Why Does the Clarinet Jump a Twelfth Between Registers?

On many woodwinds, pressing a register key produces a note about an octave above with a nearly related fingering. The clarinet usually jumps a twelfth—an octave plus a perfect fifth. This is not a fingering system made difficult for its own sake. It follows from the resonances available to the air column.

The clarinet is approximately cylindrical, acoustically closed at the reed end and open at the bell. An ideal cylinder closed at one end has strong resonances near one, three and five times the fundamental rather than one, two and three times. The register key does not merely shorten the tube. Its small opening weakens the lowest resonance and helps the reed and air column settle into the next strong mode. At roughly three times the frequency, the interval is a twelfth.

A real clarinet is not an ideal pipe. Tone holes, the bell, the reed and small changes in bore shape adjust pitch and colour. Nor does the sound contain only odd harmonics, because the reed generates a fuller spectrum. Even so, the bore’s preference for the odd sequence explains the broad low register, the difficulty of crossing the “break”, and why upper-register fingering cannot simply copy the lower register.

When listening, notice not only the leap in pitch but also the change in colour and resistance across the registers. The player coordinates air, embouchure and fingers through an acoustic transition. A pianist in an ensemble should hear it not as the same colour shifted upwards, but as the instrument entering a different organisation of resonance.

https://www.phys.unsw.edu.au/jw/clarinetacoustics.html
https://www.phys.unsw.edu.au/jw/woodwind.html