Coupled Modes and Time-Domain Simulations of a Twelve-String Guitar with a Movable Bridge

Miguel Marques ?, Jose´ Antunes ?, Vincent Debut ?

? Applied Dynamics Laboratory, Campus Tecnologico´ e Nuclear, Instituto Superior Tecnico´ Universidade Tecnica´ de Lisboa, Estrada Nacional 10, 2695-066 Bobadela, Portugal and Mechanical Engineering Department, Instituto Superior Tecnico´ Universidade Tecnica´ de Lisboa, Av Rovisco Pais 1, 1049-001 Lisboa. Portugal

[email protected], [email protected], [email protected]

ABSTRACT model the dynamics of an archtop twelve strings acoustic guitar. For such purpose, we built a conceptual model Coupling between the different vibrating sub-systems of a for a set of twelve (six pairs of) strings coupling, at some musical instrument is an important feature in music acous- point of the strings (where the bridge would be located), tics. It is the reason why instruments of similar families to a body, through the bridge, which will be modelled as a have such different and characteristic sounds. In this work, spring with a very large stiffness. In this paper, the body we propose a model for a twelve strings (six pairs) guitar, will be simplified as a thin wood plate. Other coupling such that the strings are coupled with the instrument body techniques have already been proposed in the literature of through the moving bridge, which is the relevant compo- musical acoustics. Specifically for guitars, different mod- nent for energy transmission from the strings to the guitar els, which do not take into account the vibrations of the body and back. In this preliminary study, the guitar body is dead side of the string, have already been proposed [1–4], modelled as a simple plate, strings being assumed to dis- but to our knowledge, the sympathetic vibrations of a gui- play planar-only vertical motions. However, the coupled tar with a movable bridge, displaying a dead side of the equations thus obtained can be readily extended to cope string after the bridge, has never been addressed nor mod- with real guitar body modes and orbital string motions. Af- elled. A model of coupling between one string and the ter obtaining the coupled modes of the instrument, we il- violin body, which has the same features of our model, has lustrate the instrument time-domain coupled dynamics, by been proposed by some of the present authors, in order to considering the characteristic modal frequencies typical of address the problem of the wolf note and the nonlinear be- a Portuguese guitar. In particular we show how, when only haviour of the string/bow interaction [5]. one string alone is plucked, energy is transmitted to all other strings, causing sympathetic vibrations, which con- The Portuguese guitar (Cithara lusitanica) is a pear-shaped tribute to give this guitar its own characteristic sound iden- instrument with twelve metal strings (six courses), descen- tity. dant from the renaissance european cittern. This instrument is widely used in Portuguese traditional music, mainly 1. INTRODUCTION in Fado, and more recently also started to play a consider- able role among urban Portuguese musicians. Unlike most The acoustic guitar has been the subject of many studies, common guitars, this guitar has a bent soundboard (arched both on its theoretical description and on the experimen- top) with a bridge somewhat similar, although smaller in tal side. Yet, there still are limitations when it comes to size, to the bridge of a violin, a neck typically with 22 fixed the description of real guitars with movable bridges, i.e. metal frets and it is tuned by a fan-shaped tuning mecha- when the bridge role is not simplified as a simple enforced nism, consisting in twelve screws, acting as pegs, mounted pinned boundary condition. Overcoming this limitation, with small gliding pins where the strings are attached to could surely help improving the guitar engineering, e.g. it adjust its tension. It has the typical tuning of the Euro- could allow us to optimise the bridge position in order to pean cittern tradition, and has kept an old plucking tech- achieve the best radiative acoustic response, without loss nique, described in sixteenth century music books. The of the archtop guitar sound characteristics. first courses are composed by plain steel strings and tuned Motivated by the wish to extend our understanding of in unison, and the remaining are combinations of a plain the acoustics of the Portuguese guitar, we set ourselves to steel string and an overspun copper on steel string tuned one octave bellow. There are two different models of the Copyright: c 2013 Miguel Marques et al. This is an open-access article distributed Portuguese guitar: the Lisbon guitar and the Coimbra gui- under the terms of the Creative Commons Attribution 3.0 Unported License, which tar, named after two Portuguese cities where the two most permits unrestricted use, distribution, and reproduction in any medium, provided important Fado styles emerged. They differ in some de- the original author and source are credited. tails, such as the body measurements, the string length (the where j = 1, ..., J is related to the number of nodal lines along the xx direction, l = 1, ..., L is the number of nodal lines along the yy direction, mjl are the modal masses, ωjl are the modal angular frequencies, ζjl are the modal damping coefficients, and Fjl are the generalized forces. To simplify the formulation, these equations can be or- ganised in order to have a single modal index g, with a total of G = J × L modes, by ordering the frequencies in an ascendent form. The modal masses of the body are LxLy mg = ρ 4 , with ρ being the wood surface density, and Lx,Ly the plate dimensions. Each string will have its own set of modal dynamical equations, given as

s s 2 s s mnq¨n(t) + 2mnωnζnq˙n(t) + mnωnqn(t) = Fn(t), (3) Figure 1. The master musician Carlos Paredes, playing the where n = 1, ..., N are the modes of each string s. The Coimbra Portuguese Guitar. Photo credits: Egidio Santos. s Ls modal masses of each string are mn = µ 2 , with µ being the linear density of the string, and Ls the size of the string Coimbra guitar has a larger arm), and the Lisbon guitar is (throughout this work, we will assume that all strings have tuned (each string) a whole tone above the Coimbra guitar. the same size, which is the case for the Portuguese guitar). Strings used in these two guitars are therefore different in Furthermore, the string and the body will have modeshapes size and linear mass. More about the Portuguese guitar can respectively given by be read in [6, 7] and about experimental vibrational mea- nπx surements of this instrument in [8]. In figure1, a Coimbra ψs (x ) = sin s , n s L model of the Portuguese guitar is shown as an example. s jπx lπy ψb (x , y ) = sin b sin b , (4) jl b b L L 2. GUITAR MODEL x y and eigenfrequencies (assuming no inharmonicity) respec- Throughout this section, we will give a full description of tively given by the computational model proposed in this paper. For simplicity, in this conceptual model, we will replace the guitar s s ωn = nω0, body by a thin wood rectangular soundboard plate. Al- s " # E jπ 2 lπ 2 though it would be easy to implement in the model, we ωb = h y + , (5) jl b 2 will neglect any energy transfer occurring through the nut 12ρ 1 − vp Lx Ly or the fingerboard, as there are experimental studies reveal- where ωs is the s string fundamental frequency, h is the ing that this effect is not much relevant, and we will also 0 b plate height, E is the Young bulk modulus, and v is the assume the energy transfer occurring at the tailpiece to be y p the Poisson coefficient. Throughout the paper, we will re- negligible, due to its location on the Portuguese guitar ge- fer to ωb as ωb, and to ψb (x , y ) as ψb(r), assuming a ometry. Therefore, our fully coupled system will consist jl g jl b b g proper indexation mapping g → (j, l), according to what in a rectangular plate with fixed ends, and twelve strings, has been previously stated. Given a general force F (x, t), attached to the plate trough a set of springs, displayed in acting on a surface S of a vibrating system with mode- six pairs. For simplicity, we will only consider string vi- shapes ψ (x), the corresponding forces on the modal space brations in the direction normal to the soundboard plate, m will be although it would be relatively straightforward to imple- Z ment the other direction as well. We will decompose the Fm(t) = F (x, t)ψm(x)dS. (6) S strings and the plate displacements respectively as s b s Let us define Fc (xc, t) and Fc (rc, t) as being the force ex- ∞ erted on the bridge, as seen, respectively, from each string X s s s Ys(xs, t) = ψn(xs)qn(t), and the from body at each coupling point rc (we are as- n suming that all strings will meet the bridge at the same dis- ∞ tance from the nut, xc). We will consider that the bridge X Z (x , y , t) = ψb (x , y )q (t), (1) his thin enough such that it can be modelled as a spring b b b j,l b b j,l with very large stiffness. Therefore, these forces are j,l s s s Fc (xc, t) = Kc [δ(xs − xc)Ys(xc, t) + δ(r − rc)Z(rc, t)] with the script s = 1, ..., S, for a total of S strings; where h s s i +Cc δ(xs − xc)Y˙s(xc, t) − δ(r − r )Z˙ (r , t) Y is the displacement for each string, Z is the displace- c c s b 8 9 Ns ment of the body, ψ stand for the modeshapes and q stand < X = = δ(x − x ) ψs (x )[K qs (t) + C q˙s (t)] for the modal amplitudes. In this modal formulation, the s c m c c m c m :m=1 ; plate dynamics is governed by ( G ) X h i +δ(r − rs) ψb (rs) K qb (t) + C q˙b (t) b b 2 b b c m c c m c m mjlq¨jl(t) + 2mjlωjlζjlq˙jl(t) + mjlωjlqjl(t) = Fjl(t), m=1 (2) (7) where Kc,Cc are, respectively, the (very large) stiffness given the matrices constant and the damping constant of the bridge, and  s  ψ (xc)  1  F b(rs, t) = −F s(x , t), ∀s ∈ [1,S] (8) s . s s c c c c [Φc] = . . ψ1(xc) . . . ψn(xc) ,  s  ψn(xc) The modal projections forces (7) are given by  s   ψ1(xc)  s−b  .  b s b s N Φc = − . . ψ1(rc) . . . ψg(rc) , s X s s s s F (t) = [Kcq (t) + Ccq˙ (t)] ψ (xc)ψ (xc)  s  n m m m n ψn(xc) m=1  ψb(rs)  G  1 c  X b b b s s s−b  .  s s − Kcqm(t) + Cbrq˙m(t) ψm(rc)ψn(xc), Φc = − . . ψ1(xc) . . . ψn(xc) , m=1  b s  ψg(rc) S G X X  b s  b b b b s b s ψ1(rc) Fg (t) = Kcqm(t) + Ccq˙m(t) ψm(rc)ψg(rc) S   b X  .  b s b s s m=1 Φc = . . ψ1(rc) . . . ψg(rc) . S Ns s=1  b s  X X ψg(rc) − [K qs (t) + C q˙s (t)] ψs (x )ψb(rs). c m c m m c g c (14) s m=1 (9) The eigenvalues of (12) are related to the modal frequencies and modal damping coefficients as We can rewrite this equations as the linear system of ODEs q ¯ ¯ ¯2 [M]{Q¨(t)} + [C]{Q˙ (t)} + [K]{Q(t)} = {F(t)}, (10) λk = −ω¯kζk ± iω¯k 1 − ζk . (15) with the matrices and vectors built from the model param- From this relationship, we can deduce the guitar modal eters from all strings and the body damped frequencies, modal damping coefficients and modeshapes to be s s b b [M] = diag(m1, . . . , mN , m1, . . . , mG), ¯ ω¯d k = =(λk), (16) s s s s s s b b b b b b [C] = 2 · diag(m1ω1ζ1 , . . . , mN ωN ζN , m1ω1ζ1, . . . , mGωGζG), s s 2 s s 2 b b 2 b b 2 [K] = diag(m1ω1 , . . . , m ω , m1ω1 , . . . , m ω ), ¯ N N G G <(λk) n oT ζ¯ = − , (17) s s b b k ¯ {Q(t)} = q1(t), . . . , qN (t), q1(t), . . . , qG(t) , (11) λk T n s s b b o {F(t)} = F1 (t),...,FN (t),F1 (t),...,FG(t) . and

Notice that, each string will have its own, independent, S Ns ¯ X X k s ψk(x) = q¯ Θ((s − 1)Ls < x < sLs)ψ (x) modal family Ns, and therefore, the dimension of the ma- m m PS s m=1 trices and vectors in (12) will be D = G + s Ns. G X k b + q¯mΘ(x > SLs)ψm(r), (18) 2.1 Frequency Domain Analysis m=1

λ¯ t Considering the eigensolutions qk(t) =q ¯ke k , we can where Θ(α) is an Heaviside-like step function, such that transform the system (10) in an eigenvalue problem, be- it is equal to one when the argument α is true, and zero coming otherwise (notice that no subsystems are being summed), k th th q¯m is the m term of the k eigenvector, and x is defined „» – «  ff [0] [I] ¯ q¯k as −1 −1 − λk [I] ¯ = {0}, −[M] [K¯ ] −[M] [C¯] λkq¯k (12) x if Θ ((s − 1)L < x < sL ) = 1 ¯ ¯ x = s s s (19) where [K] and [C] are the effective stiffness and damping r if Θ(x > SLs) = 1 matrices, given by [K¯ ] = [K] + Kc[Φc] and [C¯] = [C] + Cc[Φc], and [Φc] is the coupling matrix, In short, the coupled system eigenfunctions (18) correspond to the modeshapes of the S coupled strings and the mode-  s1 s1−b  [Φc ] [0] [0] ... [Φc ] shapes of the coupled soundboard, while (19) corresponds s2 s2−b  [0] [Φc ] [0] ... [Φc ]  to the coordinate of each string when dealing with the cou-    .. .  pled strings modeshapes, and corresponds to the coordi- [Φc] =  [0] [0] . .  ,   nates of the soundboard when dealing with the coupled  . . .. .   . . . .  soundboard modeshapes. As for the original decoupled b−s1 b−s2 b Φc [Φc ] ...... [Φc] modeshapes, we normalized (18) such that the maximum (13) amplitude of its absolute value is one. 2.2 Time Domain Analysis 3. RESULTS 2.2.1 Initial Conditions Throughout this section, we will consider simulations us- We will assume that the initially, all strings are static, hav- ing the typical values of a Lisbon Portuguese guitar. All 44 cm ing zero initial displacements as well as zero initial veloci- strings have a length of from the nut to the bridge, 17.5 cm ties, and there will rather be an initial excitation force act- and from the bridge to the stop tailpiece (total 61.5 cm 1 ing on one or more strings (i.e. a finger or a nail plucking length is ). Table contains the strings notes, the the string, or a pick striking the string) during a short initial corresponding frequencies according to the standard tun- time interval. We will model the excitation force in such a ing of the Lisbon Portuguese guitar, and the correspond- way that its modal projection (over the excited string modal ing linear masses. Notice that the frequencies shown in 1 space) will be table represent the fundamental frequency at which the active part of the string, i.e. the length between the nut f n ˆ ˜ h ˙ ˙ io s Fn (t) = Kf Zf (t) − Ys(xf , t) + Cf Zf (t) − Ys(xd, t) ψn(xf ), and the bridge, should be vibrating; the actual frequency at (20) which the full-length string is tuned will be given by mul- 44 where Kf ,Cf are respectively the finger/nail/pick stiff- tiplying the presented value by the factor 44+17.5 . Based ness constant (here taken to be very large) and damping on the average results obtained in experimental identifi- constant, Zf (t) is the finger/nail/pick displacement, and cations, all strings will have equal damping coefficients s Ys(xf , t) is the string displacement at the point where it is ζn = 0.05% ∀n ∈ Ns. Each string s will have a total num- plucked/struck. This displacements are, respectively, ber of degrees of freedom Ns such that the frequency of the highest mode will be f ∼ 10kHz. As for the body, we Z Ns ˙ max tmax Zf (t) = Zf t = t|tmin , (21) tmax strings pair 1st 2nd 3rd 4th 5th 6th and notes b4 b4 a4 a4 e4 e4 b4 b3 a4 a3 d4 d3 Ns frequency 493.88 440 329.63 493.88 440 293.66 X s s (Hz) 493.88 440 329.63 246.94 220 146.83 Ys(xf , t) = qm(t)ψm(xf ). (22) linear density 3.78 3.94 6.20 3.78 3.94 11.30 m=1 (10−4 kg/m) 3.78 3.94 6.20 14.48 21.22 35.36 This choice allow us to easily perform simulations in which number of 28 32 42 28 32 27 modes used 28 32 42 57 64 95 different strings are plucked/struck at different times. 2.2.2 Time-step integrating procedure Table 1. Notes and corresponding frequencies of the Lis- We start by reducing the ODEs system (10) to a system bon Portuguese guitar standard tuning. of first order ODEs, based on the unconstrained subsystem modal responses will assume the soundboard plate to be squared with 30 cm by side, surface density ρ = 0.5 kg/m2, ζb = 1% ∀g ∈ Q˙ (t) [0] [I] Q(t) g = −1 ¯ −1 ¯ G, and we adjust the parameters Ey, vp and hb such that Q¨(t) −[M] [K] −[M] [C] Q˙ (t) b the first body frequency will be f1 = 275 Hz (which is 0 one of the values measured in [8] for a Portuguese guitar + . (23) Ff (t) body); also, we have considered 36 modes, such that the maximum body frequency is ∼ 10KHz. The total num- We will refer to (23) as ber of modes considered for the system is therefore 543. p˙(t) = A p(t) + Fe(t). (24) The strings of each pair will be separated by a distance of 4mm, and each pair will be separated by a distance of The analytical solution of (23) is 8mm (these are the typical values chosen by portuguese Z t guitar luthiers). A(t−t0) At −Aτ p(t) = p(t0)e + e e Fe(τ)dτ. (25) t0 3.1 Frequency Domain Results Assuming a very short time step ∆t = t − t0 → 0, it becomes possible to approximate Fe(t) as being constant In Figure2, we show the modeshape of the first fully cou- during each ∆t. We can then discretize equation (25), ob- pled system mode, which is dominated by the 11th string, taining the numerical solution the lowest tuned frequency of the system. Except for the 11th string, all other strings will have qualitatively equiva- pti+1 = pti eA∆t + A−1 eA∆t − I F ti , (26) e lent vibrations to that of the 1st string in this system mode, where I is the identity matrix. Notice that at each step as detailed in the figure. In Figure3, we show the mode- ti−1 ti, one must recompute the vector Fe , using the results shape of the fifteenth system mode, which is dominated by pti−1 , according to what as previously stated in (20). This the 1st string. In this mode, we observe significant vibra- method is quite stable (for the considered external forces tions in all strings tuned to b4 or b3. We find that the first Fe(t)) if given an accurate calculation of the state tran- pair of strings has a rather different phase than the b4 string sition matrix eA∆t. As to accuracy, this method provide of the fourth pair, and on the other hand, the b3 string of highly accurate results up to the time step bound ∆t ≤ the fourth pair, which frequency corresponds to the second Tmin/10, where Tmin is the smallest period of the system mode of the b4 strings, share the same phase as the first (the period of the highest considered mode). pair, and has an amplitude considerably smaller than that Figure 2. First mode of the coupled system. The red dots Figure 3. Fifteenth mode of the coupled system. The red represent the bridge position. Upper plot, string modal re- dots represent the bridge position. Upper plot, string modal sponses; medium plot, detail of the 1st string modal re- responses; medium plot, detail of the 11th string modal sponse; lower plot, detail of the body modal response. response; lower plot, detail of the body modal response. of the b4 strings. The eleventh string (which is the d3 string Although typically twelve-string guitar players would pluck of the sixth pair), is interestingly vibrating in its fifth mode a string pair rather than a single string, we will focus in with a very low amplitude; the fourth mode of this string analysing the most simple possible scenarios. The string has a frequency of about 525Hz, which is almost 30Hz excitation lasts for 0.01s, and the simulation will last for higher than the fifteenth system mode. We also notice that 3s; however, we find that in the first scenario, almost all there are small discrepancies between the frequency of the energy of the system has been damped after 1.5 seconds, system modes and the frequency of the dominating strings. which is remarkably similar to what we typically hear when This suggests that the coupling of the different subsystems, plucking a portuguese guitar string. In Figures5 and6, we as well as the dead side of the strings, adds some inhamor- show the plots of the energy evolution in each string, in the nicity to the system, which would account for the audible body, and the total energy of the system, respectively for differences between different musical instruments. the scenarios (i) and (ii). In Figure4, we show the average energy of each subsystem, relative to the average total 3.2 Time Domain Results energy of the coupled system. As expected, in the first scenario, the first string is the We performed time domain simulations of the situations in subsystem which has the most significant amount of en- which: (i) the musician will only excite the first (b4) string, ergy, but interestingly, its energy will practically vanish (ii) the musician will only excite the eleventh (d3) string. after the first 0.6 seconds, while the coupled system still has a fair amount of energy. Shortly after 0.2 seconds have 5. REFERENCES passed, the energy of the second, seventh and eight strings [1] Woodhouse, “On the Synthesis of Guitar Plucks”, in (all the strings tuned to the same fundamental frequency Acta Acustica United With Acustica Vol. 90, 2004, pp. b4, and the string tuned to the second harmonic b3) will be 928–944. comparable in magnitude with the energy of the first string, while the total energy of the system will have a larger mag- [2] G. Derveaux, A. Chaigne, P. Joly, and E. Becache,´ nitude from this moment on. Particularly the second string, “Time-domain simulation of a guitar: Model and at the time in which it achieves its maximum energy (when method”, in J. Acoust. Soc. Am. Vol. 114, No.6, 2003, close to 0.3 seconds), has almost the same energy as the pp. 3368–3383. excited string, which is not surprising given that they are tuned to the same frequency and they are very closely lo- [3]E.B ecache,´ G. Derveaux, A. Chaigne, and P. Joly, cated at the bridge. The body displays an energy profile “Numerical Simulation of a Guitar”, in Computers which reveals that there is energy being transferred back Structures Vol. 83, 2005, pp. 107–126. and forth between itself and the twelve strings. All the [4] A. Nackaerts, B. Moor, and R. Lauwereins, “Coupled strings of the system will display some amount of energy, string guitar models”, in Proceedings of the WSES In- even if they have been tuned in frequencies rather lower ternational Conference on Acoustics and Music: The- than that of the excited strings. This will give this guitar its ory and Applications 2001, Skiathos, 2001, p.119. own distinguishable sound. In the second scenario, there is not a very efficient trans- [5] O. Inacio,´ J. Antunes, and M.C.M. Wright, “Computa- fer of energy from the excited string to the other subsys- tional modelling of string-body interaction for the vio- tems (comparing with the first scenario), and the only sub- lin family and simulation of wolf notes”, in J. Of Sound systems which will receive a relevant amount of energy And Vibration Vol. 310, 2008, pp. 260–286. are the soundboard, the d4 string (which is within the same [6] P. C. Cabral, A Guitarra Portuguesa. Ediclube, 1999. pair, and tuned to the second harmonic of the excited string), and the strings of the ”neighbour” pair(a3, a4). Based in [7] ——, The History of The Guitarra Portuguesa, (chap- our observations, the most efficient transfer of energy oc- ter in The Lute In Europe 2). Menziken, 2011. curs when exciting strings with more than one subsystem tuned to a multiple frequency, due to the fact that this sub- [8] O. Inacio,´ F. Santiago, and P. C. Cabral, “The systems will experience sympathetic vibrations. portuguese guitar acoustics: Part 1 - vibroacoustic measurements”, in Proc. of the 4th ibero-American Congress Acustica´ , Guimaraes,˜ 2004, pp. 14–17.

4. CONCLUSIONS

We have developed a conceptual model to accurately perform, in both frequency and time domain, analysis of twelve- string guitars as a fully coupled system. The formulation considered is sufficiently versatile to be also applicable to model any other plucked string instrument, regardless of the number of strings or the geometry in which they are disposed on the instrument body. In this preliminary analysis, we have found results which corroborate that the body, through the bridge, will account for a significant part of the energy transmission across the multiple subsystems. We also show the relevance of the ”dead side” of the string (i.e. the continuation of the string after the bridge) for the wave propagation across the string. We stress that this results might be a significant contribution for works on the optimisation of guitar characteristics and radiation.

Acknowledgments The authors warmly thank Professors Pedro Serrao,˜ Antonio´ Relogio´ Ribeiro and Octavio´ Inacio´ for interesting discussions on modelling issues, as well as Pedro Caldeira Cabral Figure 4. Average Energy of each subsystem; upper plot: for very important discussions and valuable information on excitation of the first (b4) string, lower plot: excitation of guitar building and performance issues. This work was the eleventh (d3) string. supported by FCT - Fundaçao˜ para Cienciaˆ e Tecnologia´ - Portugal under the project PTDC/FIS/103306/2008. Figure 5. Energy plots for an excitation of the first (b4) string. Figure 6. Energy plots for an excitation of the eleventh (d3) string.