Loudness Perception

Abstract

Loudness perception, which MeSH classifies under auditory perception, is the subjective magnitude of a sound — how strong or intense it seems to a listener. It is not the sound's physical level: loudness depends jointly on intensity, frequency, bandwidth, and duration, and it grows compressively, so a tenfold rise in intensity sounds only about twice as loud. Fletcher and Munson's equal-loudness contours map the sound-pressure level needed at each frequency to match a reference tone, defining the phon. Stevens's sone scale and power law then fix the rule that loudness roughly doubles for every 10-phon rise, while modern models compute loudness from the pattern of excitation across the cochlea's critical bands. These models extend the account to impaired hearing, where lost compression produces the steep, abnormal loudness growth called recruitment.

Keywords: loudness, phon, sone, equal-loudness contours, loudness recruitment

What Loudness Perception Is

Loudness is the perceptual attribute of a sound that orders it on a scale from quiet to loud. It is the subjective correlate of physical intensity, but it is not identical to intensity, and the gap between the two is the whole subject. A sound's physical level is measured in decibels of sound-pressure level (dB SPL), a logarithmic ratio against a fixed reference pressure; loudness is what the listener actually hears, and the two are related by a psychological transformation that is neither linear nor logarithmic but a compressive power function.

Three facts make loudness more than a relabelling of the decibel. First, loudness depends on frequency: a 20 dB SPL tone at 3 kHz is clearly audible, while the same level at 100 Hz is below threshold and heard as nothing at all, because the ear is far more sensitive in the mid-frequency range where speech lives. Second, loudness depends on bandwidth: a sound whose energy is spread across a wide range of frequencies is louder than a pure tone of the same total energy confined to one frequency, once the spread exceeds a critical band. Third, loudness grows compressively with intensity: a large increase in physical power yields a much smaller increase in perceived loudness, which is why a tenfold increase in intensity sounds only about twice as loud.

Because loudness is a private, subjective magnitude, measuring it means asking listeners to compare, match, or scale what they hear. The history of the field is the history of turning those judgments into numbers — first by matching sounds in loudness to define the phon, then by scaling loudness directly to define the sone, and finally by modelling the cochlear processing that produces both.

The Phon and the Equal-Loudness Contours

The first systematic measurement of loudness fixed a reference and asked how every other sound compared with it. Fletcher and Munson had listeners adjust the level of a test tone at each frequency until it sounded equally loud as a 1 kHz reference tone, and plotted the sound-pressure level required at every frequency to achieve each match (Fletcher & Munson, 1933). The resulting curves are the equal-loudness contours: each contour joins all the frequency-level combinations that sound equally loud.

The contours define the phon, the unit of loudness level. A sound has a loudness level of N phons if it sounds as loud as an N-dB SPL tone at 1 kHz; by construction the phon and the decibel coincide at 1 kHz and diverge everywhere else. The contours sag in the middle and rise steeply at the low-frequency end, which captures the ear's poor sensitivity to bass: to match a 1 kHz tone at 40 phons, a 100 Hz tone must be delivered at a far higher physical level. The contours also flatten as level rises, so the frequency dependence of loudness is strongest at low levels and weaker at high ones — the reason music sounds bass-thin when played quietly. These curves, refined over decades, are now standardised internationally as ISO 226, whose 2023 revision drew on fresh measurements across many laboratories (Suzuki et al., 2024).

The Sone Scale and the Power Law

The phon orders sounds by loudness but does not say how much louder one sound is than another: 80 phons is not twice 40 phons in any perceptual sense. To capture ratios, Stevens had listeners scale loudness directly — halving it, doubling it, or assigning numbers in proportion to what they heard — and defined the sone as the unit of loudness itself (Stevens, 1955). One sone is fixed as the loudness of a 1 kHz tone at 40 dB SPL (equivalently, 40 phons); two sones is twice as loud, half a sone half as loud.

Mapping sones onto phons gives the field's central quantitative law: loudness roughly doubles for every 10-phon increase in loudness level. Above about 40 phons, loudness in sones is well described by S = 2^{(P − 40)/10}, where P is the loudness level in phons. This is a special case of Stevens's power law, which holds that subjective magnitude grows as a power of stimulus intensity (Stevens, 1957). For loudness the exponent is about 0.3 in sound intensity, so a tenfold (10 dB) rise in intensity multiplies loudness by 10^0.3 ≈ 2. The compressive exponent is what lets the ear span the enormous range from a whisper to a jet engine — a trillionfold range of intensity — without the loudness scale running away.

Table 1

Loudness Level in Phons and the Corresponding Loudness in Sones

Note. Each 10-phon step doubles the loudness in sones, the signature of the compressive power law. The relation holds above about 40 phons; below it loudness falls off more steeply toward the threshold of hearing (Stevens, 1955; Stevens, 1957).

Loudness Models and Critical Bands

Scaling experiments describe loudness; loudness models predict it from the physics of a sound. The foundational insight is that the ear does not sum energy uniformly across frequency but within overlapping filters, the critical bands. Zwicker and Scharf's model computes a specific loudness — a loudness density — within each critical band from the excitation the sound produces there, then integrates specific loudness across all the bands to give total loudness (Zwicker & Scharf, 1965). This is why bandwidth matters: spreading a fixed energy across more critical bands recruits more of the summation and yields a louder sound, once the spread crosses a band boundary. Loudness also integrates over time: a brief sound grows louder as its duration lengthens up to roughly 100–200 milliseconds, after which loudness levels off, and the rate of this temporal integration itself depends on level (Florentine, Buus, & Poulsen, 1996).

Marks placed the scaling and the modelling on a common theoretical footing, relating the loudness of simple and complex sounds to the numerical judgments listeners make of them and clarifying how the power law for a single tone generalises to spectrally complex sounds (Marks, 1979). Moore and colleagues then developed the modern Cambridge loudness models, which compute an excitation pattern along the cochlea, transform it to specific loudness, and integrate — reproducing the equal-loudness contours, the sone scale, and the effects of bandwidth and duration from a single mechanism, and crucially extending the computation to impaired ears (Moore, 2014). The most recent versions handle time-varying sounds such as speech and music and incorporate binaural inhibition, so that a sound presented to both ears is not simply twice as loud as to one (Moore et al., 2016).

Loudness Recruitment and Hearing Loss

The clinical importance of loudness models lies in recruitment, the abnormal growth of loudness that accompanies cochlear hearing loss. A listener with cochlear damage has an elevated threshold — faint sounds are inaudible — yet at high levels the same listener perceives loudness almost normally. Loudness therefore grows abnormally fast across the reduced range between the raised threshold and normal loudness: sounds leap from inaudible to uncomfortably loud over a far smaller change in level than in a healthy ear.

Buus and Florentine argued that recruitment is not a separate pathology but the direct consequence of losing the cochlea's active compression (Buus & Florentine, 2002). A healthy cochlea amplifies faint sounds through the active mechanics of the outer hair cells and compresses the response as level rises, which is exactly what produces the compressive (roughly 0.3-exponent) loudness growth of normal hearing. When outer-hair-cell function is lost, that amplification and compression go with it, the threshold rises, and loudness growth above threshold becomes steep and linear — recruitment. This reframing matters for hearing aids: the aim is not simply to amplify but to restore compression, applying more gain to faint sounds than to loud ones so that the full range of input is mapped back onto the listener's reduced range of comfortable loudness (Oetting et al., 2018). The standard synthesis of this physiology, measurement, and modelling is the Loudness monograph of the Springer Handbook of Auditory Research (Florentine et al., 2011).

Worked Example

Consider a 1 kHz tone, so that loudness level in phons equals sound-pressure level in dB. The tone is presented first at 50 phons and then raised by 20 phons to 70 phons. By how much does its perceived loudness increase?

Use the sone relation S = 2^{(P − 40)/10}. At 50 phons the loudness is S = 2^{(50 − 40)/10} = 2¹ = 2 sones. At 70 phons it is S = 2^{(70 − 40)/10} = 2³ = 8 sones. The ratio is 8 / 2 = 4, so the 20-phon increase makes the tone four times as loud, not 40% louder as the decibel figures might suggest.

The same answer follows from the power law. A 20-phon rise is a 20 dB rise, a hundredfold increase in intensity (10^20/10 = 10² = 100). With a loudness exponent of 0.3 on intensity, loudness scales by 100^0.3 = 10^0.6 ≈ 3.98, which rounds to the factor of 4 the sone arithmetic gives directly. Two routes — doubling per 10 phons, and the 0.3 power law — agree because they are two statements of the same compressive transformation. These worked numbers match the Sone Scale demonstration above.

Current Directions

The most active practical frontier is the application of loudness models to hearing devices. Because recruitment is a loss of compression, a well-fitted hearing aid must restore a listener's perceived loudness to normal across the whole range of everyday sounds, and recent work has turned the loudness model into an explicit fitting target: adjust the aid's multi-band compression until the impaired ear's computed loudness matches a normal ear's, rather than merely making sounds audible (Oetting et al., 2018). Extending loudness models to realistic, time-varying and binaural listening — speech in a room, music over headphones — is the companion modelling effort, since everyday loudness is rarely the steady pure tone of the classical experiments (Moore et al., 2016).

A second direction is the international re-standardisation of the equal-loudness contours themselves. The 2023 revision of ISO 226 pooled equal-loudness measurements made since the 2003 edition, tightening the reference curves on which loudness meters, audiometric calibration, and model validation all depend (Suzuki et al., 2024). That a quantity first mapped in 1933 is still being re-measured and re-standardised nearly a century later reflects both how fundamental loudness is to acoustics and how much its precise form depends on the population and method of measurement.

Discussion

Loudness perception is the clearest worked example of the central problem of psychophysics: relating a private sensory magnitude to a physical quantity. The decibel tempts the intuition that loudness is simply level on a logarithmic axis, but a century of measurement shows otherwise. Loudness depends on frequency, so the same level is heard very differently across the spectrum; it depends on bandwidth, so energy spread across critical bands is louder than energy concentrated in one; and it grows compressively with intensity, doubling only for each tenfold rise in power. The phon, the sone, and the critical-band model are the three instruments the field built to pin that transformation down.

The deepest lesson is that the compressive growth of normal loudness is not a quirk of judgment but a signature of cochlear mechanics — the active amplification and compression supplied by the outer hair cells. That is why its loss in cochlear hearing damage produces recruitment, and why the modern aim of amplification is to restore compression rather than merely add gain. Loudness thus sits at the junction of perception, physiology, and clinical practice: a subjective magnitude that a model can now compute from a waveform, diagnose from an audiogram, and partly restore with a well-fitted device.

Common Misconceptions

Loudness is just the decibel level of a sound.

The decibel is a physical ratio of sound pressure; loudness is the subjective magnitude the listener hears. They coincide only for a 1 kHz tone, and diverge with frequency, bandwidth, and level: a 100 Hz and a 1 kHz tone at the same dB SPL are not equally loud (Fletcher & Munson, 1933).

Doubling the sound's power doubles its loudness.

Loudness grows compressively, as roughly the 0.3 power of intensity. Doubling the power (a 3 dB rise) increases loudness by only about 23%; it takes a tenfold increase in intensity, a 10 dB rise, to roughly double the loudness (Stevens, 1957).

A person with hearing loss just needs everything made louder.

Cochlear loss raises the threshold but leaves loudness near-normal at high levels, so loudness growth is abnormally steep (recruitment). Uniform amplification would make loud sounds intolerable; the fix is compression — more gain for faint sounds than for loud ones (Buus & Florentine, 2002; Oetting et al., 2018).

A sound presented to both ears is twice as loud as to one.

Binaural loudness is greater than monaural but not double. Each ear partly inhibits the other's contribution, so modern loudness models incorporate binaural inhibition rather than simply summing the two ears (Moore et al., 2016).

Glossary

Auditory perception.

The perception of sound, the broader faculty within which loudness is the attribute of perceived intensity.

Bandwidth.

The range of frequencies a sound spans; once it exceeds a critical band, spreading the same total energy across a wider bandwidth increases loudness.

Binaural loudness.

The loudness of a sound heard with both ears, which exceeds monaural loudness but by less than a factor of two because of binaural inhibition.

Critical band.

The frequency bandwidth of a cochlear filter within which energy is integrated together; loudness is summed across critical bands, so bandwidth beyond one band increases loudness.

Decibel (dB SPL).

A logarithmic measure of sound-pressure level relative to a fixed reference pressure; a physical quantity, distinct from the perceptual quantity loudness.

Equal-loudness contour.

A curve joining all frequency-and-level combinations that sound equally loud; the set of contours defines the phon and is standardised as ISO 226.

Excitation pattern.

The distribution of neural excitation a sound evokes along the cochlea; modern loudness models transform it to specific loudness and integrate to predict loudness.

Loudness level.

The level in phons of a 1 kHz tone judged equally loud as the sound in question; it orders sounds by loudness but does not express loudness ratios.

Loudness summation.

The growth of loudness as a sound's energy is spread across more than one critical band, modelled by integrating specific loudness across frequency.

Loudness.

The subjective magnitude of a sound, ordering it from quiet to loud; the perceptual correlate of intensity, measured in sones.

Phon.

The unit of loudness level; a sound is at N phons if it is as loud as an N-dB SPL tone at 1 kHz. Equal to the decibel at 1 kHz only.

Power law.

Stevens's rule that subjective magnitude grows as a power of stimulus intensity; for loudness the exponent is about 0.3 in intensity, giving compressive growth.

Recruitment.

The abnormally rapid growth of loudness above an elevated threshold in cochlear hearing loss, caused by the loss of the cochlea's active compression.

Sone.

The unit of loudness itself, fixed so that 1 sone is the loudness of a 40-phon tone; loudness in sones doubles for roughly every 10-phon increase.

Specific loudness.

The loudness density within a single critical band; total loudness is the integral of specific loudness across all bands in the Zwicker and Cambridge models.

Key Researchers

Harvey Fletcher

(1884–1981). American physicist at Bell Telephone Laboratories, often called the father of stereophonic sound. With W. A. Munson he measured the equal-loudness contours and defined the phon and the loudness level, giving acoustics its first quantitative map of how perceived loudness depends jointly on intensity and frequency. See his Wikipedia biography.

Mary Florentine

(living). Matthews Distinguished Professor at Northeastern University, specialising in psychoacoustics and the perception of loudness. Her work reconsidered loudness recruitment as a consequence of lost cochlear compression, and she edited the standard Loudness monograph in the Springer Handbook of Auditory Research. See her Wikipedia biography.

Lawrence E. Marks

(living). Psychophysicist at the John B. Pierce Laboratory and Yale University, known for his theoretical account of loudness and loudness judgments and for research on cross-modal perception and synaesthesia. ORCID 0000-0002-0457-260X.

Brian C. J. Moore

(living). Emeritus Professor of Auditory Perception at the University of Cambridge and architect of the Cambridge loudness models, which compute loudness from a cochlear excitation pattern for both normal and impaired hearing and extend the computation to time-varying and binaural sounds. ORCID 0000-0001-7071-0671.

Stanley Smith Stevens

(1906–1973). American psychophysicist who founded the Harvard Psycho-Acoustic Laboratory, defined the sone scale of loudness, and formulated the psychophysical power law of which loudness is the archetypal case. See his Wikipedia biography.

Eberhard Zwicker

(1924–1990). German acoustician at the Technical University of Munich, central to the theory of critical bands and to the loudness-summation model that integrates specific loudness across frequency to compute total loudness. See his Wikidata record.

Frequently Asked Questions

What is loudness perception?

Loudness perception is how strong or intense a sound seems to a listener — its subjective magnitude, ordered from quiet to loud. It is the perceptual correlate of physical sound intensity, but it is not the same thing: loudness also depends on the sound's frequency, its bandwidth, and its duration, and it grows with intensity according to a compressive power law rather than in step with the decibel.

What is the difference between a phon and a sone?

Both are loudness units, but they answer different questions. The phon is a unit of loudness level: a sound is at N phons if it is as loud as an N-dB tone at 1 kHz, which orders sounds by loudness but says nothing about ratios. The sone is a unit of loudness itself: 2 sones is twice as loud as 1 sone, so sones express how much louder one sound is than another. The two connect through the rule that loudness roughly doubles, adding one doubling of sones, for every 10-phon step.

Why does a tenfold increase in sound intensity only double the loudness?

Because loudness grows compressively, as about the 0.3 power of intensity. A tenfold (10 dB) rise in intensity multiplies loudness by 10^0.3, which is about 2. This compression lets the ear span the roughly trillionfold range of intensities between the faintest audible sound and the threshold of pain without the loudness scale becoming unusably large.

What are equal-loudness contours?

They are curves that join all the combinations of frequency and sound level that sound equally loud. Measured first by Fletcher and Munson and now standardised as ISO 226, they show that the ear is most sensitive in the mid-frequency range and much less sensitive to very low and very high frequencies, especially at low levels — which is why quiet music sounds thin in the bass.

Why does bandwidth affect loudness?

The cochlea integrates energy within frequency filters called critical bands. As long as a sound's energy stays within one critical band, spreading it out does not change its loudness; but once the energy spreads beyond a critical band, loudness increases, because loudness is summed across bands. A wide-band sound is therefore louder than a pure tone carrying the same total energy.

What is loudness recruitment?

Recruitment is the abnormally rapid growth of loudness in people with cochlear hearing loss. Their threshold is raised, so faint sounds are inaudible, but loud sounds are perceived almost normally, so loudness climbs steeply across a compressed range. It is caused by the loss of the cochlea's active compression, and it is why simply turning up the volume can make sounds jump from inaudible to uncomfortable.

How do hearing aids address recruitment?

By compression rather than uniform amplification. Because the impaired ear has lost its own compression, a hearing aid applies more gain to faint sounds than to loud ones, mapping the wide range of real-world sound levels back onto the listener's reduced range of comfortable loudness. Modern fitting methods use loudness models to set the compression so that the impaired ear's computed loudness matches that of a normal ear.

Can loudness be calculated from a sound's waveform?

Yes. Loudness models, beginning with Zwicker and Scharf and developed into the modern Cambridge models, compute the excitation the sound produces along the cochlea, convert it to a specific loudness in each critical band, and integrate across bands to predict total loudness. Current versions handle time-varying sounds such as speech and music and account for the contribution of both ears.

References

Buus, S., & Florentine, M. (2002). Growth of loudness in listeners with cochlear hearing losses: Recruitment reconsidered. Journal of the Association for Research in Otolaryngology, 3(2), 120–139. https://doi.org/10.1007/s101620010084

Fletcher, H., & Munson, W. A. (1933). Loudness, its definition, measurement and calculation. The Journal of the Acoustical Society of America, 5(2), 82–108. https://doi.org/10.1121/1.1915637

Florentine, M., Buus, S., & Poulsen, T. (1996). Temporal integration of loudness as a function of level. The Journal of the Acoustical Society of America, 99(3), 1633–1644. https://doi.org/10.1121/1.415236

Florentine, M., Popper, A. N., & Fay, R. R. (Eds.). (2011). Loudness (Springer Handbook of Auditory Research, Vol. 37). Springer. https://doi.org/10.1007/978-1-4419-6712-1

Marks, L. E. (1979). A theory of loudness and loudness judgments. Psychological Review, 86(3), 256–285. https://doi.org/10.1037/0033-295X.86.3.256

Moore, B. C. J. (2014). Development and current status of the “Cambridge” loudness models. Trends in Hearing, 18, 2331216514550620. https://doi.org/10.1177/2331216514550620

Moore, B. C. J., Glasberg, B. R., Varathanathan, A., & Schlittenlacher, J. (2016). A loudness model for time-varying sounds incorporating binaural inhibition. Trends in Hearing, 20, 2331216516682698. https://doi.org/10.1177/2331216516682698

Oetting, D., Hohmann, V., Appell, J.-E., Kollmeier, B., & Ewert, S. D. (2018). Restoring perceived loudness for listeners with hearing loss. Ear and Hearing, 39(4), 664–678. https://doi.org/10.1097/AUD.0000000000000521

Stevens, S. S. (1955). The measurement of loudness. The Journal of the Acoustical Society of America, 27(5), 815–829. https://doi.org/10.1121/1.1908048

Stevens, S. S. (1957). On the psychophysical law. Psychological Review, 64(3), 153–181. https://doi.org/10.1037/h0046162

Suzuki, Y., Takeshima, H., & Kurakata, K. (2024). Revision of ISO 226 “Normal equal-loudness-level contours” from 2003 to 2023 edition: The background and results. Acoustical Science and Technology, 45(1), 1–8. https://doi.org/10.1250/ast.e23.66

Zwicker, E., & Scharf, B. (1965). A model of loudness summation. Psychological Review, 72(1), 3–26. https://doi.org/10.1037/h0021703