Your reed being off (on either side) by a millimeter can totally mess up your sound on sax.
Other adjustments, like how far the ligature is up on the reed also make a difference.
Even how tight the ligature is can change your sound.

If your reed seems a little bit too stiff, then put the tip of the reed very slightly below the top of the mouthpiece. And you can also tighten the ligature at the bottom-most part of the reed.
If it’s a little soft, then put the tip of the reed very slightly above the top of the mouthpiece. And you can also tighten the ligature at the top-most part of the reed.
These small adjustments affect the physical movement of the reed and thus the sound.
If you place a ruler on the edge of the table with a large portion of it hanging over the side, it will vibrate if you make it. However, if most of it is on the table and only a small section over the edge, it won’t vibrate very much.
There are equations for how this stuff works, if you’re interested I can explain them.


