|Acoustics of the clarinet
Notes are the written pitch.
At high frequency, things get very complicated. High frequency waves can travel further past open tone holes (the air in the tone hole doesn't have much time to move at high frequencies) and so the pitch of the high frequency peaks are thus affected more than the low frequencies by the subtle effect of reflections near the open tone holes at the end of the bore. Yes, we could think of this as a somewhat flat seventh harmonic of E4. Or a very sharp ninth harmonic of A#3. For notes so far above the cut-off frequency, the entire bore is involved and one could say that any open tone holes operate not only as register holes, but as minor adjustments to the frequency of the desired resonance. No problem for Catherine, however. She just plays it (sound spectrum and file below).
Sound spectrum of a Bb clarinet played using fingering for C#7.
For more explanation, see Introduction to clarinet acoustics
You can hear C#7 played.