Cornwall IV vs. Volti Rival, Razz; Razz v. 1, 2, 3 -- what changed?

I can say owning a pair of horns has changed what I feel is important about hifi. The highly dynamic speakers just sound more real, as in less reproduced. I struggle to enjoy my other speakers now. 
 

I think the thing to understand with horns is the difference vs standard speakers is the controlled directivity, not so much the efficiency. Efficiency comes with the controlled directivity as less energy is spilled into the room and directed at the listener. The dispersion differences greatly affects the sound stage and it will be important if you enjoy it. 
 

other types of speakers can achieve similar results in dynamics/controlled directivity without a horn (line source etc). I know for me I will probably never let the JBLs go (lol…I know…I know). I might add a wider dispersion speakers sometime to rotate in and out. 

I am also a fan of horn speakers because of the superior dynamics, vivid sound and ability to deliver a soundfield that is large an enveloping.  Most horn systems, even though they are large in size, can fit well into smaller rooms because of their controlled directivity.  The higher efficiency of horn systems allow one to use the very best sounding amplifiers which are, in my opinion, low-powered tube amps.  The combination is magical.

Horns are more efficient not just because of their directivity, but primarily because they allow for more efficient transfer of the force of the movement of the diaphragm to the air.  Air is very light and offers very little resistance so energy cannot easily be transferred (think of how hard it is to throw a feather).  The mechanical resistance of the air is increased by confining it in a space to be compressed by the diaphragm.  Compression drivers utilize a small chamber behind the diaphragm to greatly increase this resistance and improve transfer of the energy (not all "horn" drivers are compression drivers, some have horns to act primarily as a wave guide).  Higher efficiency means lower amount of current running through the crossover and voice coil for any given volume level.  This reduces heating of these components.  Heating results in "thermal compression" where the higher temperature means increased electrical resistance and less current flow, and therefore less speaker movement than if the wire is subject to less heating.

There are other types of high efficiency speakers that share many of the good attributes of horn-based systems.  One can find single driver systems in quarter wave backloaded horn cabinets that are remarkably close to delivering a full spectrum sound and reasonably high volume levels.  I have heard, and like very much systems by Songer Audio, Charney Audio, and Voxativ.  I have also heard impressive systems utilizing fullrange drivers in multi-way systems so that another driver, or two, handles the extreme lows and/or highs.  The Cube Audio Nenuphar Basis is a good example (fullrange for mids and highs, powered woofer for the lows).

As for the choices mentioned my the OP, I like all three choices.  The Rival, in particular sounds good.  The Volti speakers are particularly nice sounding, although the Razz does not quite do integration of the woofer and the upper range quite as well as I would like (still, a minor issue to me given its virtues).  The Cornwall can be a touch rough sounding, but, it, like all three of these speakers, can be made to sound very good when using the right amplifiers (meaning good, low-powered tube amplifiers).  These are not speakers to stint on the quality of the amp.

I really appreciate the thoughtful and detailed responses and explanations about why and how horns work as they do. Thank you so much for thinking, writing, and posting!

Haven't heard Greg's speakers but intimately familiar with Cornwalls and other things Klipsch. 

It's my experience that three things will improve the performance of Klipsch horn speakers, especially the larger offerings.

Sit far enough away for the drivers to integrate. Horns are not near field, or even mid field speakers.

Do NOT toe the speakers enough so that they are firing directly at you. Even the newer speakers like the CW IV sound better when the axis crosses well behind your head.

SET amps bring out the best in Klipsch speakers that are efficient enough to accommodate them. A good 300B amp with stout transformers and power supplies will handle that 15" driver of the Cornwall with no problems.

. Modern klipsch speakers have advanced quite a bit over their predecessors, some of which could run you from the room. Even as good as the new ones are, they aren't for everyone.

FWIW Volti has a new model called Lucera which fits between the Razz and Rival.