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RE: you've never heard of "cupped hands" distortion?

Not sure where to begin, there are several misunderstandings here.
First, "non-flat" amplitude response is not distortion, it is non-flat amplitude response.

Distortion in the first order is sound the speaker adds not present in the input signal and normally 2,3,4, 5 etc times the input frequency and is caused by non-linearity.

A horns response is governed by the drivers acoustic power response AND it's directivity both are a function of frequency.

Due to issues I outlined in the loudspeaker chapter in "Handbook for Sound Engineers 5th edition" horn drivers do not have a flat power response and can only measure flat driven directly when on a horn who's radiation pattern narrows with increasing frequency.

For a horn that has the same pattern angles up high, a CD or EQ slope is applied yielding flat response and at the same time as correcting the amplitude, it also corrects the phase being a mostly minimum phase device.

Another issue with horns is they are too large to coherently combine with the lower device and I have only found one approach in 20+ years that allows making a multidriver / multi way horn radiate as if it only had one driver and no crossover phase shift.
When you do that, there is no cupped hand horn sound at all.

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