A Real-Time Synthesis Oriented Tanpura Model
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Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016 A REAL-TIME SYNTHESIS ORIENTED TANPURA MODEL Maarten van Walstijn, Jamie Bridges, and Sandor Mehes Sonic Arts Research Centre School of Electronics, Electrical Engineering, and Computer Science Queen’s University Belfast, UK {m.vanwalstijn,jbridges05,smehes01}@qub.ac.uk cotton ABSTRACT thread finger Physics-based synthesis of tanpura drones requires accurate sim- tuning bridge nut ulation of stiff, lossy string vibrations while incorporating sus- bead tained contact with the bridge and a cotton thread. Several chal- lenges arise from this when seeking efficient and stable algorithms 0 xc xb → x xe L for real-time sound synthesis. The approach proposed here to address these combines modal expansion of the string dynamics Figure 1: Schematic depiction of the tanpura string geometry (with with strategic simplifications regarding the string-bridge and string- altered proportions for clarity). The string termination points of the thread contact, resulting in an efficient and provably stable time- model are indicated with the vertical dashed lines. stepping scheme with exact modal parameters. Attention is given also to the physical characterisation of the system, including string damping behaviour, body radiation characteristics, and determi- the reliance on iterative solvers and from the high sample rates nation of appropriate contact parameters. Simulation results are needed to alleviate numerical dispersion. presented exemplifying the key features of the model. This paper aims to formulate a leaner discrete-time tanpura string model requiring significantly reduced computational effort, but retaining much of the key sonic features of the instrument. 1. INTRODUCTION Two aspects that distinguish this challenge somewhat from other cases of string-barrier interaction are (a) the sustained nature of Among mechanically-induced sound effects naturally afforded by the impactive interaction (with a high potential for audible high- musical instruments, the generation of overtones in tanpura drone frequency artefacts, including aliasing) and (b) the sensitivity of playing is one of the more spectacular and intriguing examples. In the jvari to some of the system parameters and to discretisation er- Indian musical tradition, the phenomenon is known as jvari (mean- rors. The key features of the proposed model, presented in Section ing ‘life-giving’), and arises from the impactive interaction of the 2, can be summarised as follows: vibrating string with a hard-surfaced bridge. The player fine-tunes the effect by carefully positioning a thin thread between the bridge the spatially distributed string-bridge collision forces are sup- and the string (see Fig. 1). • pressed to a single variable, which - in conjunction with neglect- As a vibrational phenomenon, the jvari effect has attracted ing contact damping and using a unity exponent in the contact scientific interest for almost a century, starting with the musical law - allows updating the numerical system without the use of acoustics poineering work by Raman [1]. Several ways of analy- an iterative solver; sing and modelling the vibrations of the tanpura and other ‘flat- the thread interaction, which effects a ‘softer’ string termina- bridge’ instruments such as the sitar, veena, and biwa have been • proposed since, with the aims ranging from theoretical understand- tion, is explicitly modelled as a local spring-damper connection; ing (usually relying on simplifying assumptions regarding the na- a modal expansion approach is utilised, which allows formulat- ture of the interaction [2, 3, 4]) to more practical discrete-time sim- • ing a numerical model with exact modal frequencies and damp- ulation [5, 6] including several synthesis oriented studies [7, 8]. ing; The problem also naturally bears some resemblance to various other cases involving collisions, including string-fingerboard con- discretisation is performed on a first-order partial derivative form • tact in the guitar [9, 10], violin [11], and bass guitar [12, 13, 14]. of the modal differential equations, which facilitates the use of a Despite these advances, efficient and realistic synthesis of the two-point discrete gradient for discretisation of the bridge con- sound of flat-bridge string instruments appears to have remained a tact force; somewhat elusive goal. One of the original difficulties, namely that numerical stability is independent of the system parameters and of potential instability when incorporating collision forces, has re- • the temporal step; the only numerical constraint is that the mode cently been addressed more widely within a finite-difference con- series is truncated at Nyquist in order to avoid mode aliasing. text, by construction of time-stepping schemes that respect the en- ergy balance inherent to the underlying continuous-domain model For realistic synthesis of tanpura drones, one also needs to de- [15, 16, 17, 18]. Tanpura models based on such energy meth- termine appropriate system parameters, including those related to ods can reproduce the jvari effect by simulating distributed string- string damping, bridge and thread contact, and sound radiation; bridge collisions [17, 19, 20]. However the algorithms that im- this is discussed in Section 3. Exemplifying simulation results are plement these tanpura models are not particularly suited to sound then presented and discussed in Section 4, followed by concluding synthesis because of the high computational burden resulting from remarks and perspectives in Section 5. DAFX-175 Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016 40 6 2. TANPURA STRING MODEL m] µ 1 20 string bridge 5 2.1. Model Equations 0 force 0 4 The transversal displacement of the string depicted in Fig. -20 y(x; t) -1 axial position [mm] displacement [ 1, with the spatial domain defined as x [0;L] and t denoting -40 force density [kN/m] 3 time, may be described by: 2 2 4 6 8 10 0 5 10 axial position [mm] time [ms] @2y @2y @4y @y Figure 2: Left: string motion snapshot obtained with a model sim- ρA = T EI γ(β) @t2 @x2 − @x4 − @t ulating distributed bridge contact [28]. The profile of the orange + c(x; t) + b(x; t) + e(x; t); (1) surface indicates force density, and the dash-dot line indicates the F F F corresponding instantaneous central contact point. Right: Varia- in which ρ, A, T , E, and I are mass density, cross-sectional area, tion of the central contact point over the first 12ms. The flat dashed tension, Young’s modulus, and moment of inertia, respectively. lines indicates periods of no contact. Assuming simply supported ends, the boundary conditions are @2y where both height constants h and h are normally zero to en- y(x; t) = 0; = 0: (2) c b x=0;L @x2 x=0;L sure grazing contact at equilibrium. From the second equation it is straightforward to derive that Frequency-dependent string damping is incorporated by defining @Vb the parameter γ(β) in (1) as: Fb(t) = : (10) − @yb h 2 i γ(β) = 2ρA σ0 + σ1 + σ3β β ; (3) The forces exerted by the string at the left-end termination (‘o’) j j and the nut end (‘n’) are where β is the wave number and σ0;1;3 are fit parameters. The @y @3y interactions with the cotton thread, the bridge and a plucking fin- Fo(t) = T EI ; (11) @x x=0 − @x3 x=0 ger are modelled using the force densities c(x; t), b(x; t) and F F @y @3y e(x; t), respectively. These are defined here in a simplified form F (t) = T + EI : (12) n 3 byF pre-determining their spatial distributions, hence modelling each − @x x=L @x x=L as (z = c; b; e): Since Fo(t) is generally much smaller than Fc(t) and Fb(t), the total force exerted by the string on the bridge can be calculated as z(x; t) = z(x)Fz(t); (4) F Fd(t) = Fc(t) Fb(t): (13) − − where z(x) are spatial distribution functions of the form Approximations to the emitted sound can be found by filtering Fd(t) and Fn(t), where the filters have transfer functions that ap- ( π h π i 2w cos w (x xz) : x Dz proximate measured body radiation responses (see Section 3.3). z(x) = z z − 2 (5) 0 : otherwise The plucking force is specified here in highly simplified form: in which D = [x 1 w ; x 1 w ] denotes a spatial domain of z z 2 z z 2 z a sin2 [(π/τ ) h (t t )] : t width w and centre− position −x . Equation (5) is a good approx- F (t) = e e e e e ; (14) z z e 0− : otherwise2 T imation to the force profile typically observed in the initial vibra- tions as computed with distrubuted contact models (see the left where e = [te; te + τe] and with T plot of Fig. 2). In addition this form provides a convenient way of 1 τe sinh(βet/τe) 1 he(t) = t + : (15) exciting mainly the first mode of the string (by setting xe = L, 2 2 sinh(βe) w = L). The impactive contact with the bridge is modelled here e As seen in Fig. 3, the parameter β > 0 controls the attack and using a lossless contact law with unity exponent and elasticity con- e release slopes of the plucking function, which allows mimicking stant k : b the gentle style in which tanpura strings are generally plucked. Fb(t) = kb hb yb(t) ; (6) b − c The other control parameters are the amplitude ae and the over- in which y denotes u(y) y, where u(y) is the unit step function. all pluck signal timespan τ . More sophisticated plucking models b c · e Given that the string never detaches from the cotton thread, this have been proposed (see, e.g.