What do saxpophone ligatures do?
A ligature clamps the reed to the mouthpiece. Think of this a bit like when you hold a ruler on the edge of a table and flick the bit sticking over so it goes “boing”. You need to hold the part of the ruler firmly, or the end sticking over does not vibrate well. The same principle applies to the way a reed vibrates on a mouthpiece. Think of the ligature as the pressure that holds the butt of the reed against the table of the mouthpiece, so the blade (= the boingy bit of the ruler) can vibrate nicely.
Saxophone ligatures have developed a long way since the days when clarinetists would just wind a piece of cord or leather thong around the reed to secure it to the mouthpiece (but some people still swear by this method).
Ligature, Reed & Mouthpiece
If mouthpiece table is flat:
Provided the reed is not distorted, any good fitting ligature will work, and there should be no significant difference in sound between one ligature and another. If the reed swells due to moisture, you should adjust the reed with a blade or reed tool.
If mouthpiece table is concave:
Some mouthpiece tables are concave by design. One theory is that because of the resulting slight gap between reed and table the reed will swell into the concavity and close the gap, however it’s probably best to just make sure you flatten the reed in this case. If the curve of the concavity is from front to back, the tightness and position of the ligature can be critical. If the concavity is from side to side, the way the ligature applies pressure can be critical: if it applies pressure to the centre of the reed, it will push the reed down into the concavity and so create a good seal, but if it applies pressure at the edges, then it could cause the gap to remain which means the reed does not seal well and can cause squeaks or deadening of the sound. Both of these situations explain why many people think that a ligature affects the sound.
Note that if the reed swells into the concavity, or if the reed is flat and has formed itself to the concavity, then this reed is unlikely to perform well when tried on a different mouthpiece. For this, and other reasons, I believe a flat table is preferable to a concave table.
These days some very sophisticated ligatures are being made by manufacturers with all kinds of claims that various enhancements can improve your sound. How much of that is true and how much is marketing hype to get you to part with your hard-earned cash is open to debate, more on that later.
Most ligatures are tightened by one or two screws, either laterally (from the side) to tighten a band around the mouthpiece and reed, or from the top directly down onto the reed (as with the Otto Link type). Do not overtighten the screws, they should be tight enough to just stop the reed from slipping or to take up any slight warpage in the reed or table of the mouthpiece. Of course, there should never be any warpage of the mouthpiece table, and you should always prepare your reeds by sanding or scrapingto make them flat, but sometimes a previously even reed can warp very slightly while playing, and if a ligature is tight enough it will apply pressure to flatten a slightly warped reed against the table. If the surface of the reed is not held against the table then the sound can be slightly muffled or harder to produce. You will probably notice this most often when trying to play quiet low notes.
Does a ligature affect the sound of the saxophone?
Some people consider that the type of ligature affects the sound, however this topic has been hotly debated. I’ve never noticed this as long as the ligature is a good fit for the mouthpiece and is holding the reed firmly, and neither the reed or mouthpiece table are uneven. It is very important that the ligature is not too distorted or badly fitting or the tone will definitely suffer. For this reason the more flexible textile based ligatures may sound better, but only if compared to adamaged or poorly fitting metal one, as they do not get damaged by dropping them, sitting on them or most other kinds of abuse you can find for them. But the bog standard 2-screw type will sound as good as anything provided it is the right size and is in good working order.
How Different Ligatures affect less than perfect reeds:
even reed good ligature
A reed with an even surface and a ligature which fits the back of the reed. This could be either a metal lig that is correct for the mouthpiece or any fabric ligature which molds to the reed. Pressure from just the top centre or from the sides will hold the reed.
swollen reed pressure from centre
When a reed is swollen in the middle, a ligature which applies pressure from the middle may flatten it slightly, but still leave gaps at the edges
swollen reed pressure from edges
A ligature which applies pressure from the edges to a reed can actually help with closing the gaps, but could be causing further distortion of the reed.