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The Clarinet Tonehole as a Two-Port Junction
Figure 6.6:
Lumped-parameter description of the
clarinet tonehole.
|
The clarinet tonehole model developed by Keefe [225] is
parametrized in terms of series and shunt resistance and reactance, as
shown in Fig. 6.6. 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) |
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|
|
(closed-hole shunt impedance) |
|
|
(7.17) |
(open-hole series impedance) |
|
|
|
(closed-hole series impedance) |
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|
|
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 [225]
where is the physical tonehole chimney height at its center.
Note that the specific resistance of the open tonehole, , is the
only real impedance and therefore the only source of wave energy loss at
the tonehole. It is given by [225]
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
and is the real part of the propagation wavenumber (or minus the imaginary part
of complex spatial frequency ). In [224], for the large-tube limit (i.e.,
when the tube radius is large compared with the viscous boundary layer), is given
by
where
is the adiabatic gas constant for air
[298], is the thermal conductivity of air, and
is the specific heat of air at constant pressure. In [224], the
following values are given for air at Kelvin (
C),
and valid within degrees of that temperature:
where
can be interpreted as times the ratio of the tonehole radius
to the viscous boundary layer thickness [224]. The constant
is referred to as the Prandtl number, and is the shear
viscosity coefficient [224]. In [68], it is noted that
is greater than under practical conditions in musical acoustics, and so it
is therefore sufficient to keep only the first and second-order terms in the
expression above for .
The open-hole effective length , assuming no pad above the hole,
is given in
[225] as
See [225] for the case in which a pad lies above the open
hole. In [378], a unified tonehole model is
given which supports continuous opening and closing of the tonehole.
For implementation in a digital waveguide model, the lumped parameters
above must be converted to scattering parameters. Such formulations of
toneholes have appeared in the literature: Vesa Välimäki
[481,477] developed tonehole models based on a ``three-port''
digital waveguide junction loaded by an inertance, as described in
Fletcher and Rossing [133], and also extended his
results to the case of interpolated digital waveguides. It should be
noted in this context, however, that in the terminology of
Appendix G, Välimäki's tonehole representation is a loaded
2-port junction rather than a three-port junction. (A load can be
considered formally equivalent to a ``waveguide'' having wave
impedance given by the load impedance.) Scavone and Smith
[375] developed digital waveguide tonehole
models based on the more rigorous ``symmetric T'' acoustic model of
Keefe
[225], using general purpose digital filter design techniques to
obtain rational approximations to the ideal tonehole frequency
response. A detailed treatment appears in Scavone's CCRMA
Ph.D. thesis [379]. This chapter, from
[437], considers an exact
translation of the Keefe tonehole model, obtaining two one-filter
implementations: the ``shared reflectance'' and ``shared
transmittance'' forms. These forms are shown to be stable without
introducing an approximation which neglects the series inertance terms
in the tonehole model.
By substituting
in (6.19) to convert spatial
frequency to temporal frequency, and by substituting
for , into (6.17) 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. 6.7, where
|
(7.22) |
is the reflectance of the tonehole (the same from either direction), and
|
(7.23) |
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) [404, p. 221].
Figure 6.7:
Frequency-domain, traveling-wave
description of the clarinet tonehole.
|
The approximate forms in (6.23) and (6.24) 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 (6.19).
In a manner analogous to converting the four-multiply Kelly-Lochbaum (KL)
scattering junction
[231]
into a one-multiply form (cf. (G.60) and
(G.62) on page ), we may pursue a ``one-filter'' form of the waveguide
tonehole model. However, the series inertance gives some initial trouble,
since
instead of zero as in the KL junction. In the scattering formulas
(G.83) and (G.85) on page
for the general loaded waveguide junction, the reflectance seen on any
branch is always the transmittance from that branch to any other
branch minus . I.e., if denotes the transmittance from
branch to all other branches meeting at the junction, then
is the reflectance seen on branch . Substituting
into the basic scattering relations (6.22), and factoring out ,
we obtain, in the frequency domain,
and, similarly,
The resulting tonehole implementation is shown in Fig. 6.8.
We call this the ``shared reflectance'' form of the tonehole junction.
In the same way, an alternate form is obtained from the substitution
which yields the ``shared transmittance'' form:
shown in Fig. 6.9.
Figure 6.8:
``Shared-reflectance'' implementation of
the clarinet tonehole model.
|
Figure 6.9:
``Shared-transmittance''
implementation of the clarinet tonehole model.
|
Figure 6.10:
``Shared-reflectance'' tonehole
model with unstable allpasses pulled out to the inputs.
|
Figure 6.11:
``Shared-transmittance''
tonehole model with unstable allpasses pulled out to inputs.
|
Since
, it can be neglected to first order, and
, reducing both of the above forms to an approximate
``one-filter'' tonehole implementation.
Since
is a pure negative reactance, we have
|
(7.28) |
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. 6.8 or
Fig. 6.9. Using elementary manipulations, the
unstable allpasses in Figs. 6.8 and
Fig. 6.9 can be moved to the configuration shown
in Figs. 6.10 and
6.11, 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
(6.29),
the phase delay of can be computed as
Thus, the negative delay correction goes to zero with frequency
, series tonehole length , tonehole impedance , or
main bore admittance
.
In practice, it is common to combine all delay corrections into a single
``tuning allpass filter'' for the whole bore [404,194].
Whenever the desired allpass delay goes negative, we simply add a sample of
delay to the desired allpass phase-delay and subtract it from the nearest
delay. In other words, negative delays have to be ``pulled out'' of the
allpass and used to shorten an adjacent interpolated delay line. Such
delay lines are normally available in practical modeling situations.
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