### 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.*,

`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].**Next Section:**

Singing Voice

**Previous Section:**

Modeling the Lips and Mouthpiece