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| [[Image:Mel-Hz_plot.svg|right|thumb|450px|Plots of pitch mel scale versus Hertz scale]]
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| [[Image:A440.png|thumb|A440 {{audio|A440.mid|Play}}. 440 Hz = 549.64 mels]]
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| The '''mel scale''', named by [[Stanley Smith Stevens|Stevens]], [[John Volkman|Volkman]] and Newman in 1937<ref name=stevens1937>
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| {{cite journal
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| | journal = Journal of the Acoustical Society of America
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| | title = A scale for the measurement of the psychological magnitude pitch
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| | author = Stevens, Stanley Smith; Volkman; John; & Newman, Edwin B.
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| | volume = 8
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| | issue = 3
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| | publisher =
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| | pages = 185–190
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| | issn =
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| | year = 1937
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| | url = http://asadl.org/jasa/resource/1/jasman/v8/i3/p185_s1
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| |bibcode = 1937ASAJ....8..185S |doi = 10.1121/1.1915893 }}</ref>
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| is a perceptual [[Scale (music)|scale]] of [[pitch (music)|pitch]]es 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 [[Hertz|Hz]] tone, 40 [[decibel|dB]] above the listener's threshold. Above about 500 Hz, larger and larger [[interval (music)|interval]]s are judged by listeners to produce equal pitch increments. As a result, four [[octave]]s 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.
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| A popular formula to convert <math>f</math> hertz into <math>m</math> mel is:<ref>
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| {{cite book
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| | title = Speech communication: human and machine
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| | author = Douglas O'Shaughnessy
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| | publisher = Addison-Wesley
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| | year = 1987
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| | isbn = 978-0-201-16520-3
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| | page = 150
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| | url = http://books.google.com/books?ei=rhFfSpa-BJOCNsTT4IUG&id=mHFQAAAAMAAJ&dq=Speech+Communications:+Human+and+Machine&q=2595#search_anchor
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| }}</ref>
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| :<math>m = 2595 \log_{10}\left(1 + \frac{f}{700}\right)</math>
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| ==History and other formulas==
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| There is no single mel-scale formula.<ref>
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| {{cite book
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| | title = Foundations of Modern Auditory Theory
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| | editor = Jerry V. Tobias
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| | publisher = Academic Press
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| | volume = 1
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| | author = W. Dixon Ward
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| | chapter = Musical Perception
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| | year = 1970
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| | page = 412
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| | quote = no one claims yet to have determined 'the' mel scale.}}</ref> The popular formula from O'Shaugnessy's book can be expressed with different log bases:
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| :<math>m = 2595 \log_{10}\left(1 + \frac{f}{700}\right) = 1127 \log_e\left(1 + \frac{f}{700}\right) \ </math>
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| The corresponding inverse expressions are:
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| :<math>f = 700(10^{m/2595} - 1) = 700(e^{m/1127} - 1) \ </math>
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| There were published curves and tables on psychophysical pitch scales since Steinberg's 1937<ref>
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| {{cite journal
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| | journal = Journal of the Acoustical Society of America
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| | title = Positions of stimulation in the cochlea by pure tones
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| | author = John C. Steinberg
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| | volume = 8
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| | issue = 3
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| | publisher =
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| | pages = 176–180
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| | issn =
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| | year = 1937
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| | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000008000003000176000001
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| |bibcode = 1937ASAJ....8..176S |doi = 10.1121/1.1915891 }}</ref>
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| curves based on [[just-noticeable difference]]s of pitch. More curves soon followed in Fletcher and Munson's 1937<ref>
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| {{cite journal
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| | journal = Journal of the Acoustical Society of America
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| | title = Relation Between Loudness and Masking
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| | author = Harvey Fletcher and W. A. Munson
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| | volume = 9
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| | pages = 1–10
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| | year = 1937
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| |bibcode = 1937ASAJ....9....1F |doi = 10.1121/1.1915904 }}</ref>
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| and Fletcher's 1938<ref>
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| {{cite journal
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| | journal = Journal of the Acoustical Society of America
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| | title = Loudness, Masking and Their Relation to the Hearing Process and the Problem of Noise Measurement
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| | author = Harvey Fletcher
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| | volume = 9
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| | pages = 275–293
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| | year = 1938
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| | issue = 4
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| | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000009000004000275000001
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| |bibcode = 1938ASAJ....9..275F |doi = 10.1121/1.1915935 }}</ref>
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| and Stevens' 1937<ref name=stevens1937/> and Stevens and Volkmann's 1940<ref>
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| {{cite journal
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| | journal = American Journal of Psychology
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| | title = The Relation of Pitch to Frequency: A Revised Scale
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| | author = Stevens, S., and Volkmann, J.
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| | volume = 53
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| | issue = 3
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| | pages = 329–353
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| | year = 1940
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| | url = http://www.jstor.org/pss/1417526
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| }}</ref>
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| papers using a variety of experimental methods and analysis approaches.
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| In 1949 Koenig published an approximation based on separate linear and logarithmic segments, with a break at 1000 Hz.<ref>
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| {{cite journal
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| | journal = Bell Telephone Laboratory Record
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| | title = A new frequency scale for acoustic measurements
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| | author = W. Koenig
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| | volume = 27
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| | pages = 299–301
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| | year = 1949
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| }}</ref>
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| Gunnar Fant proposed the current popular linear/log formula in 1949, but with the 1000 Hz corner frequency.<ref>
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| Gunnar Fant (1949) "Analys av de svenska konsonantljuden : talets allmänna svängningsstruktur",
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| LM Ericsson protokoll H/P 1064
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| </ref>
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| An alternate expression of the formula, not depending on choice of log base, is noted in Fant (1968):<ref>Fant, Gunnar. (1968). Analysis and synthesis of speech processes. In B. Malmberg (Ed.), ''Manual of phonetics'' (pp. 173-177). Amsterdam: North-Holland.</ref><ref>
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| {{cite book
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| | title = Techniques in speech acoustics
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| | author = Jonathan Harrington and Steve Cassidy
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| | publisher = Springer
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| | year = 1999
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| | isbn = 978-0-7923-5731-5
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| | page = 18
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| | url = http://books.google.com/books?id=E1SyZZN8WQkC&pg=PA18
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| }}</ref>
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| :<math>m = \frac{1000}{\log(2)} \log(1 + \frac{f}{1000}) \ </math>
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| In 1976, Makhoul and Cosell published the now-popular version with the 700 Hz corner frequency.<ref>
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| {{citation
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| | work = ICASSP 1976
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| | title = LPCW: An LPC vocoder with linear predictive spectral warping
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| | author = John Makhoul and Lynn Cosell
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| | volume = 1
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| | publisher = IEEE
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| | pages = 466–469
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| | year = 1976
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| | url = http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1170013
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| }}</ref>
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| As Ganchev et al. have observed, "The formulae [with 700], when compared to [Fant's with 1000], provide a
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| closer approximation of the Mel scale for frequencies below 1000 Hz, at the price of higher inaccuracy for frequencies higher than 1000 Hz."<ref>
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| {{citation
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| | work = Proceedings of the SPECOM-2005 | |
| | title = Comparative evaluation of various MFCC implementations on the speaker verification task,
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| | author = T. Ganchev, N. Fakotakis, and G. Kokkinakis
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| | pages = 191–194
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| | year = 2005
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| | url = http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.75.8303
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| }}</ref> Above 7 kHz, however, the situation is reversed, and the 700 Hz version again fits better.
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| Data by which some of these formulas are motivated are tabulated in Beranek (1949), as measured from the curves of Stevens and Volkman:<ref>Beranek, Leo L. (1949). ''Acoustic measurements''. New York: McGraw-Hill.</ref>
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| {{Table
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| |type = class = "wikitable"
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| |title = Beranek 1949 mel scale data from Stevens and Volkman 1940
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| | row1 = '''Hz''' {{!!}} 20 {{!!}} 160 {{!!}} 394 {{!!}} 670 {{!!}} 1000 {{!!}} 1420 {{!!}} 1900 {{!!}} 2450 {{!!}} 3120 {{!!}} 4000 {{!!}} 5100 {{!!}} 6600 {{!!}} 9000 {{!!}} 14000
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| | row2 = '''mel''' {{!!}} 0 {{!!}} 250 {{!!}} 500 {{!!}} 750 {{!!}} 1000 {{!!}} 1250 {{!!}} 1500 {{!!}} 1750 {{!!}} 2000 {{!!}} 2250 {{!!}} 2500 {{!!}} 2750 {{!!}} 3000 {{!!}} 3250
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| }}
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| A formula with a break frequency of 625 Hz is given by Lindsay & Norman (1977);<ref>Lindsay, Peter H.; & Norman, Donald A. (1977). ''Human information processing: An introduction to psychology'' (2nd ed.). New York: Academic Press.</ref> the formula doesn't appear in their 1972 first edition:
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| :<math>m = 2410 \log_{10}(1.6\times10^{-3} f + 1)</math>
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| 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 [[Equivalent rectangular bandwidth|ERB-rate]] scale of Glasberg & Moore (1990) uses a break point of 228.8 Hz,<ref>B.C.J. Moore and B.R. Glasberg, "Suggested formulae for calculating auditory-filter bandwidths and excitation patterns" Journal of the Acoustical Society of America 74: 750-753, 1983.</ref> and the cochlear frequency–place map of Greenwood (1990) uses 165.3 Hz.<ref>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.</ref>
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| 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:<ref>
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| {{citation
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| | journal = Proc. ICASSP 1999
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| | title = Fitting the mel scale
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| | author = Umesh, S. and Cohen, L. and Nelson, D.
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| | publisher = IEEE
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| | pages = 217–220
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| | isbn = 0-7803-5041-3
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| | year = 1999
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| | url =
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| }}</ref>
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| {{Table
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| |type = class = "wikitable"
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| |title = Umesh et al. 1999 mel scale data from Stevens and Volkman 1940
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| | row1 = '''Hz''' {{!!}} 40 {{!!}} 161 {{!!}} 200 {{!!}} 404 {{!!}} 693 {{!!}} 867 {{!!}} 1000 {{!!}} 2022 {{!!}} 3000 {{!!}} 3393 {{!!}} 4109 {{!!}} 5526 {{!!}} 6500 {{!!}} 7743 {{!!}} 12000
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| | row2 = '''mel''' {{!!}} 43 {{!!}} 257 {{!!}} 300 {{!!}} 514 {{!!}} 771 {{!!}} 928 {{!!}} 1000 {{!!}} 1542 {{!!}} 2000 {{!!}} 2142 {{!!}} 2314 {{!!}} 2600 {{!!}} 2771 {{!!}} 2914 {{!!}} 3228
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| }}
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| 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:<ref>http://lists.mcgill.ca/scripts/wa.exe?A2=ind0907d&L=auditory&P=389</ref>
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| {{quote|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 non-musicians to search for any differences in performance that may be governed by musician/non-musician differences or subject differences generally.}}
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://users.utu.fi/jyrtuoma/speech/Mel2Hz.html Hz–mel, mel–Hz conversion] (uses the O'Shaughnessy equation)
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| *[http://scitation.aip.org/dbt/dbt.jsp?KEY=JASMAN&Volume=8&Issue=3 J. Acoust. Soc. Am. table of contents] for Stevens et al. paper
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| *[http://www.sfu.ca/sonic-studio/handbook/Mel.html Handbook for Acoustic Ecology]
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| ==See also==
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| *[[Bark scale]]
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| *[[Mel-frequency cepstrum]]
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| *[[Fletcher–Munson curves]]
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| {{Acoustics}}
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| [[Category:Scales]]
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| [[Category:Psychoacoustics]]
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