CONICAL...

The bore of a wind instrument harbours a sound wave and the length of the bore governs the wavelength. Now, how long actually is the bore of a conical woodwind?

a cone without an apex..

Saxophones are conical woodwinds. For this type of instrument the acoustical length reaches to where the cone–diameter would be zero. However, you cannot blow a cone on a zero–diameter and that's why a part of the top was cut off: this is the so called 'truncated length' (see drawing below). So actually, a saxophone is lot longer than it seems to be. For a soprano, the difference is already around 15 to17 centimeters; for a baritone approximately 35 centimeters. For a soprano, the wavelength of the lowest possible note, a written B^b, is around 83 centimeter; yet a soprano is only about 65 to 66 centimeters long. The truncated length plays an important role in the saxophone's acoustical behaviour.
In practice, this truncated length is as if replaced by its volume, the so called 'truncated volume'. Again, for a soprano this volume is easily about 5 to 5½ cc; for a baritone it certainly is around 17 cc. This volume we effectively find back in the volume of the mouthpiece chamber (which can be measured simply with a measuring glass) plus an imaginary volume that is induced by the dampened reed motion. Any conical woodwind obeys this rule, (the oboe–family as well!) and when an instrument is well tuned, the sum of both volumes must approximately amount to the said magnitude.

Now, what are the limits for this truncation? The German instrument maker and acoustician Otto Steinkopf states in his 'Zur Akustik der Blasinstumente' that these limits are on the one hand an admittance to the tube which is so small that a uselessly feeble and thin sound is produced; on the other hand a truncation that is so large that half the wavelength doesn't fit into the tube anymore (or, to be a little bit more precise: that the pressure–antinode of the half wavelength comes outside the entrance to the instrument). The saxophone just fulfills this last requirement: the pressure–antinode of the note that is generated by hole no 25 (written high F^#) – which is, by the way, never used in the first register – is right inside the entrance of the neck. At the same time, this is the reason why a neck on its own doesn't behave like a saxophone and sounds differently as well: the neck by itself doesn't fulfill this requirement, it is too short. Furthermore, Steinkopf says, a small truncation will favor the higher partials and make the instrument respond with more ease in the higher ranges and the other way around: as far as that's concerned the saxophone aims at the lower range (!) and at a maximum sound volume.

The imaginary volume that is generated by the dampened reed motion is not that easy to measure. It can be calculated, though, and we can make an estimate of it on the basis of the truncated volume minus the measured mouthpiece volume. Values in the order of a couple of cubic centimeters are very well possible and that is a substantial part of the total truncated volume. That this value depends both on reed motion and reed damping is known from practice. After all, it is all too well known that the pitch of the saxophone tone rises when playing softly: reed movement gets smaller and so diminishes the generated imaginary volume and in the same ratio the connected acoustical length. In the same vein it is well known that the embouchure can influence pitch: influence on damping = influence on the volume that is generated = influence on the acoustical length. In the oboe, which uses practically no real mouthpiece volume, the working reed generates the entire truncated volume and this explains why the oboe is that choosy on its reeds.
Mouthpieces with narrow or wide chambers differ mainly in the internal distribution of volume: so called narrow chambers are especially narrow at the reed side or have a high baffle. However, when you take the internal volume with a mearuring glass in the position in which you can play in tune, it appears that both 'narrow' and 'wide' mouthpieces bring practically the same amount of liquid into the cylinder.

..and a wave

Just as with the clarinet, the tube of a saxophone is as if closed at the mouthpiece–side. This means that pressure differences are possible here: near the moutpiece tip there must be a pressure–antinode. And just as with the clarinet the bell–side behaves like an open end and harbours a displacement–antinode. In the first register the half–open half–closed tube of a clarinet contains a quarter of a wavelength. Nonetheless, with the same length a saxophone sounds an octave higher. The likewise half–open half–closed tube of a saxophone thus contains half of a wavelength.
The difference in behaviour of course is caused by the tube being conical. Yet, about the precise mechanism behind it acousticians are not really unanimous. The behaviour of the conical tube apparently is less obvious than it seems to be. Here we restrict ourselves to the conclusion that the conical bore behaves like a double–open tube and in the first register (just like the flute) harbours half a wavelength. Another aspect of the surprising behaviour of the conical bore is that the sound wave little by little is compressed at the mouthpiece–side. In other words, acoustical length is slowly but surely getting shorter than the length in centimeters. Nodes and antinodes are gradually getting closer and closer to one another.
For this, see the article on the neck as well.

In the second register, reckoned to the acoustical top, a full wavelength fits inside the tube. Ideally, there is a register hole halfway the standing wave; it is then located in a place where there are no pressure differences and it prevents the forming of the first–register note. In practice however, this demand is fulfilled for only two notes quite perfectly. After all a saxophone has only two register holes, which are placed, each of them, more or less midway the range for which they are operative.
In the third register, reckoned to the acoustical top, a wavelength and a half fits inside the tube. In this case two register holes are possible, if not necessary. Here a register hole often is supplied by some peculiar fingering, in which discontinuous rows of closed keys occur.

intonation

Tuning of registers can be influenced by playing a subtle game with bore diameters. As the diagram shows, in a standing wave there are spots with and without pressure differences. By combining these spots with a narrowing or a widening of the bore, the buildup of pressure can be influenced: when a narrowing of the bore coincides with a place where pressure differences are high, this results in even higher pressure differences, thus enhancing the resilience of the system with a rise in pitch as a result, and the other way around. The opposite will almost automatically be the case for a next higher register: in those places where in the one register pressure differences are high, these diffrences are small in the next.

Variations in the bore profile also influence the speedier components of the sound wave (partials) in their pitch and these, in turn, influence the fundamental in pitch and sonority. A partial which, because of a well–proportined bore profile, acquires a tendency to go well with its fundamental, will sound easily and relatively loud. This makes that a perturbation of the bore profile will have quite a series of consequences for sonority and intonation and these consequences will be different in character for the different ranges of the instrument.