Idealized digital models for conical reed instruments, with focus on the internal pressure waveform

Abstract

Two models for the generation of self-oscillations of reed conical woodwinds are presented. They use the fewest parameters (of either the resonator or the exciter), whose influence can be quickly explored. The formulation extends iterated maps obtained for lossless cylindrical pipes without reed dynamics. It uses spherical wave variables in idealized resonators, with one parameter more than for cylinders: the missing length of the cone. The mouthpiece volume equals that of the missing part of the cone, and is implemented as either a cylindrical pipe (first model) or a lumped element (second model). Only the first model adds a length parameter for the mouthpiece and leads to the solving of an implicit equation. For the second model, any shape of nonlinear characteristic can be directly considered. The complex characteristics impedance for spherical waves requires sampling times smaller than a round trip in the resonator. The convergence of the two models is shown when the length of the cylindrical mouthpiece tends to zero. The waveform is in semiquantitative agreement with experiment. It is concluded that the oscillations of the positive episode of the mouthpiece pressure are related to the length of the missing part, not to the reed dynamics.

Continuation of periodic solutions of various types of delay differential equations using asymptotic numerical method and harmonic balance method

“…The waveform of each unknown over one period is a by-product of the method and is available for each value of γ, i.e for each point of the bifurcation diagram 5. The waveform of p is compared to the one obtained with the numerical integration algorithm presented in [18] for γ = 0.5 and is in very good agreement as can be seen in figure 6. On the same figure, the pressure obtained by integration with the built-in solver ddensd [19] of MATLAB is presented.…”

“…This example shows the robustness of our approach. The fourth and last example is a neutral model of saxophone [18,9] using transcendental functions in its writing. A solution is compared to a finite-difference scheme from the literature [18] and a bifurcation diagram is drawn, showing an expected behaviour for this type of system according to [11].…”

“…In this section, a neutral system that is a dimensionless model of saxophone taken from [18] is studied. It has also been the subject of the conference talk [9].…”

“…The derivative of Eq. (40) in [18] is taken to get rid of the integrals. This will necessitate to be taken into account when considering the Fourier developments of the unknowns.…”

“…The dimensionless delay τ can be expressed in term of the length of the conical resonator ℓ of the saxophone, truncated at its input to plug the mouthpiece, and the length x 1 of the missing part of the cone (from the apex to the input of the truncated cone). This is not our purpose here to understand the refinements of this model, the interested reader can find the details in [18]. Below, the value of τ is set to 2…”