11.2 Loudness perception and equal-loudness contours

Loudness Perception Fundamentals

Loudness perception sits at the intersection of physics and psychology. The physical intensity of a sound wave is one thing; how loud it sounds to you is something else entirely. Your brain doesn't process volume the way a microphone does, and understanding that gap is central to psychoacoustics.

This matters for practical work in audio engineering, hearing protection design, noise regulation, and hearing aid fitting. The core question is: how do we connect objective measurements of sound to the subjective experience of "how loud is that?"

Loudness and Sound Pressure Level

Loudness is the subjective perception of how intense a sound seems. It's a psychological experience, not a physical measurement, and it varies from person to person.

Sound pressure level (SPL), on the other hand, is an objective measurement of a sound wave's pressure amplitude, expressed in decibels (dB). The decibel scale is logarithmic, which compresses the enormous range of pressures the ear can detect (from the threshold of hearing to the threshold of pain) into a more workable range of roughly 0–130 dB.

The relationship between SPL and perceived loudness is not linear. A rough but useful rule: an increase of about 10 dB in SPL corresponds to a doubling of perceived loudness. So a 70 dB sound doesn't seem "a little louder" than a 60 dB sound; it seems about twice as loud.

Frequency dependence adds another layer of complexity. Your ear is not equally sensitive at all frequencies:

• The ear is most sensitive between about 2–5 kHz, a range that overlaps heavily with speech frequencies. This is no coincidence; evolutionary pressure favored sensitivity in the range most useful for communication.
• At low frequencies (below ~500 Hz) and high frequencies (above ~8 kHz), sensitivity drops off. A 50 Hz tone needs significantly more SPL to sound as loud as a 3 kHz tone at the same perceived volume.

Equal-Loudness Contours and Hearing

Equal-loudness contours are curves plotted on a graph of frequency (x-axis) vs. SPL (y-axis). Each curve connects all the frequency-SPL combinations that a listener perceives as equally loud. They give you a map of how human hearing sensitivity changes across the frequency spectrum.

The unit used to label these contours is the phon. A phon value is defined by the SPL in decibels of an equally loud 1 kHz tone. For example, if a 100 Hz tone at 60 dB SPL sounds just as loud as a 1 kHz tone at 40 dB SPL, that 100 Hz tone has a loudness level of 40 phons.

A Brief History

• Fletcher and Munson (1933) published the original equal-loudness contours, which became the foundation of psychoacoustic research on loudness.
• Robinson and Dadson (1956) revised these curves with improved methodology.
• The current international standard is ISO 226:2003, which reflects more recent data and corrects some inaccuracies in the earlier curves.

What the Contours Tell You

The shape of these curves reveals several things about human hearing:

• The curves dip lowest (meaning the ear needs the least SPL) around 3–4 kHz. This is where your hearing is sharpest.
• At low frequencies, the curves rise steeply. You need much more SPL at 50 Hz than at 3 kHz to perceive the same loudness. This is why bass-heavy music can "disappear" at low playback volumes.
• At higher loudness levels, the contours flatten out. The difference in sensitivity across frequencies becomes less dramatic as overall level increases. In other words, at high volumes, your hearing response becomes more uniform.

Practical Applications

• Audio equipment design: Speaker and headphone frequency responses are shaped with these contours in mind, ensuring that reproduced sound matches perceived loudness expectations.
• A-weighting in noise measurement: The A-weighting filter applied to sound level meters approximates the inverse of the equal-loudness contour at low levels, giving readings that better reflect how loud a noise seems.
• Noise regulations and hearing protection: Standards for workplace noise exposure and hearing protector ratings account for frequency-dependent loudness perception.

Loudness Scaling in Psychoacoustics

Equal-loudness contours tell you when two sounds seem equally loud, but they don't tell you how much louder one sound is than another. For that, you need a loudness scale.

The Sone Scale

The sone is a unit of perceived loudness designed to be proportional to the listener's experience.

• 1 sone is defined as the loudness of a 1 kHz tone at 40 dB SPL (which is also 40 phons).
• 2 sones means "twice as loud as 1 sone." 4 sones means "four times as loud."
• Because of the ~10 dB doubling rule, each 10 dB increase roughly doubles the sone value.

This linear relationship to perception is what makes the sone scale useful. If you're comparing the loudness of two vacuum cleaners rated at 3 sones and 6 sones, the second one genuinely sounds about twice as loud.

Stevens' Power Law

The sone scale is grounded in Stevens' Power Law, which models the general relationship between a physical stimulus and its perceived magnitude:

$L = k \cdot I^{n}$

• $L$ = perceived loudness
• $I$ = sound intensity
• $k$ = a constant
• $n$ = the power exponent (approximately 0.3 for loudness)

The exponent of 0.3 means perceived loudness grows much more slowly than physical intensity. A tenfold increase in intensity produces only roughly a doubling in perceived loudness.

Applications

• Hearing aid fitting: Loudness scaling helps audiologists map amplification to a patient's specific perception, rather than just boosting dB levels uniformly.
• Product sound quality: Manufacturers of cars, appliances, and electronics use sone ratings and loudness models to evaluate and improve how their products sound.
• Audio compression algorithms: Perceptual coding (like MP3 and AAC) uses psychoacoustic models, including loudness scaling, to discard audio data that listeners are unlikely to notice.

Loudness vs. Sound Intensity

It's worth being precise about the distinction between these two quantities, since they're easy to conflate.

Sound intensity is a purely physical measure: the amount of sound energy flowing through a unit area per unit time, measured in watts per square meter ($\text{W/m}^2$). It's proportional to the square of sound pressure. A microphone can measure it; it doesn't change depending on who's listening.

Loudness is perceptual. Two people exposed to the same sound intensity may report different loudness, and the same person will judge the same intensity differently depending on frequency, duration, and context.

Weber-Fechner Law and the JND

The Weber-Fechner law states that perceived sensation is proportional to the logarithm of stimulus intensity. This is the theoretical basis for why we use the decibel (a logarithmic unit) to describe sound levels: it roughly tracks how we actually experience changes in intensity.

The just noticeable difference (JND) is the smallest change in a stimulus that a listener can reliably detect. For sound intensity, the JND is typically around 1 dB for moderate-level broadband sounds, but it varies with:

• Frequency: Sensitivity to changes is better in the 1–4 kHz range.
• Overall level: At very low or very high SPLs, the JND tends to increase (you need a bigger change to notice it).

Other Factors That Affect Loudness Perception

Loudness isn't determined by intensity and frequency alone. Several other factors play a role:

• Duration: Very short sounds (under ~200 ms) are perceived as quieter than longer sounds of the same intensity. This is called temporal integration; the auditory system effectively sums energy over a brief window.
• Spectral content: A broadband sound (spread across many frequencies) generally sounds louder than a pure tone at the same SPL, because it excites a wider region of the basilar membrane. This effect is called loudness summation.
• Temporal patterns: Amplitude-modulated or fluctuating sounds can be perceived differently in loudness compared to steady-state sounds at the same average level.
• Spatial factors: Sound arriving from multiple directions or sources can affect perceived loudness due to binaural summation and localization cues.