Why Does a Clarinet’s Register Key Leap a Twelfth?

On a flute, a similar fingering in the upper register often gives the octave. Press a clarinet’s register key and the note usually rises by a twelfth. The key has not somehow “added” a fifth. The difference begins with which standing waves the air column readily supports. A clarinet has an approximately cylindrical bore and a reed-controlled end, so it behaves acoustically rather like a pipe closed at one end. Its resonance pattern strongly favours odd harmonics. When the fundamental gives way to the next principal mode, the third harmonic appears first: a twelfth above, not the second harmonic at the octave.

Opening the small register vent does not simply shorten the tube to the length of a particular high note. It weakens the low-register resonance and helps the higher mode establish itself. The same family of fingerings therefore reappears a twelfth higher. This is also why the clarinet needs additional keywork to complete its scale rather than copying its lower fingerings neatly at the octave.

Listen to two sustained notes produced from the same air column, first in the low register and then in the upper one. The second is not merely a brighter copy. Its attack is more concentrated, while the balance of harmonics and the player’s sense of resistance change. Breath, reed and bore must settle into a different resonance; the register key only helps the system cross that threshold.

https://www.phys.unsw.edu.au/jw/clarinetacoustics.html
https://www.britannica.com/art/clarinet