This topic relates to the issue common to saxophones in which the neck suffers some degree of "pull down" which results in the area of the tube at the bend becoming elliptical. The essential question is whether this has a measurable or significant effect upon the soundwave that passes through this elliptical area.
When a neck suffers some degree of "pull down", it is reduced in volume to some extent where the bend occurred. This has measurable and significant effects.
The shape of the cross-section (circular, elliptical, triangular, squared, ...) do not matter.
The curvature of the neck has negligible effects (contrarily to the curvature of the elbow).
On an ideal neck a cross section taken perpendicular to the center axis of the neck as it curves would be circular...
No. A circular cross section is not acoustically superior to any other shape such as ellipsis or even squares, they will behave the same if they have the same area.
Were the neck made so that it had an oval cross section in the first place with the same cross sectional area as the cylindrical/ circular neck what would the effects be?
This is actually the case. Saxophone necks have non-cylindrical cross sections where they are bent. Some years ago, I unsoldered all parts from an alto neck (Selmer Series II, never pulled down) to measure the dimensions:
View attachment 28112
as an example, the external diameters, as measured top-down versus right-left, for one cross-section are 19.5mm diameter and 20.4mm diameter. The actual shape of the cross-section is probably not an ellipse but some non-circular shape, more or less symmetric, a bit egg-shaped or something else, as are most real things, non-ideal.
This does not matter, what matters is the cross-sectional area as a function of the distance along the spine of the instrument.
If you "pull-down" your neck, the area will be reduced, this will have acoustic consequences.
My thinking at this point is that it would be interesting to make a garden hose woodwind with no tone holes and clamp the tube halfway closed at different areas along its length and listen for changes in the pitch and/or timbre of the sound.
It would be great if Antoine would weigh in to share his insight on this topic.
Well, I don't think it is worthwhile to make a garden-hose experiment. It would be difficult to relate any finding with what would occur on a saxophone, plus, clamping half-way is very radical.
Why not do the taste test? Take some well-chewed gum and reduce the diameter at the bend of the neck by varying amounts and see what happens. Nederveen's book contains a formula for figuring the acoustic effects of toroidal bends, and IIRC neither the bends in a normal neck nor a slight increase due to pulldown are significant acoustically.
I think this is a much better approach for testing the effect. I already did many such experiments. I encourage anyone to test this out, in some cases of bad neck design, you may even improve the response of your horn. Isn't it well known anyway that you can change (for the best or the worse) the response of your instrument with small modifications to the volume of the neck different positions?
Also, why not take two necks with very similar response and "pull-down" on one? Then, you can compare what happen. Ideally, you need to have a few good players to test it independently, we should never trust ourself too much, we have a tendency to only listen to what we already believe in....
Making a round neck oval must decrease the cross-sectional area of the affected region, so it alters the pitch for the established reasons.
Yes.
Thank you for that clarification. In light of Kymarto's observation that it is the area, not the shape that is the most significant factor, a logical conclusion might be that the notes which contain pressure anti-nodes in the area of reduced volume will experience a slight rise in pitch.
Not exactly. If the area is reduced in the vicinity of the bend, so is the volume of the neck. Reducing the volume of the neck raises the pitch of the first register, so that you will pull out your mouthpiece to tune properly. This will lower the pitch of the second register. If the "pull-down" is relatively important, you'll notice it in the palm-keys.
Also, such volume perturbations will affect every notes to a varying extent, not only those with an anti-node located exactly at the perturbation. See Eq, (35.10) from Nederveen's book, there is a cosine relation to the distance of a pressure node.