Mel scale

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The mel scale, proposed by Stevens, Volkman and Newman in 1937 (J. Acoust. Soc. Am 8(3) 185--190) 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 equating a 1000 Hz tone, 40 dB above the listener's threshold, with a pitch of 1000 mels. Above about 500 Hz, larger and larger 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.

Many musicians and psychologists prefer a two-dimensional representation of pitch by tone color (or chroma) and tone-height, or a three-dimensional one such as the helical structure advocated by Roger Shepard, as more representative of other properties of musical hearing.

To convert $f$ hertz into $m$ mel use:

$m = 1127.01048 \log_e(1+f/700) \$

And the inverse:

$f = 700(e^{m/1127.01048} - 1) \$

An alternate formula is noted in Fant (1968):

$m = (1000/\log_{10}2) (\log_{10}(1+f/1000)) \$

Others are in Beranek (1949), Lindsay & Norman (1977), O'Shaughnessy (1987).

[edit] Bibliography

Beranek, Leo L. (1949). Acoustic measurements. New York: McGraw-Hill.
Fant, Gunnar. (1968). Analysis and synthesis of speech processes. In B. Malmberg (Ed.), Manual of phonetics (pp. 173-177). Amsterdam: North-Holland.
Lindsay, Peter H.; & Norman, Donald A. (1977). Human information processing: An introduction to psychology (2nd ed.). New York: Academic Press.
O'Shaughnessy, Douglas. (1987). Speech communication: Human and machine. Reading, MA: Addison-Wesley.
Stevens, Stanley Smith; Volkman; John; & Newman, Edwin. (1937). A scale for the measurement of the psychological magnitude of pitch. Journal of the Acoustical Society of America, 8 (3), 185-190.