inharmonicity is dependent on the dimensions of
the frustum. Attempts at robust synthesis using a
model which correctly simulates the behavior of a
truncated conical section may in turn be hindered.
3,
e.5
0.0
-----'. ~ --
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(cone) = ro/re (cylinder)
Figure 5: Modal frequencies for a closed-open conic
frustum normalized by the fundamental frequency
of an open pipe of the same length as the frustum.
The dotted curves indicate integer relationships to
the first mode.
Benade (1976) reports that the effects of
truncation can be reduced by utilizing a
reed/mouthpiece cavity with an equivalent volume
equal to that of the missing conical section. This
constraint is based on a lumped characterization of
the reed/mouthpiece cavity, which is only appropriate for low-frequency modes whose wavelengths
are large in comparison to the dimensions of the
cavity. Higher-frequency modes are less likely to
benefit from such a change because they are more
directly affected by changes in waveguide shape.
Figure 6 plots the mode ratios for a cylindercone compound horn designed so that the cylindrical section volume is equal to the truncated conic
section volume. For /3 = 1, the structure is of infinite length and all its modes converge to zero. In
comparison with Fig. 5, the compound horn displays nearly harmonic mode ratios out to values of
/3 in the range 0.2-0.3.
Another property of conic frusta can be directly attributed to the input inertance element,
M0, in the equivalent circuit (Fig. 4). The inertance, whose magnitude varies with the parameter
/3, tends to "Lshunt" low-frequency wave components, thus imposing a "high-pass" characteristic
on the resulting air column mode structure. For
longer conic sections, the lowest modes can be significantly attenuated, which in turn destabilizes oscillatory regimes dependent on these modes. This
behavior is often apparent in the lowest notes of
saxophones, which tend to be difficult to control
under soft playing conditions. Figure 7 shows an
0 0.1 0.2 0.3 0.4 005 0.6 0.7 0.8 0.0
(cone) = Tl (cylinder)
Figure 6: Modal frequencies for a closed-open,
cylinder-cone compound horn in which the cylindrical section volume is equivalent to the missing
conic section volume. The dotted curves indicate
integer relationships to the first mode.
example conic section input impedance in which
this effect is demonstrated. The smooth curve indicates the combined influence of the conicity inertance and the open-end load impedance.
Frequency (kHz)
Figure 7:
impedance.
An example conic frustum input
4 Model Approaches
In order to implement the equivalent circuit
of a conical waveguide using digital waveguide
techniques, it is necessary to express the lumped
impedance elements of Fig. 4 in terms of travelingwave parameters and then convert these expressions to discrete-time filters. The impedance of
the input inertance, given in terms of a Laplace
transform, is M0(s) = (px0/A~,) s, where p is the
mass density of air, A0 is the area of the spherical wavefront at the waveguide input, and s is
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