The mel scale (after the word melody) 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.
Formula A formula (O'Shaughnessy 1987) to convert f hertz into m mels is
m = 2595 log 10 ( 1 + f 700 ) . {\displaystyle m=2595\log _{10}\left(1+{\frac {f}{700}}\right).}
History and other formulas The formula from O'Shaughnessy's book can be expressed with different logarithmic bases:
m = 2595 log 10 ( 1 + f 700 ) = 1127 ln ( 1 + f 700 ) . {\displaystyle m=2595\log _{10}\left(1+{\frac {f}{700}}\right)=1127\ln \left(1+{\frac {f}{700}}\right).}
The corresponding inverse expressions are
f = 700 ( 10 m 2595 − 1 ) = 700 ( e m 1127 − 1 ) . {\displaystyle f=700\left(10^{\frac {m}{2595}}-1\right)=700\left(e^{\frac {m}{1127}}-1\right).}
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):
m = 1000 log 2 log ( 1 + f 1000 ) . {\displaystyle m={\frac {1000}{\log 2}}\log \left(1+{\frac {f}{1000}}\right).}
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 does not appear in their 1972 first edition:
m = 2410 log 10 ( 0.0016 f + 1 ) . {\displaystyle m=2410\log _{10}(0.0016f+1).}
For direct comparison with other formulae, this is equivalent to
m = 2410 log 10 ( 1 + f 625 ) . {\displaystyle m=2410\log _{10}\left(1+{\frac {f}{625}}\right).}
Most mel-scale formulas give exactly 1000 mels at 1000 Hz. The break frequency (e.g. 700 Hz, 1000 Hz, or 625 Hz) is the only free parameter in the usual form of the formula. Some non-mel auditory-frequency-scale formulas use the same form but with much lower break frequency, not necessarily mapping to 1000 at 1000 Hz; for example the ERB-rate scale of Glasberg and Moore (1990) uses a break point of 228.8 Hz, and the cochlear frequency–place map of Greenwood (1990) uses 165.3 Hz. Other functional forms for the mel scale have been explored by Umesh et al.; they point out that the traditional formulas with a logarithmic region and a linear region do not fit the data from Stevens and Volkmann's curves as well as some other forms, based on the following data table of measurements that they made from those curves:
Slaney's MATLAB Auditory Toolbox agrees with Umesh et al. and uses the following two-piece fit, though notably not using the "1000 mels at 1000 Hz" convention:
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