Refinements to the Model of a Single Woodwind Instrument Tonehole

Using the Finite Element Method (FEM), a single unflanged tonehole was simulated for a wide range of heights and diameters in order to improve the accuracy of transmission-matrix calculations for instruments with toneholes of large diameter and short height, as found on saxophones and concert flutes. These calculations confirm the validity of existing models for toneholes of smaller diameter and longer height, as found on clarinets. Revised one-dimensional transmission-matrix models of open and closed toneholes are presented to extend the validity of the models based on the FEM results. Further, these tonehole models are verified to be valid for use with both cylindrical and conical (flare angles up to 6 degrees) air columns. For open and closed toneholes, new formulas for the low frequency values of the shunt and series length correction are developed as a function of t/b and δ. Discrepancies with current theories are particularly apparent in the series length correction term. At higher frequencies, the open shunt equivalent length increases faster than previously predicted, corroborating recent experimental data (Dalmont et al., 2002). This effect is more important for short toneholes. These results do not take into account any possible internal or external interactions between the toneholes on an instrument, which may have an important effect for large-diameter toneholes