Why can a clarinet jump into a higher register?

A clarinet player can use a register key to move from a low note to a much higher one. A useful first explanation treats the bore as a cylindrical air column, nearly closed acoustically by the reed at one end and open at the other.

In that ideal tube, acoustic pressure variation can be largest at the closed end but must be nearly zero at the open end. The longest standing wave fits one quarter-wavelength into the tube. The next possibilities fit three quarters, five quarters and so on. Their frequencies are f, 3f and 5f, where f = c/(4L). For L = 0.6 meters and supplied sound speed c = 343 meters per second, the first two are about 142.9 and 428.8 hertz.

A small register hole weakens and detunes the lowest resonance, helping the instrument sustain a higher playing regime. The near-tripling corresponds roughly to a musical twelfth: an octave plus a fifth. It differs from simply doubling the frequency.

A real clarinet has a moving reed, tone holes, a bell and a bore that is not a perfect cylinder. These alter its resonances and played pitches. The ideal pressure curves explain the odd-resonance pattern; they do not calculate an actual performance.