structural:reformatting
Head joint shape

The head-joint tube is tapered slightly towards the closed end. Theobald Boehm described the shape of the taper as parabolic. Examination of his flutes did not reveal a true parabolic curve, but the taper is more complex than a truncated cone. The head joint is the most difficult part to construct because the lip plate and tone hole have critical dimensions, edges, and angles that vary slightly between manufacturers and in individual flutes, especially where they are handmade.

Head joint geometry appears particularly critical to acoustic performance and tone, but there is no clear consensus on a particular shape amongst manufacturers. Acoustic impedance of the embouchure hole appears to be the most critical parameter. Critical variables affecting this acoustic impedance include: chimney length (the hole between lip-plate and head tube), chimney diameter, and radiuses or curvature of the ends of the chimney. Generally, the shorter the hole, the more quickly a flute can be played; the longer the hole, the more complex the tone. Finding a particularly good example of a flute is dependent on play-testing. Head joint upgrades are usually suggested as a way to improve the tone of an instrument.

Cheaper student models may be purchased with a curved head to allow younger children with shorter arms to play them.