I know there are bleach shades out but I wondering if the B1 tooth shade is noticeably white as well?

They're are a few factors to consider when choosing a shade for your crowns. Since you know your looking for something whiter than A2 and wanting to look natural, you need to take into consideration your coloring and skin tone. The darker your skin, the whiter the shade will look.
B1 is one of the whitest natural shades, and is whiter than A2. However, there are a few shades between B1 and the bleach shade guide. 
You should consider 1M1 or 0.5M1. Both of these are beautiful white, natural colors. 

B1 is considered the lightest and brightest white shade under the "normal" shade guide.  There is a bleaching shade guide as well.  You can find them online. 

Hi:B1 shade is noticably whiter than A2 for sure.  However, in order to pick proper color the adjacent teeth and your skin tone plays a very important role.Best,Dr. Maddahi