#### The Clarinet Tonehole as a Two-Port Junction

The clarinet tonehole model developed by Keefe [240] is parametrized in terms of series and shunt resistance and reactance, as shown in Fig. 9.43. The

*transmission matrix*description of this two-port is given by the product of the transmission matrices for the series impedance , shunt impedance , and series impedance , respectively:

where all quantities are written in the frequency domain, and the impedance parameters are given by

(open-hole shunt impedance) | |||

(closed-hole shunt impedance) | (10.51) | ||

(open-hole series impedance) | |||

(closed-hole series impedance) |

where is the wave impedance of the tonehole entrance,

*i.e.*, that of an acoustic tube of cross-sectional area ( is air density and is sound speed as usual), is the tonehole radius, is the wavenumber (radian spatial frequency), is the open-tonehole effective length (which is slightly greater than its physical length due to the formation of a small air-piston inside the open tonehole), is the ``specific resistance'' of the open tonehole due to air viscosity in and radiation from the hole, is the closed-tonehole height, defined such that its product times the cross-sectional area of the tonehole exactly equals the geometric volume of the closed tonehole. Finally, and are the equivalent

*series*lengths of the open and closed tonehole, respectively, and are given by

where is the radius of the main bore. The closed-tonehole height can be estimated as [240]

ln

where is the radius of curvature of the tonehole, is the
viscous boundary layer thickness which expressible in terms of the shear
viscosity of air as
*i.e.*, when the tube radius is large compared with the viscous boundary layer), is given by

(10.52) | |||

(10.53) |

for , into (9.51) to convert physical variables to wave variables, ( is the bore wave impedance), we may solve for the outgoing waves in terms of the incoming waves . Mathematica code for obtaining the general conversion formula from lumped parameters to scattering parameters is as follows:

Clear["t*", "p*", "u*", "r*"] transmissionMatrix = {{t11, t12}, {t21, t22}}; leftPort = {{p2p+p2m}, {(p2p-p2m)/r2}}; rightPort = {{p1p+p1m}, {(p1p-p1m)/r1}}; Format[t11, TeXForm] := "{T_{11}}" Format[p1p, TeXForm] := "{P_1^+}" ... (etc. for all variables) ... TeXForm[Simplify[Solve[leftPort == transmissionMatrix . rightPort, {p1m, p2p}]]]The above code produces the following formulas:

Substituting relevant values for Keefe's tonehole model, we obtain, in matrix notation,

We thus obtain the scattering formulation depicted in Fig. 9.44, where

is the

*reflectance*of the tonehole (the same from either direction), and

is the

*transmittance*of the tonehole (also the same from either direction). The notation ``'' for reflectance is chosen because every reflectance is a

*Schur function*(stable and not exceeding unit magnitude on the unit circle in the plane) [428, p. 221]. The approximate forms in (9.57) and (9.58) are obtained by neglecting the negative series inertance which serves to adjust the effective length of the bore, and which therefore can be implemented elsewhere in the interpolated delay-line calculation as discussed further below. The open and closed tonehole cases are obtained by substituting and , respectively, from (9.53). In a manner analogous to converting the four-multiply Kelly-Lochbaum (KL) scattering junction [245] into a one-multiply form (cf. (C.60) and (C.62) on page ), we may pursue a ``one-filter'' form of the waveguide tonehole model. However, the series inertance gives some initial trouble, since

*I.e.*, if denotes the transmittance from branch to all other branches meeting at the junction, then is the reflectance seen on branch . Substituting

(10.58) |

and, similarly,

(10.59) |

The resulting tonehole implementation is shown in Fig. 9.45. We call this the ``shared reflectance'' form of the tonehole junction. In the same way, an alternate form is obtained from the substitution

(10.60) | |||

(10.61) |

shown in Fig. 9.46.

In this form, it is clear that is a first-order

*allpass*filter with a single pole-zero pair near infinity. Unfortunately, the pole is in the right-half-plane and hence

*unstable*. We cannot therefore implement it as shown in Fig. 9.45 or Fig. 9.46. Using elementary manipulations, the unstable allpasses in Figs. 9.45 and Fig. 9.46 can be moved to the configuration shown in Figs. 9.47 and 9.48, respectively. Note that is stable whenever is stable. The unstable allpasses now operate only on the two incoming wave variables, and they can be implemented implicitly by slightly reducing the (interpolated) delay-lines leading to the junction from either side. The tonehole then requires only one filter or . We now see precisely how the negative series inertance provides a

*negative, frequency-dependent, length correction*for the bore. From (9.63), the phase delay of can be computed as

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Tonehole Filter Design

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Clarinet Synthesis Implementation Details