Psychoacoustics

Abstract

Psychoacoustics is a branch of psychophysics that studies how physical properties of sound — intensity, frequency, and timing — map onto what a listener actually hears. It asks quantitative questions: how much must a sound's level rise before it seems twice as loud, how finely the ear resolves neighbouring frequencies, and why one tone can render another inaudible. The field grew from the telephone laboratory and the auditory-physiology bench into a mature measurement science, producing the equal-loudness contours, the loudness and pitch scales, and the critical-band and auditory-filter models that describe frequency selectivity. Those constructs rest on the mechanics of the cochlea, where a travelling wave sorts frequencies by place along the basilar membrane. This article surveys loudness, pitch, critical bands and auditory filters, and masking, and closes with the neural-coding questions that remain open.

Keywords: psychoacoustics, loudness, pitch

Psychoacoustics is the experimental study of auditory sensation: the rules that relate a measurable acoustic stimulus to a reportable percept. Where acoustics describes sound in the air and auditory physiology describes the machinery of the ear, psychoacoustics occupies the middle ground, treating the listener as an instrument whose judgments can be measured with the same rigour as any physical quantity (Moore, 2013). Its methods are those of classical psychophysics — thresholds, difference limens, and scaling — applied to the one modality in which the physical stimulus is unusually well specified, which is why hearing became the proving ground for so much of quantitative perception.

What Psychoacoustics Is

Psychoacoustics is the quantitative study of how sound is heard. Its subject is the mapping between the physical description of a stimulus — sound-pressure level, frequency, spectral shape, and temporal envelope — and the subjective attributes a listener reports, chiefly loudness, pitch, and timbre. The discipline took its modern form in the 1930s, when Harvey Fletcher and his colleagues at Bell Telephone Laboratories set out to specify how the telephone should reproduce speech, and in doing so measured the loudness of tones across the whole audible field (Fletcher & Munson, 1933). At almost the same time, Georg von Békésy was opening the cochlea to observation, showing that a sound sets up a travelling wave whose peak falls at a frequency-dependent place along the basilar membrane — the physical substrate of the perceptual regularities the psychophysicists were charting (Békésy, 1960).

Two features distinguish psychoacoustics from perception research in other modalities. First, the stimulus is exceptionally well controlled: a pure tone is defined by two numbers, and complex sounds can be synthesised to specification, so the input side of the psychophysical function is rarely in doubt. Second, the auditory system imposes strong, lawful transformations that are stable across listeners, which makes the percept predictable enough to model. The result is a field in which perceived magnitudes obey mathematical laws — loudness grows as a power function of intensity (Stevens, 1957) — and in which the ear's frequency analysis can be captured by a compact set of filters. Table 1 lists the principal quantities the field relates.

Loudness and the Equal-Loudness Contours

Loudness is the perceptual correlate of sound intensity, but the two are far from proportional, and the relationship depends on frequency. Fletcher and Munson established this by having listeners match the loudness of tones at different frequencies to a 1000 Hz reference, tracing out the equal-loudness contours: curves joining the sound-pressure levels that sound equally loud across the spectrum (Fletcher & Munson, 1933). The contours sag in the middle and rise at the edges, showing that the ear is most sensitive between roughly 2 and 5 kHz and needs far more physical energy to reach the same loudness at low frequencies. The loudness level of any sound is read off these curves in phons: a sound has a level of 40 phons if it is as loud as a 1000 Hz tone at 40 dB sound-pressure level.

Phons index equal loudness but not how much loudness there is; for that the field uses the sone, a ratio scale on which 2 sones is twice as loud as 1. The mapping from intensity to sones follows Stevens's power law, under which perceived magnitude grows as a compressive power of stimulus intensity — for loudness, an exponent near 0.3, so that a tenfold (10 dB) rise in intensity yields only about a doubling of loudness (Stevens, 1957). Predicting the loudness of an arbitrary sound from its spectrum requires integrating this compressive growth across the ear's frequency channels, the task of modern loudness models used in standards and hearing aids (Moore, Glasberg, & Baer, 1997).

Pitch and the Frequency Scales

Pitch is the attribute that lets a listener order sounds from low to high, and it tracks frequency, but not in equal steps of hertz. Stevens, Volkmann, and Newman asked listeners to adjust tones so that pitch intervals felt equal, and from those judgments built the mel scale, on which equal distances correspond to equal perceived pitch (Stevens, Volkmann, & Newman, 1937). The scale is anchored so that 1000 Hz equals 1000 mels; below about 500 Hz mels run roughly parallel to hertz, but above it the scale compresses, so that equal pitch steps span ever-wider frequency ranges. A closely related index, Zwicker's Bark scale of critical-band rate, divides the audible range into 24 successive critical bands and so ties the pitch axis directly to the ear's frequency resolution (Zwicker, 1961). The demonstration below maps a chosen frequency onto both scales at once.

Two mechanisms underlie pitch, and their relative contributions are still debated. Place coding reads pitch from which part of the basilar membrane is most excited, while temporal coding reads it from the timing of neural firing locked to the waveform's periodicity (Oxenham, 2018). That periodicity carries pitch at all is shown most directly by the missing fundamental: a harmonic complex tone whose lowest component has been removed is still heard at the pitch of that absent fundamental, because the waveform repeats at the fundamental's period whether or not energy is present there — a percept no purely spectral, place-based account predicts on its own (Oxenham, 2018). Recent work has kept the question live: individual listeners differ markedly in whether they hear pitch from spectral or temporal cues (McPherson & McDermott, 2018), and experiments dissociating place from timing show that the perception of frequency modulation depends on cochlear place coding in ways earlier temporal accounts did not predict (Whiteford, Kreft, & Oxenham, 2020).

Critical Bands and Auditory Filters

The single most productive idea in psychoacoustics is that the ear analyses sound through a bank of overlapping band-pass filters, one centred on every audible frequency. Fletcher introduced the critical band to explain why a tone is masked only by noise falling within a limited frequency region around it: the ear appears to sum energy within a band and ignore energy outside it (Fletcher, 1940). Zwicker quantified these bands across the spectrum, dividing the audible range into 24 contiguous critical bands whose widths define the Bark scale (Zwicker, 1961).

Modern work replaced the rectangular idealisation with a realistically shaped auditory filter, most often measured with the notched-noise method, in which the threshold for detecting a tone is tracked as a spectral notch in a masking noise is widened (Glasberg & Moore, 1990). The filter's sharpness is summarised by its equivalent rectangular bandwidth (ERB) — the width of the ideal rectangular filter passing the same power — for which Moore and Glasberg gave a simple formula relating bandwidth to centre frequency (Moore & Glasberg, 1983). Because the filters widen with frequency, the ear's frequency resolution is finer, in proportional terms, at low frequencies than at high, a fact that organises much of hearing (Zwicker & Fastl, 2007). The demonstration below draws the auditory filter at a chosen centre frequency and reports its ERB.

Masking and Cochlear Tuning

Masking — the raising of one sound's threshold by the presence of another — is the everyday audible consequence of the auditory filter, and the method by which the filter is measured. When a masker and a target fall in the same critical band, their energies compete within one filter and the target is hidden; when they fall in different bands, masking is slight (Fletcher, 1940). The pattern is strikingly asymmetric: a low-frequency masker masks higher frequencies far more effectively than lower ones, the upward spread of masking, which grows with masker level as the excitation it produces spreads toward the base of the cochlea (Zwicker & Fastl, 2007). The demonstration below shows this excitation pattern and its asymmetry as the masker level is changed.

How sharp the underlying cochlear filter really is proved contentious. Behavioural masking and physiological otoacoustic measurements converged on the conclusion that human cochlear tuning is substantially sharper than that of the laboratory animals long used as models, revising the accepted picture of frequency selectivity (Shera, Guinan, & Oxenham, 2002). That the same filter shape can be estimated from a listener's detection thresholds and from sound emitted back out of the ear is a measure of how tightly the psychoacoustics of masking is bound to cochlear mechanics (Oxenham, 2018).

Worked Example

Consider how the ear's frequency resolution changes with frequency, using the ERB formula of Glasberg and Moore, ERB = 24.7 × (4.37 F + 1), where the centre frequency F is in kilohertz and the bandwidth is in hertz (Glasberg & Moore, 1990). At F = 1 kHz the filter width is 24.7 × (4.37 × 1 + 1) = 24.7 × 5.37 = 132.6 Hz. At F = 0.25 kHz it is 24.7 × (4.37 × 0.25 + 1) = 24.7 × 2.0925 = 51.7 Hz. The absolute bandwidth is thus about 2.57 times larger at 1 kHz than at 250 Hz.

In proportional terms the story reverses. At 1 kHz the filter spans 132.6 / 1000 ≈ 13% of its centre frequency; at 250 Hz it spans 51.7 / 250 ≈ 21%. The ear's resolving power, expressed as a fraction of frequency, is therefore finer at 1 kHz than at 250 Hz, even though the low-frequency filter is narrower in hertz. Converting to the companion ERB-number scale, N = 21.4 × log₁₀(4.37 F + 1), places 250 Hz at N ≈ 6.9 and 1000 Hz at N ≈ 15.6, so the two-octave step from 250 to 1000 Hz crosses about 8.8 ERBs — roughly a third of the 24 or so critical bands that tile the whole audible range (Moore & Glasberg, 1983). The auditory-filter demonstration above reproduces these widths as the centre frequency is varied.

Discussion

Psychoacoustics succeeded because it treated a subjective quantity as a measurable one and found that the measurements obeyed laws. The equal-loudness contours, Stevens's power law, the mel scale, and the critical band are all instances of the same programme: pin down the psychophysical function, then explain it mechanistically (Stevens, 1957). The explanatory payoff came when the perceptual regularities turned out to mirror cochlear mechanics — the critical band corresponds to a roughly constant distance along the basilar membrane, and the frequency-to-place map first seen in the travelling wave underwrites both pitch and frequency selectivity (Békésy, 1960). This tight coupling between percept and physiology is what lets a single auditory-filter model predict masking, loudness summation, and aspects of pitch from one small set of parameters.

The programme also has limits that continue to define the research front. Loudness and pitch are not fully reducible to intensity and frequency: loudness depends on bandwidth and duration, and pitch on periodicity as well as place, so single-number scales are useful approximations rather than complete theories. The long dispute over the true sharpness of human cochlear tuning shows how far a purely behavioural estimate can be improved by converging physiological evidence (Shera, Guinan, & Oxenham, 2002). For cognitive psychology the field offers a model of how a perceptual system can be reverse-engineered from behaviour, and a reminder that the mapping from world to experience is lawful, compressive, and frequency-dependent rather than a faithful copy of the physical signal.

Current Directions

Contemporary psychoacoustics is increasingly concerned with individual differences and with the neural code for pitch. Large-sample work shows that listeners genuinely differ in the cues they weight — some hearing pitch chiefly from resolved spectral components, others from temporal fine structure — which complicates the assumption of a single universal listener that classical scaling relied on (McPherson & McDermott, 2018). Experiments designed to pull place and timing cues apart are sharpening the account of what frequency modulation and pitch actually depend on, with evidence that cochlear place coding carries more of the burden than temporal theories assumed (Whiteford, Kreft, & Oxenham, 2020).

A second front connects the classical measures to hearing loss and its remediation. Loudness models built on the auditory filter are embedded in hearing-aid fitting and in standards for loudness prediction (Moore, Glasberg, & Baer, 1997), and reviews synthesising the perception and neural coding of sound are framing the next questions about how the healthy and impaired ear encode frequency and level (Oxenham, 2018). The open problem across both fronts is the same one Fletcher faced: to predict, from the physics of an arbitrary sound and the state of a particular ear, exactly what that listener will hear.

Glossary

Auditory Filter.

A band-pass filter the ear applies at each frequency; the bank of such filters accounts for frequency selectivity and masking.

Bark Scale.

Zwicker's scale of critical-band rate, dividing the audible range into 24 successive critical bands.

Basilar Membrane.

The frequency-sorting structure of the cochlea, stiff at the base and compliant at the apex, along which a travelling wave peaks by frequency.

Critical Band.

The frequency region within which sounds interact in one auditory filter, so that only energy inside it masks a target tone.

Equal-Loudness Contour.

A curve joining the sound-pressure levels that sound equally loud across frequency, indexed in phons.

Equivalent Rectangular Bandwidth (ERB).

The width of the ideal rectangular filter that passes the same power as a real auditory filter; a standard measure of its sharpness.

Excitation Pattern.

The distribution of activity a sound produces across the array of auditory filters; its spread underlies masking.

Loudness.

The perceptual magnitude of sound intensity, measured in sones and growing as a compressive power of intensity.

Masking.

The raising of the threshold for one sound by the presence of another; the audible signature of the auditory filter.

Mel Scale.

A perceptual pitch scale on which equal distances correspond to equal perceived pitch, anchored at 1000 mels = 1000 Hz.

Missing Fundamental.

The pitch heard at a complex tone's fundamental frequency even when no energy is present there; evidence that pitch draws on periodicity, not spectral place alone.

Phon.

The unit of loudness level: a sound's level in phons equals the dB SPL of an equally loud 1000 Hz tone.

Pitch.

The attribute by which sounds are ordered from low to high; correlated with frequency and periodicity.

Place Coding.

The encoding of frequency by which location on the basilar membrane is most excited.

Psychoacoustics.

The branch of psychophysics that relates the physical properties of sound to auditory sensation.

Sone.

The ratio unit of loudness magnitude, defined so that 2 sones is twice as loud as 1 sone.

Temporal Coding.

The encoding of frequency by the timing of neural firing locked to the waveform's periodicity.

Travelling Wave.

The wave of displacement that moves along the basilar membrane and peaks at a frequency-dependent place, described by Békésy.

Upward Spread of Masking.

The tendency of a masker to hide higher frequencies more than lower ones, increasing with masker level.

Key Researchers

Georg von Békésy (1899-1972). Harvard University; he mapped the travelling wave along the basilar membrane and established the place principle of cochlear frequency analysis, work recognised with the 1961 Nobel Prize in Physiology or Medicine.

Harvey Fletcher (1884-1981). Bell Telephone Laboratories; he founded quantitative psychoacoustics, co-deriving the equal-loudness contours and introducing the critical band and the auditory filter.

Brian C. J. Moore (b. 1946). University of Cambridge; he derived the notched-noise auditory-filter shapes and the ERB scale and wrote the standard textbook of hearing. ORCID

Andrew J. Oxenham (contemporary). University of Minnesota; he studies pitch, masking, and the neural coding of sound, and helped revise estimates of human cochlear tuning. ORCID

Christopher J. Plack (contemporary). University of Manchester and Lancaster University; he works on pitch, temporal coding, and cochlear synaptopathy, linking psychoacoustic measures to auditory physiology. ORCID

Eberhard Zwicker (1924-1990). Technical University of Munich; he systematised the critical-band concept, defining the Bark scale and the loudness and sharpness models of engineering psychoacoustics.

Frequently Asked Questions

What is psychoacoustics in simple terms? It is the science of how physical sound becomes what we hear, relating measurable quantities such as intensity and frequency to loudness, pitch, and timbre (Moore, 2013).

Why does loudness not simply equal volume in decibels? Because sensitivity varies with frequency and loudness grows compressively: the equal-loudness contours show the same level sounds louder in the mid-range, and a 10 dB rise only about doubles loudness (Fletcher & Munson, 1933; Stevens, 1957).

What is a critical band? It is the frequency region the ear treats as a single channel, so that only sounds falling within it interact and mask one another (Fletcher, 1940; Zwicker, 1961).

What is the ERB of an auditory filter? The equivalent rectangular bandwidth is the width of the ideal rectangular filter passing the same power as the real one; it widens with frequency and summarises the ear's resolving power (Glasberg & Moore, 1990).

How is pitch related to frequency? Pitch rises with frequency but not in equal steps of hertz; perceptual scales such as the mel and Bark capture the compressive mapping (Stevens, Volkmann, & Newman, 1937).

Why can one sound make another inaudible? When a masker and target share an auditory filter their energies compete, and masking spreads asymmetrically toward higher frequencies as level rises (Zwicker & Fastl, 2007).

Is pitch coded by place or by timing? Both contribute, and their balance is still debated; recent work shows listeners differ in the cues they weight and that place coding matters more than some temporal accounts assumed (McPherson & McDermott, 2018; Whiteford, Kreft, & Oxenham, 2020).

How sharp is human frequency tuning? Converging behavioural and otoacoustic evidence indicates that human cochlear tuning is sharper than earlier animal-based estimates suggested (Shera, Guinan, & Oxenham, 2002).

References