Three dimensions rectangular (x, y, z) polar (r, Θ, Φ)
Θ = horizontal azimuth in the xy plane, Φ = incline from xy plane
x = r cos Θ cos Φ y = r sin Θ cos Φ z = r sin Φ r = √(x² + y² + z²)
sin Θ = y / √(x² + y²) cos Θ = x / √(x² + y²) tan Θ = y / x = sin Θ / cos Θ
sin Φ = z / √(x² + y² + z²) cos Φ = √(x² + y²) / √(x² + y² + z²) tan Φ = z / √(x² + y²) = sin Φ / cos Φ
cos Θ cos Φ = x / √(x² + y² + z²) sin Θ cos Φ = y / √(x² + y² + z²)
When Φ = 0 then cos Φ = 1 and sin Φ = 0 and z = 0 and r = √(x² + y²)
Two dimensions rectangular (x, y) polar (r, Θ)
Θ = horizontal azimuth in the xy plane
x = r cos Θ y = r sin Θ r = √(x² + y²)
sin Θ = y / √(x² + y²) cos Θ = x / √(x² + y²) tan Θ = y / x = sin Θ / cos Θ
MS sum and difference
Sum M = (L + R) / √2
Difference S = (L - R) / √2
Left L = (M + S) / √2 = (L + R) / 2 + (L - R) / 2
Right R = (M - S) / √2 = (L + R) / 2 - (L - R) / 2
When L = R then S = 0 Θ = 0°
When R = 0 then M = S Θ = 45°
When L = 0 then M = -S Θ = -45°
When L = -R then M = 0 Θ = ±90°
M = cos Θ M > 0 90° > Θ > -90° Stereophonic space is limited to the front half space
M < 0 directions 90° to 270° are heard Θ + 180° = 270° to 450°. 180° = -180° 270° = -90° 360° = 0° 450° = 90°
The values of L and R may vary continuously and independently. L=R, R=0, L=0 and L=-R are specific directions. Multiple phantom stereophonic illusions may be produced in any direction simultaneously.
Interference between L and R speakers
Stereo channel level and time differences are not natural spatial cues. Speakers are sound sources. Sound from left and right speakers is heard by both left and right ears. Natural spatial cues Head Related Transfer Function HRTF, Interaural Level Difference ILD, Interaural Time Difference ITD, and head motion locate speakers in three dimensional space.
Stereo microphones are "in phase", the same polarity. Mixing consoles pan mono channels continuously from R = 0 through L = R to L = 0.
Speakers must be less than 60° apart to produce a phantom center stereophonic illusion when L = R.
Ears are left and right. Natural spatial cues Interaural Level Difference ILD and Interaural Time Difference ITD locate speakers. Phantom stereophonic illusions can be heard between left and right speakers in front, behind or above the listener. No phantom stereophonic illusion can be heard on the side between front and back speakers or between up and down speakers.
For the listener equal distance between left and right speakers and facing the center between left and right speakers, sounds from the left and right speakers arrive at left and right ears simultaneously. Sound from the left speaker arrives at the left ear first and the right ear second after natural Interaural Time Difference ITD due to the distance between the ears. Sound from the right speaker arrives at the right ear first and the left ear second after natural Interaural Time Difference ITD due to the distance between the ears.
Speakers L R
Ears left right
1st arrival L R
2nd arrival R L natural spatial cues ITD and ILD
When stereo channel R = 0 natural spatial cues locate the left speaker
Speakers L R
Ears left right
1st arrival L 0
2nd arrival 0 L
When stereo channel L = 0 natural spatial cues locate the right speaker
Speakers L R
Ears left right
1st arrival 0 R
2nd arrival R 0
When stereo channels L = R then (L-R) = 0 phantom center stereophonic illusion between L and R speakers
Speakers L R
Ears left right
1st arrival (L+R)/2 (L+R)/2
2nd arrival (L+R)/2 (L+R)/2 natural ILD reduces comb filter interference
When stereo channels L = -R then (L+R) = 0 (L-R)/2 + (R-L)/2 = 0 destructive interference dipole null
Speakers L R
Ears left right
1st arrival (L-R)/2 (R-L)/2
2nd arrival (R-L)/2 (L-R)/2
Stereo headphones isolate L and R stereo channels to left and right ears, eliminating interference between left and right. Stereo channel level and time differences are not natural spatial cues. Stereo channel level and time differences may exceed natural Interaural Level Difference ILD and Interaural Time Difference ITD. Headphones move with the listener's head. The stereophonic illusion is heard internal to the headphone listener's head.
The head is too small to create natural head shadow Interaural Level Difference ILD for long wavelengths (low frequencies). Mixing console panpots can create stereo channel level differences that exceed natural Interaural Level Difference ILD at low frequencies.
Spaced microphones that exceed the distance between left and right ears create stereo channel time differences that exceed natural Interaural Time Difference ITD spatial cues. When stereo channel time differences exceed natural ITD, sound is heard from the direction of the earlier speaker.
Audiophile Heretic, July 24, 2026 Stereophonic Space II Menu