I have been searching and I can’t find an answer to this. Am I correct in assuming the pitch of a regular tube with a square cross section will be determined by it’s corner to corner measurement, i.e. longest measure?
I thought that what mattered was the cross-sectional area of each point in the bore, regardless of the shape.
But I look forward to reading a post from someone who actually knows!
Shouldn’t the pitch be determined by the length of the tube, regardless?
For all practical purposes, the resonant frequencies of a square pipe will be the same as a round pipe of the same length and similar cross sectional area.
Just like baking (from The Cheesecake Cookbook, by Kevin Krell)
“Most cheesecakes are made in an 8”, 9" or 10" springform pan (a round pan with removable sides). The 9" is the most common. A 1/2 recipe can usually be prepared in a similar sized pie pan. Sometimes, however, you will not have the correct size pan available. Rather than purchasing a pan of the proper size, it is possible to substitute one of a different size and shape according to the list below…
All pan sizes are measured inside from rim to rim.
8" ROUND = 50 SQUARE INCHES
9" ROUND = 64 SQUARE INCHES
10" ROUND = 79 SQUARE INCHES
8" X 8" SQUARE = 64 SQUARE INCHES
9" X 9" SQUARE = 81 SQUARE INCHES
11" X 4 1/2" RECT. = 50 SQUARE INCHES
11" X 7" RECT. = 77 SQUARE INCHES
16" X 5" RECT. = 80 SQUARE INCHES
It is important that the surface area of the substitute pan be very close to that of the pan originally required. Therefore, these pans may be used interchangeably:
8" ROUND = 11" X 4 1/2" RECTANGULAR
9" ROUND = 8" X 8" SQUARE
10" ROUND = 9" X 9" SQUARE
10" ROUND = 11" X 7" RECTANGULAR
10" ROUND = 16" X 5" RECTANGULAR"
Yes, assuming the pans are all of equal depth. 
I searched for 25 minutes and found nothing. The trick will be finding the correct search term that gets to the answer.
Maybe save 7 hours of searching by contacting Nick Metcalf (?), who has made some square tube whistles.
Certainly, length is the dominant effect.
However, for equal tube area, a square tube will have about 13% more surface area. So, more area for viscous friction.
My hunch is that a square tube would have a slightly lower pitch. How much lower, I do not know.
Another hunch: the difference might be so small that fancy lab gear would be needed to measure it.
So I guess the answer comes down to “experiment and figure out” 
! Thanks for all the thoughts! Results will be forth coming.
A square tube should support a range of frequencies, but that’s masked by them averaging out to a single frequency with degraded tone. My first prototype resin quena had an octagonal bore (as I wasn’t rotating it with a motor at that stage when experimenting with resin - you need that rotation to keep it circular while the liquid resin gradually hardens and sets) and it sounds horrible as a result of those different length paths at different alignments. The frequency will not be determined by the corner to corner separation distance, or the side to side separation distance, but the average separation distance. Exactly how to average it I’m not sure, but perhaps you could divide it be 360 degrees and work out the separation distance at each of those (one degree at a time) before averaging them. This is a guess though and it may be wrong.
I think your hunches could be right. I made an uilleann “C” chanter with a square bore based on a C19th original and the pitch is where I would expect to be at about 10 cents sharp of modern C
Yes, please keep us up to date on progress. And tell us what’s prompted this mischief making?
Square whistles, eh? Who’d ever have thought of such a thing? Young people these days, I don’t know…
Or…
The lengths some people are prepared to go to to stop their whistle rolling off the table at the pub…
I’m inclined to agree that cross-sectional area matters.
After all, bore-perturbations and tapered bores influence tuning.
Imagine the fun (work!) of “tapering” or “perturbing” a square tube !
Hmmm . . . once a jig-fixture is made, repeats + mods might not be too bad . . . compared to delicate lathe-work on fragile tubes . . .
Hmmmx2: Use a pattern-following mill for flat tube sections. Tweak the pattern and all the repeats are easy . . . Hmmm . . .
Some whistle makers seem to achieve their bore compression in the head of a whistle (to balance octaves) by crushing an otherwise cylindrical bore into a squared off shape (perhaps by using a press). If the resulting square cross section retains the same diagonal distance as the original round bore’s diameter, then the cross-sectional area of the new square bore will be less than the original round one. This will effectively taper the bore at the head. If you look closely at whistles from Goldie you can see this effect, at least on three of the four sides. The bottom is still round. Superficially, this looks like an external decorative/stylistic effect, but a peak inside the bore shows that there is a lot more to it than that. And a study of acoustics makes it all the more interesting! 