Mel scale

Mel scale
Plots of pitch mels versus hertz

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, 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.

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

There is no single mel-scale 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 just-noticeable differences of pitch. More curves soon followed in Fletcher and Munson's 1937[5] and Fletcher's 1938[6] and Steven's 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(1 + \frac{f}{1000}) \

In 1976, Makhoul and Cosell published the now-popular 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 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 & 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

References

  1. ^ a b Stevens, Stanley Smith; Volkman; John; & Newman, Edwin (1937). "A scale for the measurement of the psychological magnitude pitch". Journal of the Acoustical Society of America 8 (3): 185–190. http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000008000003000185000001. 
  2. ^ Douglas O'Shaughnessy (1987). Speech communication: human and machine. Addison-Wesley. p. 150. ISBN 9780201165203. http://books.google.com/books?ei=rhFfSpa-BJOCNsTT4IUG&id=mHFQAAAAMAAJ&dq=Speech+Communications:+Human+and+Machine&q=2595#search_anchor. 
  3. ^ 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." 
  4. ^ John C. Steinberg (1937). "Positions of stimulation in the cochlea by pure tones". Journal of the Acoustical Society of America 8 (3): 176–180. http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000008000003000176000001. 
  5. ^ Harvey Fletcher and W. A. Munson (1937). "Relation Between Loudness and Masking". Journal of the Acoustical Society of America 9: 1–10. 
  6. ^ 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. http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000009000004000275000001. 
  7. ^ Stevens, S., and Volkmann, J. (1940). "The Relation of Pitch to Frequency: A Revised Scale". American Journal of Psychology 53 (3): 329–353. http://www.jstor.org/pss/1417526. 
  8. ^ W. Koenig (1949). "A new frequency scale for acoustic measurements". Bell Telephone Laboratory Record 27: 299–301. 
  9. ^ Gunnar Fant (1949) "Analys av de svenska konsonantljuden : talets allmänna svängningsstruktur", LM Ericsson protokoll H/P 1064
  10. ^ Fant, Gunnar. (1968). Analysis and synthesis of speech processes. In B. Malmberg (Ed.), Manual of phonetics (pp. 173-177). Amsterdam: North-Holland.
  11. ^ Jonathan Harrington and Steve Cassidy (1999). Techniques in speech acoustics. Springer. p. 18. ISBN 9780792357315. http://books.google.com/books?id=E1SyZZN8WQkC&pg=PA18. 
  12. ^ John Makhoul and Lynn Cosell (1976), "LPCW: An LPC vocoder with linear predictive spectral warping", ICASSP 1976 (IEEE) 1: pp. 466–469, http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1170013 
  13. ^ T. Ganchev, N. Fakotakis, and G. Kokkinakis (2005), "Comparative evaluation of various MFCC implementations on the speaker verification task,", Proceedings of the SPECOM-2005: pp. 191–194, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.75.8303 
  14. ^ Beranek, Leo L. (1949). Acoustic measurements. New York: McGraw-Hill.
  15. ^ Lindsay, Peter H.; & Norman, Donald A. (1977). Human information processing: An introduction to psychology (2nd ed.). New York: Academic Press.
  16. ^ 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.
  17. ^ 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.
  18. ^ Umesh, S. and Cohen, L. and Nelson, D. (1999), "Fitting the mel scale", Proc. ICASSP 1999 (IEEE): 217–220, ISBN 0780350413 

External links

See also


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Mel — may refer to: Contents 1 Places 2 People 3 Other uses 4 See also Places Australia MEL, the IATA airport code for …   Wikipedia

  • Mel-frequency cepstrum — In sound processing, the mel frequency cepstrum (MFC) is a representation of the short term power spectrum of a sound, based on a linear cosine transform of a log power spectrum on a nonlinear mel scale of frequency. Mel frequency cepstral… …   Wikipedia

  • Mel — Einheit Norm Hilfsmaßeinheit Einheitenname Mel Einheitenzeichen mel Beschriebene Größe(n) Tonheit Größensymbol(e) Z,z …   Deutsch Wikipedia

  • Mel-Skala — Einheit Norm Hilfsmaßeinheit Einheitenname Mel Einheitenzeichen mel Beschriebene Größe(n) Tonheit Größensymbol(e) Z,z …   Deutsch Wikipedia

  • Mel Bernie Company — The Mel Bernie Company, trading as 1928 Jewelry (and sometimes referred to simply as 1928 Jewelry ), is a manufacturer and wholesaler of costume jewelry and novelties.[1] They also distribute their products directly to consumers via their web… …   Wikipedia

  • Mel Edwards — Melvin Edwards Personal details Born May 4, 1937 (1937 05 04) (age 74) Houston, Texas Nationality American Profession Sculptor Website Official website …   Wikipedia

  • Musical scale — For psychoacoustic scale, see bark scale and mel scale. Pattern of whole and half steps in the Ionian mode or major scale on C   …   Wikipedia

  • Bark scale — The Bark scale is a psychoacoustical scale proposed by Eberhard Zwicker in 1961. It is named after Heinrich Barkhausen who proposed the first subjective measurements of loudness [Zwicker, E. (1961),… …   Wikipedia

  • Acoustic scale — For the concept of scale in acoustics, see Acoustics and Scale (ratio). Acoustic scale on C  Play …   Wikipedia

  • Harmonic major scale — In music theory, the harmonic major scale is a musical scale which found occasional use during the common practice era and is now often employed, particularly in jazz. It may be considered as a major scale with the sixth degree flattened, or the… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”