On 2022-04-06, Fons Adriaensen wrote:

I ask because theory-wise pantophonic and periphonic soundfields shouldn't be captured the same way,

There is no such thing as a 'pantophonic sound field'. We don't live in Flatland. All microphones operate in 3D space.

Some more than others. Interferometric boom mics try to operate in 1D land. You know, the ones which attenuate everything off-axis.

The ideal pantophonic mic array does the same in 2D. It cuts off everything above horizon, sharply. Then the ideal pantophonic array reproduces cylindrical waves, matched.

That's the only way the recording and reproduction conditions ever truly match. Faller's math showed that on blackboard, and the NFC-HOA work made it even clearer, if between the lines.

The way *I* interpret the mess is that pantophony is a mirage; there can be only periphony within the ambisonic framework. We might argue about its order, or about how truly regular the capture and reproduction rigs need to really be, but in the end, the spherical harmonical framework underlying ambisonic just doesn't pan out except in full 3D.


Unless you mean a sound field with all sources in the horizontal plane of the mic. Then it is just a subset of a 3D sound field, and a conventional AMB mic will capture it correctly.

Actually it will not. It's easy to fall prey to the idea of planar symmetry, but the 3D soundfield doesn't behave like that. For infinitely far-away sources in the horizontal plane the symmetry idea holds, but only because such sources constitute planewaves when they hit the mic. Any closer sources even in the horizontal plane are near-field, and constitute a near-field component even in the third dimension. Recreating them to first order calls for Z as well. And if you cut out Z...we're back to Christoff's skeptical analysis of the system: the apparent amplitude of a horizontal source takes on a 1/r term within the reproduction rig, because some of the sound energy is leaking off the plane.

The effect was analysed in WFS work, earlier. In there they used synthesized point sources and -- planar array that they had -- actually compensated actually for the term in software. Obviously nothing like that can be done in whole ambisonic soundfields.

nor do they represent even the same encoding system.

How does a sound field 'represent an encoding system' ? Your statement may be grammatically correct but it has no meaning.

Sorry, I'm being vague as usual. The exact statement would be that pantophony is the theory of cylindrically symmetrical solutions to the wave equation, whereas periphony concerns itself with spherically symmetrical ones. The continuous symmetry group respected by the former is that of the one-sphere by a line, and the one respected by the second is that of the two-sphere. By well known topological reasoning, you cannot continuously, much less differentiably or linearly, embed the latter into the former, since it's of lower dimension. Neither can you work around the problem using any linear representation, as would be the case with circular/cylindrical harmonics and spherical harmonics.

Working down from that idea, there can *be* *no* consistent definition of what a sound source off the horizontal plane means in pantophony. There's just no way to define it without having it vanish somewhere, or suddenly flip sign. And since we know you need planewaves of all directions in order to decompose any monopole... You might think you can forget about such behavior, but in the sense of the enveloping wave equation you actually cannot: every nearby point source in the horizontal plane requires a Z-wise component. (More on that later.)

It might not be easy to see, because we're used to dealing with pointwise pressure fields only; the acoustical equation instead of the full wave equation. But then only in the case of infinitely far off sources can we assume that pressure and velocity are in phase, in the plane of symmetry. That is the far field/plane wave assumption. With near field sources, the outwards radiating field from a point source is reactive at each point. Pressure and velocity are *not* in phase, and the vector describing energy transfer (in EM I think the Poynting vector) is *not* in the plane, but outwards from the source, all round. So what happens is that while the pressure field is fully symmetric in the horizontal plane, there still has to be a Z component in order to recreate the field to full first order.

You can't capture that with a periphonic array, nor can you recreate it using a periphonic rig. The only way periphony can work is by approximating infinitely far off sources i.e. planewaves. There pantophony can work, and that's how it was analyzed by Gerzon, no less. But if you want to approximate near fields -- any fully general fields -- you run into a topological singularity (pretty much the hairy ball theorem): there's just no way to map down the 3D soundfield into a 2D compatibility signal set.

Try it out if you don't believe me. Follow the math in the NFC-HOA papers to first order, with a horizontal near source. What you'll get is a W signal with its directional velocities and pressure slighly out of phase (remember, this is not the acoustical approximation anymore, but the full soundfield, with four fully independent components per point; the stuff Angelo Farina's reactive field work incited way back). What it does is counteract from above and below that 1/r term in amplitude Faller naïvely thought would topple the whole pantophonic framework.

The same applies to decoder design. Unless your speakers are infinitely long vertical line sources and radiate only in the horizontal plane, your system is a 3D one.

...and that's precisely why pantophony is an idea born dead. We don't have infinite vertical line sources, nor microphone arrays which mimic their directional patterns. The only thing we really have is 3D mic arrays and 3D rigs.

Let's not pretend there is some 2D thingy anywhere there. Because while a noble aspiration, it's also a topological impossibility. Separate dimensions just do not and cannot mesh like that. I believe it'd be better to center ambisonic work around how to do with lesser vertical resolution -- which is topologically speaking somewhat workable -- than to think pantophonic formats would ever actually live upto Gerzon-like rigor.
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