# Re: Origin of the Mel frequency scale equation? ("James D. Miller" )

```Subject: Re: Origin of the Mel frequency scale equation?
From:    "James D. Miller"  <jamdmill@xxxxxxxx>
Date:    Mon, 10 Mar 2008 10:25:08 -0400
List-Archive:<http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>

Gunnar Fant long ago suggested the Technical Mel(TM) scale and it is in
his 1974 Book "Speech Sounds and Features (MIT Press 1974).

He gave us TM = [1000/log(2)]*log[1+f/1000],

Where the log is to the base 10 and f is in Hz.

This fits the Mel Scale of Stevens and Volkman (1940) except at
the very high frequencies. See the figure in Fant's book.

I have  developed a quadratic that fits the 1940 scale very
accurately and modification of the Fant equation that also fits the
1940 scale over its entire range.

I have no idea of where the Mel = C*log (1 + f/700) came from in
the the speech-engineering literature. With the appropriate choice of
C, I expect it doesn't differ too much from Fant's TM over the ranges
of interest to speech engineers.

Jim Miller
jamdmill@xxxxxxxx

Quoting Christine Rankovic <rankovic@xxxxxxxx>:

> It might be:
>
> Stevens, S., and J. Volkmann (1940).  The relation of pitch to
> frequency:  A revised scale.  Am. J. Psychol. 53:329-353.
>
> Sincerely,
>
> Christine Rankovic, PhD
> rankovic@xxxxxxxx
>
>
> ----- Original Message ----- From: "Arturo Camacho" <acamacho@xxxxxxxx>
> To: <AUDITORY@xxxxxxxx>
> Sent: Monday, March 10, 2008 1:09 AM
> Subject: Origin of the Mel frequency scale equation?
>
>
>> Dear members of the list,
>>
>> I am looking for the reference of first use of the equation
>>
>> m = C log(1+f/700)
>>
>> known as mel frequency scale transformation. In Wikipedia says that the
>> scale was originated by Stevens, Volkman and Newman in 1937 (J. Acoust.
>> Soc. Am 8(3) 185--190), but the paper only has tabulated data and no
>> equation. The paper by S.B. Davis & P. Mermelstein (1980), "Comparison of
>> parametric representations for monosyllabic word recognition in
>> continuously spoken sentences", IEEE Trans. on ASSP 28, 357-366 is usually
>> cited in the speech recognition community as origin of MFCCs, but the
>> equation is absent there as well.
>>
>> Thanks,
>>
>> Arturo
>>
>> --
>> __________________________________________________
>>
>> Arturo Camacho, PhD
>> Alumni
>> Computer and Information Science and Engineering
>> University of Florida
>>
>> E-mail: acamacho@xxxxxxxx
>> Web page: www.cise.ufl.edu/~acamacho
>> __________________________________________________
>>
>

--
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Electrical Engineering Dept., Columbia University