Clarinet Reed Modelling

Ye Lu2011-4-11

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

Distributed Representation of the Reed

Lumped Modelling

Euler 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

Impulse Invariant method

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

Adams-Moulton Methods

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

Adams-Moulton Methods

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.

Frequency Response

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

Weighted Sample Methods

Weighted Sample Methods

Numerical Methods

Weighted Sample Methods

Weighted Sample Methods

Weighted Sample Methods

Weighted Sample Methods

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.)

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

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

Frequency Domain Analysis

Time Domain Analysis

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

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

Time Domain Analysis

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

Time Domain Analysis

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

Limitation of the Model

Boundary Conditions

Implicit θ-Scheme

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

Implicit θ-Scheme

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

The reed tip can not exceed a certain value

The tip is not stopped suddenly but rather gradually

Simulation Result

The Reed Model

Implicit θ-Scheme

Reed Construction

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

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

Numerical Results

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

Numerical Results

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.

References

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