Re: Frequency to Mel Formula

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

With respect to Umesh ("Fitting the Mel Scale", 1999), I hadn't 
actually got hold of his paper until just now; sure enough, he 
compared all the same fits, but started with a different table, from 
Stevens and Volkman.

Here are the Stevens and Volkman numbers:
f_stevens = [40; 161; 200; 404; 693; 867; 1000; 2022; 3000; 3393; 
4109; 5526; 6500; 7743; 12000]
mel_beranek = [43; 257; 300; 514; 771; 928; 1000; 1542; 2000; 2142; 
2314; 2600; 2771; 2914; 3229;

Here are the Fant numbers that I used:
% Baranek's tabulated data that Fant said fit log(1 + f/1000):
f_baranek = [20; 160; 394; 670; 1000; 1420; 1900; 2450; 3120; 4000; 
5100; 6600; 9000; 14000];
mel_beranek = (0:250:3250)';

I've added the Stevens table points on the svg plot at
http://dicklyon.com/tech/Hearing/Mel-like_scales.svg
The Umesh curve is closer to they data they fitted, naturally.

Looks like the Fant numbers are indeed from Beranek:
http://books.google.com/books?id=yCsLAAAAMAAJ&q=mel+inauthor:beranek&dq=mel+inauthor:beranek&lr=&as_drrb_is=b&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=1950&as_brr=0&ei=FbGASsuGFZuOkQTylZStCg
and
http://books.google.com/books?id=WKM8AAAAIAAJ&q=3450+inauthor:beranek&dq=3450+inauthor:beranek&lr=&as_brr=0&ei=SLGASraCI6KKkASh0OivCg

Jim Beauchamp kindly asked the right questions that helped me clarify this.

Dick


> Don,
>
> Thanks again for your great explanations of this complicated stuff.
>
> All that notwithstanding, I'm still poking around at why we have 
> these two different mel scales, with breaks at 700 and 1000.  So I 
> got hold of Fant's book, which has Baranek's data table in it, and 
> plotted up some comparisons.
>
> See http://dicklyon.com/tech/Hearing/Mel-like_scales.svg
>
> The "Mel 1000" curve comes pretty close to the Baranek table data up 
> through about 4 kHz, then diverges far from it above that.  The "Mel 
> 700" curve misses pretty badly around 2-6 kHz, but fits better on 
> average if you count the highest frequencies.
>
> The "Umesh" curve, f / (0.741 + 0.00024*f), doesn't fit particularly 
> well, but has a good shape, so I did a "fit" and got f / (0.759 + 
> 0.000252*f).
>
> I also did a mel-type fit, and found a broad optimum for the corner 
> around 711.5 Hz (under the constraint that 1000 Hz maps to 1000, 
> which I should probably have tried relaxing, but didn't).
>
> Anyway, here's my theory:  Fant fitted to the frequency range he 
> cared about, which probably only went to 4 kHz or so.  And then 
> someone else probably did a fit to the same Baranek table over the 
> whole range, and got the 700 number (the plot shows that the 711.5 
> point are pretty much right on the 700 curve).  And that's why we 
> see Baranek referenced so much, maybe?
>
> I also looked at goodness of fit (sum squared error in mel space) 
> including all the frequencies in the Fant/Baranek table.  It turns 
> out that the Umesh type fit has only 1/8 as much error as the 
> mel-like fit, due to the Bark-like curvature at the high-frequency 
> end.
>
> So for people who like Baranek's table (assuming Fant has a true 
> copy of it), the Umesh type function should be a win.  But I don't 
> think that function extends well to the larger log-like range that 
> we find in the ERB and Greenwood type curves, which are the ones 
> that make more sense in auditory-based applications.
>
> That's my theory and I'm sticking to it.
>
> Dick