The ``cyclone'' and ``cylindrical saxophone'' models offer acoustically accurate representations of their corresponding physical manifestations.
In addition, they inherit the various potential disadvantages previously discussed for these structures, including modal inharmonicity and weak low frequency mode support. An accurate physical model does not always make the best musical instrument.
It is an interesting exercise to throw physical reality aside and to consider a more abstract approach to the design of an acoustic air column or resonator.
In order to synthesize conical bore instrument sounds using a pressure-controlled driving mechanism, we desire a stable structure with the following features:
resonance frequencies given by n c/2L, for
an input point with sufficient pressure fluctuation to drive a reed function.
An open-open cylinder possesses the desired modal characteristic. However, air pressure variations are constrained to zero at an open pipe end, or pressure node.
Another system possessing the desired resonance structure is a stretched string, fixed at both its ends. The string and open pipe share a variety of analogous properties via an exchange of mechanical and acoustic variables.
Likewise, there are similarities in the way that each system can be driven. A string can be bowed, plucked, or struck at any point along its length except near either of its ends, where the mechanical velocity is constrained to zero.
By analogy, it should be possible to drive an open pipe by applying a pressure-controlled excitation at any point along its length other than near an open end.
This possibility has not been realized because no appropriate pressure-controlled device has been developed that can be positioned inside a pipe without modifying its acoustic properties.
Digital synthesis systems, however, are not limited by the physical constraints of reality.
Inspired by the bowed string, the ``blowed string'' model incorporates an open-open cylindrical air column structure and a non-linear reed function, applied at a ``blowing'' point that can be varied along the length of the pipe.
The ``blowed string'' digital waveguide block diagram is shown in Fig. 23.
``Blowed string'' block diagram.
One end of the pipe is modeled with a lossy reflectance filter
, while the other end is represented by an ideal impedance of zero (which corresponds to the pressure wave multiplier -1).
The internal pressure at any point within the air column is calculated by summing the two traveling-wave components. The reed function uses the internal pressure value, together with the current ``blowing'' pressure, to determine an appropriate function output.
The actual ``blowed string'' implementation can be simplified with respect to the block diagram of Fig. 23. In practice, each delay-line pair can be combined into a single unit and the scalar multiplier can be commuted to a more convenient implementation point.
Thus, the entire system can be implemented using two interpolating delay lines, a simple first-order lowpass filter, an adder, and a non-linear reed reflection function. It is even possible to use just a single delay line with a fractional-delay tap output.
Variation of the blow point can provide a wide range of timbres. The relationship between bow position and harmonic content of a bowed-string sound applies to this structure as well.
When ``blowed'' at 1/nth the distance from a pipe end, modes at integer multiples of n are not excited.
By positioning the ``blow point'' at the center of the air column, the characteristic timbre of a clarinet is achieved. In addition to being the most computationally efficient model discussed, the ``blowed string'' displays exceptionally robust behavior over a wide parameter space.
Figure 24 displays the input impedance and sound spectrum produced by the ``blowed string'' waveguide model with a ``blow'' point positioned at 1/5 the distance from an open end.
Example ``blowed string'' model input impedance and synthesized sound spectrum.
The attenuation of the 5th harmonic is clearly evident. In general, there is significantly more harmonic content in this sound than was present in the previously discussed models.