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The same argument can be extended further down the sax. Go to the spreadsheet at
https://public.sheet.zoho.com/public/dave.mclau/ring-frequency, click on "Click to Edit" and try inserting the diameter of the sax at the first open tone-hole for any given note. I'm sure you'll find the critical frequency is far higher than that of the note itself.
In my view, the material of the sax body cannot affect the tone.
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Hmm. Interesting, and actually, though I used to agree with you, a thought has occurred, and I ran your spreadsheet, and now maybe I don't...
Put in the next tone-hole, say 30mm diameter - this has a ring frequency of 36871Hz, still inaudible, BUT the mixing of this with the frequency from the 25mm hole will give a resultant at 44245-36871=7374Hz - THAT's audible.
NOW change the material to copper (speed of sound 3901m/s instead of 3475) and the resultant changes to 8278Hz - that will sound different.
However, until someone (probably in Japan) builds twenty dimensionally identical saxes (probably to an accuracy of a few microns), ten each out of any two materials of their choice, and proves by rigorous laboratory testing that they do emit different sounds when played identically, then proves by similar experiments that they can consistently reproduce this effect with normal production methods, and publishes a peer-reviewed scientific paper with the results...
...then I'll believe the material itself makes an audible difference.
I'd be very happy to be told this experiment has been done, or any along similar lines, but I rather suspect it hasn't, and all assertions as to the merits of brass, copper, bronze, phosphor bronze, nickel silver, acrylic etc. etc. etc. are hype, speculation, or wishful thinking on the part of the owners.
Cryogenics? Just possible, on similar grounds. But I don't think so, somehow...
Cheers
Jonathan