Thanks Tom, that clears things up.
Ambisonics is basically Michael Gerzon's a direct extension of Alan Blumlein's coincident stereo Mid/Side technique to three dimensions. It's a beautifully elegant and mathematically rigorous approach which provides amazing control over manipulating coincident stereo.
I have an ambisonic microphone and have learned a lot from recording with it and playing around with various ways of decoding the resulting recordings. I didn't mention it since it's not something immediately available to the OP and the discussion gets complex, but I can see it being useful for this particular situation for a few reasons:
First the single microphone is far less imposing than some of these other suggestions, it would hardly impose much on the social situation placed in the center of the circle. Second, after the recording has been made, the resulting four channel output can be manipulated after the fact to point virtual microphones of any standard first-order pattern in any direction. That doesn't mean that a pair of back-to-back cardioids might not be the best sounding choice and that a recording made with an ambisonic microphone would necessarily be better, but it does mean that you gain control over re-pointing those cardioids however you'd like after the fact. You also can vary the pickup pattern of those two virtual microphones from cardioid towards figure-8 or the other way towards omni. So you can play around with microphone arrangements after the fact to find what works best, then as the makeup of players changes around the circle from song to song and the focus changes from one area of the circle to another, you could re-point the microphones as appropriate. That might all be more involved than necessary or appropriate, but it's entirely possible.
Ambisonics does have important limits though: First, it's limited to coincident techniques, so unless used in combination with other microphones, any time-based stereo techniques are out. Second, first-order ambisonics, which is the limit of current ambisonic microphones, works well for 2-channel stereo output and up to at most 4 channel surround. It cannot provide the channel separation required for robust surround sound for music recording with more than 4 playback channels. You need space between microphones in a array to achieve that. It does Blumlein very nicely with the ability to tilt and point the array though, and could also do four cardioids in an 'X', just without any spacing between them.
I've only played with ambisonic stereo decoding and some limited 4 and 5 channel surround output. I'd love to hear a true ambisonic reproduction system, which places speakers at the vertices of a geometric solid and doesn't mess around with pointing virtual microphones this way and that for stereo decodes or typical multi-channel playback, but instead aims to reproduce the sound-field as sampled by the single-point microphone during the recording at the point in space located at the center of the playback array. At a minimum, that requires 8 speakers located at the corners of a cube in 3-dimensions. A circle of pickers with people milling about behind them laughing and having a good time while the trees rustle overhead in the breeze may well be the perfect demo for that!