Rather than starting here by explaining the basics of ambisonics (which is what the 2nd and 3rd post slots above are intended for, and the linked threads touch on in part), I'm thinking we might just jump right into it and continue discussion of a couple interesting topics brought up by questions kuba e was asking in the recently locked threads. Specifically these two separate discussions: proximity effect or the absence of it in ambisonic microphones, and ways of altering pattern with frequency in 1st order ambisonic systems.
I'll start with the second topic-
kuba e asked [my paraphrase here] Can we modify pattern such that it varies by frequency by EQ'ing the (omni) W-channel? In other words might we be able to specify, say, an omni at low frequecies transitioning to a supercard at high frequencies? Or say, a subcardioid at very low frequencies smoothly morphing to a hyper-cardioid through the midrange at some transitional frequency and then to a cardioid at some high frequency, or whatever.
The easiest way of altering pattern by frequency will be with the use of more advanced ambisonic plugins. This is essentially what the Schoeps PolarFlex system does (originally done using hardware now through software) except applied to ambisonics. But this being a thread about how ambisonics works in general, lets discuss what's going on "under the hood" and kuba e's specific question. The part below in italics is mostly from a PM I sent to kuba e earlier today, copied here with his permission and mine (to honor, in spirit, the rule regarding no public posting of PM discussions).
Yes, in 1st order ambisonics, EQing the W channel will alter the pickup pattern by frequency (without altering the axis). Your understanding of the fundamentals of 1st order Ambisonics as an extension of Mid/Side that you posted the other day in the OctMic Blumlein thread is correct. Altering pattern by frequency is essentially the same as with Mid/Side, where EQing either Mid or Side channel differently than the other will alter pattern in a way that corresponds to the difference in EQ. Same goes for altering axis. But a significant difference from Mid/Side is that those two aspects are no longer inseparably linked when deriving about multiple outputs, such as a stereo pair. Pattern and angle can be adjusted separately and need not affect each other. With basic sum/difference combination you can vary pattern with frequency without change of angle if both microphones are oriented in the same direction, or change angle without change of pattern using two crossed figure 8's. If you want a stereo output from a Mid/Side sum/difference arrangement change of pattern and change of angle always remain linked.
Consider this- in addition to being able to change pattern with frequency, one could also change angle with frequency. Imagine a pair of cardioids (or any 1st order pattern) that remain cardioid across all frequencies but the angle between them changes by frequency. That might be useful. With ambisonics, one can vary both aspects (pattern and angle) by frequency, separately from each other. Mid/Side varies both at the same time, but only on its own terms, not separately, and in a way that is inverse to what is typically useful for a stereo pair (pattern getting wider as angle gets narrower, instead of the opposite).
My advice is to think of the fundamental workings of ambisonics separately from the practical implementations of ambisonic micrphones. The fundamental workings are B-format manipulations as extension of Mid/Side, or rather sum/difference, which is the basis upon which both Mid/Side and Ambisonics work. "Native" ambisonic micrphone arrays using 2 or 3 figure-8 microphones and an omni (all as close as possible to the same point in space) seek to derive that directly from the outputs of the microphones, along with level and frequency matching. In that case the microphones used correspond directly to the W,X,Y,Z b-format channels. Think about the basics of ambisonics in that way, and not as mics on a sphere and the implementation specifics of how that gets converted to W,Z,Y,Z b-format components, which is more complicated.
Dual Mid/Side is a good conceptual bridge between 2ch Mid/Side and ambisonics. A horizontal only "native ambisonic" microphone array such as the setup Nimbus uses (you've probably seen photos of it), or what TS member EMRR did a few times using Sennheisers (pictured in the OMT thread) is fundamentally the same as a dual mid/side setup. The more typical and practical Dual M/S implementation is to use a single figure 8 Side and two cardioid Mids. It's done that way for a few practical reasons: Its easier for people who already understand regular Mid/Side (just adds a rear facing cardioid Mid); more folks have 2 cardioids and a single fig-8 than an two fig-8's and an omni; and reportedly Schoeps tried it both ways with their capsules and doing it with two cards and a singe Fig-8 worked somewhat better for them. Regardless, all of these physical arrangements can be more or less easily understood as using the same fundamental sum/difference basis as Mid/Side.
Where things get more complicated are ambisonic microphones with capsules arrayed over the surface of a sphere. That's done with 1st order ambisonic microphones for a few practical reasons: one can't really get all 3 or 4 microphones in the same spot but this arrangement gets closer to doing so, and an ambisonic microphone using four cardioid elements is easier to implement and less costly for a manufacturer than using figure-8s. Its done that way for higher orders because its the only way, one cannot make a "native 2nd order ambisonic array" because it would require a the addition of native 2nd order microphones (which do not physically exist) to an existing native 1st order arrangement, which already has a hard time achieving close coincidence.
Where I think your summary in the other thread goes somewhat astray is in considering 2nd order entirely different from 1st order. On a fundamental level it is not. It adds complexity by introducing additional spherical harmonics which do not correspond to native 1st order patterns, but the basis upon which it works in B-format is the same, only further extended. Where it is very different is in practical implementation (mics arrayed across the surface of a sphere), yet in the same way that a 1st order ambisonic microphone made using 4 cardioids on the faces of a tetrahedron (4 mics on a sphere) differs from a "native array" ambisonic arrangement of 3 figure-8's + omni.
Ambisonics in terms of basic B-format is complicated enough just in understanding the theory. The engineering of practical A-format implementations used to derive b-format is a much greater complication and best left out of this discussion. The two locked threads which inspired this one are intended to be about how well the engineering of that particular microphone has been implemented. The OT discussion over there was more concerned with concerned ambisonic theory in general (universal b-format rather than specific a-format).