I've done a bit more looking for where these guys got their formulae:
At 3:55 PM -0400 7/15/09, Dan Ellis wrote:
I think Fant is the more appropriate reference (for log(1+f/1000)) and
O'Shaugnessy for log(1+f/700).
The "700" version appears in a couple of papers
before O'Shaughnessy's book, and he tells me
that got it from some place that he can't
recall, but definitely did not make it up
himself. Here are the two that I've found:
Ananthapadmanabha, T. V. (1980) "Formant ratios
on mel scale for male/female and male/child
speakers", Acoustics Letters, UK, vol 4.
and
John Makhoul and Lynn Cosell (1976) "LPCW: An
LPC Vocoder with Linear Predictive Spectral
Warping" ICASSP'76 466-469.
It appears that Makhoul may have made it up to
fit (and Doug says he was at that ICASSP, so
that may be where he got it). John Makhoul says
in that paper:
This relation is similar to those of critical
band masking effects and equal intelligibility
curves [8]. The mel?frequency relation can be
approximated by the following equation
m = 2595 log10(1 + f/700)
where f is the frequency in Hz and m is the
pitch in mels. The mel scale is adjusted such
that m=1000 mels corresponds to f10OO Hz.
In response to my inquiry, John reviewed his notes and said:
... In my notes, I have pasted a copy of Fig. 48 from The Speech
Chain by Peter Denes, which shows the plot of mel scale versus
frequency. I remember distinctly reading off the mel values from that
plot because Denes did not include a table of values in his book. I
also remember that earlier formulas divided f by 1000, after Fant. I
have the equation with f divided by 700 in my notebook, along with a
hand-drawn plot of the mel scale versus frequency (with values taken
from the Denes plot) and the comment: "This equation is almost a
perfect fit to the above curve." I frankly do not remember if I came
up with the 700 number or I got it from somewhere else. But, if I had
gotten it from somewhere else, why didn't I reference that work?
After all, I reference other things related to it. My guess is that I
must have tried a few values and found 700 to give the best visual
fit.
(one may wonder why at Bolt, Beranek, and
Newman, he had to read values off a plot instead
of using Beranek's 1949 table, but that's how
life was before Google)
This makes perfect sense, since the 700 fits
better than the 1000, even for the tabulated
data from Berakek that Fant lists in his 1959
paper, which is the likely source of Denes's
plot. The 1000 Hz fits better if the domain is
restricted to 4 kHz on the high end, but 700 Hz
fits better overall if the full range is
considered, as was illustrated in the plot that
I sent around before:
http://dicklyon.com/tech/Hearing/Mel-like_scales.svg
The Denes plot can be seen here (later edition, presumably same plot):
http://books.google.com/books?id=ZMTm3nlDfroC&pg=PA104
It only goes up to 10,000 Hz, which is 3000 mel.
Makhoul's 700 Hz formula goes right through that
point.
As for Fant and the 1000 Hz version, he cites
his own 1949 paper in Swedish, saying "This
formula, discussed in more detail earlier (Fant,
1949), is a better approximation than the Koenig
scale..."
This line is found in his 1959 paper, which is
what's reprinted in the usually cited 1973 book:
G. Fant, "Acoustic description and
classification of phonetic units", Ericsson
Technics, No. 1, 1959
reprinted in G. Fant, Speech Sounds and
Features, MIT Press, Cambridge, MA, 1973, pp.
32-83
Dan Ellis had pointed out the 1973 book's
reference to the 1949 Swedish source, and had
pointed out Davis & Mermelstein (1980)'s
reference to Fant's 1959 English paper, but
seems to have missed, as I had, the fact that
the 1973 book chapter was a reprint of that 1959
paper. Steve Greenberg just pointed that out to
me this weekend. It's easy to miss the small
note at the bottom of the first page of the book
chapter.
There's also a mel formula with 625 Hz offset
(expressed as the reciprocal, 1.6e-3 s), a good
fit to the full 14000 Hz of Beranek's data
table, in Lindsay and Norman, 1977:
"mels = 2410 log (1.6x10^{-3} f + 1)"
Human Information Processing: An Introduction to Psychology
Peter H. Lindsay and Donald A. Norman
Edition 2
Academic Press, 1977
http://books.google.com/books?id=6d9OAAAAMAAJ&q=%22mels+2410+log%22
(I haven't checked first edition)
the same is also in
Sensation and Perception
Stanley Coren, Clare Porac, and Lawrence M. Ward
Academic Press, 1979
http://books.google.com/books?id=AN9qAAAAMAAJ&q=%22mels+2410+log%22
This suggests that Makhoul's 700 Hz was not yet
widely known and used in the late 1970s (not
surprisingly, as it was only in an ICASSP paper,
not likely noticed in the psychology field), and
that others were fitting similar values, finding
Fant's 1000 Hz unsatisfactory, perhaps. I'll
Don Norman what he recalls.
I'd still like to see that 1949 paper (or
perhaps it's a book, at 139 pages). I've asked
the National Library of Sweden, who have a copy,
if they can make me a copy, but it seems
unlikely. Anyone in Stockholm want to go take a
look?
In the mean time, it seems reasonable to cite
Fant 1959 or 1949 as the source of the first mel
formula, and Makhoul 1976 as the source of the
modern 700 Hz version. I'll update wikipedia.
Of course, this is just for the formulas. As
Jont Allen points out, Fletcher and Munson had
plots of all this in 1937 in JASA, but didn't
name it like Stevens did. And as Don Greenwood
points out, the data are all seriously flawed,
and a better formula is one with an offset of
165 Hz (for a human cochlea map), or as Glasberg
and Moore point out, 228 Hz for an ERB-rate
scale.
Dick