"Register Hole Design for Cone Woodwinds" by Dr. Arthur H. Benade, Case Western Reserve University, Cleveland, Ohio 44106.
(This is an abstract of Dr. Benade's paper, given as part of the musical acoustics section of the Spring 1973 meeting of the Acoustical Society of America. For copies of the entire paper, write to Dr. Benade directly.)
In a woodwind the register hole produces a transition from low to second playing register in two ways. At low dynamic levels the hole must increase the damping of the first mode sufficiently that oscillation cannot be sustained. Oscillation then takes place on the basis of the second resonance peak. At high dynamic levels this effect of the hole is insufficient intermode cooperation can still sustain the low register oscillation, so that the first mode frequency must be suitably shifted relative to that of the second mode to make their cooperation impossible. Optimum hole reactance raises the first mode frequency 25 per cent. For an instrument whose low register half-wavelength is L, having a bore taper 2(dR/dx)=T, and provided with a (centered) register hole of radius b and effective length te=(t + 2b), the desired frequency shift is produced when (b2/te)=( /4)2LT2
The damping requirement is best approximated by making the physical length t as small as is mechanically practical (1 mm on a saxophone or bassoon). Oboes and English Horns traditionally have vent hole proportions in agreement with this design. Small modifications invariably give trouble. Most saxophones are dramatically improved by reworking the vent holes to the present specification. Similar principles apply to bassoons (and clarinets), but here the multiple use of the vent somewhat compromises the design.