What is loudness?

Context

There are differences between sound parameters we measure physically and the sound perception we have as humans. For instance, we are used to describe the loudness of a sound using the Decibel scale, but how to explain that two sounds with the same dB level but at two different frequencies can be perceived at two different levels ? To answer that question was first created the A-weighting, but it is not very convenient to use precisely.

The dB values do not accurately reflect the human perception of loudness because they just consider the physical properties of a sound and do not take into account human hearing subtleties. The science aiming to understand the human perception of sound and improve sound quality is called psychoacoustics.

Definition

Loudness is a psychoacoustic metric which provides a numerical measure of a sound volume based on human perception.
Its unit is “sone”, a 1 sone loudness corresponding to a level of 40 dB for a 1 kHz tone by definition. It provides a visualization on a linear scale : double the volume, double the loudness.

Its calculation is described in DIN 45631 and ISO 532 B, first published in 1975. Both are depending on sound duration, frequency and band-width. The latter integrates a very detailed transfer function which takes into account the masking effect of the human hearing system proposed by Eberhard Zwicker.

Calculation principle

Loudness can not be considered as an independent sum of spectral components loudnesses. In fact, in a complex sounds, two separated components may influence each other, especially if their frequency separation is small. Consequently the frequency selectivity of our hearing system, modeled by the critical bands, plays an important role in loudness calculation.

This leads to the notion of critical-band intensities. But, if we leave the strict framework of critical bands to consider the actual slope produced in our hearing system, we have an intermediate value called excitation. Overall loudness is equivalent to the area under the curve of whatever excitation regions are active, which biologically correspond to the excited area on the basilar membrane in the inner ear. Mathematically, it is the integral of specific loudnesses over critical-band rate:

N = ∫ N’ dz

The influence of level on the excitation slope is shown on the figure below from Zwicker and Fastl in [1]. The excitation slopes widen towards higher frequency with the level increasing. This procedure thus takes into account the human ear sensitivity on frequency.

Excitation slopes for kHz critical-band wide noise at different dB levels

Excitation level versus frequency bark scale for a 1 kHz critical-band wide sound at different dB levels.

Equal loudness curves

To match the human perception, this metric has been developed with human juries. Two sounds would be played :

  • 1kHz tone at a particular dB level
  • a different frequency tone

The jury would adjust the second sound level until both seem to be equally loud. By implementing this methods with numerous frequencies and levels the following curves of equal loudness have been drawn, originally by Fletcher and Munson in 1933.

Threshold in quiet

According to the tone frequency, the dB level evolves to match the 1 kHz perceived loudness. For example, a 100 Hz tone has to be played at 60 dB to be perceived like a 1 kHz tone played at 50 dB.

Illustration

See also

Source

[1] H. Fastl, E. Zwicker: Psychoacoustics. Springer, Berlin, 1990.

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