Brasses

Brass instruments consist of a mouthpiece enclosing the lip-valve, an acoustic tube, and finally a rapidly flaring bell. Additionally, most brass instruments have valves which switch among different tube lengths.

The mouthpiece defines the precise geometry of the lip opening as well as providing partial terminations against which the lips work. For lip-valve modeling, one highly regarded model is that of S. Adachi [2,3,61] in which a mass modeling a lip follows an orbit in 2D, with different eccentricities at different frequencies. Such motion of brass players' lips has been observed by Copley and Strong [42] and others. One- and two-mass models were also developed by Rodet and Vergez [128,191]. An early one-mass, ``swinging door'' model that works surprisingly well is in Cook's HosePlayer [38] and and TBone [37]. It is also used in the Brass.cpp patch in the Synthesis Tool Kit [40]; in this simplified model, the lip-valve is modeled as a second-order resonator whose output is squared and hard-clipped to a maximum magnitude.

The acoustic tube can be modeled as a simple digital waveguide [11,15,155,176,188], whether it is cylindrical or conical. More complicated, flaring horn sections may be modeled as two-port digital filters [17,23,30,53,142,188]. It is known, however, that nonlinear shock waves develop in the bore for large amplitude vibrations [79,117]. A simple nonlinear waveguide simulation of this effect can be implemented by shortening the delay elements slightly when the instantaneous amplitude is large. (The first-order effect of air nonlinearity in large-amplitude wave propagation is an increase in sound speed at very high pressures [24].)

It is also known that the bores of ``bent horns,'' such as the trumpet, do not behave as ideal waveguides [148]. This is because sharp bends in the metal tube cause some scattering, and mode conversion at high frequencies. Such points along the horn can be modeled using high-frequency loss and reflection.

As in the case of woodwinds, the flaring bell of a horn cannot be accurately modeled as a sparse digital waveguide, because traveling pressure waves only propagate without reflection in conical bores (which include cylindrical bores as a special case) [122].²¹ Digital waveguides are ``sparse'' (free of internal scattering) only when there are long sections at a constant wave impedance.

Since the flare of most brass instruments occurs over a significant distance, the reflection impulse response of the bell is typically quite long. For example, the impulse response of a trumpet bell is hundreds of samples long at typical audio sampling rates [188]. Perhaps the most straightforward approach is to model the bell as a finite-impulse-response (FIR) digital filter. However, such an FIR filter requires hundreds of ``taps'' (coefficients), and is thus prohibitively expensive compared with everything else in a good synthesis model. It is well known that infinite-impulse response (IIR) filters can be much less expensive [115,123], since they use poles and zeros instead of only zeros, but bell impulse responses turn out to pose extremely difficult IIR filter design problems [188]. The main source of difficulty is that the initial impulse response, corresponding to building ``back-scatter'' as the horn curvature progressively increases, appears to be rising quasi-exponentially over hundreds of samples. The natural solution for this provided by many filter design programs is an unstable digital filter. Methods that guarantee stability have been observed to suffer from poor numerical conditioning. The most cost-effective solution to date appears to be the use of truncated IIR (TIIR) digital filters [195]. These filters use an unstable pole to produce exponentially rising components in the impulse response, but the response is cut off after a finite time, as is needed in the case of a bell impulse response. By fitting a piecewise polynomial/exponential approximation to the reflection impulse response of the trumpet bell, very good approximations can be had for the computational equivalent of approximately a 10th order IIR filter (but using more memory in the form of a delay line, which costs very little computation).

In more detail, the most efficient computational model for flaring bells in brass instruments seems to be one that consists of one or more sections having an impulse response given by the sum of a growing exponential and a constant, i.e.,

$$y(n) = \left\{\begin{array}{ll} a e^{c n} + b, & n=0,1,2,\ldots,N-1 \\ [5pt] 0, & \mbox{otherwise}. \\ \end{array}\right.$$

The truncated constant $b$ can also be generated using a one-pole TIIR filter, with its pole set to $z=1$. The remaining reflection impulse response has a decaying trend, and can therefore be modeled accurately using one of many conventional filter design techniques. In [188,189], the Steiglitz-McBride IIR filter design algorithm [102] yielded good results, as shown in Fig. 18.

Figure 17: Example of a TIIR filter for generating a growing exponential or constant segment (from [188]).

Figure 18: Impulse-response and driving-point-impedance fit for the trumpet bell using two offset exponentials and two biquads designed as a 4th-order IIR filter using the Steiglitz-McBride algorithm (from [188]). The dot-dashed lines show the model segment boundaries.

In summary, a cost-effective synthesis model for brasses includes a careful finite-difference lip-valve model, a digital waveguide bore, and a TIIR bell filter.