When the general concepts are clear, it is fairly easy to grasp what influence the different parts of an instrument have on sonority. It all boils down to a mutaul influence between the form of the tube and the wave inside and the way in which the reed closes on the mouthpiece tip. We will scrutinize the actors in this play one by one.
As the instrument is more reactive, it will feed back a wave with stronger peaks to the mouthpiece. Stronger peaks in the mouthpiece result in more partials inside the tube. An instrument is more reactive when it has a higher 'cut–off' frequency: greater tone holes, keys hanging over them higher and less disruptions like things projecting into the bore, edges and ridges and dirt clogging to the wall.
Generally, narrowness of the mouthpiece chamber leads to a load onto the reed. Pressure differences, caused by the standing wave inside the mouthpiece chamber, are enhanced by this narrowness. Keep in mind that the reed tip is at a spot where pressure differences are high by their very nature. Both the high pressure gets higher and the low pressure gets lower, just like the tidal extremes of high tide and low tide grow higher and lower in the narrowing of a river mouth. The more these extremes are boosted by this narrowness, the more abrubt the opening and closing behaviour of the reed will get. The result naturally will be an air admittance to the tube with stronger peaks.
Reed strength also has an influence. When the reed – and especially the reed tip – gets more supple, it will, following the conditions of alternating high and low pressure in the mouthpiece chamber, close more readily on to the mouthpiece tip or be pushed away from it. Here too the effect will be that the opening and closing behaviour of the reed gets more abrubt and with again the same effect that the air admittance to the tube will have stronger peaks. Whether you narrow a mouthpiece chamber or use a more supple reed, in a way amounts to the same thing. The difference between playing a softer or a harder reed remains as it is, though. Both things are intertwined and must be weighed together.
We can in the meantime guess what the tip will do: a small opening will make the reed to close more easily than a bigger one, with again as a result the same more abrubt opening and closing behaviour of the reed with the same more peakish air admittance to the tube. In fact, it is as well and for a major part a matter of the tip–angle, the angle between the tangent to the tip and the table in degrees, than that of the sheer opening in itself. The tip–angle mostly governs the ease of the reed tip closing to the mouthpiece. Here, at the very tip, facing design meets reed meets mouthpiece design and makes instrument behaviour.