A register hole is a compromise. The smaller it is, the less disruptive influence it has, but, at the same time, the less effective it is, especially for those notes for which it is not in the ideal place. And since mechanical simplicity plays quite an important role, we will have to deal with only two register holes, which together serve the second register. Saxophones with up to four register holes are definitely history.
As is well known, a mechanism that is operated by key no. 12 automatically shifts from the lower to the upper register hole. This means that the lower hole operates the notes D2 (C#2) up to and including G#2 and the upper hole the notes A2 up to and including F3 (F#3). The lower hole thus has a working range of 7 semitones; the upper hole covers 9. As far as the distance in centimetres between the outermost ideal postions is concerned, both register holes cover an approximately equally large range.
When a register hole sits in the ideal place – which is right at the half–wavelength pressure node – it has no influence whatsoever on the intonation of that note. There are no pressure differences here and opening an hole to the outside world makes no difference to standing wave inside the tube (except that the formation of the wave in the first register is prevented). Unfortunately, this situation can be realized ideally with the two holes in only two cases. The lower hole sits roughly in the ideal place for E; the upper for C#. The register holes won't influence the pitch of both these notes. In all other cases registerholes do cause some deviation of pitch, which deviation is bigger as the location of the hole is farther away from the ideal spot. Whether a register hole sits too low or too high for the note concerned, does not make any difference for this influence on intonation: in both cases there is a rise in pitch.
The graph following below on various intonation influences for an alto saxophone originates from C. J. Nederveen's thesis. On the vertical axis we find the deviation of intonation; on the left side in relative magnitude, on the right side in cents; on the horizontal axis we find the range of the instrument. Different types of deviation are depicted. The deviation caused by the register holes, in which we are interested here, is named Dreg and is coloured.
Nederveen has both measured and calculated the deviation caused by the register holes. The picture you get is twice a kind of a festoon, one for each register hole, the lowest point of which is on the zero line – for the note for which the register hole is in the ideal position – and the highest peaks rise up to somewhat 20 cts. sharp. As was to be expected, the greatest deviations are to be found for the upper register hole as this hole acoustically serves the larger range and relatively shows the greater deviations from the ideal spot.
A somewhat different approach to the same problem you find with Curt Altarac, who plotted intonation influences by a row of register holes in different positions along the neck. At least as fascinating is the experimental alto saxophone with 12 octave vents, that was built by Michael Brockman, and which thus has a separate octave vent for every single note...
To minimize the unwanted but inevitable deviations to an acceptable level, register holes are kept small. There is nonetheless a limit to that: when a hole is too small, it loses its effectiveness, especially when it is not in the ideal place. The tone gets the tendency to fall an octave, especially when blown forte.
Register holes are made in a variety of remarkable forms. A short tube, which projects a little way into the bore of the instrument, traditionally is part of the configuration. This is not only the case in saxophones; in fact the saxophone shares this shape with the clarinet. The thought behind it is to add some acoustical resistance to the hole.
Yet, there are acousticians who doubt this kind of shape, and maybe on good grounds. The working of register holes indeed can be improved. Even so, such scientific doubts hardly influence the way instruments are built. Apart from that, the tube of course plays an important role in keeping condensation away from the tiny register hole, which might otherwise easily clog with moisture.
There is another way to get round any disruptive influence of a "wrongly" placed register hole and that is to make use of a tone hole as a register hole. This can be effective when we have a tone hole available that really is in the right spot. In the saxophone, there are only few such holes, like holes no. 19 (palm D) and 20 which can be used as a register hole for the notes D2 or Eb2 or holes no. 17 or 16 (Tc according to Londeix) with (the long) C#2.
For the 'third' register this is in fact the appropriate way. The reason that toptones or flageolet tones have such peculiar fingerings, is caused by the necessity for extra register holes in specific places.
An aspect of register holes that up till here has not yet been discussed, is their influence on sonority. Register holes do display the tendency to 'hiss' and to add a kind of a hoarse damping to the tone. This is especially noticeable with notes for which these holes are far removed from the ideal place: D2 (and C#2) for the lower register hole, sometimes A2 but certainly F3 and F#3 for the upper one. This flaw goes hand–in–hand with a more difficult response in the pianissimo. Hissing can be reduced by the shape of the register pips.
Another and less desired side–effect of a register hole is an instability that we sometimes find around G2 and G#2. These notes show a tendency, especially in tenors and in combination with a looser embouchure, to produce a spontaneous multiphonic. This instability is connected with the size of the lower register hole.