key with a perforated center. Pointed arms – arms connecting the keys to the rods, which are pointed and extend to the keys' centers. These are found on more expensive flutes. French model – a flute with pointed French-style arms and open-hole keys, as distinguished from the plateau style with closed holes. Inline G – the standard position of the left-hand G (third-finger) key – in line with the first and second keys. Offset G – a G key extended to the side of the other two left-hand finger keys (along with the G key), making it easier to reach and cover effectively. Split E mechanism – a system whereby the second G key (positioned below the G key) is closed when the right middle-finger key is depressed, enabling a clearer third octave E; omitted from many intermediate- and professional flutes, as it can reduce the tonal quality of the third-octave F (F6). Trill keys – two small, teardrop shaped keys between the right-hand keys on the body; the first enables an easy C–D trill, and the second enables C–D. A–B lever or "trill" key is located in line directly above the right first-finger key. An optional C trill key that facilitates the trill from B to C is sometimes found on intermediate- and professional flutes. The two trill keys are also used in playing the high B and B. Foot joint – the last section of the flute (played farthest towards the right). C foot – a foot joint with a lowest note of middle C (C4); typical on student flutes. B foot – a foot joint with a lowest note of B below middle C (B3), which is an option for intermediate and professional flutes. D roller – an optional feature added to the E key on the foot joint, facilitating the transition between E/D and D/C, and C. "Gizmo key" – an optional key on the B foot joint that helps play C7. 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