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Re: frequency to mel formula

I just checked Beranek's book: Acoustic Measurements. Beranek cites Stevens and Volkman as the source of his plot on page 203 (Beranek provides no equation) . The full reference provided by Beranek is: S.S. Stevens and J. Volkman, "The relation of pitch to frequency: a revised scale," Amer. J. Psychol., vol. 53, 329 (1940).

Christine Rankovic

----- Original Message ----- From: "Dan Ellis" <dpwe@xxxxxxxxxxxxxxx>
To: <AUDITORY@xxxxxxxxxxxxxxx>
Sent: Wednesday, July 15, 2009 3:55 PM
Subject: Re: frequency to mel formula

I'm not sure if this is worth discussing on the full list, but...

After the discussion last year I actually got a hold of the Beranek
1949 book from our library's cold storage, and the reference is wrong.
In the book, Beranek gives empirical values for the Mel scale, but no
equation.  Clearly, this reference got mangled somewhere along the
way: there may be a different early Beranek reference, but it isn't
this one.

I think Fant is the more appropriate reference (for log(1+f/1000)) and
O'Shaugnessy for log(1+f/700).


On Wed, Jul 15, 2009 at 3:34 PM, James D. Miller<jamdmill@xxxxxxxxxxx> wrote:
As Dan explained last time this was discussed, the correct reference to the formula cited by Beauchamp is

LL Beranek, Acoustic Measurements, Wiley, New York, 1949), p.329.

as the source for mel(f) = 1127 ln(1 + f/700)

Jim Miller

-----Original Message-----
From: AUDITORY - Research in Auditory Perception [mailto:AUDITORY@xxxxxxxxxxxxxxx] On Behalf Of Dan Ellis
Sent: Wednesday, July 15, 2009 3:04 PM
To: AUDITORY@xxxxxxxxxxxxxxx
Subject: Re: [AUDITORY] frequency to mel formula

We discussed this last year. See
and the surrounding thread.

I think the actual origin is Fant in a paper in Swedish from 1949,
summarized in his 1973 book:

Fant, C G M "Analys av de svenska konsonantljuden" LM Ericsson
protokoll H/P 1064, 1949: 139pp.

referenced on p. 48 of
Fant, G "Speech Sounds and Features", MIT Press, 1973.

but Fant uses log(1+f/1000). The log(1+f/700) was attributed to

O'Shaughnessy, D. (1978) Speech communication: Human and machine.
Addison-Wesley, New York, page 150.


On Wed, Jul 15, 2009 at 2:11 PM, James W.
Beauchamp<jwbeauch@xxxxxxxxxxxxxxxxxxxxxx> wrote:
Dear List,

On the Wikipedia page

a formula for computing frequency in terms of mels is given as:

mel = log(1 + fr/700)*1127 .

It is easily inverted to fr = 700*exp(mel/1127 - 1) .

My question is: Where do these formulas come from? I.e., I need
a journal reference for these formulas.

Thanks much,

Jim Beauchamp
Univ. of Illinois at Urbana-Champaign