2.1 Two ears, two signals

Sounds from the front-and-center reach both ears at the same time and with the same intensity. Sounds from anywhere else do not. A source 30° to your right is closer to your right ear than your left, so it arrives at the right ear first. It is also partially shadowed by your head, so it arrives at the left ear with reduced intensity. The brain uses both differences.

Interaural Time Difference (ITD)

The geometry is straightforward. A source at azimuth $\theta$ relative to the head’s midline, at distance much larger than the head’s radius $r$ (about 8.75 cm in an average adult), creates an extra path length to the far ear of approximately

At $\theta = 90°$ (sound directly to the side), ITD $\approx 640\,\mu\text{s}$. At $\theta = 30°$, ITD is about 250 μs. At $\theta = 0$ (front), ITD is zero.

The auditory system’s measurable threshold for ITD discrimination is on the order of 10 μs. That is, the difference in arrival time between two ears that the brain can reliably detect is about ten microseconds. The implications for the neural circuitry that has to perform this measurement are immense — we will pick that up in movement 7.

Interaural Level Difference (ILD)

The head acts as a barrier to sound waves, but only at frequencies whose wavelength is comparable to or smaller than the head’s diameter. For 20 cm head diameter, this means wavelengths $\lesssim 20$ cm, or frequencies $\gtrsim 1500$ Hz. Below 1.5 kHz, sound waves diffract around the head and arrive at both ears with similar intensity. Above 1.5 kHz, the contralateral ear sits in the head’s acoustic shadow, receiving an attenuated signal.

At 3 kHz, a sound directly to one side shows an ILD of about 10 dB. At 8 kHz, ILDs can exceed 20 dB. The brain can resolve about 1 dB.

The duplex theory of sound localization, due to Lord Rayleigh in 1907, states that ITDs dominate at low frequencies (where ILD is small because of diffraction) and ILDs dominate at high frequencies (where ITDs cannot be measured because phase information is lost — phase locking fails above 5 kHz, as we will see in movement 6). At every frequency, the brain uses whichever cue is available.

The interactive below shows the duplex theory in motion. The top diagram shows a source at the chosen azimuth around the head, with the wavefronts at the current frequency drawn around the source — when those wavefronts are large compared to the head, sound diffracts smoothly around it; when they’re small, the head casts a clean acoustic shadow. The two readouts to the right give the actual ITD (in μs) and ILD (in dB) at this configuration. The bottom plot shows where each cue is reliable as a function of frequency: ITD reliability falls off through 1.5–5 kHz as phase-locking dies in the auditory nerve, while ILD reliability grows through the same band as the head starts to shadow. The cursor on the plot marks the current frequency; the labelled regime (“ITD-dominated”, “ILD-dominated”, or “both cues contribute”) follows from where the cursor sits.

Drag the frequency to 200 Hz and the wavefronts around the source dwarf the head — diffraction wraps the sound around to the far ear, ILD collapses, ITD is the only cue worth listening to. Drag to 8 kHz and the wavefronts shrink to a couple of centimetres; the head casts a real shadow, ILD swings hard with azimuth, and ITD has dropped out because the auditory nerve can no longer phase-lock to track it. Between roughly 1 kHz and 3 kHz the two regimes overlap, and the brain uses both. That is the duplex theory, made physical.

With actual stereo audio

The interactive below renders the same scene with audio playback. Drag the sound source around the head, then press play noise (use headphones) to hear stereo pink noise with the calculated interaural delays and gains applied. The audio version doesn’t separate ITD-regime from ILD-regime — both are baked into the playback at the same time — but it is the closest substitute for the lived experience of moving a source around your head.