Okay this gets more technical. Apologies in advance for the thread-jack. Feel free to ignore, but at least checkout the photos of the cool high-tech mics at the bottom!
Almost all the microphones we use are first order microphones in the technical, mathematically described sense as mentioned above. First order
describes the set of all pickup patterns ranging from omnidirectional to figure-8 along a continuum, generated by combination of the basic omni and bi-directional first order components. Various combination ratios produce pattern shapes we refer to as subcardioid, wide-card, cardioid, supercardioid and hypercardioid, all of which are within the set. Microphones with these patterns are available with various degrees of quality of course, roughly corresponding to price. Besides other engineering/cost choices made by the manufacturer, those quality issues concern in part how closely the achieved pickup pattern matches the mathematical ideal, as well as how closely that pattern is maintained across the entire frequency range. These are attributes important for microphones used in stereo pairs for high-quality stereo recording, both for good imaging and for natural sounding ambient reverberant sound. Better behaved mics in those areas are of a top order
in the sense that they are higher-quality than mics which don't work as well for stereo recording, but because of this important meaning of first order
acoustic mathematical confusing and The mistake is to misconstrue the term "first order" as a synonym for "top quality"
What microphones aren't first-order? All basic microphones capsules are first order. But we can use them in ways in which their pickup pattern is modified to intentionally no longer approximate a mathematical first order ideal shape. Any boundary mounted microphone is no longer behaving strictly as a basic first order pattern, typically that's done with omnis mounted to a wall or the floor, but other patterns can be boundary mounted too.
Two mics on either side of a jecklin-disk or similar baffle no longer behave in a simple first order fashion. And a binaural recording made with omni capsules placed in one's ears or in that of a dummy head has a complex response which is no longer well approximated and cannot be synthesized using "first order" mathematics. An interference tube microphone (shotgun mic) doesn't exhibit a first-order pickup pattern since the increased directivity imparted by the interference tube is greater than a supercard can achieve alone (supercard being the most unidirectional first-order pattern along the continuum from omni to bidirectional).
A shotgun mic uses a supercardioid capsule (first order) with an interference tube attached to the front of it which increases directionality at high frequencies at the expense of greatly reduced off-axis polar pattern quality and consistency. The off-axis response of a shotgun is both more rough overall and varies dramatically by frequency. That means shotgun mics don't perform very well as stereo pairs, despite a long history of tapers using them in pairs to record the Grateful Dead from locations far back in the audience. That worked for a few reasons, yet in a stereo configuration, shotguns suffer greatly from exactly the problem Moke mentions with regards to sound arriving from off-axis. To help cover their inherent raggedy off-axis and overall response flaws, Dead tapers typically ran them in 3-mic configurations with a single omni in the center, and those 3 channels were combined to two before being recorded to 2-channel tape. The singe omni extended the frequency response as well as smoothing some of the flaws of the shotguns. I've argued here at TS for other techniques which utilize the positive attributes directional attributes of shotgun mics and covers for their flaws in a similar way, but without trying to use them as a stereo pair, such as a single forward-facing center mic mixed with two wide spaced omnis or with a wider than normal spaced/angled directional pair. Those configurations are a more appropriate application for modern outdoor fests and arena shows in which the PA mix is almost always primarily monophonic, unlike 80's era Dead, which was PA reinforced in stereo), yet the taping position is still quite distant, and they avoid the "stereo shotguns" problem altogether while achieving a richer stereo quality across a wider frequency spectrum, with strong potential to produce recordings which are more enveloping.
There are some microphone arrays which can produce higher than first order microphone patterns by way of carefully matrixing the output from multiple microphones, Schoeps does that with their CMIT shotgun which uses a second rear-facing supercardioid capsule and clever DPS programing within the microphone to produce a mic with the increased directivity of a shotgun though the full frequency spectrum with minimal off-axis problems. They'd probably not call it a higher-order microphone but it is certainly a move in that direction and uses similar techniques. I'd love to be able to use one of those! True higher-order microphones typically consist of an array of numerous small omnis arranged in a specific pattern. In that way second, third, fourth, fifth or higher order microphone patterns can be generated. The irony is that most higher-order microphones have thus far not been very "musical" and have terrible pattern behavior quality in the sense of the measures which interest us as music tapers. They achieve greater directionality but just don't sound very good. Below is an example of a second order cardioid pattern- notice that the front lobe is tighter than a supercard, and that there is not a single secondary lobe but three- a rear lobe with the same polarity as the front, and two side lobes with inverse polarity-
Now we are no longer simply combining one directionless omni pressure component with a bi-directional component pointed in a particular direction. To achieve greater than first-order directional microphone patterns, we need to generate "higher order components" and combine them in various ways. Below is a computer generated image of mathematically defined shapes arranged from top to bottom in increasing orders of "spherical harmonics" which can theoretically be extended to any nth-order. Ambisonics deals directly with such spherical harmonics. The sphere at the top of the pyramid of shapes represents the omni pressure component, think of it as the basic omnidirectional "zero order" stating point. The second row down represents the first order pressure-gradient bi-directional component, shown as three separate bi-directional fig-8 patterns pointing in the three cartesian directions (left/right, front/back, and up/down). By various matrix combinations of the omni component and these 3 bi-directional components, we can generate any first order microphone pattern pointing in any direction we like. That's why any first order ambisonic microphone requires four recording channels. By recording those four channels live, we can come back later and create as many virtual first order microphone patterns as we like in that point in space and point them in any direction we care to once we get down to mixing. That's the principle behind how first order ambisonic microphones such as the Soundfield microphones and the Core-Audio Tetramic work. As the order complexity is increased further, one needs to include all the shapes of the next layer down to produce patterns of the next higher order. 2nd order ambisonics requires 9 channels of information instead of 4. 3rd order requires 16 channels.
Below is a mic capable of 5th order which uses 32 microphone channels, those are omnidirectional pressure capsules flush-mounted into the surface of a sphere-
And here's another that is capable of 5th order in some directions, restricted to the horizontal plane only (no up/down virtual microphone adjustment, no virtual mics pointing at the ceiling) but is capable of much higher quality and suitable for music recording, using in this example 8 Schoeps omnis in a specially designed mount. The 8 recorded channels are then processed by heavy DSP routines in a dedicated computer control box to create the virtual microphone patterns-