Re: frequency to mel formula

["Richard F. Lyon" <DickLyon@xxxxxxx>]

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