Here's a simplified explanation of how the "8-channel Magic Array" pictured above works, lets assume using we are using cardioid microphones all the way around.
For mono, only the center forward-most microphone is used, producing single channel mono from a single forward facing cardioid.
For 2-channel stereo, only the +/- 45 degree forward angled Left/Right pair close to the center, on either side of the forward most center mic are used, producing a DIN-like cardioid pair with a 90 degree SRA pickup angle.
For 3 channel stereo (LCR) the center microphone is used along with the L/R microphone pair. The center microphone is positioned relatively far in front of the L/R pair, and is time-delayed to bring it into alignment with them. That does several things: the imaging between the +/- 45 degree L/R mic pair in the L/R speaker pair is unchanged from the 2-channel stereo configuration. The center mic is positioned far enough away from that pair to have it's diffuse-field pickup (the ambience, reverb, audience reaction, everything outside the SRA 'playback image' window) decorrelated from what the pair is picking up. This avoids the problem of "too many mics, positioned too close together, with the intend of mixing them" which I've posted about here at TS. That's a common problem for tapers running 4-channels consisting of two typical stereo pairs mounted on the same stand with the intent of mixing them all together. In this case the combination of the extended spacing, the delay, and the angle difference between the center and Left microphones forms an SRA segment which images only between the Left and Center speakers, and links across the center with the SRA segment between the Right and Center speakers. Because of that, the imaging cues in the L/R segment, the L/C segment and the C/R segment do not conflict with each other upon playback. And therein lies some of the "magic".
Playback with more than 3-channels uses various combinations of what I describe above around the array. For four channels only the central four microphones are used, which produces a 90 degree SRA between each quadrant. For five channel playback, the forward center microphone is added to those four in the same way it is added to the stereo pair in the 3-channel LCR example above. And so on up to 8 channels total.
[edit- BTW, a Williams "Magic array" needn't have the full 8 channels pictured and described here. The simplest version, recording content for 1, 2 or 3 channel playback, uses just 3 microphones as described in the 3-channel stereo example above, and would consist of like a NOS-like pair plus the single center mic extending forward. The 4 channel version which works for 2-channel stereo (re-pointable to each of 4 directions) and also works for 4 channel quad, is identical to a setup quite commonly used for location ambience recording called the IRT-cross. Partly because it is manageable and compact, IRT-cross is popular for recording location sound for 5.1 television and film, in that situation feeding all but the center channel which is reserved for dialog). A 5-channel magic array consists of the IRT-cross plus the well-forward placed center mic. So it's incremental, and doesn't ahve to be overly complex with smaller overall channel counts.]
Williams intends this as a way for production to economically record stereo and surround content simultaneously, using one array. I share that goal, and although I don't need every combination, the concepts behind this are invaluable. I'd love to try variants on this and in some ways I have been doing so already for the last few years. In particular, since I originally came across this a few years ago I've been scheming on ways to mount a single cardioid 1.5 to 2 meters forward of the other mics, sort of like how I now fly the wide omnis out to either side. Imagine the photo above with only the four center mics and the one extending forward (okay, I'd keep the wide-spaced omnis, so it would look like the photo above without the backwards extending microphone). The main problem in doing that is anything extending forward or aft is difficult to manage at outdoor festival events where things are typically setup in rows of chairs, folks may pass carrying flag poles, totems, inflatable balloon critters on a leash or whatever. The omnis extending to the sides are in-line with the row of chairs and thus far less vulnerable than a forward (or aft) arm extending over the next row of chairs. I could possibly setup a second stand two rows forward of my main stand, or clamp to another taper's stand setup there, but getting the spacing adjusted correctly that way would be more difficult. Anyone here should be able to relate to those issues, even if running a simple 2-channel setup.
Anyway, I just came across the image this morning (Posthorn is selling the mounting system for doing this using Schoeps mics, as well as doing 4 and 5 channel "star arrays"), which fired me up again about these Williams' Magic Arrays in particular, and although I've gone on an on about Stereo Zoom and MMAD at TS numerous times, I don't think I've ever really discussed "Magic Arrays" in much detail.