DI = 10 log [180° / arcsin (sin α/2 · sin β/2)]

where α is the wall angle in one plane and β is the wall angle in the other.

If the sound source is placed at the apex of a corner, then sound radiation is confined to eighth-space. Energy is radiated π/2 steradians into the room in the only direction possible, and the maximum radiation pattern is fixed and defined by the wall angles. Because of this directionality, there is 9dB DI increase over omnidirectional radiation.

Beyond that, I don't mean to be rude, but your position seems overly argumentative.

About crossover points, the transition from low to mid in a π cornerhorn is 250Hz and the midrange is a straight-sided horn. Crossover to the tweeter is at 1.6kHz. As you can see, the wavelengths at the crossover points make it pretty easy to position adjacent radiators within a 1/4 wavelength of each other.

Regarding directivity matching, the hardest range to control is the bass, which is bound by the room's walls. The midrange is placed close enough to the corner that the lowest edge of its range is aided by the walls, so where its directivity control begins to fail, the room walls begin to act as flare extensions. At higher frequencies, the horn flare alone sets its directivity. By using this approach, directivity throughout the audio spectrum is maintained within the range of 9 to 11

• Measurements