Mel scale

The mel scale, named by Stevens, Volkmann, and Newman in 1937, is a perceptual scale of pitches judged by listeners to be equal in distance from one another. The reference point between this scale and normal frequency measurement is defined by assigning a perceptual pitch of 1000 mels to a 1000 Hz tone, 40 dB above the listener's threshold. Above about 500 Hz, increasingly large intervals are judged by listeners to produce equal pitch increments. As a result, four octaves on the hertz scale above 500 Hz are judged to comprise about two octaves on the mel scale. The name mel comes from the word melody to indicate that the scale is based on pitch comparisons.

A popular formula to convert f hertz into m mels is:

There is no single mel-scale formula. The popular formula from O'Shaugnessy's book can be expressed with different logarithmic bases:

The corresponding inverse expressions are:

There were published curves and tables on psychophysical pitch scales since Steinberg's 1937 curves based on just-noticeable differences of pitch. More curves soon followed in Fletcher and Munson's 1937 and Fletcher's 1938 and Stevens' 1937 and Stevens and Volkmann's 1940 papers using a variety of experimental methods and analysis approaches.

In 1949 Koenig published an approximation based on separate linear and logarithmic segments, with a break at 1000 Hz.

Gunnar Fant proposed the current popular linear/logarithmic formula in 1949, but with the 1000 Hz corner frequency.

An alternate expression of the formula, not depending on choice of logarithm base, is noted in Fant (1968):

In 1976, Makhoul and Cosell published the now-popular version with the 700 Hz corner frequency. As Ganchev et al. have observed, "The formulae [with 700], when compared to [Fant's with 1000], provide a closer approximation of the Mel scale for frequencies below 1000 Hz, at the price of higher inaccuracy for frequencies higher than 1000 Hz." Above 7 kHz, however, the situation is reversed, and the 700 Hz version again fits better.

Data by which some of these formulas are motivated are tabulated in Beranek (1949), as measured from the curves of Stevens and Volkmann:

A formula with a break frequency of 625 Hz is given by Lindsay & Norman (1977); the formula doesn't appear in their 1972 first edition:

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