Morse taper

I'm intrigued to know how the machine tool taper rules work. I have a drill whose chuck is a taper fit on the spindle. It appears to me that the spind le is a Morse taper 1 (I may be wrong). If it is indeed this size, I assume d that the chuck (with a female taper hole) would also be a Morse taper 1. But the manufacturers are calling it a B12. In fact, when I told them it wa s a Morse 1 it threw them until I sent a photo. So my question is, what is the difference between a Morse taper and a B taper (in principle, I mean, n ot in actual size, because I know there are several different sizes). Surel y the male and female would be the same fit, would they not?