Sound Level

The amount of noise is often measured in dB(A), but this turns out to be a not-so-precise unit of measurement. The amount of noise highly depends not only on the amount of energy (the sound pressure), but also on the frequency and the harmony of the sound.

Psychoacoustics
To start with the harmony: A Beethoven concert at 60 dB SPA may sound a lot better than yours truly singing in the shower at the same sound level. Or a heavily distorted recording of the Beethoven concert playing on a cheap speaker for that matter. This applies to all sorts of sounds, and is out of scope of this document.

Sound Pressure Level
Sound Pressure is the amount of pressure that sounds generates. A physicist may refer to it as the amount of energy per volume. Pressure levels are often measured in micropascal (µPa).

20 µPa (only 0.00002 Pascal, or 0.0000000002 bar) is often regarded as the lowest sound pressure that a human ear can detect. The human roughly detects loudness not linear, but logarithmic. So for a human, the difference between 100 µPa and 1000 µPa is about the same as the difference between 1000 µPa and 10000 µPa.

Sound Pressure Level (SPL) is the logarithmic scale of the root mean square of the sound pressure, and is often expressed in decibel (dB). Since a logarithmic scale only describes the ratio between two values, it is required to agree on an arbitrary base value. For sound pressure level, the base of 0 dB SPL is equal to 20 µPa, the threshold for human hearing. Here is a list of relation between sound pressure level and sound pressure:

Sound Power (or Accoustic Power) is the energy that is required to create a given sound pressure. The power is equal to the square root of the sound pressure times the acoustic impedance (about 420 Pa·s/m for air), divided by the area of sound pressure (P = p²∙Z/A). Without going in the details, this is the reason that the Sound Pressure Level is related to the log of the square root of the sound pressure, rather than simply the log of the sound pressure.

Loudness
The human ear is only receptacle for frequencies between roughly 20 Hertz and 20 kiloHertz, and is most sensitive for sounds around 3000 Hertz. So a sound at 3000 Hertz is perceived as louder than a sound with the same sound pressure level at 100 Hertz or at 10.000 Hertz.

The relation between the perceived loudness level and frequency is usually described in Equal-loudness contours.

The perceived Loudness Level is measured in phons. A sound with a loudness level L has the same perceived loudness level by the human ear as an 1 kHz tone with the same sound pressure level in dB SPL. Thus to a human ear, a given sound of 40 phon has the same loudness as a 1 kHz tone generator at 40 dB SPL.

Equal-loudness contours
No human ear is the same, and the ability to hear higher frequencies degenerates with age. Hence, the equal-loudness contours is different for each person. Multiple Equal-loudness contours have been measured and published.


 * Fletcher-Munson (1933):Early (though not the first) and often quoted research on this work, created by averaging the result for 11 people.
 * Robinson-Dadson (1956):A repeat of the original research for 30-90 people.
 * ISO/R 226:1961:First international standard, based on data by Robinson-Dadson.
 * ISO 226:1987:Revision of ISO R226:1961 with the same data.
 * ISO 226:2003:Complete revision of ISO 226:1987 based on much more research. It turned out that the Robinson-Dadson research was flawed, especially in the low frequencies.
 * dB(A):Mathematically simple contour comparable to an equal-loudness contour, often used to translate between loudness and sound levels. While this a moderately good approximation for frequencies below 1 kHz at 60 phon, it overestimates sounds around 10kHz with about 10 dB. It is considered suitable only for sounds in the 0-60 phon range, and is often used in official regulations. (The fact that it is only suitable for sounds in the 0-60 phon range has not stopped anyone from applying dB(A) sound contours to measure sounds levels of airplanes, which well exceed 60 phon).
 * dB(B):Simple contour developed for loudness corrections in the 55-85 phon range. It is not really used anymore.
 * dB(C):Simple contour developed for loudness corrections for loud noise, and is similar to the 100 phon contour in the low frequencies. It is frequently used in official regulations.
 * dB(D):Equal-loudness contour comparable to dB(C), but with an extra peak around 6 kHz to take into account that the human ear is sensitive at those values. It's intended use was to correct for (military) aircraft noise. It is little used anymore in favour of dB(C).
 * ITU-R BS.468-4:Standard developed by the BBC, with a sharper cut-off at high frequencies than dB(A), and according to some better suited than dB(A) for that reason. I am not convinced, and considered it roughly as good as dB(A), and rather see ISO 226:2003 if accurate measurements are required.
 * dB(Z) (unweighed):Also known as dB(Zero) for unweighed (uncorrected) sound pressure levels.

This is a visual comparison of the equal-loudness curves published by Fletcher-Munson, Robinson-Dadson and ISO 226:2203. The file was based on work by Peter Skirrow, published at Wikipedia:

One thing you will notice is that in the ISO 226 standard, the equal-loudness contour differs for the loudness of the sound. It lists the contours for 20, 40, 60, 80 and 100 phon. In other words, if you are playing music and turn the volume up or down, the music changes: at higher volume, the bass will sound louder compared to the treble.

This is a visual comparison of the acoustic weighing curves for dB(A), dB(B), dB(C), dB(D), ITU-R BS.468 and ISO 226:2003:

The reports associated with the 2003 revision of ISO 226 provide a good overview of the different equal-loudness contours:
 * Full Revision of International Standards for Equal-Loudness Level Contours (ISO 226)
 * Precise and Full-range Determination of Two-dimensional Equal Loudness Contours by Yôiti Suzuki, et al.
 * Equal-loudness-level contours for pure tones by Yôiti Suzuki and Hisashi Takeshima (doi:10.1121/1.1763601)

For those interested in formulas, is an Excel spreadsheet with data parameters and formulas for for equal-loudness contours, including Fletcher-Munson, ISO 226:2203, ITU-R BS.468, and IEC 61672 (dB(A), dB(B), dB(C), dB(D)). Unfortunately, I could not find a public source with data for Robinson-Dadson/ISO 226:1987/ISO 226:1961.

Limitations of Equal-Loudness contours
While a corrected loudness level (dB(A) or ISO 226:2003) is certainly a better quantity than the simple sound level in dB (SPL), it by no means is perfect. It does not take into account the following effects:


 * Duration of sound. Sounds shorter than 200 ms are perceived as softer.
 * If the sound is measured directly (free field) or mostly through reflections (diffuse field). The diffuse field equal-loudness contours are a few dB flatter for high frequencies than the (often used) free field equal-loudness contours.
 * the above equal-loudness contours are aimed at an average human ear. The fluctuation between humans is easily 5 dB, and is larger for some humans. Other mammals, such as cats and dogs have a different frequency reach (cats as wide as 45 Hz to 64 kHz), and may be affected by noised in the infrasound and ultrasound regions.
 * if only one ear is used, the minimum audible field (the hearing treshold) is about 10 dB higher than listed above.

Further reading: by Angélique A. Scharine, Kara D. Cave and Tomasz R. Letowski
 * Auditory Perception and Cognitive Performance

Comfortable Loudness Levels
The following table lists a few comfortable loudness levels

In musical terms, the levels are:

Sone
Sone is a measure of Loudness, just as Pone is a measure of the Loudness Level. Unfortunately, the relation between sone and phon is not as simple as that between Sound Pressure and Sounds Pressure Level (which is just a simple logarithmic relation).

1 sone is the loudness at 40 phon. Experimental evidence has shown that to a human ear, the loudness doubles every 10 phon for sounds above 40 phon. So 2 sone corresponds to 50 phon, 4 sone corresponds to 60 phon. Using the same scale, you would expect that 0.5 sone corresponds to 30 phon, but in reality this is 31 phon. For loudness levels lower than 40 phon, the relation is not exponential, but a power of 2.86.

Since sones are an even less precise unit of measurement than a phon (since it is related to the subjective hearing of a human ear), I recommend that if you ever encounter a measurement in sone, you should first convert it to phons, and secondly question the source why it used such subjective units of measurements.

The relations to convert sones to phons are: