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Ambisonics attempts to record and reconstitute a spherical-harmonic expansion of the soundfield about a given listening point.

A spherical-harmonic expansionB.37is closely related to a multipole expansion.B.38

The Ambisonics order is equal to the order of the multipole expansion, which is the maximum number of spatial derivatives used to create each term. This number is 0 for the monopole term, 1 for the three dipole terms, 2 for the five quadrupoles, and so on.B.39 Thus, first-order Ambisonics consists of a monopole and three dipoles. (Ordinary stereo L,R can be viewed as a monopole L+R and one dipole L-R superimposed at the listening point.) Second-order Ambisonics adds five quadrupole terms, and third-order Ambisonics adds seven octupole terms (and so on, but third-order seems to be the highest used in practice).

Legendre polynomials

Any source-free soundfield, being analytic, can be reconstructed completely from any series expansion about any point in the space. It's not at all obvious to me, however, what the audible error would be as one leaves the expansion-point zone. Both physical and psychoacoustics come into play.

Panning is essentially equivalent to first-order Ambisonics since any stereo presentation can be modeled as some combination of $ L+R$ and $ L-R$ . For example, suppose your recording is made with two microphones separated so far that they contain independent signals, and you play one out of each speaker. Then the listener hears $ aL +
bR$ where $ a$ and $ b$ depend on seating, channel gains, etc. (Relative delay is ignored.) Then the signal at the listening point can be rewritten as $ \alpha (L+R) + \beta (L-R)$ , where $ \alpha =
(a+b)/2$ and $ \beta = (a-b)/2$ . So they are precisely equivalent when the speed of sound is infinity.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University