I just made some experiments panning around a test signal with the 3rd-order B-panner provided by York University (leaving the 3rd-order to zero, of course), and employing "manual" decoding in Audiomulch (a bit tricky, but it works).
Of course, it was necessary to employ gain controls (which work in dB), instead of absolute gains, but the conversion of the gain coefficients in dB is easy.
It resulted advisable to change slightly the decoding coefficients, for avoiding any "rear" radiation when the sound is panned straight to one loudspeaker direction (I prefer the opposite one is really muted).
Furthermore, I changed the order of the output channels in a circular order (there was an ordering error in my previous post):
FL, FR, RR, RL.
The "new" optimized decoding coefficients are as follows:
FL = W + 0.680*X + 0.680*Y + 0*U + 0.257*V
FR = W + 0.680*X - 0.680*Y + 0*U - 0.257*V
RR = W - 0.680*X - 0.680*Y + 0*U + 0.257*V
RL = W - 0.680*X + 0.680*Y + 0*U - 0.257*V The coefficients are designed for creating smooth cardioid microphones, certainly "narrower" than first order cardioids, but not much narrower than the first order hypercardioids actually employed.
I made an Excel spreadsheet showing the curves, you can download it from:
http://pcfarina.eng.unipr.it/Public/B-format/Software/2nd-order-decoder/
Looking at the curves, you will see that the "panning law" of this virtual 2nd-order cardioid is only slightly narrower than that of the 1st-order hypercardioid. For a sound panned straight in the direction of a loudspeaker, the two loudspeakers just on the sides of it will still radiate a significant amount of sound, namely they will have a gain of 0.234 (-12.6 dB) instead of 0.333 (-9.54 dB) for the first order hypercardioid.
This still provides a good guard against the "speaker detent" effect. In fact, panning the sound around with the small Audiomulch patch which I created (and which can also be downloaded from the web site above), I do not get any speaker detent effect, while obtaining a much sharper image (due to the lack of out-of-phase sound from the opposite loudspeaker, and the slightly steeper panning law).
Final remarks - definitely 4 loudspeaker are a bit too few for 2nd-order rendering. But, as we in practice are just employing one of two 2nd-order components, this is not really 2nd-order: it 1s 1.5 order... In fact we are just using 4 channels....
And it sound worth the effort of an extra channel, to my ears (furthermore, 3-channels is an odd number, 4 channels is much better managed on a computer).
So I definitely suggest that Bruce also add support for the traditional square layout also on his new 2nd-order VST decoder plugin....
Angelo Farina, 10 May 2006