I bought a semi expensive ligature for my clarinet, and I would like to know if it is a good idea to use it on an Alto Sax. I've tried using a cheap one on my alto sax, and it seemed to work alright (they're not that different in size). I hope I can use my good ligature on my alto sax because it makes quite a difference on my clarinet. Is this in any way bad for the instruments? If it isn't, is it worth it getting an Alto Sax ligature?
I am surprised that the clarinet ligature fits on the alto saxophone mouthpiece (I play both.) What do you mean by "fit"? The ligature should fit snugly on the mouthpiece under the "u" part of the reed. It should secure the reed just where the reed and mouthpiece naturally meet. If the ligature is too high, the reed will not vibrate properly.