# Mel

Physical unit
Unit name Mel
Unit symbol ${\ displaystyle \ mathrm {mel}}$
Physical quantity (s) Tonality
Formula symbol ${\ displaystyle Z, z}$
dimension ${\ displaystyle {\ mathsf {1}}}$
Named after english melody

The Mel (from the English word melody ) is the unit of measurement for the psychoacoustic size tonality with the symbols  Z (or  z ) and describes the perceived pitch of pure tones , so the pitch perception. The Mel scale was proposed by Stanley Smith Stevens , John Volkman, and Edwin Newmann in 1937 .

## Definitions

In general, the following applies to the pitch: a tone that is perceived twice as high receives twice the tone value, a tone that is perceived as half as high receives half the tone value. The tone scale can be determined with the help of psychoacoustic experiments .

There are two definitions for the Mel scale, which differ in the reference value :

• The basis of the original definition according to Stanley Smith Stevens is the tone with the frequency  f = 1000  Hertz , it is assigned the tone class  Z = 1000 mel.
This Mel scale can be roughly described with the following formula:
{\ displaystyle {\ begin {aligned} Z & = 2595 \ cdot \ log _ {10} (1 + f / 700) \\\ Leftrightarrow Z & = 1127 \ cdot \ ln (1 + f / 700) \\\ Leftrightarrow f & = 700 \ cdot \ left [\ exp \ left ({\ frac {Z} {1127}} \ right) -1 \ right] \ end {aligned}}}
• Eberhard Zwicker later defined a Mel scale based on the Bark scale with the musical note  c as the base. The pitch Z = 131 mel is assigned to this tone with the frequency  f = 131 Hertz  . Later it was changed again to 125 Hz. The rest of the article refers to this definition.

## Relation to the frequency

Relationship between perceived pitch (pitch in mel) and frequency

The following applies to the relationship between pitch and frequency:

• For frequencies up to approx. 500 Hz, the frequency scale and the Mel scale are almost proportional . A doubling of the frequency from 100 Hz to 200 Hz leads to a doubling of the pitch from 100 mel to 200 mel. I.e. a musical interval of one octave corresponds to a doubling of the perceived pitch.
• For frequencies greater than 500 Hz, there is a non-linear relationship between frequency and pitch (1000 Hz = 850 mel, 8000 Hz = 2100 mel). So z. B. the frequency of a tone can be increased from 1500 Hz to 10,000 Hz to achieve a doubling of the pitch from 1100 mel to 2200 mel; a musical interval of more than 2.5 octaves is required here for a doubling of the perceived pitch. I.e. In this frequency range, sound intervals are perceived smaller than they are musically seen.
• The perceived pitch of complex tones differs from that of the purely harmonic sine tones described so far . Up to a frequency of 5 kHz it is - with minor deviations - proportional to the logarithm of the frequency. In this range, the “just noticeable difference” (JND, just noticable difference) is approximately constant for complex tones.

## Auditory mechanisms for determining the tone level

Relationship between location on the basilar membrane, tonality in mel and frequency of a tone

The hearing uses different mechanisms to perceive pitch:

• At frequencies below 500 ... 800 Hz, the time structure of the ear signals is evaluated and used for pitch perception. Here the pitch perception follows the musical pitch very precisely.
• At frequencies above 1600 Hz, the hearing is no longer able to follow the time structure of the ear signals. Here, the pitch perception is derived from the position of the maximum excitation on the basilar membrane of the inner ear by assuming a linear relationship between the pitch and the location of the maximum oscillation amplitude of the basilar membrane: the same distances on the basilar membrane correspond to the same pitch differences.
• at frequencies between about 800 and 1600 Hz, the two mechanisms described overlap.

## Levels of sensation

Over the entire audible range from 16 Hz to 19,000 Hz, 620 levels of tone can be distinguished with a constant width of 3.9 mel. The listening area thus covers 2400 mel.

Another measure of the pitch is the bark: 1  bark = 100 mel 1 mel = 0.01 bark. It is the basis of the Bark scale for critical bands ( frequency groups ). Another frequency group scale is the ERB scale . ${\ displaystyle \ Leftrightarrow}$

## literature

• Ernst Terhardt: On the pitch perception of sounds I, psychoacoustic basics ; Acustica 26, (1972). Pp. 173-186
• Stanley Smith Stevens, John Volkman, Edwin Newman: A scale for the measurement of the psychological magnitude of pitch . In: Acoustical Society of America (Ed.): The Journal of the Acoustical Society of America . 8, No. 3, 1937, pp. 185-90.