91 (2005)

[32] M Karjalainen V Valimaki T Tolonen Plucked-stringmodels From the Karplus-Strong algorithm to digitalwaveguides and beyond Computer Music Journal 22(1998) 17ndash32

[33] M Karjalainen BlockCompiler Efficient simulation ofacoustic and audio systems Proc 114th AES ConventionAmsterdam The Netherlands 22-25 March 2003

[34] L Hiller P Ruiz Synthesizing musical sounds by solvingthe wave equation for vibrating objects Part II Journal ofthe Audio Engineering Society 19 (June 1971) 542ndash551

[35] A Chaigne On the use of finite differences for musi-cal synthesis Application to plucked stringed instrumentsJournal drsquoAcoustique 5 (1992) 181ndash211

[36] M Karjalainen 1-D digital waveguide modeling for im-proved sound synthesis Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Pro-cessing Orlando Florida USA May 13-17 2002 1869ndash1872

[37] C Erkut M Karjalainen Virtual strings based on a 1-D FDTD waveguide model Stability losses and travel-ing waves Proceedings of the Audio Engineering Society22nd International Conference Espoo Finland June 15-17 2002 317ndash323

[38] H Jarvelainen V Valimaki Audibility of initial pitchglides in string instrument sounds Proceedings of the In-ternational Computer Music Conference Havana Cuba17-23 September 2001 282ndash285 Available on-line athttplibhutfiDiss2003isbn9512263149article3pdf

[39] J C Strikwerda Finite difference schemes and partial dif-ferential equations Wadsworth Brooks amp Cole CaliforniaUSA 1989

[40] A Chaigne V Doutaut Numerical simulations of xylo-phones I Time-domain modeling of the vibrating barsJournal of the Acoustical Society of America 101 (January1997) 539ndash557

325