### Bell Models

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)
[357].^{10.24} Digital waveguides are ``sparse''
(free of internal scattering) only when there are long sections at a
constant wave impedance.

The most cost-effective bell filters (and, more generally, ``flare
filters'') to date appears to be the use of
*truncated IIR* (TIIR) digital filters
[540]. 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.*,

The C++ class in the Synthesis Tool Kit (STK) implementing a basic
brass synthesis model is called `Brass.cpp`.

#### Literature Relevant to Brasses

A nice summary of the musical acoustics of brass instruments is given by Campbell [70].

Research by Cullen et al. [98] and Gilbert et al. [159,161] has been concerned with artificial mouth experiments to verify a theoretical model of vibrating lips, and to calibrate the model to realistic playing conditions. These data can be used in the construction of improved lip-valve models for virtual brass instruments. Additional literature relevant to brass instruments includes [187,186,188,247,529,319,331,336,534,535,536,445,93].

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Singing Voice

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Modeling the Lips and Mouthpiece