Editing Mel scale (section)

==History and other formulas==

The formula from O'Shaughnessy's book can be expressed with different logarithmic bases:
<math display="block">m = 2595 \log_{10}\left(1 + \frac{f}{700}\right) = 1127 \ln\left(1 + \frac{f}{700}\right).</math>

The corresponding inverse expressions are
<math display="block">f = 700\left(10^\frac{m}{2595} - 1\right) = 700\left(e^\frac{m}{1127} - 1\right).</math>

There were published curves and tables on psychophysical pitch scales since Steinberg's 1937<ref>
{{cite journal
 | journal = Journal of the Acoustical Society of America
 | title = Positions of stimulation in the cochlea by pure tones
 | author = John C. Steinberg
 | volume = 8
 | issue = 3
 | pages = 176–180
 | year = 1937
 | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000008000003000176000001
 | bibcode = 1937ASAJ....8..176S
 | doi = 10.1121/1.1915891 | url-access = subscription
 }}</ref> 
curves based on [[just-noticeable difference]]s of pitch.  More curves soon followed in Fletcher and Munson's 1937<ref>
{{cite journal
 | journal = Journal of the Acoustical Society of America
 | title = Relation Between Loudness and Masking
 | author1 = Harvey Fletcher
 | author2 = W. A. Munson
 | volume = 9
 | issue = 1
 | pages = 1–10
 | year = 1937
 | bibcode = 1937ASAJ....9....1F |doi = 10.1121/1.1915904 }}</ref> 
and Fletcher's 1938<ref>
{{cite journal
 | journal = Journal of the Acoustical Society of America
 | title = Loudness, Masking and Their Relation to the Hearing Process and the Problem of Noise Measurement
 | author = Harvey Fletcher
 | volume = 9
 | pages = 275–293
 | year = 1938
 | issue = 4
 | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000009000004000275000001
 | bibcode = 1938ASAJ....9..275F
 | doi = 10.1121/1.1915935 | url-access = subscription
 }}</ref>
and Stevens' 1937<ref name=stevens1937/> and Stevens and Volkmann's 1940<ref>
{{cite journal
 | journal = American Journal of Psychology
 | title = The Relation of Pitch to Frequency: A Revised Scale
 | author1 = Stevens, S.
 | author2 = Volkmann, J.
 | volume = 53
 | issue = 3
 | pages = 329–353
 | year = 1940
 | doi=10.2307/1417526
 | jstor = 1417526
 }}</ref>
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&nbsp;Hz.<ref>
{{cite journal
 | journal = Bell Telephone Laboratory Record
 | title = A new frequency scale for acoustic measurements
 | author = W. Koenig
 | volume = 27
 | pages = 299–301
 | year = 1949
 }}</ref>

[[Gunnar Fant]] proposed the current popular linear/logarithmic formula in 1949, but with the 1000&nbsp;Hz corner frequency.<ref>
Gunnar Fant (1949) "Analys av de svenska konsonantljuden : talets allmänna svängningsstruktur", LM Ericsson protokoll H/P 1064.</ref>

An alternate expression of the formula, not depending on choice of logarithm base, is noted in Fant (1968):<ref>Fant, Gunnar. (1968). Analysis and synthesis of speech processes. In B. Malmberg (ed.), ''Manual of phonetics'' (pp.&nbsp;173–177). Amsterdam: North-Holland.</ref><ref>
{{cite book
 | title = Techniques in speech acoustics
 | author1 = Jonathan Harrington
 | author2 = Steve Cassidy
 | publisher = Springer
 | year = 1999
 | isbn = 978-0-7923-5731-5
 | page = 18
 | url = https://books.google.com/books?id=E1SyZZN8WQkC&pg=PA18
 }}</ref>
<math display="block">m = \frac{1000}{\log 2} \log\left(1 + \frac{f}{1000}\right).</math>

In 1976, [[John Makhoul|Makhoul]] and Cosell published the now-popular version with the 700&nbsp;Hz corner frequency.<ref>
{{Cite book
 | title = ICASSP '76. IEEE International Conference on Acoustics, Speech, and Signal Processing
 | chapter = LPCW: An LPC vocoder with linear predictive spectral warping
 | author1 = John Makhoul  
 | author2 = Lynn Cosell
 | volume = 1
 | publisher = IEEE
 | pages = 466–469
 | year = 1976
 | author1-link = John Makhoul
 | doi = 10.1109/ICASSP.1976.1170013
 }}</ref>
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&nbsp;Hz, at the price of higher inaccuracy for frequencies higher than 1000&nbsp;Hz."<ref>
{{citation
 | work = Proceedings of the SPECOM-2005
 | title = Comparative evaluation of various MFCC implementations on the speaker verification task
 | author1 = T. Ganchev
 | author2 = N. Fakotakis
 | author3 = G. Kokkinakis
 | pages = 191–194
 | year = 2005
 | citeseerx = 10.1.1.75.8303
 }}</ref> Above 7&nbsp;kHz, however, the situation is reversed, and the 700&nbsp;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:<ref>Beranek, Leo L. (1949). ''Acoustic measurements''. New York: McGraw-Hill.</ref>

{| class="wikitable" style="text-align:center"
|+ Beranek 1949 mel scale data from Stevens and Volkmann 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&nbsp;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 does not appear in their 1972 first edition:
<math display="block">m = 2410 \log_{10}(0.0016 f + 1).</math>

For direct comparison with other formulae, this is equivalent to
<math display="block">m = 2410 \log_{10}\left(1 + \frac{f}{625}\right).</math>

Most mel-scale formulas give exactly 1000 mels at 1000&nbsp;Hz. The break frequency (e.g. 700&nbsp;Hz, 1000&nbsp;Hz, or 625&nbsp;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&nbsp;Hz; for example the [[Equivalent rectangular bandwidth|ERB-rate]] scale of Glasberg and Moore (1990) uses a break point of 228.8&nbsp;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&nbsp;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>

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:<ref>
{{cite conference
 | conference = Proc. ICASSP 1999
 | URL=https://www.researchgate.net/publication/3793925_Fitting_the_Mel_scale
 | doi=10.1109/ICASSP.1999.758101
 | title = Fitting the mel scale
 | author1 = Umesh, S.
 | author2 = Cohen, L.
 | author3 = Nelson, D.
 | pages = 217–220
 | isbn = 978-0-7803-5041-0
 | year = 1999
 }}</ref>

{| class="wikitable"  style="text-align:center"
|+ Umesh et al. 1999 mel scale data from Stevens and Volkmann 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
|}

[[Malcolm Slaney|Slaney]]'s MATLAB Auditory Toolbox agrees with Umesh et al. and uses the following two-piece fit, though notably not using the "1000&nbsp;mels at 1000&nbsp;Hz" convention:<ref>Slaney, M. Auditory Toolbox: A MATLAB Toolbox for Auditory Modeling Work. Technical Report, version 2, Interval Research Corporation, 1998., translated to Python in [https://librosa.org/doc/0.10.0/_modules/librosa/core/convert.html#hz_to_mel librosa] ([https://librosa.org/doc/0.10.0/generated/librosa.mel_frequencies.html librosa documentation]).</ref>
<math display="block">
 m(f) = \begin{cases}
  \dfrac{3f}{200}, & f < 1000, \\
  15 + 27 \log_{6.4} \left(\dfrac{f}{1000}\right), & f \geq 1000.
 \end{cases}
</math>