Wanted to bump up this thread because I thought of something else to post.
I am my SAI chapter's new president and we've been talking pretty informally about plans for the next year. We are considering starting a committee with reps from all three music groups (SAI, TBS, PMA) so we can meet, talk about things our groups are doing, and plan occasional group things.
One thing a girl from TBS suggested that I think is a FABULOUS idea is a big group service project! Pick something like VH1's "Save the Music" or maybe find a local music program in need. Just anything! And plan a big joint fundraiser or project to support it. This is an awesome thing, because if you think about it, all music GLOs have different purposes, but our core common goal--supporting the art of music--is the same. So there's no reason NOT to work together.
Honestly, communication is the key!!! I can't stress that enough. Keep good relations between the other groups, and respect what they do. You might even drop them a card signed by all your members that says something like: "We appreciate all that you do to support music!" (I was looking through our old scrapbooks, and there was a card just like that from our TBS chapter to us. It's something really thoughtful!)
I hope some of this helps. Reply or PM me if you want any more ideas!