The loudness of a sound is a perceptual measure of the effect of the energy content of sound on the ear. It is related to the decibel (dB) which is an logarithmic scale used to quantify the power of a sound. Doubling the sound power of a sound does not lead to a doubling on the decibel scale but instead to an increase of 3dB. Perceived loudness, however, is also dependent on the frequency content of a sound so, for example a very low frequency sound such as a 20Hz tone at 40dB would be perceived to be quieter to a normal hearing person than a 1kHz tone at 40dB. This is explored in more detail below. Some examples of the use of loudness would include; in examining sound quality in the car industry, measuring car interior noise, engine noise, exhaust noise etc. To examine earth-moving machinery noise emissions. It has also been used in research to quantify the sound quality of some domestic appliances, vacuum cleaners, or refrigerator noise for example, and in the calculation of an unbiased annoyance metric.

A definition of the loudness of tones can be constructed from the results of experiments such as loudness ‘magnitude estimation’. The ‘loudness level’ of a sound (introduced in the twenties by Barkhausen) is defined as 'the sound pressure level of a 1 kHz tone in a plane wave and frontal incident that is as loud as the sound; its unit is “phon”.’ (Zwicker & Fastl 1990)

Fig 1. Sketch showing the basic shape of the equal loudness contours

So a sound that is as loud as a 1kHz tone with a sound pressure level of 40dB (for example) is said to have a loudness level of 40 phon. This principle can be used to define the loudness of tones by comparing them with an equivalently loud 1kHz tone.

BS ISO 226 (published 2003) makes use of this idea of a loudness level and constructs 'equal loudness contours' using data from twelve references including work done in Denmark, Germany, and Japan published from 1983 to 2002. (This standard is a revision of the 1987 version which was based largely on the work of Robinson and Dadson (1956).) A sketch of the basic shapes of these contours can be found in fig 1. It shows how the low frequency tones have to be have more sound power to sound as loud as the 1kHz tone.

However, this is not the end of the story, calculation of the loudness of more complex sounds requires further thought as the 'critical band-width' comes into play. Critical band width is a measure of the frequency resolution of the ear. For example two tones less than one critical bandwidth apart will not be heard as two separate sounds instead the sounds will partially mask each other, making loudness summation a more complex process. Third octave bands can be used as an approximation to critical bands and BS 4198 (published 1967 as an equivalent to ISO 532/R) provides a graphical method (Method B) for the calculation of the loudness of complex sounds from third octave bands.

So while it is more usual in acoustics to see the "loudness" of a signal expressed in dB(A), a better measure of the perceived loudness can be found by proper application of the critical bandwidths

A specific loudness can be calculated from the dB level for each third octave band using the assumption that

' a relative change in loudness is proportional to a relative change in intensity.' (Zwicker and Fastl 1990)

So values of specific loudness (N') in sone per Bark can be calculated using a power law. Masking curves can then be constructed around these levels representing the effect of critical bands. The final value for loudness (N) is then calculated as the integral (i.e. the area) under the curve and is presented in sones.

\[N = \int_0^{24Bark} N' dz\]

Where z is the critical band rate (measured in Bark). For ease of use, however, a computer programme implementing this model has also been produced. The structure of Zwicker's 1984 programme to calculate loudness in this way is presented here. The loudness of a 93.85 dB 1000Hz tone is shown in fig 2. However, the validity of this method has been questioned for impulsive sounds (i.e. <200ms duration) [4] because of dependence of subjective loudness on sound duration for short bursts of sound. A method for loudness calculation using octave bands (Method A) is also presented in BS 4198 (1967) but this calculation relies on the assumption that the sound in question has an approximately continuous spectrum.

Fig 2. The specific loudness of a 93.85dB, 1000Hz tone.

However, method B also has limitations. Moore, Glasberg and Baer (1997), for example, are critical of this method on a number of levels and have produced a model of loudness calculation of their own. The BS 4198 (1967) calculation relies on the equal loudness contours presented in an earlier version of BS ISO 226, subsequently BS ISO 226 has been updated (to the 2003 version) whereas BS 4198 (1967) has not. Another criticism comes from work cited by Fastl in 'Psychoacoustics: Facts and Models' (Zwicker & Fastl 1990). In this work the measurement of thresholds of tone complexes used to determine the size of the critical band width suggests that thresholds for complex sounds are lower than for pure tones. BS ISO 226 (2003) acknowledges that equal loudness contours can also be determined for bands of noise but that in the absence of sufficient data for a rigorous definition BS ISO 226 (2003) could be applicable to one-third-octave-bands of noise.

To overcome the threshold problem Moore, Glasberg and Baer (1997) propose using a non-zero value for loudness at the threshold. They also suggest that using BS 4198 (1967) to calculate of binaural loudness does not adequately take into account factors such as the relationship between monaural and binaural loudness, or that binaural thresholds are lower than monaural thresholds.

[1] BS ISO 226, 'Acoustics - Normal Equal Loudness Level Contours', (2003)

[2] BS 4198, 'Method for Calculating Loudness', (1967)

[3] ISO 532/R, 'Acoustics- Method for Calculating Loudness Level' (1975)

[4] Blommer M., Otto N., Wakefield G., Feng B. J., Jones C., ‘Calculating the Loudness of Impulsive Sounds’, SAE Transactions 104/6, Pt2, pp. 2302-2308, (1996)

[5] Fastl H, ‘Subjective duration and Temporal masking patterns of broadband noise impulses’, Journal of the Acoustical Society of America, Vol 61, No 1, P162-168 (1977)

[6] Gabriel B, Kollmeier B, Meller V ‘Influence of various measurement procedures on the equal –loudness level contours’ , Contributions to Psychological Acoustics, P223-231 (1993)

[7] Hellman R, Broner N, ‘Assessment of the Loudness and Annoyance of Low-Frequency Noise from a Psychoacoustical Perspective’, Internoise (1999)

[8] Kachur M, ‘A survey of Sound Quality jury evaluation correlations: Loudness vs. A-weighted Sound Level'’, Sound Quality Symposium (1998)

[9] Launer S, Hohmann V, Kollmeier B, ‘Categorical Loudness Summation – Experiments and Models’ , Contributions to Psychological Acoustics P233-249 (1993)

[10] Moore B C J, Glasberg B R, Baer T, ‘A Model for the Prediction of Thresholds, Loudness and Partial Loudness', Journal of the Audio Engineering Society P224- 240, Vol 45, No 4 (1997)

[11] Moore B C J, Glasberg B R, ‘A Revision of Zwicker’s Loudness Model'’, Acustica, Vol 82, P333-345 (1996)

[12] Moore B C J, Glasberg B J ‘Suggested formulae for calculating auditory-filter bandwidths and excitation patterns’, Journal of the Acoustical Society of America, Vol 74(3), P750-753 (1983)

[13] Robinson D W, Dadson M A, 'A redetermination of the equal-loudness relations for pure tones' British Journal of Applied Physics, Vol 7, P166-181, (1956)

[14] Tuami O, Zacharov N, ‘A real-time binaural loudness Meter’ (139th Meeting of the ASA)

[15] Verhey J L, Kollmeier B, ‘Spectral Loudness Summation as a function of duration’, Journal of the Acoustical Society of America, Vol 111(3), P1349-1358, (2002)

[16] Zwicker E et al, ‘Program for calculating loudness according to DIN 45631 (ISO 532B)’, Journal of the Acoustical Society of Japan Vol 12, part 1 (P39-42) (1991)

[17] Zwicker E, Fastl H, 'Psychoacoustics: Facts and Models’ (Springer 1990)

[18] Zwicker E et al, 'BASIC-Programme for calculating the loudness of sounds from their 1/3-oct band spectra according to ISO 532B', Acustica Vol 55, (Letters to the editor) P63 (1984)

[19] Zwicker E, ‘Dependence of post-masking on masker duration and its relation to temporal effects in loudness’, Vol 75, No 1, P219-223 (1984)

[20] Zwicker E ‘Procedure for calculating loudness of temporally variable sounds’ , Journal of the Acoustical Society of America, Vol 62, No.3, P675-682 (1977)