In my humble opinion rather a lot of variables affect what sound eventually emerges when we introduce some wind into the sharp end of our horns:
1. The material the reed is made of (cane or man-made).
2. Reed 'strength' (1.5, 2.0, 2.5, etc.).
3. Flexibility of reed generally (how damp it is or becomes).
4. General condition of reed.
5. Amount of saliva that adheres to underside of reed as one plays.
6. How the reed is attached to mp (type of lig, material of lig, how securely).
7. What the mp is made of.
8. How the mp has been faced ('accuracy', tip gap, etc.).
9. Mouthpiece dimensions (window size/shape, table, rails, etc.).
10. Saliva accumulating in mp barrel as one plays.
11. How the reed is positioned on the mp (in relation to mp tip, straightness, etc.)
12. Embouchure of player (position of bottom lip, etc.).
13. Playing style being attempted.
14. Frequency of note being attempted.
15. Condition of rest of horn (pads, springs, etc.).
16. Frame of mind the player is in (!).
17. Player confidence for piece.
18. Time elapsed since player last ate or drank anything.
19. Air temperature around performer.
20. Acoustics of playing area.
21. ... and probably several other things!
None of the above are straightforeward 'yes' or 'no' variables unfortunately so any attempt to come up with the answer to the 'perfect blow' won't get us very far. In this thread the above item number 6 "how the reed is attached to the mp" is being discussed. So I'll say a bit more about that:
An alto reed is approximately 7cms long. Half that length is designed to waft around in the breeze while the other half (the thick end) is supposed to be held down somehow. The ideal is some sort of fixing that firmly holds the thick end over its entire 3.5cm because, if that doesn't happen, transient vibrations occur when different notes are attempted ... which of course affects the sound that mostly comes out at the other end of the horn. I use the word 'mostly' because sound comes out of lots of other places too ... but that's another story!
To continue ...
Most of us have done the following at some time or other:
Take a 12" thin metal ruler and place it over the edge of a desk at its halfway point, hold the other end down and give it a 'twang'. A note comes out. Move it further out and a lower note happens, further in a higher note. Now alter the 'holding down' in different ways. Five fingers spaced along the part of the ruler on the desktop gives a pretty pure note because its well held down and can't move. Removing fingers in various combinations will cause other notes to appear too (harmonics I suppose) as bits of the ruler held down earlier vibrate in sympathy with the main note. That's what I think different ligs do if they don't hold the reed absolutely firmly on the table for half of its length. Item 12 (embouchure, particularly what the player's bottom lip does both in terms of position and firmness) also messes with the reed's ability to vibrate as it would otherwise.
So, yes, different ligatures, with different 'hold down' patterns, will likely have different affects on the resulting sound. I don't personally think what they are made of makes any significant difference, its their mechanical efficiency that matters most. Tightly wound on elastic bands, used in emergencies, do the job pretty well for a mechanical reason rather than any other, as do several other fixing methods.
That's my take on all this. In an attempt to be amusing I was tempted to conjure up a spoof formula, using all the above variables, that would reveal the 'perfect sax sound'. I'll leave that to someone else!!!!!