Plots of pitch mel scale versus Hertz scale
A440
. 440 Hz = 549.64 mels
The mel scale, named by Stevens, Volkman and Newman in 1937^{[1]} 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 mel is:^{[2]}

m = 2595 \log_{10}\left(1 + \frac{f}{700}\right)
Contents

History and other formulas 1

References 2

External links 3

See also 4
History and other formulas
There is no single melscale formula.^{[3]} The popular formula from O'Shaugnessy's book can be expressed with different log bases:

m = 2595 \log_{10}\left(1 + \frac{f}{700}\right) = 1127 \log_e\left(1 + \frac{f}{700}\right) \
The corresponding inverse expressions are:

f = 700(10^{m/2595}  1) = 700(e^{m/1127}  1) \
There were published curves and tables on psychophysical pitch scales since Steinberg's 1937^{[4]} curves based on justnoticeable differences of pitch. More curves soon followed in Fletcher and Munson's 1937^{[5]} and Fletcher's 1938^{[6]} and Stevens' 1937^{[1]} and Stevens and Volkmann's 1940^{[7]} 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.^{[8]}
Gunnar Fant proposed the current popular linear/log formula in 1949, but with the 1000 Hz corner frequency.^{[9]}
An alternate expression of the formula, not depending on choice of log base, is noted in Fant (1968):^{[10]}^{[11]}

m = \frac{1000}{\log(2)} \log\left(1 + \frac{f}{1000}\right) \
In 1976, Makhoul and Cosell published the nowpopular version with the 700 Hz corner frequency.^{[12]} 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."^{[13]} 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 Volkman:^{[14]}
Beranek 1949 mel scale data from Stevens and Volkman 1940
Hz

20

160

394

670

1000

1420

1900

2450

3120

4000

5100

6600

9000

14000

mel

0

250

500

750

1000

1250

1500

1750

2000

2250

2500

2750

3000

3250

A formula with a break frequency of 625 Hz is given by Lindsay & Norman (1977);^{[15]} the formula doesn't appear in their 1972 first edition:

m = 2410 \log_{10}(1.6\times10^{3} f + 1)
Most melscale 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 nonmel auditoryfrequencyscale formulas use the same form but with much lower break frequency, not necessarily mapping to 1000 at 1000 Hz; for example the ERBrate scale of Glasberg & Moore (1990) uses a break point of 228.8 Hz,^{[16]} and the cochlear frequency–place map of Greenwood (1990) uses 165.3 Hz.^{[17]}
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 Volkman's curves as well as some other forms, based on the following data table of measurements that they made from those curves:^{[18]}
Umesh et al. 1999 mel scale data from Stevens and Volkman 1940
Hz

40

161

200

404

693

867

1000

2022

3000

3393

4109

5526

6500

7743

12000

mel

43

257

300

514

771

928

1000

1542

2000

2142

2314

2600

2771

2914

3228

Donald D Greenwood, a student of Stevens who worked on the mel scale experiments in 1956, considers the scale biased by experimental flaws, and posted in 2009 to a mailing list:^{[19]}
I would ask, why use the Mel scale now, since it appears to be biased? If anyone wants a Mel scale they should do it over, controlling carefully for order bias and using plenty of subjects  more than in the past  and using both musicians and nonmusicians to search for any differences in performance that may be governed by musician/nonmusician differences or subject differences generally.
References

^ ^{a} ^{b} Stevens, Stanley Smith; Volkman; John; & Newman, Edwin B. (1937). "A scale for the measurement of the psychological magnitude pitch". Journal of the Acoustical Society of America 8 (3): 185–190.

^ Douglas O'Shaughnessy (1987). Speech communication: human and machine. AddisonWesley. p. 150.

^ W. Dixon Ward (1970). "Musical Perception". In Jerry V. Tobias. Foundations of Modern Auditory Theory 1. Academic Press. p. 412.
no one claims yet to have determined 'the' mel scale.

^ John C. Steinberg (1937). "Positions of stimulation in the cochlea by pure tones". Journal of the Acoustical Society of America 8 (3): 176–180.

^ Harvey Fletcher and W. A. Munson (1937). "Relation Between Loudness and Masking". Journal of the Acoustical Society of America 9: 1–10.

^ Harvey Fletcher (1938). "Loudness, Masking and Their Relation to the Hearing Process and the Problem of Noise Measurement". Journal of the Acoustical Society of America 9 (4): 275–293.

^ Stevens, S., and Volkmann, J. (1940). "The Relation of Pitch to Frequency: A Revised Scale". American Journal of Psychology 53 (3): 329–353.

^ W. Koenig (1949). "A new frequency scale for acoustic measurements". Bell Telephone Laboratory Record 27: 299–301.

^ Gunnar Fant (1949) "Analys av de svenska konsonantljuden : talets allmänna svängningsstruktur", LM Ericsson protokoll H/P 1064

^ Fant, Gunnar. (1968). Analysis and synthesis of speech processes. In B. Malmberg (Ed.), Manual of phonetics (pp. 173177). Amsterdam: NorthHolland.

^ Jonathan Harrington and Steve Cassidy (1999). Techniques in speech acoustics. Springer. p. 18.

^ John Makhoul and Lynn Cosell (1976), "LPCW: An LPC vocoder with linear predictive spectral warping", ICASSP 1976 (IEEE) 1: 466–469

^ T. Ganchev, N. Fakotakis, and G. Kokkinakis (2005), "Comparative evaluation of various MFCC implementations on the speaker verification task,", Proceedings of the SPECOM2005: 191–194

^ Beranek, Leo L. (1949). Acoustic measurements. New York: McGrawHill.

^ Lindsay, Peter H.; & Norman, Donald A. (1977). Human information processing: An introduction to psychology (2nd ed.). New York: Academic Press.

^ B.C.J. Moore and B.R. Glasberg, "Suggested formulae for calculating auditoryfilter bandwidths and excitation patterns" Journal of the Acoustical Society of America 74: 750753, 1983.

^ Greenwood, D. D. (1990). A cochlear frequency–position function for several species—29 years later. The Journal of the Acoustical Society of America, 87, 2592–2605.

^ Umesh, S. and Cohen, L. and Nelson, D. (1999), "Fitting the mel scale", Proc. ICASSP 1999 (IEEE): 217–220,

^ http://lists.mcgill.ca/scripts/wa.exe?A2=ind0907d&L=auditory&P=389
External links

Hz–mel, mel–Hz conversion (uses the O'Shaughnessy equation)

J. Acoust. Soc. Am. table of contents for Stevens et al. paper

Handbook for Acoustic Ecology
See also
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