Why does a clarinet's register key actually produce a note a 12th higher, instead of an octave?

both Clarinet and Alto Sax do this. From what little I know, it has to do with the mouthpiece shape and valve structure. See, the oboe and boon use a very different type of reed than sax or clarinet, and the alternate s that exist for sax and clarinet do not exist for the other two. In short, they are two different types of woodwinds. ....So, yea. I don't know. Thanks for reading though.