# 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 are perceived at two different levels ? A-weighting was created to consider the frequency dependance of the human hear perception of levels. But human hearing has way more subtleties. That’s why the psychoacoustics metric called Loudness have been introduced to consider the complexity human perception of noise level.

### 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 the ISO 532-1 standard. The result depends on sound duration, frequency and band-width, and integrates a very detailed transfer function of the human hearing system proposed by Eberhard Zwicker.

### Calculation principle

The computation of the Loudness requires several steps, each corresponding to the simulation of an aspect of the hearing system.

1. The time history of the signal is first conditioned for the computation. It is divided into elementary time segments (typically 200 ms). The third octave band spectrum of each segment is then computed.

2. A transfer function is applied to each spectrum to produce the core loudness for each time step. This transfer function considers the human ear sensitivity on frequency. Summarily, it simulates the curves of equal loudness, originally proposed by Fletcher and Munson in 1933. According to those curves, 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.

3. The non-linear temporal decay of the hearing system is then simulated.

4. Lastly, the frequency masking and time masking effect are simulated. The total Loudness and the specific Loudness over critical band rate (in Bark) or frequency (in Hz) are returned

### Illustration

The figure below shows the fine band spectrum of an electric motor noise at constant speed. The corresponding audio file is available on the right. The noise is composed of many harmonics. At lower frequency, the harmonics are linked to the slotting effect (H12 hamronic around 500 Hz, H24 harmonic around 1 kHz and H36 around 1.5 kHz). At higher frequency, the high order slotting harmonics are mixed with PWM harmonics. Although the H12 harmonic have a slightly higher level in dBA, it is difficult to clearly identified the most annoying harmonic from the spectrum.

On the other hand, the plot of the specific loudness over critical band shown below allows to clearly identify the H12 harmonics (in the critical band 5) as the most annoying in term of perceived noise level.