in, e.g.), it is more straightforward to treat the
tonehole as a lumped load along the bore, and most modeling efforts
have taken this approach.
Toneholes in woodwind instruments are essentially cylindrical holes in
the bore. One modeling approach would be to treat the tonehole as a
small waveguide which connects to the main bore via one port on a
three-port junction. However, since the tonehole length is small
compared with the distance sound travels in one sampling instant (
in, e.g.), it is more straightforward to treat the
tonehole as a lumped load along the bore, and most modeling efforts
have taken this approach.
The musical acoustics literature contains experimentally verified models of tone-hole acoustics, such as by Keefe [240]. Keefe's tonehole model is formulated as a ``transmission matrix'' description, which we may convert to a traveling-wave formulation by a simple linear transformation (described in §9.5.4 below) [467]. For typical fingerings, the first few open tone holes jointly provide a bore termination [38]. Either the individual tone holes can be modeled as (interpolated) scattering junctions, or the whole ensemble of terminating tone holes can be modeled in aggregate using a single reflection and transmission filter, like the bell model. Since the tone hole diameters are small compared with most audio frequency wavelengths, the reflection and transmission coefficients can be implemented to a reasonable approximation as constants, as opposed to cross-over filters as in the bell. Taking into account the inertance of the air mass in the tone hole, the tone hole can be modeled as a two-port loaded junction having load impedance equal to the air-mass inertance [144,513]. At a higher level of accuracy, adapting transmission-matrix parameters from the existing musical acoustics literature leads to first-order reflection and transmission filters [240,409,406,407,467]. The individual tone-hole models can be simple lossy two-port junctions, modeling only the internal bore loss characteristics, or three-port junctions, modeling also the transmission characteristics to the outside air. Another approach to modeling toneholes is the ``wave digital'' model [531] (see §F.1 for a tutorial introduction to this approach). The subject of tone-hole modeling is elaborated further in [409,506]. For simplest practical implementation, the bell model can be used unchanged for all tunings, as if the bore were being cut to a new length for each note and the same bell were attached. However, for best results in dynamic performance, the tonehole model should additionally include an explicit valve model for physically accurate behavior when slowly opening or closing the tonehole [408].
