The hyperbolic contour is a variety of the exponential, and is the most efficient type. A hyperbolic horn by comparison is pretty much like a trumpet stretched out, a tube that flares very little until it gets to the end where it flares suddenly. The trade-off is more distortion in the deep bass region. The problem with such a tight flare is that as sound pressures increase, the restricted passage for the air causes it to begin to compress. This causes distortion - not good.
Hyperbolic horns are also somewhat longer than exponential horns.
The tractrix is sometimes called the tractory or equitangential curve. It is a curve well-known in the world of mechanics. Later, Leibniz, Johann Bernoulli and others studied the curve. The mouth size of a horn determines the lowest frequency at which there still is any significant reinforcement of the sound. This together defines the absolute available low-end cutoff and it is obvious from what is said, that a small horn cannot go very low.