Numerical Simulation of Tanpura String Vibrations M
ISMA 2014, Le Mans, France
Numerical Simulation of Tanpura String Vibrations M. Van Walstijna and V. Chatziioannoub aQueen’s University Belfast, Sonic Arts Research Centre, Queen’s University Belfast, BT7 1NN Belfast, UK bUniversity of MPA Vienna, Institute of Music Acoustics, Anton-von-Webern-Platz 1, 1030 Vienna, Austria [email protected]
609 ISMA 2014, Le Mans, France
A numerical model of a tanpura string is presented, based on a recently developed, stability-preserving way of incorporating the non-smooth forces involved in the impactive distributed contact between the string and the bridge. By defining and modelling the string-bridge contact over the full length of the bridge, the simulated vibrations can be monitored through the force signals at both the bridge and the nut. As such it offers a reference model for both measurements and sound synthesis. Simulations starting from different types of initial conditions demonstrate that the model reproduces the main characteristic feature of the tanpura, namely the sustained appearance of a precursor in the force waveforms, carrying a band of overtones which decrease in frequency as the string vibrations decay. Results obtained with the numerical model are used to examine, through comparison, the effect of the bridge and of the thread on the vibrations.
1 Introduction
The tanpura is a fretless string instrument producing lively sounding drones typical of the musical cultures of the Indian subcontinent. It usually has four metal strings stretched over a resonant body connected to a long, hollow neck [1, 2]. Like various other Eastern string instruments, its specific overtone-rich sound results from the interaction of its strings with a slightly curved bridge, but with the additional feature of having a thin thread placed between the string and the Figure 1: Side-view of a tanpura bridge. bridge (see Figure 1). The musician adjusts the position of the thread - which is commonly made of silk or cotton - in ff search of a desired javari, i.e. the drone-like sound e ect. better approach to constructing time-stepping schemes is the The main mechanical principles involved in the appearance of artefacts in the extracted signals, such as the generation of a javari have been understood for some time. ‘spikes’ visible in the nut force signals presented in [11]. Raman [1] already noted in 1921 that the string vibrations of This paper describes a numerical model of tanpura string the tanpura and other Indian instruments will contain a full vibrations, based on a recently developed energy method series of overtones regardless of where the string is plucked for modelling distributed contact in musical instruments or mechanically held still, and attributed this phenomenon [12]. The proposed numerical formulation is derived by correctly to the impactive interaction between the bridge and discretising equations governing the transversal vibrations the string. Burridge [3] provided a more detailed analysis of a stiff string stretched over a bridge as depicted in Figure of sitar string vibrations, identifying two main stages, both 2. While previous studies on the tanpura have defined one quasi-periodic. For the specific configuration of a tanpura, of the two terminations at the point where the string meets Valette et al. [4] showed that the placement of a thread the thread (A), in our model this falls at the position of can be considered as creating a “two-point bridge”, which the connection with tuning bead (B). The rationale is to periodically reinforces high frequency waves that travel explicitly model contact with the thread and with the bridge slightly ahead of waves of lower frequencies due to string over its full length, which allows the computation of the total ff sti ness. force on the bridge. In addition, it is of help in assessing Neverthless, time-domain simulations of the tanpura and where the lower bound of the contact elasticity constants other “flat-bridge” string instruments have yet to advance should lie. to a level that allows a detailed quantitative comparison The structure of the paper is as follows: a dissipative with experimental data; at the same time, physics-based form of the power law defining a collision force and its sound synthesis models do not display the same level discretisation is briefly discussed in Section 2. The equations of realism as those of various other string instruments. governing the system of geometry (B) in Figure 2 are then One possible reason for this is that vibrations of a string presented in Section 3, followed by the formulation of interacting with a one-sided bridge constraint have mostly the numerical scheme in Section 4. The main results are been studied and synthesised under various simplifying presented and discussed in Section 5. assumptions. For example, by defining the string-bridge collisions as fully inelastic [3, 5, 6], or completely lossless [7, 8], either of which may be a too severe simplification. (A) In addition, string stiffness and losses are often omitted, but adding these to an existing formulation is relatively y simple. The complementary extension of defining the x contact as semi-elastic is less straightforward though, (B) because modelling impactive contact with repelling forces - which are necessarily non-analytic functions of the transversal displacement of the string - generally poses considerable challenges with regard to the construction of stable, convergent time-stepping schemes [9]. While various x =0 x = xc x = L simulation results have successfully been obtained (e.g. for tuning thread nut the sitar [10]), provably stable formulations of this kind bead position have yet to appear. A further indication of the need for a Figure 2: Tanpura string model configurations.
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2 Impactive contact with damping exponents. Some justification for the choice α = 1 can be found in the measurement of elastic deformation of strings, A well-known dissipative form of a power law for the contact such as carried out by Taguti for the silk string of the biwa force Fb between an object positioned at y with a barrier [7]. A further consideration is that choosing α = 1 with positioned below it at yb is that by Hunt and Crossley [13]: sufficiently high k values is the simplest way of avoiding unrealistically deep compression. For a string of length L, ∂y F = k (y − y)α 1 − r , (1) the boundary conditions are: b b ∂t ∂2 , = , y = , where α is the exponent, k is a stiffness-like term, and y y(0 t) yTB 2 0 (7a) ∂t = denotes h(y) · y, where h is the Heaviside step function. The x 0 ∂2y amount of damping is controlled through the coefficient r. y(L, t) = 0, = 0, (7b) ∂ 2 As explained in [14], the lossless version of this power law t x=L can be discretised in stable form by first writing the force as where yTB denotes the vertical position of the left-end a derivate of the potential energy, which equals support (see (B) in Figure 2). The bridge profile yb(x)is taken here as measured by Guettler [15], with its maximum k α+ V(y) = (y − y) 1 . (2) positioned at x = 46.5mm, y = 0mm. The terms y (x) α + 1 b c and yf(x) represent the shapes of the cotton thread and the This approach is readily extended to the above Hunt-Crossley plucking finger, respectively, both of which are defined as form, by writing (1) as parabolic. At the start of any simulation, the initial state of the string is first solved by letting the initial vibrations due ∂V ∂V F = − + r , (3) to contact forces decay. The string is then excited by setting b ∂y ∂t the finger spring constant kf to zero. Using T to denote the and discretising using mid-point-in-time derivative kinetic component, the system energy is: approximations: L H = T (p) + Vt(u) + Vs(v) + Vb(y) + Vc(y) + Vf(y) dx n+1 n n+1 n n+ 1 V(y ) − V(y ) V(y ) − V(y ) 0 F 2 = − + r , (4) (8) b yn+1 − yn Δt where the energy densities are zero outside the respective where n is the time index. This equation can be written into a domains and otherwise form suitable for use in a numerical model simulating vibro- p2 1 T (p) = , V (u) = τu2, impact phenomena after subsitution of (2). In application to 2ρA t 2 modelling distributed contact, as in the following section, the 1 kb force term in these equations is replaced with force density, V (u) = EIv2, V (y) = (y − y)2 , (9) s 2 b 2 b potential energy is replaced with potential energy density, k k and stiffness becomes stiffness per unit length. V (y) = c (y − y)2 , V (y) = f (y − y)2 , c 2 c f 2 f and where 3 Governing equations ∂y ∂y ∂2y p = ρA , u = , v = . (10) ∂t ∂x ∂x2 Considering a stiff, lossy vibrating tanpura string with external force terms due to contact with the bridge (supscript We may now, following a path similar to that of the lossless ‘b’), a cotton thread (supscript ‘c’), and a plucking finger formulation in [12], rewrite the system dynamics in terms of its energy densities and losses per unit length as follows (supscript ‘f’), the equation of motion in space-time coordinates (x, t) can be stated as ∂ ∂ ∂V ∂2 ∂V ∂V ∂V ∂V p = τ − s − b − c − f ∂t ∂x ∂u ∂x2 ∂v ∂y ∂y ∂y ∂2y ∂2y ∂4y ∂ ∂y ρA = τ − EI 1 − η − ρAγ 3 5 ∂ 2 ∂ 2 ∂ 4 ∂ ∂ ∂ y ∂ y ∂y t x x t t − τη + EIη − ρAγ ∂t∂x2 ∂t∂x4 ∂t + Fb(x, t) + Fc(x, t) + Ff(x, t), (5) ∂Vb ∂Vc ∂Vf + rb + rc ++rf , (11a) where ρ, A, τ, E and I denote string mass density, cross- ∂t ∂t ∂t ∂y ∂T section, tension, Young’s modulus, and moment of inertia, = . (11b) respectively, and where η and γ represent loss parameters. ∂t ∂p The respective force densities acting upon the string are, The total force on the bridge is within their specific spatial domains, L = F , + F , . ∂y Fb(t) b(x t) c(x t) dx (12) F (x, t) = k y (x) − y(x, t) 1 − r , (6a) 0 b b b b ∂t This expression can be used as a first approximation to ∂ F , = − , − y , the sound produced by the instrument. Another signal that c(x t) kc yc(x) y(x t) 1 rc ∂ (6b) t is useful to extract from simulations - as it allows direct ∂ comparison with earlier studies on tanpura string vibrations F , = − , − y , f(x t) kf yf(x) y(x t) 1 rf ∂ (6c) [4, 2, 11] - is the force that the string exerts on the nut: t ∂3y ∂y and zero outside their domains. These expressions F (t) = EI − τ . (13) n ∂ 3 ∂ essentially represent Hunt-Crossley contact laws with unity x x=L x x=L
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4 Numerical formulation With the added contact forces, (15) becomes Denoting the spatial and temporal step with Δx and γΔt 2η F = 1 + I + 1 + D s + 2 (Dyn − qn) Δt, respectively, the string state is discretised as 2 Δt n y ≡ y(mΔx, nΔt). Applying difference operators as in n+ 1 n+ 1 n+ 1 m − f 2 − f 2 − f 2 . (22) [12], the system without contact forces can be written in b c f matrix form as where the added force terms are non-linear functions of s s + 2η + 2η , and where the respective interpolated versions of are qn 1 − qn = −D 1 + yn 1 + 1 − yn y Δt Δt computed in the same way as for : γΔ t n+1 n s = I s, s = I s, s = I s. − y − y (14a) ¯b b ¯c c ¯f f (23) 2 + + yn 1 − yn = qn 1 + qn, (14b) Equation (22) can be solved at each time step with the multi- dimensional Newton-Rhapson method, using the Jacobian where qn =Δt/(2ρA)pn, and where D is a square symmetric matrix representing all spatial differentiation involved [12], γΔt 2η ∗ ∗ ∗ J = 1 + I + 1 + D +I C I +I C I +I C I , with numerical versions of the boundary conditions defining 2 Δt b b b c c c f f f the elements at and near the corners. Note that in this case (24) the condition in (7a) of a ‘lowered left support’ has to be where Cb, Cc, and Cf are diagonal matrices with elements incorporated. The system can be solved at each time step by n+ 1 n+ 1 n+ 1 finding the root s of the function ∂ f¯ 2 ∂ f¯ 2 ∂ f¯ 2 = b,i , = c, j , = f,k . cb,i,i ∂ cc, j, j ∂ cf,k,k ∂ γΔt 2η s¯b,i s¯c, j s¯f,k F = 1 + I + 1 + D s + 2 (Dyn − qn) (15) (25) 2 Δt and subsequently updating the string displacement vector 5 Simulation results and scaled momentum vector with
+ + To obtain representative string parameters, the diameter yn 1 = yn + s, qn 1 = s − qn. (16) (d = 0.3 mm) and length (L = 668 mm) of the third string Now adding contact forces to the system, force density terms of a small travelling tanpura were measured. Taking into at the bridge, thread, and finger are computed at separate account the fundamental frequency of the speaking length grids for each element, using the spatial stepsizes Δx , Δx , of the string as well as the mass density and Young’s b c ff and Δx , respectively. Hence a way of translating between modulus of steel, the tension and sti ness terms were set f τ = . −1 = . × −5 the string grid and each of these separate grids is required. accordingly to 31 47 N m and EI 8 35 10 2 ρ = . × −4 −1 To this purpose, third-order Lagrange interpolation matrices Nm, with A 5 58 10 Kg m . The damping γ = . −1 η = × −9 are used as follows: parameters were set to 0 6s and 7 10 s, which results in a frequency-dependent decay pattern which yn = I yn, yn = I yn, yn = I yn. approximately matches that observed when the string is left ¯b b ¯c c ¯f f (17) in free vibration (i.e. without string-bridge interaction). The Note that interpolation is a necessary tool in accurate bridge and thread contact elasticity coefficients are chosen 8 −2 positioning of the thread, the bridge, and the finger. The as kb = kc = 1 × 10 Nm , which ensures that the effective (scaled) contact forces are formulated at these points, for compression does not exceed 5% of the string diameter. example for the bridge we have, from (4): The contact damping coefficients are set to rb = rc = 0.1 and and rf = 1. The numerical parameters are as follows: n 2 n 2 1 , − − , − , − 1 n+ (yb i y¯b,i s¯b i) (yb i y¯b,i) Δt = 1/176.4 ms, Δx = 3 mm, Δx = 0.18 mm, Δx = 0.15 f¯ 2 = −β 3 b c b,i b mm, and Δx = 1 mm. s¯b,i f + ζ − n − 2 − − n 2 , b (yb,i y¯b,i s¯b,i) (yb,i y¯b,i) (18) 5.1 Quasi-Helmholz motion with β = (k Δt2)/(2ρA) and ζ = (k r Δt)/(2ρA), and where b b b b b Figure 3 shows snapshots of the string motion for an initial n+1 n condition that matches the shape of the first mode of the s¯b,i = y¯ , − y¯ , . (19) b i b i string; this is achieved by re-defining the shape of the The forces can be translated back to the string spatial finger. In the plots, the more recent states are represented coordinates using a corresponding downsampling by colour-intensive curves, with the colour-tone fading out interpolant: for the earlier string states. It can be observed that the n+ 1 ∗ n+ 1 bridge collisions force the string to gradually take on a more f 2 = I f¯ 2 . (20) b b b triangular shape, indicating the excitation of the other modes This can be done without affecting energy conservation of vibration. A Helmholz-like motion emerges, as illustrated for zero damping if the scaled conjugate is used as the by the appearance of a kink that travels along the string, downsampling interpolant [9]: as indicated by the arrows in Figure 3(c). Similar findings have been presented in [11] and in studies of various other ∗ Δx I = b I t . (21) string-bridge configurations [3, 8]. b Δx b
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(a) (b) (c)
2 2 2
1 1 1
0 0 0 y [mm] y [mm] y [mm] −1 −1 −1
−2 −2 −2
0 200 400 600 0 200 400 600 0 200 400 600 x [mm] x [mm] x [mm]
Figure 3: Snapshots of the string vibration, with the first mode initial condition, for the 1st (a), 17th (b), and 33rd (c) period of oscillation. The arrows in (c) indicate the movement of the kink, indicative of a Helmholtz-like motion.
0.5 0.5 0.5 0.5
[N] 0 0 0 0 n F
−0.5 −0.5 −0.5 −0.5
7 8 9 10 11 60 62 64 112 113 114 115 116 164 166 168
0.5 0.5 0.5 0.5
[N] 0 0 0 0 n F
−0.5 −0.5 −0.5 −0.5
7 8 9 10 11 60 62 64 112 113 114 115 116 164 166 168 time [ms] time [ms] time [ms] time [ms]
Figure 4: Evolution of the nut force signal for two different initial conditions: top: first mode shape, bottom: finger pluck shape.
Figure 5: Spectral evolution for different initial conditions and bridge configurations.
613 ISMA 2014, Le Mans, France 5.2 Precursive wave generation 6 Conclusions
Figure 4 plots instances of the nut force signal for the A numerical model for simulation of tanpura string ‘first mode shape’ and the ‘finger pluck’ cases. For both vibrations has been presented. The results generated initial conditions, a gradual development toward having one with the simulations are qualitatively in agreement with sustained precursor per cycle is observed. As explained in measurements and findings from earlier studies, but they [4], this precursor is a packet of high-frequencies arriving also reveal that the model fails to predict the effect of at the nut before the lower frequencies due to the string removing the thread. Further research could establish which ff sti ness. The precursor keeps being ‘fed’ once per cycle refinements and extensions would be needed to address with high-frequency components through the non-linear this discrepancy. Examples of nut and bridge force signals interaction at the bridge end, which destroys the symmetry of are supplied in audio format on the accompanying website the force waveform periods. In the case of the plucked finger, (www.somasa.qub.ac.uk/˜mvanwalstijn/isma14). the precursor that originally appears in the other half of the waveform gradually fades out due to frequency-dependent losses. References
[1] C. Raman, “On some indian stringed instruments,” in Proc. of 5.3 Influence of the bridge and the thread the Indian Assoc. for the Cultivation of Science, 1921, vol. 7, pp. 29–33. Figure 5 shows the spectral evolution of the nut force signal obtained from the simulation for both initial conditions, and [2] D. Bertrand, Les chevalets ”plats” de la lutherie de l’Inde, for three different bridge configurations. For the two plots on Editions de la Maison des sciences de l’homme, 1992. ff the left, the bridge stiffness per unit length kb is set to zero for [3] R. Burridge, J. Kappra , and C. Morshedi, “The sitar string, x > xc, where xc = 40mm is the position of the the thread. a vibrating string with a one-sided inelastic constraint,” SIAM This means that the speaking length of the string is free to Journal on Applied Mathematics, vol. 42, no. 6, pp. pp. 1231– vibrate, resulting in independently decaying modes, with no 1251, 1982. energy conversion from the fundamental to any of the other [4] C. Valette, C. Cuesta, C. Besnainou, and M. Castellengo, “The modes. tanpura bridge as a precursive wave generator,” Acustica, vol. With the bridge and thread in place (i.e. the full model), 74, pp. 201–208, 1991. all modes are excited for both initial conditions, and the [5] J. Ahn, “A vibrating string with dynamic frictionless impact,” precursor can be observed as a formant region with a spectral Appl. Numer. Math., vol. 57, no. 8, pp. 861–884, 2007. centroid that varies over time. For the plucked finger case, [6] A. Krishnaswamy and J. Smith, “Methods for simulating at first the formant frequency decays, then briefly stays string collisions with rigid spatial obstacles,” in Proc. IEEE approximately constant, followed by a period of slower WASPAA, New York, 2003. decay. The appearance of these distinct regimes in the [7] T. Taguti, “Dynamics of simple string subject to unilateral formant centre frequency pattern is in accordance with the constraint: A model analysis of sawari mechanism,” Acoust. analysis of experimentally obtained nut force signals in [4]. Sci. Technol., vol. 29, no. 3, pp. 203–214, 2008. Finally, the two plots on the right-hand side show the [8] D. Kartofelev, A. Stulov, H. Lehtonen, and V. Valim¨ aki,¨ spectral evolution when no cotton thread is present in the “Modeling a vibrating string terminated against a bridge with system. In the real instrument, the removal of the thread arbitrary geometry,” in Stockholm Musical Acoustics Conf., or even a small adjustment of its position will result in a Stockholm, 2013. much reduced formant/precursor effect, with the sound being [9] S. Bilbao, Numerical Sound Synthesis, Wiley & Sons, 2009. more similar to that of a string freely vibrating (i.e with no [10] C. Vyasarayani, Transient Dynamics of Continuous Systems impactive bridge interaction). However, as seen in the plots, with Impact and Friction, with Applications to Musical the model still produces vibrations with a strong formant- Instruments, Ph.D. thesis, University of Waterloo, Canada, like feature in the spectrum. A possible explanation is that - 2009. unlike in the presented model - the vibrations of a real string [11] C. Cuesta C. Valette, M´ecanique de la corde vibrante, are not restricted to the vertical plane. That is, a level of Hermes, Paris, 1993. coupling between the two transversal polarisations, either at [12] V. Chatziioannou and M. van walstijn, “Sound synthesis end points or through distributed non-linear coupling (i.e. for contact-driven musical instruments via discretisation of ‘string whirling’) is common to all string instruments. The Hamilton’s equations,” in Proc. of the Int. Symp. Musical vertical string motion does not contribute to collisions with Acoustics, Le Mans, France, 2014. the bridge, so with the added plane of motion, there may [13] K. Hunt and F. Crossley, “Coefficient of restitution interpreted be more tight conditions for triggering a strong javari. In as damping in vibroimpact,” J. Applied Mech., pp. 440–445, the light of this notion it is worthwhile noting that Raman’s 1975. studies include photographic evidence of the string deflection [14] V. Chatziioannou and M. van Walstijn, “An energy conserving being equally prominent in both planes for string vibrations finite difference scheme for simulation of collisions,” in with a javari [16]. Stockholm Musical Acoustics Conf. / Sound and Music Computing Conf., Stockholm, 2013. [15] K. Guettler, “On the string-bridge interaction in instruments with a one-sided (bridge) constraint,” available from knutsacoustics.com, 2006. [16] B. Chaitanya Deva, The Music of India, Munshiram Manoharlal Publishers, 1980.
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