The saxophone is something of an oddity among instruments, since it was designed "from scratch" to produce its musical sounds. When Adolphe started, the 'rithmetic was reasonably well known so he could predict in advance what kinds of noises his new device would make.

The rather large diameter of the bore, and the conical taper, make it produce "overtones" (harmonics) that are, at least theoretically, at pure enharmonic intervals with the note being played. It's the very accurate and even placement of the overtones that are responsible for the "velvety smooth" tone when the sax is played so as to emphasize them.

Most of the vaguely similar instruments evolved from "open finger holes" by addition of keypads and levers to extend the capabilities of the starting instrument (exept the pennywhistle, which has thus far failed to evolve much ;>)). When you open a small hole in a long tube, there's an "edge effect" that causes the acoustic length to be slightly longer than the physical length of the air column between the mouthpiece and the hole. The length of the edge effect depends on the frequency of the sound and on the details of the hole, so the second harmonic of the note you're playing has a different natural "edge distance" than the note itself, and for a simple tube with tiny holes, the "octave note" cannot be made exactly in "just relationship" with the fundamental. Higher harmonics are "stressed" even more out of tune.

With the saxophone, the large pads were intentionally incorporated to minimize the fingerhole edge effect, and compared to other similar instruments, the finger holes are ENORMOUS. This was on purpose so that the "velvet tone" would continue to be there when you played on up the scales.

It's not clear whether it was intentional on the part of the inventor, but the very large diameter for the saxophone's bore length relative to its length makes the air column "quite flexible." With proper lip control (and a few other manipulations internal to the player) the sax's note can be pulled in either direction from the natural pitch. A good player should be able to "droop" most notes by a semitone, or "bend" them up by about a quarter tone. Beginners quite frequently aren't aware of how often they use this effect unintentionally and are playing out of tune. The flexibility of the instrument means that you have to play in tune on purpose, and expecially as a beginner you should strive to do so, so as not to give the instrument an undeserved bad rep. The effect can be used very effectively by competent players to apply "nuance" to the tune. It's used (hopefully on purpose) quite a lot by blues, swing, and jazz saxophonists.

John