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The reason why loudspeakers should not be placed in the corners of a room are connected with the acoustic phenomenon of standing waves. Standing waves are also known as 'room resonances' or 'eigentones'. They occur at frequencies where any of the dimensions of the room are equal to half of the wavelength, and at integer multiples of these frequencies.
A room that is 3 meters wide therefore will have resonances at around 57 Hz, 114 Hz, 171 Hz etc., plus resonances caused by the length and height dimensions.
Resonances are bad things acoustically. They emphasize certain frequencies, which causes boominess in the bass end, and unpleasant effects in the low mid range. In general, room resonances do not cause any significant problems in the upper mid range frequencies and above.
Firstly, this sounds unpleasant, generally more so in smaller rooms. Secondly, it will affect your judgement of any recording or mix you are creating and cause you to change frequencies that are not problematical in the recording, only in your room.
All rooms suffer from resonances, unless they have been acoustically treated. The problem is worse however near the boundaries of the room. The standing waves that cause room resonances form high sound pressure regions near the walls, floor and ceiling. And in the corners, then the resonances from two or all three dimensions of the room combine.
Obviously, you are not likely to listen from the corner of the room, but it works in the other direction too - a loudspeaker placed close to a boundary, or in a corner, will excite standing waves easily. So in general the best advice is to pull out the loudspeakers from the corners and walls. If you ever see wall-mounted loudspeakers in a recording studio, you can bet that they have used special acoustic treatment to reduce the formation of standing waves, and the speakers will certainly never be in the corners of the room.
But there is another way of looking at loudspeaker placement. The standard away-from-the-wall positioning results in a fairly narrow area of coverage. You can only be within quite a small area of the room and hear good sound. Elsewhere, you will be away from the direct line-of-fire of the speakers, and the sound will be muddy.
But if you place your loudspeakers, contrary to common advice, in the corners (preferably midway between floor and ceiling) then you will find that they fill the room with sound! You can stand anywhere and the sound will be great. Apart from the standing waves that is.
The solution is to provide bass trapping behind the loudspeakers, and you will need quite a lot of it. The alternative is to cut the low frequencies going to the speaker so that they don't excite the standing waves so much. You will still get plenty of bass, this is just a compensatory measure.
Acousticians will be horrified by this advice. It is contrary to every law of physics, yet it can be a very acceptable compromise between coverage and sound quality. If you want to fill the room with sound, this technique is worth a try.