11.3 Pitch perception and critical bands

Pitch Perception

Pitch is how your brain interprets the frequency of a sound. It's not a physical property of the sound wave itself but a subjective experience, which is why two people can sometimes disagree about whether a tone sounds higher or lower. Understanding pitch perception reveals how the ear and brain collaborate to turn vibrations into meaningful auditory experience.

Critical bands, covered in the second half of this guide, describe how the auditory system groups nearby frequencies together for processing. They explain phenomena like auditory masking and are foundational to modern audio technology.

Pitch and Its Relationship to Frequency

Pitch is the psychological correlate of a sound's fundamental frequency. A higher frequency generally produces a higher perceived pitch, but the relationship isn't perfectly linear.

Several other factors shift pitch perception:

• Intensity: Very loud tones can sound slightly different in pitch than quiet ones at the same frequency.
• Duration: Extremely short tones (under ~10 ms) don't give the auditory system enough time to establish a clear pitch.
• Spectral content: The overtone structure (timbre) of a sound can influence which frequency your brain latches onto as "the pitch."

Simple vs. Complex Tones

• A simple tone (pure tone) consists of a single frequency, produced by a pure sine wave. A tuning fork is the classic example.
• A complex tone contains multiple frequencies: a fundamental plus harmonics. Musical instruments and the human voice produce complex tones.

With complex tones, your brain typically perceives the pitch as matching the fundamental frequency, even when the fundamental is weak or missing entirely. This phenomenon is called virtual pitch (or the missing fundamental), and it's a key reason why pitch perception can't be reduced to simple frequency detection.

Place Theory of Pitch Perception

Place theory explains pitch perception through the physical structure of the cochlea:

1. Sound enters the cochlea and creates a traveling wave along the basilar membrane.
2. Different frequencies cause maximum displacement at different locations: high frequencies peak near the base, low frequencies peak near the apex.
3. This tonotopic organization is preserved all the way up the auditory pathway, from the cochlea to the auditory cortex. Neurons at each stage are tuned to specific frequency ranges.

Place theory works well for mid-to-high frequencies but struggles to explain pitch perception below about 500 Hz, where the basilar membrane's frequency resolution becomes too coarse. At low frequencies, temporal theory (also called phase-locking) fills the gap: auditory nerve fibers fire in sync with the waveform's cycles, and the brain reads the timing pattern to determine pitch.

Most current models treat pitch perception as a combination of both place and temporal coding.

Critical Bands

Definition and Significance

A critical band is the frequency range within which sounds are processed by the same group of hair cells on the basilar membrane. It's a direct measure of the auditory system's frequency resolution: if two tones fall within the same critical band, your ear has difficulty separating them.

Critical bands matter for several reasons:

• Auditory masking: A loud tone masks nearby quiet tones most effectively when they share the same critical band. This is why you can't hear a soft whisper that's close in frequency to a loud hum.
• Loudness perception: Two tones within the same critical band don't sound as loud together as two tones of equal intensity spread across different critical bands.
• Complex sound analysis: Your ability to pick out a single voice in a crowded room (the cocktail party effect) depends partly on how critical bands separate overlapping frequency content.

Critical Bands and the Basilar Membrane

The basilar membrane varies in stiffness and width along its length, which creates its tonotopic frequency map. Each critical band corresponds to approximately 1.3 mm of distance along the membrane, and this physical distance stays roughly constant regardless of frequency.

What changes is the frequency range each band covers:

• At low frequencies (e.g., around 100 Hz), a critical band spans a relatively wide range, roughly 100 Hz.
• At high frequencies (e.g., around 10 kHz), a critical band spans a much wider range in Hz terms (over 1000 Hz), but the ratio of bandwidth to center frequency stays more consistent.

This means your ear has finer frequency resolution at low frequencies than at high frequencies, which aligns with how important fine pitch distinctions are for speech and music in the lower registers.

The Bark Scale

The Bark scale is a psychoacoustic scale that maps frequency to perceptual pitch based on critical bands. One Bark corresponds to approximately one critical bandwidth.

• The scale divides the audible range (20 Hz to 20 kHz) into 24 Bark units.
• It's non-linear relative to Hz: low frequencies are stretched out (more Bark units per kHz), while high frequencies are compressed.

The standard approximation formula:

$z = 13 \arctan(0.00076f) + 3.5 \arctan\!\left(\frac{f}{7500}\right)^2$

where $z$ is the Bark value and $f$ is frequency in Hz.

The Bark scale has direct applications in audio engineering and signal processing. MP3 and other lossy audio codecs use psychoacoustic models based on critical bands to decide which frequency components can be removed without noticeable quality loss. Speech recognition systems also use Bark-scale (or the related Mel-scale) frequency analysis to process audio in a way that mirrors human perception.