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Ye Lu 2011-4-11

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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

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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

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Linear multistep methods used for the numerical solution of ordinary differential equations

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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.

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Used for a generic linear system and are based on a polynomial interpolation of the system input

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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.)

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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

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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

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

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For all the digital reeds, pt converges to the dynamic estimate pressure1802 1-step methods exhibit robustness with respect to the sampling rate

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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

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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

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http://www.tandfonline.com/doi/abs/10.1080/10236190802385298#preview

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The reed tip can not exceed a certain value The tip is not stopped suddenly but rather gradually

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The cause of the discontinuity in the one- dimensional case is not the omittance of the reed’s torsional motion

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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

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The closer the lip is positioned towards the free end of the reed, the stronger the non-linear behavior of S becomes

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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. Acta Acustica 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. Acta Acustica 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.

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