The properties of horns that are suitable for use in woodwinds are deduced from first principles. The cylindrical pipe and complete cone are shown to be the only shapes which satisfy these requirements exactly. The behavior of nearly perfect cylinders and almost complete cones is described, the influence of closed finger holes on the effective bore of an instrument is discussed, and the effect of the mouthpiece cavity is analyzed. Damping of the normal modes by the walls of the bore is shown to play a dominant role in the playing behavior and tone color of woodwinds, and various consequences are deduced.