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Clarinet Reed Modelling

Clarinet Reed Modelling. Ye Lu 2011-4-11. Distributed Representation of the Reed. A bar with non-uniform cross-sectional area Clamped to the mouthpiece at one end Additional constraints provided by the mouthpiece profile and interaction with the lip. Lumped Modelling. Euler Method.

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Clarinet Reed Modelling

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  1. Clarinet Reed Modelling Ye Lu 2011-4-11

  2. Distributed Representation of the Reed • A bar with non-uniform cross-sectional area • Clamped to the mouthpiece at one end • Additional constraints provided by the mouthpiece profile and interaction with the lip

  3. Lumped Modelling

  4. Euler Method

  5. Impulse Invariant method • Compute the Inverse Laplace transform to • get impulse response of the analogue filter • Sample the impulse response (quickly enough to avoid aliasing problem) • Compute z-transform of resulting sequence

  6. Adams-Moulton Methods • Linear multistep methods used for the numerical solution of ordinary differential equations

  7. Adams-Moulton Methods • Linear multistep methods used for the numerical solution of ordinary differential equations A. M. Schneider, J. T. Kaneshige, and F. D. Groutage. Higher Order s-to-z Mapping Functions and their Application in Digitizing Continuous-Time Filters. Proc. IEEE, 79(11):1661–1674, Nov. 1991.

  8. Frequency Response

  9. Weighted Sample Methods • Used for a generic linear system and are based on a polynomial interpolation of the system input

  10. Weighted Sample Methods

  11. Numerical Methods

  12. Weighted Sample Methods

  13. Weighted Sample Methods

  14. Weighted Sample Methods

  15. Weighted Sample Methods

  16. Frequency Domain Analysis • Typical resonance frequency lie in the high frequency region, non-critical in helping self-sustained oscillations • the reed resonance has a role in adjusting pitch, loudness and tone color, and in helping transitions to high regimes of oscillation (S. C. Thompson. The Effect of the Reed Resonance on Woodwind Tone Production. J. Acoust. Soc. Am., 66(5):1299–1307, Nov. 1979.)

  17. Frequency Domain Analysis • Euler Method provide poor accuracy even with Fs=44100Hz • Results for the AM methods are in good agreement with theoretical predictions • the magnitude of AM2 amplifies the magnitude of the resonance • the methods becomes unstable at Fs = 190000Hz

  18. Frequency Domain Analysis • Results for the WS methods are in excellent agreement with theoretical predictions, even at low sampling rates • Numerical dissipation is introduced, the amplitude responses is smaller • The phase responses are well preserved by both methods • WS methods better approximate the reed frequency response than AM methods

  19. Time Domain Analysis The quasi-static estimated value underestimates the true pt (D. H. Keefe. )

  20. Time Domain Analysis • For all the digital reeds, pt converges to the dynamic estimate pressure1802 • 1-step methods exhibit robustness with respect to the sampling rate

  21. Time Domain Analysis

  22. Time Domain Analysis • the clarion register can be produced without opening the register hole, if the reed resonance matches a low harmonic of the playing frequency and the damping is small enough • an extremely low damping causes the reed regime to be produced

  23. Time Domain Analysis

  24. The Model

  25. Limitation of the Model • One-dimensional: assume no torsional modes in the reed • No attempt to model the air flow in the reed channel or to simulate the acoustical resonator

  26. Boundary Conditions

  27. Implicit θ-Scheme http://www.tandfonline.com/doi/abs/10.1080/10236190802385298#preview

  28. Implicit θ-Scheme http://www.tandfonline.com/doi/abs/10.1080/10236190802385298#preview

  29. Simulation Result • The reed tip can not exceed a certain value • The tip is not stopped suddenly but rather gradually

  30. The Reed Model

  31. Implicit θ-Scheme

  32. Reed Construction

  33. Numerical Results • The cause of the discontinuity in the one-dimensional case is not the omittance of the reed’s torsional motion

  34. Numerical Results • The reed-lay interaction exhibits a stronger non-linearity when (1) The player’s lip moves towards the free end of the reed (2) When a thinner reed is used

  35. Numerical Results • The closer the lip is positioned towards the free end of the reed, the stronger the non-linear behavior of S becomes

  36. References • F. Avanzini. Computational Issues in Physically-based Sound Models. Ph.D. Thesis, Dept. of Computer Science and Electronics, University of Padova (Italy), 2001. • M. van Walstijn and F. Avanzini. Modelling the mechanical response of the reed-mouthpiece-lip system of a clarinet. Part II. A lumped model approximation. ActaAcustica united with Acustica, 93(3):435-446, May 2007. • F. Avanzini and M. van Walstijn. Modelling the Mechanical Response of the Reedmouthpiece- lip System of a Clarinet. Part I. A One-Dimensional Distributed Model. ActaAcustica united with Acustica, 90(3):537-547 (2004). • A. M. Schneider, J. T. Kaneshige, and F. D. Groutage. Higher Order s-to-z Mapping Functions and their Application in Digitizing Continuous-Time Filters. Proc. IEEE, 79(11):1661–1674, Nov. 1991. • C.Wan and A. M. Schneider. Further Improvements in Digitizing Continuous-Time Filters. IEEE Trans. Signal Process., 45(3):533–542, March 1997. •   V. Chatziioannou and M. van Walstijn. Reed vibration modelling for woodwind instruments using a two-dimensional finite difference method approach. In International Symposium on Musical Acoustics, Barcelona, 2007.

  37. Thank You!

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