Correct intonation is quite a serious matter, both for the musician and for the instrument maker. The plain fact that the saxophone is a conical woodwind is decisive. It discriminates the saxophone in an essential feature from the clarinet, which has, as it were, the 'same' mouthpiece, but reacts in an entirely different way to it.
The acoustical situation of a truncated cone combined with some kind of a mouthpiece volume, such as described under the heading 'conical', has great influence on the tuning of the registers between themselves. In principle it is possible to arrange tone holes in such a way as to obtain an equal tempered tuning for the first register. The consequence for the second register would be desastrous, though. In practice we need to choose such a compromise that both registers can be used: some notes should be chosen flat in the one register in order not to be too sharp in the other.
At a certain proportion between the truncated length and the total length things graudually grow out of hand: pitch rises to an unacceptable level. As the truncated length is always the same, this means that the instrument length here defines a limit. This instrument length gets smalles as we open more keys. At the written note C#3 we reach a critical proportion. This note has always proved to be a problem note, which problem is usually solved by some kind of a correction mechanism in the sopranos (and sometimes in the altos too). The other saxophones remain behind with an intonation problem. The compromise that the instrument maker can offer to the musician is a slightly flat C#2 and a sometimes annoyingly sharp C#3. Higher up we need separately spaced tone holes for the same reason.
the vertical schale gives deviation of intonation in cents.
The diagram above sketches a picture of the typical intonation behaviour of a saxophone. The bundle gives four different postions of the mouthpiece on the cork. The first and the second register are shown with a break. The second break actually is not a change of regiser, but gives the split that's caused by the separate tone holes from C#3 upward. These lines can be at any height, dependent on the proportions of the holes; the fact that they diverge is inevitable, though. In these curves, no account was taken of the influence of individual tone holes. The development in the first register is not surprising. However, the dip in the second register and the subsequent remarkable rise in pitch is a compelling singularity of conical woodwinds with a truncation. The dotted lines show the natural tendency of the intonation and at the same time the necessity to use separate tone holes above C#3. The rise in pitch at F3 amounts to approximately ¾ of a semitone. For this, compare the pitches of the notes as obtained from hole no. 22 in both registers.
Both increasing and reducing the length of the truncation has an influence on the position of these lines. Increasing the truncation will make the pitch rise in the bottom end of both registers. But since the lower register is a third (a fourth for a modern baritone) longer than the second – namely down to Bb (A), this sharpening is much less noticable in the second register. Through history the truncated volume in saxophone design was reduced somewhat. As said, a larger truncated volume sharpens tuning in the bottom end of both registers, but also increases the problems around C#3 and higher up. The dip in the second register thus gets deeper. So, a somewhat smaller truncated volume has advantages for good intonation between both registers.
This smaller truncated volume was obtained by relatively small shifts in saxophone design. Recently however, since the introduction of sopranos with separate necks, the design of the soprano did change drastically in the sense that the truncation got quite a bit smaller and the length of the instrument increased to the same extend. Since the late 80s - early 90s sopranos got about a centimeter or two longer. The limit to that of course is posed by what a mouthpiece in its dimensions conventionally allows for. The greatest acoustical length – belonging to the lowest note of the instrument, a written Bb – has remained the same of course. This length is the same for all older and newer instruments and is linked to the wavelength of the lowest playable note.
So, the proportion between the real length of the instrument and its truncated length plays a role in instrument behaviour. We have met with a shift in design. A larger truncation, as we find it in the older instruments (upto the 20s – 30s), makes an insrument a wee bit harder to play well in tune according to modern standards. A matching mouthpiece with a so called 'wide' chamber gives such instruments a better balance, though.
Here we also find a relation with the distribution of volume inside a mouthpiece. This distribution exerts some influence on tuning. The reason for this must be found in that a mouthpiece chamber behaves in a similar way as the instrument itself. Bore diameter variations inside the mouthpiece influence the buildup of the standing wave within. For this, also see the article on bore perturbations. Bringing the actual volume of the mouthpiece chamber and the imaginary volume that's caused by the working reed close to one another makes good intonation in the practice of playing slightly better within reach. That's why (older) saxophones usually profit from these wide mouthpiece chambers.