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Correct, but over simplified. If a key is not open far enough to "vent" the note properly, the timbre suffers and the pitch is lower. There are some occasions such as on a D2 where the note is so sharp that it sometimes helps to compromise by lowering the low C key. This brings the pitch down to a more manageable level, but the trade off is that the D sounds even more stuffy, and D1 may be too flat. Making the keys higher than about 30% the diameter of the tonehole makes no further difference in either the pitch or the tone and I find it highly questionable as to whether there is a perceptible increase in volume or projection. What I am certain about is that the increased distance in key travel makes fast playing even more challenging.
Makes sense.
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