Loudness does not scale linearly with sound pressure, so volume controls should step by loudness, not pressure

What it is

When physical sound pressure level goes up by a fixed amount (say, 10 dB), the subjective sense of loudness does not grow by a matching proportional amount. The decibel scale is already a logarithmic transform of sound pressure, and loudness perception stacks a further layer of compression on top of decibels. This means "sound pressure level" and "loudness" are two distinct quantities — one a physical measurement, the other a subjective percept — related nonlinearly, and neither can substitute for the other.

It is easy to oversimplify this as "decibels are already a log scale, so they already match the ear." In fact the decibel transform only solves the problem of sound pressure spanning an enormous physical range; loudness perception has its own independent layer of nonlinearity on top of decibels, and that layer is what this entry is about.

Why it happens

Loudness nonlinearity comes from the auditory system's compressive response: the cochlea's basilar-membrane mechanical response to sound intensity is itself compressive, and neural firing rates also saturate as intensity rises. Together these two mechanisms squeeze an enormous physical sound-pressure range into a much narrower range of usable neural response, which is exactly what lets the ear handle both very quiet and very loud sounds within one system. The cost of that compression is that the perceived quantity and the physical quantity no longer correspond in a simple linear or even logarithmic way, but through a relationship somewhere in between, commonly approximated empirically by a power function — loudness grows roughly as sound pressure raised to a power less than one, more slowly than sound pressure itself.

This is also why, when loudness rather than decibels is used as the unit, doubling perceived loudness requires a roughly fixed increase in sound pressure level (on the order of ten-odd decibels), rather than a simple one-to-one correspondence between loudness and decibels.

Studying it

The main methods are magnitude estimation and magnitude production: listeners either directly rate sounds at different sound pressure levels (e.g., "how many times louder is this sound than the reference"), or adjust a sound's level until it reaches a specified multiple of a reference loudness. Fitting a curve to many such sound-pressure-versus-loudness-rating data points yields an empirical growth curve, from which the sone loudness scale is defined (one sone is typically anchored to loudness at a specific reference sound pressure level and frequency).

Typical independent variables: physical sound pressure level, test frequency and bandwidth. Typical dependent variables: subjective loudness ratings or adjustment results.

In interface and product sound design, this method is used to check whether the perceived step size between adjacent settings in a volume control scheme is uniform — simply making the sound pressure level increase evenly does not make the steps feel even.

Where it stops holding

• The loudness-versus-pressure relationship described here is measured at a specific reference frequency and moderate level range; the degree of compression shifts with frequency and absolute level range, so there is no single power exponent that applies universally across all frequencies and all levels.
• Individual loudness growth curves vary; certain types of hearing loss in particular cause loudness to grow abnormally fast once a certain level is crossed (loudness recruitment). Such listeners' loudness-versus-pressure relationship clearly departs from the general population's average curve and should not be applied directly.
• This entry addresses only the relationship between loudness and sound pressure itself. Loudness differences caused by frequency (the same sound pressure level sounding louder or quieter at low versus high frequencies) is a separate, independent mechanism covered elsewhere in this group.

Applying it

• If a volume control (a slider, plus/minus buttons) is graded evenly in sound pressure level or linear amplitude, users will clearly feel that "the first few steps barely change anything, and the last few steps suddenly get very loud" — this is exactly the effect of loudness's compressive relationship to sound pressure, not a flaw in the control itself.
• The correct approach is to make the perceived loudness increment roughly equal between adjacent volume steps, rather than the physical sound pressure increment. In practice this usually means the volume mapping curve itself is not linear — the sound pressure at each step needs to be derived by working backward from the loudness scale, rather than assuming a percentage scale maps directly onto pressure or amplitude.
• How to check: in a blind test, have users subjectively rate the loudness difference between adjacent volume steps (e.g., judging "how much louder is this step than the last one"), confirming whether the subjective differences across steps are roughly uniform — rather than only checking whether the sound pressure level values themselves increase evenly.

Related

• Same group: A3.02.2 Ambient noise sets the usable loudness floor · A3.02.3 Equal-loudness contours curl up at the low and high ends, making bass and treble harder to notice at low volume · A3.02.4 A brief sound's perceived loudness is lower than a sustained sound at the same sound pressure · A3.02.5 A uniform volume number across devices does not guarantee uniform perceived loudness
• Nearby: A3.01 Audible frequency range
• Search terms: loudness perception · sone scale · Stevens power law · magnitude estimation