Buy 100 bottles of gatorade for 100 band students using booster funds for a football game. If 5 students refuse the gatorade, does that mean that the booster fund didn't cover "all" students? No. You offered it to everyone like you were supposed to, and some refused.
I think you can use the same approach here. You can OFFER the scholarship fund to ALL band students, and you give funds to ALL who accept the offer (you'll have to adjust what's dolled out based on how much is available against the number of kids who accept it). I don't believe that'll mean you have to split the fund among 100 students though....There are a lot of families who are too stubborn to accept help, and plenty of other kids who just aren't interested in taking lessons, even if it's free (I highly advise against making it free through the scholarship though). Realistically, probably only 30-50 will express interest in accepting the offer. And then you can decide if it'll cover half the cost of lessons, 1/3 of the cost, only for the fall competitive semester, etc...