Horns are bad. I became personally acquainted with this fact by ownership of
a Klipsch center channel, which had their "Tractrix" horn on the tweet, the
design of which supposedly mitigates the badness of horns. Supposedly,
making the horn contour according to the geometric figure known as the
cycloid provides the efficiency boost, without the penalty.

The reason horns are bad is a consequence of the physical concept of "scale
factor." Some problems are scale invariant, and some are not. The horn is an
impedance matching device, for which the physical size is defined in terms
of the wavelength of sound that is propagated through the horn. Horns are in
wide use to match a physical media with a high impedance to a physical media
with a low impedance, for both sonic, and electromagnetic applications.

However, for a broadband audio signal, there is no single scale factor. The
"size" of the horn depends upon the wavelength, but as there is no single
wavelength in an audio signal, the horn appears to be of varying "size",
depending upon the frequency under consideration. The result is that sound
propagated through a horn has less phase coherence than was present at the
source.

As a counter to the above, one might say that all speakers, except those
with first order crossovers and sloped baffles, shift phase wildly anyway.
The ear is said to be largely insensitive to the lack of time and phase
coherence between the drivers of a multidriver speaker. So why would the
phase shift induced by a horn be so damaging?

No doubt some of the dictatorial individuals on this group will cite certain
findings that absolutely settle the question. Personally, I feel that phase
shift is probably damaging in the band where the human voice is
concentrated. I have witnessed a wide degree of variation in the degree of
vocal intelligibility of speakers. Some speakers are very pleasant to
listen to, and it comes as a shock that vocal intelligibility is poor.
Others apparently attempt to restore vocal intelligibility by nonflat
response. Still other speakers, a golden few, which seem to include panels,
manage extraordinary vocal intelligibility without any emphasis. Some
dynamic speakers are also in this group, and are not limited to simple
crossover designs. Perhaps these speakers avoid the critical region in
choice of crossover.

Returning to the original point, I have not personally been impressed by
horns. But despite the teasing title of this post, I do not believe horns
are bad. Like many other very pleasant speakers, horns probably give up the
ultimate in vocal intelligibility as an innate attribute of "hornness."