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Re: AUDITORY Digest - 10 Aug 2009 to 11 Aug 2009 (#2009-180)

Hi Jont,
I think that Dick was referring to some of my comments on the Mel scales (1937) and (1940) rather than on cochlear maps. As regards the latter, I simply commented (as I had assumed until now you probably saw) on Feldtkeller's and Zwicker's 1953 construction of their critical band function by first re-plotting Steinberg's map and the map of Davis and Stevens. To condense those comments, they next arrived at a Mittelwert (average) of the two graphical derivatives (dx/ df) that they (i.e. F and Z) had obtained from the Steinberg map (based on Shower and Biddulph frequency DL data) and the Davis and Stevens cochlear map (also influenced by same DL data Steinberg used). This Mittelwert of the two dx/df curves then determined the shape (in the inverse as df/dx) of their critical band function (their CB data did NOT). You see the Mittelwert in the inverse as df/dx in Gassler (1954) where he compared it to his Frequenzgruppe (CB) estimates. In Zwicker (1955) the df/dx is now raised on the ordinate and flattened below 300 Hz ("for engineering purposes" according to Zwicker's comment at Keele in 1977) and appears also as a chart. The same curve and similar chart next are seen in Zwicker, Flottorp, and Stevens (1957). The authors' comment on the similarity of their CB curve to the frequency DL data, doesn't seem altogether surprising considering its start in life, a fact which does not preclude a deeper reality as a co-factor.
In 1958/59, knowing none of the above (except the 1957 paper), I used the 1957 chart (i.e. "flattened derivative") CB values to make the first "BARK scale" to plot all my masking data (revising/reducing the scale's slope in higher frequencies to conform to my data). That was the first and last time I used it, replacing it in 1960/61 with my own mathematical function, based on the hypothesis that CB might be an exponential function of cochlear distance. Later in 1961 Zwicker re- possessed his "critical-band-rate scale", i.e. Bark scale (as well he might seeing that it was his), and its future unfolded from there. His and Gassler's data never had their chance to determine its shape below 400 Hz or above 3 kHz, which is a pity. Above 400 Hz they agree rather well with my 1961 function (i.e. are nearly proportional to its derivative). and below 400 there are very few data.
This last paragraph was not in my earlier post, but it feels a bit like history. 

Don

On 12 Aug, 2009, at 8:07 AM, Jont Allen wrote:


Dick,
While your into this, it seems that it would be nice to also put the Fletcher and the Steinberg data on top of this. I review the cochlear map story in my review of Fletcher, and I believe (hope) you will find all the references (mostly JASA) in this article. Fletcher claimed that the cochlear map was determined by the width of the BM, and I tried to verify that, and it is a reasonable hypothesis. Galt also found that the articulation index "importance function" is also prop to the critical bands, and if you integrate critical bands, then you get the cochlear map. In fact, that was the very first way it was done, I believe.
I didn't see Don Greenwood's posting, you refer to. Was that a private one, or did I just miss it?
I hope you publish a summary of all this, it would be good to pull together all the different cochlear maps into one grand scheme, with the proper references, and all. 

Jont

REF:
,author={Allen, J. B.}
,title={Harvey {F}letcher's role in the creation of communication
       acoustics}
,journal=JASA
,year=1996
,nonomonth=apr
,volume={99}
,number={4}
,pages={1825--1839}

You can get it from my website if you like:
http://auditorymodels.org/jba/PAPERS/Allen/Allen96.pdf

AUDITORY automatic digest system wrote:

There are 2 messages totalling 144 lines in this issue.
Topics of the day:
 1. Frequency to Mel Formula (2)
----------------------------------------------------------------------
Date:    Mon, 10 Aug 2009 22:04:06 -0700
From:    "Richard F. Lyon" <DickLyon@xxxxxxx>
Subject: Re: Frequency to Mel Formula
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

------------------------------
Date:    Tue, 11 Aug 2009 11:41:07 -0500
From:    "James W. Beauchamp" <jwbeauch@xxxxxxxxxxxxxxxxxxxxxx>
Subject: Re: Frequency to Mel Formula
Unfortunately, the "Stevens and Volkman numbers" had to be inferred
by Umesh et al (ICASSP '99) from the graph published in S & V's 1940 AJP paper. Umesh et al say: 
 "As we did not have access to the numerical data we read the points
  from the graph of Stevens and Volkman. this of course produces some
errors but we believe it is accurate enought for our considerations" 
Still, the Beranek and Umesh versions of the Stevens and Volkman data
seem to fall pretty close to the same curve on Dick's plots.
Jim
Original message:

From: "Richard F. Lyon" <DickLyon@xxxxxxx>
Date: Mon, 10 Aug 2009 22:04:06 -0700
To: AUDITORY@xxxxxxxxxxxxxxx
Subject: Re: [AUDITORY] Frequency to Mel Formula
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+inau
thor:beranek &lr=&as_drrb_is=b&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=1 
950&as_brr=0&ei=FbGASsuGFZuOkQTylZStCg
and
http://books.google.com/books?id=WKM8AAAAIAAJ&q=3450+inauthor:beranek&dq=3450+in
author:beranek&lr=&as_brr=0&ei=SLGASraCI6KKkASh0OivCg
Jim Beauchamp kindly asked the right questions that helped me clarify this. 

Dick

------------------------------
End of AUDITORY Digest - 10 Aug 2009 to 11 Aug 2009 (#2009-180)
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