Characterization of woodwind instrument toneholes with the finite element method

Abstract

A method is proposed to determine the transfer matrix parameters of a discontinuity in a waveguide with the finite element method (FEM). This is used to characterize open and closed woodwind instrument toneholes and develop expressions for the shunt and series equivalent lengths. Two types of toneholes are characterized: Unflanged toneholes made of thin material, such as found on saxophones and concert flutes, and toneholes drilled through a thick material, such as found on most instruments made of wood. The results are compared with previous tonehole models from the literature. In general, the proposed expressions provide a better fit across a wide range of frequencies and tonehole sizes than previous results. For tall toneholes, the results are in general agreement with previous models. For shorter tonehole heights, some discrepancies from previous results are found that are most important for larger diameter toneholes. Finally, the impact of a main bore taper (conicity) on the characterization of toneholes was investigated and found to be negligible for taper angles common in musical instruments.

Cross fingerings and associated intonation anomaly in the shakuhachi

“…Keefe's method [5,8,9] defines a tone hole as a T-section network consisting of a series impedance Z a and a shunt impedance Z s . The TM method can estimate both the input admittance and the standing-wave pattern starting from the radiation impedance and moving up to the embouchure (by multiplying the matrices corresponding to acoustical elements such as the bore and tone hole) [8][9][10][11][12][13]. For the detailed mathematical expressions in the TM method, readers are directed to Refs.…”

“…For the detailed mathematical expressions in the TM method, readers are directed to Refs. [8][9][10][11][12][13]16] for the sake of page saving, except the following brief comments on related matters.…”

“…The transmission matrix (TM) method, which has been developed and applied to various engineering problems such as acoustical filters and mufflers [15,16], is considered to be sufficiently established to apply to woodwind instrument bores with tone holes for calculating the input impedance or admittance [8][9][10][11][12][13]16]. Keefe's method [5,8,9] defines a tone hole as a T-section network consisting of a series impedance Z a and a shunt impedance Z s .…”

“…The slight bend near the bore bottom is neglected. Note that the position x along the bore axis is taken from the bore bottom to start with acoustic radiation at the open end (radius a 0 ) of the bore and work up the air column toward the embouchure according to the transmission matrix (TM) method [8][9][10][11][12][13]. The 13th convergent conical element from the embouchure end (x ¼ 540 mm) to the position x ¼ 490 mm is commonly seen in classical shakuhachis, although its length and conicity differ individually [14].…”

Resonance modes of a flute with one open tone hole

“…A.1. Lefebvre and Scavone [16] present a matrix of a tone hole relating ðp; UÞ before and after passing a tone hole, …”