Second and Third Order Ambisonics - the Furse-Malham Set

Michael Gerzon's original article on periphony gave versions of what was later to become known as Ambisonic encoding up to the third order. However, because these were given in cartesian coordinates and all the later work was published using polar coordinates (and with different weightings), few people have ventured into the territory above the published first order set - at least for anything other than horizontal work. Moreover, the definitions, formulations and nomeculture of spherical harmonics show a considerable degree of variation from textbook to textbook depending on whether the field of application is in maths, physics, chemistry and engineering. We published one possible re-casting of the equations which, due to simplifications of the equations made for engineering reasons, had some performance limitations in the vertical plane. After that paper was written a revised set which overcomes those limitations, the Furse-Malham set, was developed. This was produced in close cooperation with Richard Furse, a mathematician and graduate of our Music Technology course. The intention of the Furse-Malham formulation is that the coefficients generated will be normalised to have maximum values which are either + or - 1.0. This ensures that the signals in the channels have optimum dynamic range. This necessitates a departure from a mathematically pure formulation of spherical harmonics, because some harmonics need to have a scaling factor applied. Further work by Jérôme Daniel established the basic third order spherical harmonics, but without the normalising factors necessary to meet the Furse-Malham criteria. The only component in FuMa that is not required to meet this criteria is the zeroth order one, W, which has a gain of .707 (i.e. -3dB). This was set in the orgiginal Ambisonic definitions developed by Gerzon et al. and, whilst it is tempting to change this to 1.0, this would cause incompatibilities with existing harware and software. To avoid confusion, the factor for the W channel has been left as is. The new set is given in Higher Order Ambisonics, a paper abstracted from my MPhil thesis. NOTE; This set is slightly changed from that published here in 2002 in that certain of the third order components have been been reordered to make the naming convention more consistent with the low orders.

Mixed Order Systems

This system, like the earlier version, works fine when all the sounds in the soundfield are specified fully, but there are major problems when the use of mixed order source materials is unavoidable.

For instance, in the mixed first/second order case, the optimum signal matrix necessary to feed the correct portions of each of the spherical harmonic components to each speaker - the decoder - is substantially different for signals with only first order components and for those with second order as well. Whilst this can be accommodated by weighting the components appropriately at the directional coding and mixing stage, the weighting is dependent on the type of decoder (first order only, or first and second) which will be used. This is unfortunate since one of the main attractions of Ambisonics is that the composer does not need to have any knowledge of the nature of the speaker array whilst producing the piece, differences between arrays being handled by the decoder. One possible solution is to provide two rather than one version of the W (or pressure) signal, one with signal component levels appropriate to the full first/second order replay situation, the other for first only, these signals being W' and W respectively. This allows the decoding matrix to be set differently for first-only and first/second order signals, whilts existing first order decoders will perform correctly within their limits This does increase the necessary number of channels to 10 for full periphony or six in the horizontal only (W,X,Y,U,V,W') case but as the nine channels used for full 1/2 order is already more than is available on low cost multitrack recorders, the extra W' channel does not result in much additional expenditure. Note that the U, V are compatible with Bamford's horizontal only formulation.

Images of these components are available on Martin Leese's pages.


  1. Gerzon, M. A. 'Periphony: With-height Sound Reproduction' Journal of the Audio Engineering Society, Vol. 21 No. 1 Jan/Feb 1973 pp.2-10
  2. Malham, D.G. 'Higher order Ambisonic systems for the spatialisation of sound' Proceedings, ICMC99, Beijing, October 1999.
  3. Bamford, J.S. 'An Analysis of Ambisonic Sound Systems of First and Second Order' MSc in Physics Thesis submitted to the University of Waterloo, Waterloo, Ontario, Canada 1995
  4. Daniel, Jérôme, "Représentation de champs acoustiques, application à la transmission et à la reproduction de scènes sonores complexes dans un contexte multimédia", Thèse de doctorat 1996-2000
  5. Daniel, Jérôme, Rault, Jean-Bernard and Polack, Jean-Dominique 1998 'Ambisonics Encoding of Other Audio Formats for Multiple Listening Conditions', preprint no. 4795, 105th Audio Engineering Society Convention, Sept. 1998,. (Corrected version available by contacting the authors at Centre Commun d'Etudes de Télé-diffusion et Télécommunications, Cesson Sévigné, France)

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Last updated; 14th. February, 2005 by Dave Malham.
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