Electronic Music Synthesis:  Physical modelling of the drum Craig Wilson Prof Clive Greated and Dr Stefan Bilbao
Overview <ul><li>Introducing physical modelling </li></ul><ul><li>The methods of finite differences </li></ul><ul><li>Resu...
Music synthesis <ul><li>1970s: Abstract synthesis </li></ul><ul><ul><li>Mathematical description of the sound </li></ul></...
The two-headed drum <ul><li>Membranes </li></ul><ul><ul><li>Wave equation with stiffness (K) </li></ul></ul><ul><li>Sound ...
Finite Difference Approximations <ul><li>Governing PDEs of the instrument are approximated using FDAs </li></ul><ul><li>Co...
Finite Differences in a circular domain <ul><li>Continuous function </li></ul><ul><ul><li>Membrane displacement  </li></ul...
FD recipe:  Laplacian operator <ul><li>Continuous – Discrete – Recurrence - Matrix </li></ul>
FD recipe:  discrete Laplacian matrix <ul><li>MATLAB implementation </li></ul><ul><li>Vector-matrix update scheme </li></u...
Results:  Single membrane in a vacuum <ul><li>Test of Laplacian </li></ul><ul><li>Frequencies are inharmonically related a...
Results:  Single membrane in a vacuum <ul><li>Deviation in discrete from analytic frequencies </li></ul><ul><li>Deviation ...
Results:  Single membrane coupled to air cavity <ul><li>Coupling constant  d  describes internal air cavity </li></ul><ul>...
Results:  Single membrane coupled to air cavity <ul><li>This form of coupling not a viable model for single-headed drums i...
Results:  Two membranes coupled via air cavity <ul><li>Upper membrane:  u </li></ul><ul><li>Lower membrane:  w </li></ul><...
Snapshots: t = 0.48ms
Snapshots: t = 1.4ms
Snapshots: t = 2.3 ms
Snapshots: t = 3.2 ms
Snapshots: t = 4.1 ms
Snapshots: t = 5 ms
Results:  Two membranes coupled: Movie & sound <ul><li>Upper: </li></ul><ul><li>Lower: </li></ul><ul><li>Both membranes: <...
Results:  Computational and real drum FFTs Computational model Recorded snare drum
Results:  Two membrane: frequency splitting <ul><li>Splitting of the eigenfrequencies </li></ul><ul><ul><li>Corresponds to...
Limitations & extensions <ul><li>Was unable to include membrane stiffness </li></ul><ul><ul><li>Produces ‘explosive’ solut...
Conclusion <ul><li>Numerical model of a two-headed drum </li></ul><ul><ul><li>Two-membranes coupled via an internal air ca...
Question Time <ul><li>Thank you all for listening! </li></ul>
Some additional slides…
Eigenfrequencies and modes <ul><li>Sound spectrum is ‘fingerprint’ of sound </li></ul><ul><ul><li>Eigenfrequencies present...
Eigenfrequencies and modes
Parameters
Matrix update scheme
Deviation tables
Deviation tables
Laser transducer:  Justifying the air coupling
Unphysical drum: drum-like timbre
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Electronic Music Synthesis

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or How to Make a Drum using Physical Modelling.

This was my final year presentation and received an A grade.

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Electronic Music Synthesis

  1. 1. Electronic Music Synthesis: Physical modelling of the drum Craig Wilson Prof Clive Greated and Dr Stefan Bilbao
  2. 2. Overview <ul><li>Introducing physical modelling </li></ul><ul><li>The methods of finite differences </li></ul><ul><li>Results of the numerical model </li></ul><ul><ul><li>Laplacian operator and eigenfrequencies </li></ul></ul><ul><ul><li>Single membrane in a vacuum </li></ul></ul><ul><ul><li>Single membrane coupled to internal air cavity </li></ul></ul><ul><ul><li>Two membranes coupled via air cavity </li></ul></ul><ul><li>Limitations and extensions </li></ul><ul><li>Conclusion & Questions </li></ul>
  3. 3. Music synthesis <ul><li>1970s: Abstract synthesis </li></ul><ul><ul><li>Mathematical description of the sound </li></ul></ul><ul><ul><ul><li>Replicate the frequency spectrum </li></ul></ul></ul><ul><li>1990/2000s: Direct synthesis </li></ul><ul><ul><li>Physical description of the instrument </li></ul></ul><ul><ul><ul><li>Reflects the underlying acoustics (set of PDEs) </li></ul></ul></ul><ul><ul><ul><li>Improved ‘playability’ (material and dimensions) </li></ul></ul></ul><ul><ul><ul><li>Allows modular construction </li></ul></ul></ul>
  4. 4. The two-headed drum <ul><li>Membranes </li></ul><ul><ul><li>Wave equation with stiffness (K) </li></ul></ul><ul><li>Sound radiation </li></ul><ul><ul><li>Frequency-independent loss (sigma) </li></ul></ul><ul><li>Enclosed air cavity </li></ul><ul><ul><li>Couples the two membranes (d) </li></ul></ul><ul><li>Shell </li></ul><ul><ul><li>Assumed rigid </li></ul></ul>Upper membrane Enclosed air cavity Lower membrane Shell Snares Excitation
  5. 5. Finite Difference Approximations <ul><li>Governing PDEs of the instrument are approximated using FDAs </li></ul><ul><li>Continuous domain is restricted to a grid of discrete points </li></ul><ul><li>Spatial derivatives are approximated using differences from nearby grid values </li></ul><ul><li>Temporal derivatives are approximated using recurrence relations derived from PDEs </li></ul>
  6. 6. Finite Differences in a circular domain <ul><li>Continuous function </li></ul><ul><ul><li>Membrane displacement </li></ul></ul><ul><li>Circular grids less common </li></ul><ul><li>Stability conditions </li></ul><ul><ul><li>Grid spacing, time step, wave speed </li></ul></ul><ul><ul><li>Determines resolution of grid </li></ul></ul>
  7. 7. FD recipe: Laplacian operator <ul><li>Continuous – Discrete – Recurrence - Matrix </li></ul>
  8. 8. FD recipe: discrete Laplacian matrix <ul><li>MATLAB implementation </li></ul><ul><li>Vector-matrix update scheme </li></ul><ul><ul><li>Grid function u becomes a 1D vector </li></ul></ul><ul><ul><li>Operator matrices become very sparse </li></ul></ul><ul><li>Implemented FD scheme is a solution of linear matrix relations </li></ul>
  9. 9. Results: Single membrane in a vacuum <ul><li>Test of Laplacian </li></ul><ul><li>Frequencies are inharmonically related as expected </li></ul><ul><li>Correct fundamental modes of vibration (eigenvectors) </li></ul><ul><li>Deviation from analytical frequencies </li></ul>
  10. 10. Results: Single membrane in a vacuum <ul><li>Deviation in discrete from analytic frequencies </li></ul><ul><li>Deviation mainly in higher modes </li></ul><ul><li>Drum spectrum deal mainly in lower modes </li></ul>
  11. 11. Results: Single membrane coupled to air cavity <ul><li>Coupling constant d describes internal air cavity </li></ul><ul><ul><li>Cavity depth, air density, wave speed in air, tension from the membrane experienced </li></ul></ul><ul><li>Essentially treats air as a piston (Hook-like) </li></ul><ul><ul><li>Only axisymmetric modes are modelled </li></ul></ul><ul><ul><li>Models the air in an indirect way </li></ul></ul><ul><li>Expect the lowest frequency modes to increase </li></ul>
  12. 12. Results: Single membrane coupled to air cavity <ul><li>This form of coupling not a viable model for single-headed drums in an unconfined ‘sea’ of air </li></ul>
  13. 13. Results: Two membranes coupled via air cavity <ul><li>Upper membrane: u </li></ul><ul><li>Lower membrane: w </li></ul><ul><li>Action of the internal air coupling best seen through successive snapshots of grid </li></ul>
  14. 14. Snapshots: t = 0.48ms
  15. 15. Snapshots: t = 1.4ms
  16. 16. Snapshots: t = 2.3 ms
  17. 17. Snapshots: t = 3.2 ms
  18. 18. Snapshots: t = 4.1 ms
  19. 19. Snapshots: t = 5 ms
  20. 20. Results: Two membranes coupled: Movie & sound <ul><li>Upper: </li></ul><ul><li>Lower: </li></ul><ul><li>Both membranes: </li></ul>
  21. 21. Results: Computational and real drum FFTs Computational model Recorded snare drum
  22. 22. Results: Two membrane: frequency splitting <ul><li>Splitting of the eigenfrequencies </li></ul><ul><ul><li>Corresponds to push-push / push-pull paired modes </li></ul></ul>
  23. 23. Limitations & extensions <ul><li>Was unable to include membrane stiffness </li></ul><ul><ul><li>Produces ‘explosive’ solutions </li></ul></ul><ul><ul><li>Requires derivation of a new stability condition </li></ul></ul><ul><ul><li>Without stiffness the membrane is an ideal material </li></ul></ul><ul><li>Frequency-dependent loss </li></ul><ul><ul><li>Currently acts over all frequencies equally </li></ul></ul><ul><ul><li>Could use to suppress unwanted higher modes </li></ul></ul><ul><li>Full 3D treatment of internal air </li></ul>
  24. 24. Conclusion <ul><li>Numerical model of a two-headed drum </li></ul><ul><ul><li>Two-membranes coupled via an internal air cavity </li></ul></ul><ul><li>Produces some physically meaningful results </li></ul><ul><ul><li>Response to changing physical dimensions </li></ul></ul><ul><ul><li>Accurate discrete Laplacian eigenfrequencies </li></ul></ul><ul><ul><li>Shift in membrane frequencies from vacuum to air coupling </li></ul></ul><ul><ul><li>Splitting of frequencies in two-headed drums </li></ul></ul><ul><li>Produces some unphysical results </li></ul><ul><ul><li>Cannot describe a membrane in an unconfined body of air </li></ul></ul><ul><ul><li>Fails to capture characteristic drum timbre (stiffness, loss) </li></ul></ul>
  25. 25. Question Time <ul><li>Thank you all for listening! </li></ul>
  26. 26. Some additional slides…
  27. 27. Eigenfrequencies and modes <ul><li>Sound spectrum is ‘fingerprint’ of sound </li></ul><ul><ul><li>Eigenfrequencies present in sound largely determine the timbre </li></ul></ul><ul><ul><li>Frequencies time evolution </li></ul></ul><ul><li>Waveform also important: freq-indep loss </li></ul>
  28. 28. Eigenfrequencies and modes
  29. 29. Parameters
  30. 30. Matrix update scheme
  31. 31. Deviation tables
  32. 32. Deviation tables
  33. 33. Laser transducer: Justifying the air coupling
  34. 34. Unphysical drum: drum-like timbre
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