Speaker/path/room equalization based on reciprocity?

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I am toying with the idea of equalizing the complete acoustic part of a system using the principle of reciprocity:

In principle, to determine the equalization curve, one would just need to send white or pink noise through one channel of a stereo system, and use the speaker of the other channel as a microphone/sensor.

The response curve of one channel would then be the square root of the measured response through the whole system.

The equalization could be tested/verified dynamically: if the response of the complete, equalized system is flat, this means that the response of each channel is also flat (or complementary, but that's unlikely).

I imagine that things are not that simple, and that others have already had a similar idea, so what is (are) the snag(s)?
 
I do not handle it, mostly because I do not see a simple way of taking it into account.

As I see it, the system would be not be perfect, but it would be extraordinarily simple, easy and fast: for example, with a DSP module included in the amp.
You would just have to press the "Equalize" button to get your installation calibrated in less than a minute, without microphone, test instruments, calculations or moving anything inside your listening space.

The tradeoff is a slightly imperfect equalization, but anyway very much better than no equalization at all.

If the arrangement of the room is half-sensible, the listening position will be symetrical wrt. to the speakers, not too far away, and if the speakers do not have a gross directivity problem, the end result could be pretty good
 
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