For only two notes in the second register register holes are located at the right spot; for all the others they are positioned more or less in the wrong place. This mislocation mars intonation and interferes with sonority.
The influence on intonation can be graphed out quite well. Experiments with register holes show that these holes can be given a higher acoustical resistance without them losing their functionality. In general this will in general make an instrument produce better octaves, that is to say, octaves will need less embouchure correction in order to be just. Such register holes will also make them serve a larger range of notes and to help to produce a fine pianissimo A2 or a 'long' C#2. There is an end to it, nonetheless. When a registerhole gets too high a resistance, it looses its effectiveness at the outer edges of its working range and the tone tends to fall an octave easily.
A second aspect of the dimension of the register hole is an influence on sonority. As mentioned before when discussing their behaviour under Physics, register holes show a tendency to 'hiss' – especially for notes for which these holes are far away from their ideal position – and to add a certain hoarse muting to the tone. There is a quick test to this phenomenon: play a note in the second register with the register key and drop the key without dropping the note. What do you hear?
A differently shaped register hole exerts another influence on sonority. The tone can brighten up, get more stable, which also means that it will speak more easily when played softly.
Another, and in most cases undesired side effect of a too large a register hole is an instability that occurs around G2 en G#2. These notes show a tendency, especially in tenors and in combination with a loose embouchure, to produce a spontaneous and unwanted multiphonic. This instability can disappear with a differently shaped register hole.
Register holes can easily and safely be given a higher resistance in various ways without doing any damage to the instrument. More invasive are changes to the very form of the hole, yet there is also more to be gained by improved shapes for the register hole in terms of both intonation and effects on sonority.
with kind regards to Johan Thole, who brought up the subject.