On 9/02/2012 11:02 AM, Jerry wrote:
(Good grief, people.) You want the *very famous* Bauer's Law of Sines: Benjamin B. Bauer, Phasor Analysis of Some Stereophonic Phenomena, IRE Transactions on Audio, January-February, 1962. This panning law is mentioned in many introductory books on stereo theory. Here it is, quoting from the paper: Sin theta_I (S_l - S_r) ----------- = ----------- Sin theta_A (S_l + S_r) where theta_I is the azimuth angle of the virtual image, and theta_A is the azimuth angleof the real sources. S_l and S_r, are the strengths of the signals applied to the left and right loudspeakers, respectively. This we call the “stereophonic law of sines,” and it shows that through appropriate distribution of in-phase signals to the loudspeakers, the position of the virtual image for the centrally placed observer may be adjusted anywhere relative to the loudspeaker. End of quote. The angles are "half-angles" relative to the listeners nose, i.e., for loudspeakers at +/- 30 degrees, theta_A = 30 degrees. This four-page paper is recommended reading for everyone. 8^) This panning law agrees exactly with the panning described by HRTF methods at the low frequency limit (and only there).
Just wanted to write again and say thanks for this Jerry. It is indeed what I was looking for.
Solving for S_l^2 + S_r^2 = 1 it seems to work very well for my needs. It doesn't suffer from the dip in level in the middle which Olli's previous solution did.
For my case I set the speaker angle very narrow (7.5 degrees) so that I can get extreme-antiphase gain out of the equations. Then I warped theta_I by ^4 so that 50% of my panning range pans between the speakers and the outer 25% on each end of the panning range moves into the with-antiphase region.
Thanks for everyone elses comments. Clearly there's plenty of scope to go beyond this.
Best wishes, Ross. -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
