A Modular Percussion Synthesis Environment

Stefan Bilbao

Acoustics and Fluid Dynamics Group / Music

University of Edinburgh

DAFX-09,

Como, Sept., 2009.

1. Modular synthesis2. Percussion synthesis: Components and connections3. FD schemes/computational issues and costs4. Sound examples

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…

Components: Bars

Basic parameter set:

]1,0[22 102 xuuuu txxtxxxxtt

Stiffness Freq.-ind loss

Freq.-dep loss

density

E Young’s modulus

A cross-sectional area

I moment of inertia

L length

loss parameters

Scaling…

= (EI/AL4)1/2

stiffness

loss

M Mass (ratio)

Reduced equivalent parameter set:

Linear, thin, uniform bar:

+ boundary conditions: clamped, pivoting, free, etc.

PDE model:

Can relax assumptions: non-uniform, thick, nonlinear…

Components: Plates

Basic parameter set:

]/1,0[],0[),(

22 210

222

yx

uuuu tttt

density

E Young’s modulus

H thickness

Poisson’s ratio

Lx, Ly dimensions

loss parameters

Scaling…

= (EH2/(1-2)Lx2Ly

2)1/2 stiffness

loss

Poisson’s ratio

aspect ratio

M mass (ratio)

Reduced equivalent parameter set:

Linear, thin, uniform rectangular plate:

+ boundary conditions: clamped, pivoting, free, etc.

PDE model:

Can relax assumptions: non-uniform, thick, nonlinear, geometry…

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

)()( )2()2()1()1( xx

)2()2()2(

)1()1()1(

ctt

ctt

Fu

Fu

tcF 2321

20

)2()2()2()1()1()1( xx dudu )2(

)1(

)2()1(

ccc FM

MFF

Generalized relative displacement Forces equal and opposite

Spring/damper connection:

where

Connection is dissipative (passive).

Linear spring

nonlinear spring (cubic)

damper

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

)()( )()( tFu inintt x

xxx dutto tout )(),()(

output: scales with velocity over an output distribution:

Note: output distribution can be time-varying…multiple outputs can be taken simultaneously.

Difference schemes: components

11 nmm

nmm

nm uCuBu

nmu

Unknown (current) state

Previously computed state

Consider a network of M unconnected elements, under zero-input conditions.

For the mth object, can develop an explicit difference scheme directly:

mm CB ,

NN

M

NN

M

N

M

C

C

C

B

B

B

u

u

u

0

0

0

0 11

1

1

11 nnn uCuBu

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

Difference Schemes: Connections

112114111

20

22 nnnnnnnn

c kF

tcF 2341

20

)2()2()2()1()1()1( xx dudu

Unknown appears “linearly,” when previous state is known…guarantee of existence/uniqueness of

solutions…

Many possible discretizations…

nTnTn2111 ueue

For a set of Q connections, in vector form:

111

1

Q

n

Q

nc

QQ

n

Q

n bFMη 11

N

n

NQQ

n uEη

nnc

nn bFm 1

To relate generalized relative displacements to state:

Known at time step n+1 (previously computed)

Thus:

Explicit nonlinear update form

nc

ne

nnn JFKFuCuBu

11 n

cnn JFau 1

nnc

nn bFMη 1nn Euη

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

nnn

cn EabFMEJ

When excitation/connections are present, update becomes…

previously computed/supplied externally)

Using force/displacement relations:

A compact run-time loop, at least in Matlab!

known, positive definite(diagonal for non-overlapping connections)

known

Thus: a unique update, involving a low-order linear system solution…

Numerical Energy Conservation

2/1sq fN

Under lossless conditions, this network conserves energy to machine accuracy…

Energy of PlatesEnergy of BarsEnergy of ConnectionsTotal Energy

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!

sq fN

Basic configuration

A simple set of uncoupled bars…

Can vary boundary conditions

loss

striking points

Sound is very artificial…characteristic of raw linear systems!

Representative configurations

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!