The foremost function of a tonehole of course is to shorten the tube so as to give the instrument a smaller acoustical length with the corresponding faster vibration rate of the air column. This makes it possible to generate different notes with the same tube.
The extend to which the tube behaves as really shortened to a certain length is determinded by the size of the tone hole. A certain tube length always contains a somewhat bigger wavelength, because the wave has a tendency to run a little bit past the tone hole. This difference in length between the tube and the corresponding wave is known by the name of 'endcorrection'; between the hole and the corresponding wave as 'hole correction'. Now, the bigger a tone hole is, the smaller is this correction. This means that a larger hole generates a higher pitched note than a smaller one in the same place.
The disk of a key hanging over a tone hole makes the tone hole behave as if it is a bit smaller and the lower the lift of a key is, the stronger is this downsizing effect.
The size of a tone hole also has a bearing on the recipe of the partials produced. A hole of a certain dimension seems to be smaller relatively for a higher pitch. In practice, this will mostly be a harmonic, an octave or an octave + a fifth of the fundamental, for example. This higher pitched note will thus have a tendency to be a little bit weaker (and to be a bit on the flat side!). Higher than a certain frequency, which is known as the 'cut–off' frequency, waves ignore the hole entirely and run past the tone hole. The hole and its position will not govern the pitch of these partials any longer. Because of this, bigger holes develop higher partials than smaller ones. And the other way round it means that above the 'cut–off' frequency partials won't be harmonic any longer, but that, however weak they may sound, the higher they are, the more they will have random pitch. Such out-of-tune partials dampen the resonating system, yet are definitely part of the instrument's timbre.
In this way the proportions of a tone hole play an important role in the formation of the sound and in 'getting it out', Tone holes in a saxophone typically have an opening of around a 25 to 40 per cent of the tube diameter locally and as such are fairly large for a woodwind. There is a tendency for tone holes to relatively get smaller as they are placed higher up in the tube and this tendency is stronger in sopranos than in baritones. Differences in this order of magnitude do not yield much appreciable differences in sonority, the more so because the change in size is fairly gradual. It's quite another story when tone holes are smaller than, say, 10% of the bore diameter. Such small holes sometimes are found in older instruments as extra trill keys, for example a small Tf (alternative F#) or the extra Eb for the second finger of the right hand. Of course, you can more easily make a trill with a smaller key. In the past this has been fashionable for a while and a sales argument as such. Gradually these small holes all have disappeared.
In all instruments, modern as well, hole no 18 (C#) is a small one. The reason for this is in the first place to be found in intonation problems around the notes C#2-3. The accompanying differences in sonority are to be accepted.
Another reason for variations in tone hole diameter is caused by the fact whether or not a tone hole really is an 'open end'. In a saxophone not all tone holes are truly 'open ends' in the sense that for a number of notes the first–open hole is followed by a closed one. This sometimes is also the case when the fingering itself doesn't make that clear at once. The standard closed keys no. 3, 5 and 11 cause this kind of irregularity. Beside disadvantages in the form of diameter compensations in the first next higher tone hole, which has to be larger in size and the small differences in sonority mentioned before, there is an advantage as well: a note that comes from a cross–fingering has a larger downward correction in each next higher register, sounds a little more on the flat side.
This property is happily used for a couple of notes: the octaves of D1/D2 and E1/E2 (which both are on the sharp side by law of nature) are improved by the closed tone holes 3 and 5. Compare them to the octave of D#1/D#2 (which is an open fingering). The octave of C2/C3 in the 0,2 fingering is a cross fingering too: only hole no. 17 is opened; the next two down the line are closed. By nature this octave is way too sharp: compare it to the neighbouring open fingerings C#2/C#3 and B2/B3 and especially with the C2/C3 taken as 1,Tc (which again is an open fingering). In this last fingering the C3 will clearly be on the sharp side. The fact that the 0,2 fingering is a cross–fingering makes that this octave is playable quite well witout any correction, by embouchure or otherwise.
Next, a little play and variation is possible in the size of a tone hole in relation to its place. As said, a large tone hole produces an brighter sonority and a somewhat sharper pitched tone than a smaller hole in the same place. But also the tone will be slightly more on the sharp side in the second register as compared to the first register. This makes a correction possible on both factors. As an example may serve the size of tone hole no. 4 (low C) in Selmer among others, where this hole is relatively small and as such has a lenient effect on the natural sharpness of D2. Not always is this hole that small: in other makes of instruments we sometimes find it to be up to one and a half time as large as it is in Selmer! In this case, the hole is placed a little lower in the tube. It goes without saying that such instruments react in a different way to the notes in question.
The number of variations in solving the inherent intonational problems that the saxophone poses is sheer endless. A play with minor variations in bore diameter in combination with a choice in tone hole sizes gives better or less good solutions to the problem. Especially in the older instruments you can find things which gradually have disappeared because of the strong tendency to copy the instruments of the Selmer company. Also, instruments from the heyday of the American tradition show different solutions than what the French came up with. With the vanishing of the American way of building and the copying of the French way (Selmer) by the Japanese among others the picture got more and more uniform and sonority more and more got the very same. Standardizing does not always lead to a musically more satisfying sound, especially when the interesting thing about this sound is in the quirks and deviations and the necessity for the musician to cope with that.