On register holes:
Benade is speaking about the total length of the octave register tube. Basically the pip should not be higher than 1mm above the neck wall. I am, however unclear about how the tube inserted into the neck (to avoid saliva fouling the pip) should be counted in this equation. Essentially the math is only done, in my understanding, for a side hole above the tube wall. My guess is that the length of the tubing inserted into the neck has some deleterious effect, but not as much as as the height of the pip above the neck.
OCTAVE PIP:
Read Benade's paper on this at:
http://ccrma.stanford.edu/marl/Benade/
go to "Writings">"The 70s" and look under 1973 for "Register hole design for cone woodwinds". Interesting stuff. I don't believe that placement affects the actual intonation much, but rather the efficiency of the detuning of the fundamental necessary to sound the second partial optimally. As the size of the register hole is increased, there begins to be an effect on the actual pitch of the second partial, but that is rather small at the diameters normally employed for register holes, in my understanding.
Benade points out that to be effective at all dynamics, the register hole must not only inhibit the sounding of the first peak (the fundamental), it must also detune it so that there can be no cooperation between the first and second peaks at higher dynamics. He says:
"To be effective a register hole must produce maximum noncooperation between resonance peaks if it is to displace a tone from low register to second register. It must also produce sufficient damping of the first resonance peak that its participation in an oscillation at low playing levels is disfavored. In practice, one finds that 2 or 3 % shifts in the detuned frequency of the first peak can be noticed by a player. Craftsmen are quite familiar with the problem of diagnoisng and of correcting such size of adjustments. Saxophones customarily use a fairly large-diameter vent hole of a length sufficient to produce proper detuning, but such proportioning gives inadequate damping to the detuned resonance. Vent holes of this type are known to cause all sorts of trouble, so that many saxophones practically ignore their regular keys. Furthermore, there is a great deal of difficulty with the intonation behavior of the instrument as a result of such register holes. If, however, one uses very short holes of small diameter, which optimizes both the low-level and the high-level playing requirements for the hole, once can get marked improvements in the response of the register key, and often considerably reduce the intonation difficulties as well."
BOUNDARY EFFECTS
Boundary effects only occur at ~1mm from the walls, so in a large diameter woodwind such as a sax such effects are really minimal (as compared to a recorder, for instance). Certainly it is wise to smooth any edges within a bore to reduce turbulence, but with thin walled metal instruments this is next-to-impossible while still maintaining structural integrity. You can certainly smooth tone hole edges in a clarinet (and top-end instruments such as Buffets have chamfered edges) but not in a sax or a flute.
CONICITY:
In my understanding, as the cone angle diverges from the ideal, the overtones of the note sounded deviate further and further from the harmonic series. If I am not mistaken, as the cone angle narrows the partials are spread, until at unity the first partial completely disappears and the second partial sounds at the register break. Higher cone angles compress the partials, so that the octave is sharp, the octave + a fifth is sharper still, etc. If you want the math I can probaby find the formulae that describe the function, but it is not easy math as I recall.
HTH,
Toby
