Can you cite a physics book that makes that claim specifically about a saxophone?
I'm just curious because physics was my major and my life's career and I never read that in any of my physics books. It would seem to me that other factors besides length would need to be considered as well. The sax has quite a large diameter compared to many other horns, like say a trumpet, or a clarinet. In fact, when I look at the clarinet I'm amazed at the range that instrument can play seeing that it's such a small tube and it's not even bent up in any way.
It's funny that you should post this today, because I was just playing my little pocket trumpet today and trying to imagine how long it would be if it was all straight tubing. Curled up it's only about 8 inches long, but if it could be unfurled it could potentially be 5 or 6 feet long. So now I'm starting to realize that I'm actually blowing through a very long skinny tube. It's funny because just thinking of it in that way tends to make me think differently when I'm playing it. I'm starting to think in terms of getting this whole long tube to resonate as a unit. So in my mind I'm imagining that I'm blowing through a 6 foot long horn.
This should really be posted in the tech section, perhaps Pete will move it.
1 - the basic physics behind the way the sax and other woodwinds make their notes are well understood.
2 - The length/pitch statement above is correct - the longer the tube the lower the pitch. Tube diameter has no/negligible effect on pitch. Double the length, halve the frequency and the note goes down an octave. Acoutically the tube stops at /around the first open tone hole.
Where you're probably going wrong is that the sax bore is nominally conical, and this would give a very long, spindly neck. Net result is that a shorter neck is used, the volume of the missing part of the cone is replaced by the volume of the mouthpiece. Which has the 'same' effect of the full neck length. So to calc the full acoustic length, work out the missing neck length, based on the volume of the mouthpiece and the taper of the sax... And you'll quickly realise that the mouthpiece volume is dependant on how far on the neck it is pushed. Hence it's position's disproportionate effect on the instrument's pitch.
3 - You also need to understand how the harmonics work for the higher notes. This is triggered by the octave keys on a sax. Opening the octave key causes a disturbance which prevents the air inside the sax from resonating at the fundamental frequency, and it resonates at the the second hamonic - an octave higher. (remember first harmonic is the fundamental, the base note).
4 - Clarinets are different because of the cylindrical, not conical bore and overblow at the third harmonic (frequency is three times the fundamental, again triggered by the octave key. Overblowing at the third harmonic which is an octave and a fifth above the fundamental gives the clarinet it's huge range compared tot he sax.
5 - Tonehole position is modified compared to a straight calculation to take into account end factors (end correction), imperfect venting of the hole/lower hole, tone hole height and so on.
If you want a technical explanation, there's an excellent web site at http://www.phys.unsw.edu.au/jw/basics.html which has pages on how the major instruments make their sounds, including a lot of detail on the woodwinds....
I kinda got that fuzzy feeling in my head and then the feeling of shutters closing down to protect my brain in the same way as when somebody starts one of those maths problems involving eastward travelling trains and cross winds etc
Oh, hang on, it's happening ag....................