Aalborg Universitet Real-Time Control of Large-Scale Modular Physical

Aalborg Universitet Real-Time Control of Large-Scale Modular Physical Models using the Sensel Morph Willemsen, Silvin; Andersson, Nikolaj Schwab; Serafin, Stefania; Bilbao, Stefan Published in: Proceedings of the 16th Sound and Music Computing Conference DOI (link to publication from Publisher): 10.5281/zenodo.3249295 Creative Commons License CC BY 3.0 Publication date: 2019 Document Version Accepted author manuscript, peer reviewed version Link to publication from Aalborg University Citation for published version (APA): Willemsen, S., Andersson, N. S., Serafin, S., & Bilbao, S. (2019). Real-Time Control of Large-Scale Modular Physical Models using the Sensel Morph. In I. Barbancho, L. J. Tardón, A. Peinado, & A. M. Barbancho (Eds.), Proceedings of the 16th Sound and Music Computing Conference (pp. 151-158). Sound and Music Computing Network. 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Downloaded from vbn.aau.dk on: November 24, 2020 REAL-TIME CONTROL OF LARGE-SCALE MODULAR PHYSICAL MODELS USING THE SENSEL MORPH Silvin Willemsen, Nikolaj Andersson, Stefania Serafin Stefan Bilbao Multisensory Experience Lab, CREATE, Acoustics and Audio Group Aalborg University Copenhagen University of Edinburgh Copenhagen, Denmark Edinburgh, UK fsil, nsa, [email protected] [email protected] ABSTRACT standard computing power is now approaching a level suit- able for real-time performance for simpler systems. In this paper, implementation, instrument design and con- We are interested in bridging the gap between large-scale trol issues surrounding a modular physical modelling syn- modular physical modelling synthesis and sonic interac- thesis environment are described. The environment is con- tion design [16], to be able to play with such simulations structed as a network of stiff strings and a resonant plate, in real-time. Specifically, we are interested in using the accompanied by user-defined connections and excitation expressivity of the Sensel Morph [17] to control our simu- models. The bow, in particular, is a novel feature in this lations, using both percussive and bowing excitations. Our setting. The system as a whole is simulated using finite ultimate goal is to create models that are both mathemat- difference (FD) methods. The mathematical formulation ically accurate and efficient. This goal is nowadays pos- of these models is presented, alongside several new instru- sible thanks to improvements in hardware and software ment designs, together with a real-time implementation in technologies for sound synthesis, yet it has rarely been JUCE using FD methods. Control is through the Sensel achieved. The ultimate goal is to provide a modular ef- Morph. ficient synthesizer based on accurate simulations, where real-time expressivity can also be achieved. This synthesizer has already been informally evaluated by composers 1. INTRODUCTION and sound designers, who appreciated the current sonic Physical models for sound synthesis have been researched palette. for several decades to mathematically simulate the sonic This paper is structured as follows: Section2 describes behaviour of musical instruments and everyday sounds. Var- the physical models used in the implementation and Sec- ious techniques and methodologies have developed, rang- tion3 shows a general description of the FD methods used ing from mass-spring models [1–3] to modal synthesis [4] to digitally implement these models. Furthermore, Sec- and waveguide based models [5]. The latter two techniques tion4 elaborates on the real-time implementation, Sec- may be viewed as numerical simulation techniques applied tion5 shows several different configurations of the physi- to the systems of partial differential equations (PDEs). These cal models inspired by real musical instruments, Section6 equations define the dynamics of a musical instrument, ei- will present the results on CPU usage and evaluation and ther real or imagined. discuss this and finally, in Section7, some concluding re- Mainstream time-domain simulation techniques, such as marks appear. finite difference (FD) methods, were first applied to the case of string vibration by Ruiz [6] and Hiller and Ruiz [7,8], and then later by other authors [9] including, most 2. MODELS notably Chaigne [10] and Chaigne and Askenfelt [11]. The In this section, the PDEs for the damped stiff string and general use of finite-difference schemes (FDSs) in sound plate will be presented. The notation used will be the one synthesis is described in [12]. Modularized physical mod- found in [12] where the subscript for state variable u de- elling sound synthesis, whereby the user may construct a notes a single derivative with respect to time t or space x virtual instrument using basic canonical components dates respectively. Furthermore, to simplify the presented phys- back to the work of Cadoz and collaborators [1–3]. It ical models, non-dimensionalization (or scaling) will be has been also used as a design principle in the context of used [12]. FD methods [13–15], where the canonical elements are strings and plates, with a non-linear connection mecha- nism. Though computational cost of such methods is high, 2.1 Stiff string A basic model of the linear transverse motion of a string of Copyright: © 2019 Silvin Willemsen et al. This is an open-access article distributed circular cross section may be described in terms of several under the terms of the Creative Commons Attribution 3.0 Unported License, which parameters: the total length L (in m), the material density ρ permits unrestricted use, distribution, and reproduction in any medium, provided (in kg·m−3), string radius r (in m), Young’s modulus E (in the original author and source are credited. Pa), tension T (in N), and two loss parameters σ0 and σ1. The PDE for a damped stiff string may be written as [12] The stiffness parameter κ, with dimensions of s−1, is de- q 2 2 3 2 2 2 fined by κ = D/ρHLxLy where D = EH =12 1 − ν . utt = γ uxx − κ uxxxx − 2σ0ut + 2σ1utxx: (1) As in the case of the stiff string, we chose to use clamped In this representation, spatial scaling has been employed boundary conditions: using a length L, so the solution u = u(x; t) is defined for t ≥ 0 and for dimensionless coordinate x 2 [0; 1]. Further- u = n · ru = 0 (8) more, parameters γ = pT/ρπr2L2 and κ = pEr2=4ρL4 over any plate edge with outward normal direction n and and have units s−1. where ru is the gradient of u. In this work, the string is assumed clamped at both ends, so that 2.3 Connections u = ux = 0 where x = f0; 1g: (2) Adding connections between different physical models, fur- A model of a bowed string [12] may be incorporated into ther referred to as elements, adds another term to Equation (1) as (3a), (5) or (6). Assuming that element α is a stiff string utt = ::: − δ(x − xB)FBφ(vrel); with (3a) and β is a plate, the following terms are added to the afore- mentioned equations: vrel = utj(x=xB) − vB; (3b) 2 utt = ::: + Ec,αFα; (9a) where FB = fB=Ms is the excitation function (in m/s ) with externally-supplied bowing force fB = fB(t) (in N) utt = ::: + Ec,βFβ; (9b) 2 and total string mass Ms = ρπr L (in kg). The relative ve- with force-functions Fα = Fα(t) and Fβ = Fβ(t) (in locity vrel is defined as the difference between the velocity 2 of the string at bowing point x and the externally-supplied m/s ) and distribution functions Ec,α and Ec,β which have B chosen to be highly localised in our application and reduce bowing velocity vB = vB(t) (in m/s) and φ is a dimensionless friction characteristic, chosen here as [12] to δ(x − xc,α) and δ(x − xc,β; y − yc,β) respectively, but p can be extended to be connection areas [13]. We use the −av2 +1=2 φ(vrel) = 2avrele rel : (4) implementation as presented in [13] where the connection between two elements is a non-linear spring. The forces it Furthermore, δ(x − xB) is a spatial Dirac delta function imposes on the elements it connects are defined as selecting the bowing location x = xB. The single bowing 2 4 3 point can be extended to a bowing area [12]. More detailed Fα = −!0η − !1η − 2σ×η;_ (10a) models of string dynamics, again in a bowed string context, Fβ = −MFα; (10b) have been proposed by Desvages [18]. −1 Another, and more simple way to excite the string is by where !0 and !1 are the linear (in s ) and non-linear (in −1=2 extending Equation (1) to (m·s) ) frequencies of oscillation respectively, σ× is a damping factor (in s−1), M is the mass ratio between the utt = ::: + EeFe (5) two elements and η is the relative displacement between the connected elements at the point of connection (in m). using an externally-supplied distribution function E = e Lastly, the dot above η denotes a derivative with respect to E (x) and excitation function F = F (t).

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