Hearing, Loudness & Psychoacoustic Limits

The ear is not a microphone followed by a decibel display. Outer and middle ear filter the pressure field; the cochlea separates frequency along a mechanical travelling wave; hair cells transduce motion; neural adaptation, attention, and binaural processing turn that activity into percepts. Loudness, pitch, timbre, distance, and direction are related to physical measurements but are not identical to them.

Psychoacoustic limits are distributions over listeners, stimuli, contexts, and tasks. A threshold measured with young otologically normal listeners in a frontal free field is not a universal boundary for speech, headphones, older listeners, or a noisy room.

From pressure to neural representation #

flowchart TB
  subgraph PER["Peripheral mechanics"]
    direction LR
    P["Sound field"] --> O["Outer ear"] --> M["Middle ear"] --> C["Cochlear wave"]
  end
  subgraph NEU["Neural representation"]
    direction LR
    H["Hair cells"] --> N["Timing + rate"] --> B["Binaural processing"] --> X["Percepts"]
  end
  C --> H

The basilar membrane is tonotopic: high frequencies peak near the stiff base, low frequencies nearer the apex. Auditory filters overlap rather than forming isolated FFT bins. Their bandwidth generally broadens with centre frequency and with level. Energy inside a filter competes for detectability, which is why masking depends strongly on spectral proximity.

At lower frequencies, neural firing can phase-lock to waveform timing and provide temporal pitch and binaural cues. At higher frequencies, place and envelope cues dominate as fine-structure phase locking weakens. There is no single “spectrum analyser” resolution independent of level and duration.

Threshold is frequency and context dependent #

Sound-pressure level is physical:

$$L_p=20\log_{10}\left(p_{rms}/20\ \mu\mathrm{Pa}\right).$$

Audibility threshold varies with frequency, listener, field, and method. ISO 226:2023 equal-loudness contours cover pure tones in a frontal free progressive plane wave, binaural listening, and otologically normal people aged 18–25, over 20 Hz to 12.5 kHz. Those boundary conditions are part of the data.

An equal-loudness contour connects tones judged equally loud. A low-frequency tone generally needs substantially more SPL than a mid-frequency tone for the same loudness, and contour shape changes with level. Consequently:

• equal SPL is not equal loudness;
• A-weighting is a standardized engineering weighting, not a universal loudness model;
• headphone measurements require a coupler/head-related transfer and cannot be compared directly with free-field SPL;
• individual hearing loss cannot be inferred from a programme loudness meter.

Loudness is nonlinear and time dependent #

Perceived loudness grows compressively with level over much of the usable range. Cochlear mechanics and neural adaptation contribute to that nonlinearity. Short sounds also need more level than long sounds to reach the same loudness until temporal integration saturates; an impulsive peak and a steady tone with the same RMS are not equivalent percepts.

ISO 532-1 specifies Zwicker-method calculations for stationary and time-varying sounds. ITU-R BS.1770-5 instead defines a reproducible programme-loudness and true-peak algorithm for broadcast workflows. They solve different problems: the first models psychoacoustic loudness under defined inputs; the second gives an interoperable operational meter. Neither predicts every listener's judgement.

Masking hides signals selectively #

Simultaneous masking occurs when a masker raises the threshold for another sound at the same time. The effect is strongest within nearby auditory filters and is usually asymmetric: a low-frequency masker tends to mask higher frequencies more than the reverse because excitation spreads upward along the cochlear place map.

Temporal masking extends beyond exact overlap. Forward masking persists after the masker ends and can last much longer than backward masking before it begins. The amount depends on level, spectral relation, and timing. Codec models exploit only a conservative part of this behavior because audible errors are costly and listener/content variability is large.

Masking is not deletion. A component below one masking model's threshold can become audible through headphones, another listener, a different preceding sound, repeated training, or downstream processing that removes the masker.

Pitch is not identical to one spectral line #

A pure tone has a straightforward frequency–pitch relation over ordinary ranges, but complex sounds expose multiple cues. A harmonic series can retain the pitch of a missing fundamental because spacing and temporal periodicity imply its period. Inharmonic components, transients, and unresolved harmonics can produce ambiguous or multiple pitches.

Pitch discrimination improves with duration and depends on frequency, level, and training. Musical cents are logarithmic:

$$c=1200\log_2(f_2/f_1).$$

One semitone is 100 cents, but a fixed cents difference is not equally detectable across all registers, timbres, and durations.

Localization combines binaural and spectral cues #

For a source away from the median plane, the head creates:

• interaural time difference (ITD), most useful for low-frequency timing;
• interaural level difference (ILD) from head shadow, strongest at high frequency;
• direction-dependent pinna spectral notches for elevation and front/back discrimination;
• dynamic changes from head movement that resolve ambiguity.

For head width scale $d$ and azimuth $\theta$, a crude far-field delay scale is

$$\Delta t\approx\frac{d\sin\theta}{c},$$

but real head-related transfer functions include diffraction, torso, pinna, and individual geometry. ITD alone has a cone of confusion: many positions share nearly the same delay. A binaural renderer needs both ears' complex transfer functions and often head tracking, not merely left-right amplitude panning.

Room reflections alter localization. The precedence effect tends to let the first-arriving wave dominate perceived direction while later reflections affect spaciousness and timbre. If delays become large or levels high, reflections are heard as echoes instead.

Design and measurement consequences #

1. Separate the physical measurement from the perceptual claim: calibrated SPL, spectrum, and duration first; listener outcome second.
2. State transducer and field: free-field loudspeaker, diffuse field, headphone coupler, or in-ear probe.
3. Preserve phase and inter-channel timing when spatial cues matter.
4. Evaluate codecs on difficult transients, tonal signals, stereo images, and post-processed outputs, not only average scores.
5. Measure speech intelligibility with a validated method rather than equating bandwidth or loudness with comprehension.
6. Report listener population, task, level, presentation, and statistical spread in listening tests.
7. Treat equal-loudness, auditory-filter, and masking models as bounded models; do not universalize their reference cohort.

The practical boundary is simple: acoustic measurements describe the field, psychoacoustic models predict a listener response under stated conditions, and listening tests observe a population. Reliable work never substitutes one layer for another without naming the model.