Cutoff frequencies and cross fingerings in baroque, classical, and modern flutes.

Baroque, classical, and modern flutes have successively more and larger tone holes. This paper reports measurements of the standing waves in the bores of instruments representing these three classes. It presents the frequency dependence of propagation of standing waves in lattices of open tone holes and compares these measurements with the cutoff frequency: the frequency at which, in an idealized system, the standing waves propagate without loss in such a lattice. It also reports the dependence of the sound field in the bore of the instrument as a function of both frequency and position along the bore for both simple and "cross fingerings" (configurations in which one or more tone holes are closed below an open hole). These measurements show how "cross fingerings" produce a longer standing wave, a technique used to produce the nondiatonic notes on instruments with a small number of tone holes closed only by the unaided fingers. They also show why the changes from baroque to classical to modern gave the instruments a louder, brighter sound and a greater range.