1 / 16

Cymbal Synthesis

Cymbal Synthesis. Stefan Bilbao Music Edinburgh. Shell Models Finite Difference Schemes Computational Issues in Synthesis. Cymbal Modeling. Cymbals: an interesting synthesis problem: Simple PDE decription Regular geometry Highly nonlinear Time-domain methods are a very good match

gwalls
Download Presentation

Cymbal Synthesis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Cymbal Synthesis Stefan BilbaoMusic Edinburgh • Shell Models • Finite Difference Schemes • Computational Issues in Synthesis

  2. Cymbal Modeling Cymbals: an interesting synthesis problem: • Simple PDE decription • Regular geometry • Highly nonlinear Time-domain methods are a very good match (modal methods another possibility: ASA paper by Camier et al. on Tuesday)

  3. Crashes Nonlinear effects are perceptually dominant: spontaneous generation of high frequencies (crash) A linear model is wholly insufficient!

  4. A Shell Model Parameters: • Young’s modulus E • Poisson’s ratio n • Density r • Thickness h • Radius of curvature a • Size R Spherical shell model employed by, e.g., Thomas, Touze, Chaigne in recent work is a good rough approximation; Basic assumptions: shell is thin, shallow, uniform thickness and density + various other more technical hypotheses. r 0 Radial coordinate r Angular coordinate q Transverse displacement w(r,q, t) w(r,q, t) R h q a

  5. A Nonlinear System System is an extension of that describing a thin flat, lossless, linear plate: Linear plate model: Nonlinearity: Airy stress function: Bracket operator: Shell curvature: Loss terms: Excitation:

  6. Boundary and Center Conditions Free edge condition: Center conditions: Unconstrained: Clamped: Others possible: pivoting, with collision, perhaps loss

  7. Difference Schemes in Polar Coordinates Polar cordinates a natural choice: Spacing hr in radial direction, hq in angular direction Total grid size: Approx. 2p/ hr hq points Operation count/time step will scale with this number of points… rhq hr

  8. Bi-Laplacian operator in Polar Coordinates Key operation to approximate is the bi-Laplacian:  DD A sparse operation… DD Needs to be specialized at: Grid points at or bordering the center, using center conditions: Grid points at or bordering the edge, using B.C.s: Also need approximations to the Laplacian, and to operators in bracket…

  9. Implicit Schemes • One implicit family of schemes for the shell equations is the following: Free tuning parameter Explicit disc. of nonlinear terms; can stabilize by using implicit disc.! • Solution advanced as follows: 1. Determine Fn from known wn 2. Determine wn+1 from known Fn, wn 3. Update variables and repeat…

  10. Implicit vs. Explicit The use of a good implicit scheme is essential, especially in the nonlinear case. Numerical “cutoff” Explicit a=0 Implications of low numerical cutoff: • Linear case: severe dispersion, mistuning of modes • Nonlinear case: inability of scheme to generate high-frequency energy! Energy “piles up” at cutoff…sounds terrible! Implicit a=0.2497 A problem, generally, for methods working over non-uniform grids…

  11. Excitation • In a full physical model, need a model of the stick/mallet interaction • Simpler, in practice, to use an external forcing function • Here, g represents the spatial extent of the mallet, and h(t) the time history of the gesture

  12. Effects of Curvature • The main effects of increased curvature are: • Decreased sense of a hum tone • Shifting upward of lowest frequencies, to give a brighter timbre… • Sound examples: for k=100, and for different values of the curvature parameter g, • In addition, computing time decreases with increased curvature…(fewer DOFs in audio range) g = 0 g = 40 g = 60 g = 100

  13. Stability • A stabilitiy condition follows for the above scheme:

  14. Implementation • Useful to reorder the grid functions w, F as vectors: DD  • Update becomes a pair of linear systems to solve: • where are known (previously computed) vectors

  15. Fast Linear System Solution Techniques: Structured Matrices • Various approaches to solving these systems: • Calculate inverses offline (a bad idea) • Use standard linear system solvers (Gauss-Seidel, SOR, CG, etc.) • Note: system matrices are very sparse, and possess a great deal of structure---which can be exploited: Circulant blocks (from periodicity in angular direction) Can employ DFTs to simplify structure: • Essentially a deconvolution operation in the angular direction • System is even more sparse much faster linear system solution • Only works in polar coordinates! (But can generalize to fast block-Toeplitz linear system solution in Cartesian coordinates)

  16. Conclusions • Extensions: • More complex center conditions • Variable thickness • Generalized curvature • Lumped element connections (sizzles)

More Related