Z=pUcap Z equals the fraction with numerator p and denominator cap U end-fraction

To fine-tune an instrument without shifting a hole's location on the body, makers use undercutting. This involves flaring the internal exit of the tonehole where it meets the bore.

These models are not merely theoretical. They are actively used to:

To the casual observer, a wind instrument is simply a tube with holes. Whether it is a rustic bamboo flute, a brass saxophone, or a complex bassoon, the mechanism appears rudimentary: cover a hole, the pipe gets longer; uncover it, the pipe gets shorter. But in his seminal work, Bart Hopkin reveals that this simplicity is an illusion.

Designing a functional tonehole requires balancing tuning, ergonomics, and acoustic efficiency. Diameter vs. Location Placement