Advances in Modular Percussion Synthesis: Exploring Computational Methods and Connections

1. A Modular Percussion Synthesis Environment DAFX-09, Como, Sept., 2009. Stefan Bilbao Acoustics and Fluid Dynamics Group / Music University of Edinburgh • Modular synthesis • Percussion synthesis: Components and connections • FD schemes/computational issues and costs • Sound examples

2. Modular synthesis: strategies • Goal: arbitrary connections of predefined “canonical” musical objects • Many different methodologies: • Modal • Scattering, including waveguides, WDFs • Lumped networks • Direct time-domain methods (FD, FEM, spectral, etc.) • Many distinctions, in terms of • Computability (uniqueness/existence of solutions) • Range of applicable systems • Precomputational load • Memory requirements • Stability guarantees • Today: percussion…

3. Components: Bars Linear, thin, uniform bar: Basic parameter set: PDE model: Stiffness Freq.-ind loss Freq.-dep loss Scaling… Reduced equivalent parameter set: + boundary conditions: clamped, pivoting, free, etc. Can relax assumptions: non-uniform, thick, nonlinear…

4. Components: Plates Linear, thin, uniform rectangular plate: Basic parameter set: PDE model: Reduced equivalent parameter set: Scaling… + boundary conditions: clamped, pivoting, free, etc. Can relax assumptions: non-uniform, thick, nonlinear, geometry…

5. Connections • Distributed connections between objects (1 and 2, of bar or plate type): • Connection characterized by density distributions (in simplest case delta functions) • Connection described, in terms of PDEs, as • Spring/damper connection: • where • Connection is dissipative (passive). Linear spring nonlinear spring (cubic) damper Generalized relative displacement Forces equal and opposite

6. Excitations/Output • In general, for percussion, would like a mallet model… • Because collision times are very short, sufficient to use a fixed contact distribution, and forcing function or pulse. • Maximum of force pulse  amplitude • Duration of pulse  brightness • output: scales with velocity over an output distribution: • Note: output distribution can be time-varying…multiple outputs can be taken simultaneously.

7. Difference schemes: components • Consider a network of M unconnected elements, under zero-input conditions. • For the mth object, can develop an explicit difference scheme directly: Unknown (current) state Previously computed state where: • is state of mth object, defined over Nm points • are sparse matrices, of size Nm x Nm • A good idea to concatenate these schemes… where

8. Difference Schemes: Connections • Many possible discretizations…  Unknown appears “linearly,” when previous state is known…guarantee of existence/uniqueness of solutions… Thus: Known at time step n+1 (previously computed) • To relate generalized relative displacements to state:  • For a set of Q connections, in vector form:

9. Explicit nonlinear update form • When excitation/connections are present, update becomes… previously computed/supplied externally) • Using force/displacement relations: known known, positive definite (diagonal for non-overlapping connections) • Thus: a unique update, involving a low-order linear system solution… • A compact run-time loop, at least in Matlab! for n=1:Nf … read in current excitation data … eta1 = I*u1; eta2 = I*u2; eta1sq = rvec.*(eta1).^2; temp = (mvec+eta1sq).*(IB*u1+IC*u2)+(nvec+eta1sq).*eta2; A = II+(M-diag(eta1sq))*IJ; F = A\temp; u = B*u1+C*u2+S*D+J*F; out(:,n) = Q*u; u2 = u1; u1 = u; end

10. Numerical Energy Conservation Under lossless conditions, this network conserves energy to machine accuracy… • Energy function is positive definite (not quadratic!) under the usual CFL stability conditions: • Bars: • Plates: • No further stability concerns due to connections… • a nonlinear numerical stability guarantee… • and a useful debugging feature! Energy of Plates Energy of Bars Energy of Connections Total Energy

12. Basic configuration • A simple set of uncoupled bars… • Can vary • boundary conditions • loss • striking points • Sound is very artificial…characteristic of raw linear systems!

13. Representative configurations

14. Conclusions and Perspectives Direct time/space domain methods: a flexible alternative to standard physical modeling methods… • Compared with scattering methods: • Handles multiple nonlinearities easily • No topology/delay-free loop issues • No global effects on network due to propagation of port-resistances…scheme is entirely local • Simpler stability/existence/uniqueness results • Compared with modal methods • A much better match to nonlinear problems…no linear system theory or frequency domain analysis concepts necessary • IO/connections do not require recalculation of modal coefficients if varied • Minimal precomputation (no eigenvalue problems to be solved) • Minimal storage (no modal shapes/sets of coefficients to be stored) • Multiple outputs generated at no extra cost!