A horn player recently asked tips to improve the upper register on Facebook. Micah Cooper's answer was short, sweet & to the point!
Get BE and do the exercises. I went from 4 to 5 octaves in one month.
Later, he shared this with me in a message:
I wish I had BE when I was in high school. Things would have been a whole lot easier. Also, I said 5 octaves for my range in my post. When I am practicing a lot my range pushes 5 1/2. And when I come off a multiple month layoff, I can get back my chops in a week or two with the program. Just Awesome.