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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: Re: Frequency to Mel Formula*From*: "Richard F. Lyon" <DickLyon@xxxxxxx>*Date*: Mon, 10 Aug 2009 22:04:06 -0700*Approved-by*: DickLyon@xxxxxxx*Delivery-date*: Tue Aug 11 01:19:01 2009*In-reply-to*: <20090810060259.A1B522E3C@xxxxxxxxxxxxxxxxxxxxxxx>*List-archive*: <http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>*List-help*: <http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>, <mailto:LISTSERV@LISTS.MCGILL.CA?body=INFO AUDITORY>*List-owner*: <mailto:AUDITORY-request@LISTS.MCGILL.CA>*List-subscribe*: <mailto:AUDITORY-subscribe-request@LISTS.MCGILL.CA>*List-unsubscribe*: <mailto:AUDITORY-unsubscribe-request@LISTS.MCGILL.CA>*References*: <95C3910C-83F5-46F1-A953-5C8F1D141D5D@xxxxxxx> <20090728204713.3E48F95F1@xxxxxxxxxxxxxxxxxxxxxxx> <20090729115312.D431C677C@xxxxxxxxxxxxxxxxxxxxxxx> <20090729232205.7FD8C9275@xxxxxxxxxxxxxxxxxxxxxxx> <20090730031526.4DAD98857@xxxxxxxxxxxxxxxxxxxxxxx> <20090730183411.759DC49BD@xxxxxxxxxxxxxxxxxxxxxxx> <20090810060259.A1B522E3C@xxxxxxxxxxxxxxxxxxxxxxx>*Reply-to*: "Richard F. Lyon" <DickLyon@xxxxxxx>*Sender*: AUDITORY - Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>

Here are the Stevens and Volkman numbers:

Here are the Fant numbers that I used: % Baranek's tabulated data that Fant said fit log(1 + f/1000):

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 havethese two different mel scales, with breaks at 700 and 1000. So Igot hold of Fant's book, which has Baranek's data table in it, andplotted up some comparisons.See http://dicklyon.com/tech/Hearing/Mel-like_scales.svgThe "Mel 1000" curve comes pretty close to the Baranek table data upthrough about 4 kHz, then diverges far from it above that. The "Mel700" curve misses pretty badly around 2-6 kHz, but fits better onaverage if you count the highest frequencies.The "Umesh" curve, f / (0.741 + 0.00024*f), doesn't fit particularlywell, 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 corneraround 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 hecared about, which probably only went to 4 kHz or so. And thensomeone else probably did a fit to the same Baranek table over thewhole range, and got the 700 number (the plot shows that the 711.5point are pretty much right on the 700 curve). And that's why wesee 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 turnsout that the Umesh type fit has only 1/8 as much error as themel-like fit, due to the Bark-like curvature at the high-frequencyend.So for people who like Baranek's table (assuming Fant has a truecopy of it), the Umesh type function should be a win. But I don'tthink that function extends well to the larger log-like range thatwe find in the ERB and Greenwood type curves, which are the onesthat make more sense in auditory-based applications.That's my theory and I'm sticking to it. Dick

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