Coupled string guitar models

AXEL NACKAERTS, BART DE MOOR, RUDY LAUWEREINSESAT−SISTA/ACCA, Dept. Elektrotechniek

Katholieke Universiteit LeuvenKasteelpark Arenberg 10, B−3001 Leuven

BELGIUMAxel.Nackaerts@esat.kuleuven.ac.be http://www.esat.kuleuven.ac.be/sista/

Abstract: An important aspect of the sound of the acoustic guitar is the beating caused by string interaction. Inthis paper, we present a coupled string model for the acoustic guitar that includes the body in the feedbackloop. Constraints on the model parameters are determined using Finite Element analysis of the saddle.

Key−Words: Music, Sound synthesis, Physical modeling, String Coupling, Waveguide

1 IntroductionOne important aspect that affects the natural qualityof the sound of a string instrument is the sympatheticcoupling of the strings. As one string is struck,plucked or bowed, the other strings of the instrumentstart to vibrate and interact with the excited string,changing the global decay or adding beating to somepartials of the sound. This is an important aspect as asound without some amount of beating is almostimmediately recognized as synthetic. Severalmathematical structures to simulate the sympatheticcoupling of strings have been proposed in the past.In this study, we propose a general guitar model thatincludes the guitar body. This gives the performerextra possibilities and makes the global model quiteeasy to calibrate. The paper starts with the commonwaveguide models in section 2 and builds it up to afully coupled model, including the body resonator.In section 3, rules for the coupling parameters arederived based on Finite Element analysis andphysical considerations.

2 ModelingTo build the coupled multi−string model, we startfrom the acoustic principles (2.1) and apply thecommon waveguide physical modeling technique(2.2) to build a coupled string model (2.3) and finallya fully coupled instrument model (2.4).

2.1 Acoustic principles of the guitarThe acoustic sound generation mechanism of the

classical guitar is fairly straightforward: a wave onthe string is reflected at the attachment points (thesaddle/bridge and the nut). This results in reactiveforces and a slight displacement of the attachmentpoints. The nut is fairly rigid (almost idealreflection), but the bridge is attached to the flexibletop plate, resulting in an energy transfer to the guitarbody (damping of the wave and movement of the topplate). The guitar is designed such that both thesustain (duration of the sound; lower damping resultsin longer sustain) and the loudness (more transfer tothe guitar body) are acceptable. The guitar body actsas a resonator and couples the string movement tothe surrounding air.

2.2 Waveguide modelThe most widely used physical modeling techniquefor string instruments is based on the generalsolution of the wave equation describing the stringmovement [1]. The resulting structures (digitalwaveguides) are physically relevant abstractions andcomputationally efficient models that can be used asbasic building blocks for larger models [2]. A goodoverview of the evolution of the digital waveguidetechnique applied to string instrument modeling canbe found in [3].

In digital waveguide modeling, the string isrepresented by two fractional delay lines, connectedby linear filters that model the reflection at the bridge

and nut. More recent models use only one delay line,with the the two reflectors lumped together in onepoint.

2.3 Coupled string modelAt the saddle/bridge point, all the strings inducemovement of the top plate. At the same time,movement of the top plate moves all the strings. Thismeans that the strings cannot be seen as independententities, but must be seen as parts of a larger, coupledsystem. Single delay−line (SDL) waveguideimplementations of a guitar with coupled strings havea structure as shown in figure 1. Because of the dualpolarization (vertical and horizontal movement of thestrings), we need N=12 fractional delay lines torepresent the complete guitar. In the most generalcase, the delay line input y is calculated as the sumof the system input I and the vectorized output y ofthe delay lines multiplied by a coupling matrix C.

y=C⋅y+I (1)The elements ckm of the coupling matrix represent the(lumped) transmission from delay line m to delay linek, and the elements ckk represent the reflection fordelay line k. Note that, in general, the ckm are complexand frequency−dependent and should, therefore, beimplemented as linear filters. A physically soundvalue for the elements of the coupling matrix can bederived from the bridge admittance (bridge coupledmodel) [4]. In this model, the admittance measured atthe bridge is seen by all the strings and results in a

common movement. As a result, the elements of thecoupling matrix are quite complicated expressions.

2.4 Fully coupled instrument modelThe general structure basically consists of a couplingnode and a series of resonators. We can add an extraresonator to the structure: the body itself, as can beseen on figure 2. This is fairly similar to the N−dimensional loaded waveguide junction [5], but withfeedback to the same junction. Adding the body as a"pure" resonator has a great advantage. It is nowpossible to give an independent input to the body(e.g. a slap, as in flamenco playing) that will causeall the strings to vibrate. The coupling matrix nowhas a slightly different interpretation.

It is clear from the bridge−coupled model that thecoupling matrix does not represent the bridge assuch, but the combination of body and bridge (theadmittance of the whole system measured at thebridge). Existing models include the guitar body onlyas a pre− or postfiltering operation [3]. If we includethe body as an independent resonator, we effectivelysimplify the coupling matrix as we represent only thebridge. In the acoustic guitar case, the saddle/bridge

Fig. 1 A general waveguide structure with stringcoupling through a coupling matrix M. Elements of Mare complex and frequency−dependent.

~y yId1

d2

d3

dN

M

Fig. 2 Adding the body itself as a resonator with fullcoupling to strings. The coupling M now effectivelyrepresents the saddle/bridge.

~y yId1

d2

d3

dN

Body Model

M

is a fairly rigid construction, with vibration modes ofrather high frequency compared to the body: this willlead to nearly frequency independent coefficients.We have an added degree of freedom to describe thecoupling: it is possible to couple strings throughdirect bridge coupling, without influence of the body(see figure 3). Grouping the horizontal and verticalpolarizations gives the global coupling matrixstructure shown in figure 4.

We will now build several waveguide models, byusing the complete coupling matrix and byapproximating it without sacrificing too muchquality.

3 CalibrationA valid approximation of the coupling matrix can befound if we have a better understanding of thecoupling mechanism itself. To analyze the couplingbetween the strings, we construct a finite elementmodel of the saddle of the guitar using the Aladdinsoftware package [6], and calculate the displacementand internal stresses for the static load and for unitforces in X and Y directions. Based on these results,we simplify the coupling matrix and estimate thecoefficients.

Fig. 3 The different classes of coupling possible with thefully coupled instrument model.

hor

ver

hor

ver

body

cou

plin

g

direct coupling

intr

astr

ing

intr

astr

ing

Inte

rstr

ing

Fig. 4 The coupling matrix C after inclusion of the body.The polarizations are separated. The distinction betweendirect bridge coupling and body coupling is now clear.

BH

oriz

onta

l pol

.V

ertic

al p

ol.

body

−>ve

rtic

albo

dy−>

hori

zont

al

vertical −> horizontal

ver −> ver

ver −> ver

horizontal −> vertical

hor −> hor

hor −> hor

horizontal−>body vertical−>body bb

vertical reflection

horizontal reflection

BH

oriz

onta

l pol

.V

ertic

al p

ol.

= .

Fig. 5 Schematic view of the saddle. The two lowercorners of the finite element model of the saddle are fixednodes (allow no movement in X or Y direction).

y x

Fig. 6 The static stresses inside the saddle obtained byfinite element analysis. The model is based onmeasurements of the saddle /bridge of a Yamaha C70acoustic guitar using d’Addario Pro Arte 45 stringsunder normal tension. The saddle material was assumedbone (E=1.7GN/m2). Its dimensions are 78mm× 7mm ×2.5mm. (a) σxx stress in X direction, (b) σyy stress in Ydirection.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

5

x 10−3

0

0.5

1

1.5

2x 105

(a)

σ xx

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

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(b)

σ yy

3.1 Finite element analysisFigure 5 shows a schematic view of the analyzedstructure. The lower two corners of the saddle arefixed in X and Y direction (this is needed for a validfinite element simulation). We assume that adisplacement in Y direction at the bottom of thesaddle results in movement of the top plate in the Ydirection only. This is a valid assumption because thetop plate is flexible compared to the saddle/bridge (inY direction) and as rigid as the saddle/bridge (in theX direction).

We first determined the stresses at rest with sixstrings attached. The forces caused by the strings onthe saddle were calculated for d’Addario Pro Arte 45normal tension strings. Figure 6 shows the results.The experiment was repeated for a unit force in the Ydirection (figure 7) and in the X direction (figure 8).

The results for a load in Y direction (string moving invertical polarization) show that this load results instress in both the X and Y direction and that theeffect of the load is fairly localized around the loadposition (load point x=33mm, substantial stressbetween 26mm<x<40mm). This means that almostall of the force applied by the string is transmitted tothe top plate, without directly influencing the otherstrings. Of course, the displacement of the top platedue to the load will influence all the strings. Weconclude that interstring vertical−vertical couplingproceeds almost entirely through body coupling.

A load in the X direction (string moving in horizontalpolarization) gives a different picture. Again, there isstress in both X and Y direction, but the effect is notlocalized around the load point. The whole bridgesees a stress (and the related displacement). The Xcomponent of the stress results in a direct coupling inthe X direction for all the strings. As the Ycomponent in this case is rather small compared tothe Y component when loading in the Y direction, itresults in less transmission to the top plate and thus alower damping. We conclude that interstringhorizontal−horizontal coupling happens throughdirect coupling and interstring horizontal−verticalcoupling through a combination of direct and bodycoupling.

Intrastring coupling is somewhat more difficult to

determine. A load in the Y direction results in asymmetric stress pattern in the X direction. A load inthe X direction creates an asymmetric stress in the Ydirection. We believe that this leads to strongerhorizontal−vertical intrastring coupling thanvertical−horizontal intrastring coupling.

These calculations are also valid for the dynamiccase with changing load. The time constant toachieve stress equilibrium in the saddle is very small

Fig. 7 The effect of unit force in the Y direction at thethird string. The internal stresses are symmetric in the (a)X and (b) Y direction. The influence at y=0 (on the topboard) is very small in the X direction but quite large inthe Y direction.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

2

4

6

x 10−3

0

1000

2000

3000

4000

(a)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

2

4

6

x 10−3

0

1000

2000

3000

4000

(b)

Fig. 8 The effect of unit force in the X direction. Theinternal stresses are asymmetric in the (a) X and (b) Ydirection. Note that the stress in X or Y direction is notlocalized around the load point.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

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−2000

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compared to the frequency of the load change (this isvalid up to 10kHz). This is due to the high speed ofsound in bone, combined with the small dimensionsof the saddle.

Including these results in the general structure of thecoupling matrix yields the simplified matrix shown infigure 9. Hence:

H input = M16×6⋅H output+ I hor

V input = M213×6⋅H

output

V output

Boutput

+ I ver

Binput = M313×1⋅H output

V output

Boutput

+ I body

(3)

where H, V and B are the inputs and outputs ofrespectively the horizontal polarization, the verticalpolarization and the body. The bridge displacementsare in phase with the string movement, so 0<cjk<1.

3.2 Parameter estimationWe now have to determine the numerical values for

the coupling coefficients. The first step is todetermine the reflection coefficients for the differentpolarizations. A good approximation is to calculatethe damping for the fundamental frequency of eachopen string, using either Short−Time FourierTransform or subband Hankel Singular ValueDecomposition based methods [3,7,8]. Theamplitude of each polarization of the fundamentalcan be approximated by

A=exp Ba⋅t , (3)and the corresponding reflection coefficient is

ckk=exp Ba⋅Lk ⁄F s , (4)with ckk the reflection coefficient for waveguide k, athe damping, Lk the length of the waveguide and Fs

the sampling frequency.

If we consider the bridge as lossless, the sum of the"outgoing" displacements equals the "incoming"displacement. In terms of coupling coefficients, thismeans that

∑k

ckm=1 , (5)

or the column sums of the coupling matrix are one.The coupling matrix elements ckk are the result oflumping the reflection at both string ends together.An approximation for the reflection at thesaddle/bridge only is the square root of ckk. As wealready know the value of ckk, we can easilydetermine the values for the vertical−to−bodycoupling.

cV→B=1B cV refl.. (6)

The cH→H coefficients depend on the distance of the"source" string to the "target" string as can be seenon figure 8. We model this by taking the coefficient(linearly) proportional to the distance. The sameholds for cH→V. According to the finite elementanalysis, the displacement in the horizontaldimension is about three times larger than in thevertical dimension, which leads to the ratio

cH →H=3⋅cH →V . (7)The transfer from the horizontal polarization to theinstrument body is more difficult to determine. Notethat coupling to the body is the only way ofgenerating output (the body couples the stringmovement to the surrounding air). It has been foundthat

Fig. 9 Including the results of the finite element analysissimplifies the coupling matrix.

BH

oriz

onta

l pol

.V

ertic

al p

ol.

BH

oriz

onta

l pol

.V

ertic

al p

ol.

= .

0M1

M3

M2

BH

oriz

onta

l pol

.V

ertic

al p

ol.

= .

body

−>ve

rtic

al

horizontal −> vertical

hor −> hor

hor −> hor

horizontal−>body vertical−>body bb

vertical reflection

horizontal reflection

0

0

0

~0

cH →B=0.1⋅ 1B cH refl.(8)

gives acceptable results. Given equation (8) andequation (5), cH→V and cH→H must now comply to∑

k

cH →V+∑k

cH →H=0.9⋅ 1B cH refl. (10)

The last coefficients to be determined are cB→V. Thesecan be estimated by exciting the guitar body with animpulse and recording the output. The ratio of stringsound after the attack vs. body impulse amplitudegives an estimate of the coupling. Figure 10 showsthe model output when the body is excited with animpulse.

4 ConclusionThe generation of natural acoustic guitar soundsrequires a coupled string model. In this work, weincluded the guitar body as a resonator in a coupledstring digital waveguide model of the acoustic guitar.This simplifies the coupling matrix to scalar values.Finite element analysis of the saddle/bridge structureprovided a better insight of the coupling and enabledthe determination of the coupling coefficients. Theresulting model has a very natural sound and is easierto calibrate than bridge−coupled models.

5 AcknowledgmentsAxel Nackaerts is a Research Assistant with the I.W.T.(Flemish Institute for Scientific and TechnologicalResearch in Industry) and can contacted at tel.(+32) 16 321800 or fax. (+32) 16 321970. Bart De Mooris Full Professor at the Katholieke Universiteit Leuven.Rudy Lauwereins is Full Professor at the K.U.Leuven.

This work is supported by several institutions: theFlemish Government (Research Council KUL (GOAMefisto−666, IDO), FWO (G.0256.97, G.0240.99,G.0115.01, Research communities ICCoS, ANMMM,PhD and postdoc grants), Bil.Int. Research Program, IWT(Eureka−1562, Eureka−2063, Eureka−2419, STWW−Genprom, IWT project Soft4s, PhD grants)), FederalState (IUAP IV−02, IUAP IV−24, Durable developmentMD/01/024), EU (TMR−Alapades, TMR−Ernsi, TMR−Niconet), Industrial contract research (ISMC, Data4s,Electrabel, Verhaert, Laborelec) and by TexasInstruments (TI−Elite program). ESAT−K.U.Leuven ismember of DSP Valley. The scientific responsibility isassumed by the authors.

References:[1] N.H. Fletcher and T.D. Rossing, The Physics of

Musical Instruments, New York, Springer−Verlag,1991.

[2] Julius O. Smith III, Physical Modeling using DigitalWaveguides, Computer Music Journal, Vol.16,No.4, 1992, pp. 74−91.

[3] Matti Karjalainen, Vesa Välimäki, and TeroTolonen, Plucked−String Models: From theKarplus−Strong Algorithm to Digital Waveguidesand Beyond, Computer Music Journal Vol. 22, No.3, pp. 17−32, Fall 1998

[4] J.O. Smith, Efficient Synthesis of Stringed MusicalInstruments, Proc. of the International ComputerMusic Conference, San Francisco, 1993.

[5] S.A. Van Duyne, J.O. Smith, Physical Modelingwith the 2−D Digital Waveguide Mesh, Proc. of theInternational Computer Music Conference, Tokyo,Japan, pp. 40−47, 1993

[6] M.A. Austin, X.G. Chen, and W.J. Lin, Aladdin: AComputational Toolkit for Interactive EngineeringMatrix and Finite Element Analysis, ISR TechnicalResearch Report TR95−74, University of Maryland,1995

[7] B. Banc, Physics−based sound synthesis of thepiano, M.S. Thesis, Budapest University ofTechnology and Economics, May 2000, published asReport 54 of Helsinki University of Technology,Laboratory of Acoustics and Audio SignalProcessing.

[8] A. Nackaerts, B. De Moor, and R. Lauwereins,Parameter estimation for dual−polarization pluckedstring models, Internal Report 01−17, SISTA, Dept.Elektrotechniek, Katholieke Universiteit Leuven,http://www.esat.kuleuven.ac.be/sista/

Fig. 10 (Top) Total model output when the body is excitedwith an impulse. (bottom) The sympathetic vibration ofthe strings.

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