In the 1960s Paul Peterson at the University of Toronto fitted the published mel scale function using a power series. Published in a music journal; wish I had the reference here, but I don't. He used it to compose a few pieces of music for instruments in different registers. Interesting to listen to. Doug Creelman University of Toronto AUDITORY automatic digest system wrote: ```There are 6 messages totalling 358 lines in this issue. Topics of the day: 1. frequency to mel formula (6) ---------------------------------------------------------------------- Date: Wed, 15 Jul 2009 13:11:27 -0500 From: "James W. Beauchamp" Subject: frequency to mel formula Dear List, On the Wikipedia page http://en.wikipedia.org/wiki/Mel_frequency_scale 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 ------------------------------ Date: Wed, 15 Jul 2009 14:01:02 -0500 From: "McCreery, Ryan W" Subject: Re: frequency to mel formula Hi Jim, I don't know if this is the first reference to the mel scale, but it's I have read before and seen cited: Stevens, S.S. & Volkmann, J. (1940) The relation of pitch to frequency: A r= evised scale. The American Journal of Psychology 53(3), 329- I hope this helps. Ryan McCreery -----Original Message----- From: AUDITORY - Research in Auditory Perception [mailto:AUDITORY@xxxxxxxxx= ILL.CA] On Behalf Of James W. Beauchamp Sent: Wednesday, July 15, 2009 1:11 PM To: AUDITORY@xxxxxxxxxxxxxxx Subject: frequency to mel formula Dear List, On the Wikipedia page=20 http://en.wikipedia.org/wiki/Mel_frequency_scale a formula for computing frequency in terms of mels is given as: mel =3D log(1 + fr/700)*1127 . It is easily inverted to fr =3D 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 ------------------------------ Date: Wed, 15 Jul 2009 15:03:45 -0400 From: Dan Ellis Subject: Re: frequency to mel formula We discussed this last year. See http://www.auditory.org/mhonarc/2008/msg00191.html http://www.auditory.org/mhonarc/2008/msg00189.html 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. DAn. On Wed, Jul 15, 2009 at 2:11 PM, James W. Beauchamp wrote: ``` ```Dear List, On the Wikipedia page =C2=A0http://en.wikipedia.org/wiki/Mel_frequency_scale a formula for computing frequency in terms of mels is given as: mel =3D log(1 + fr/700)*1127 . It is easily inverted to fr =3D 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 ``` ``` ------------------------------ Date: Wed, 15 Jul 2009 15:55:25 -0400 From: Dan Ellis 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). DAn. On Wed, Jul 15, 2009 at 3:34 PM, James D. Miller wrot= e: ``` ```As Dan explained last time this was discussed, the correct reference to t= ``` ```he formula cited by Beauchamp is ``` ```LL Beranek, Acoustic Measurements, Wiley, New York, 1949), p.329. as the source for mel(f) =3D 1127 ln(1 + f/700) Jim Miller -----Original Message----- From: AUDITORY - Research in Auditory Perception [mailto:AUDITORY@xxxxxxx= ``` ```CGILL.CA] 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. =C2=A0See http://www.auditory.org/mhonarc/2008/msg00191.html http://www.auditory.org/mhonarc/2008/msg00189.html 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). =C2=A0The log(1+f/700) was attributed to O'Shaughnessy, D. (1978) Speech communication: Human and machine. Addison-Wesley, New York, page 150. =C2=A0DAn. On Wed, Jul 15, 2009 at 2:11 PM, James W. Beauchamp wrote: ``` ```Dear List, On the Wikipedia page =C2=A0http://en.wikipedia.org/wiki/Mel_frequency_scale a formula for computing frequency in terms of mels is given as: mel =3D log(1 + fr/700)*1127 . It is easily inverted to fr =3D 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 ``` ` ` ``` ------------------------------ Date: Wed, 15 Jul 2009 17:29:47 -0400 From: Christine Rankovic Subject: 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" To: 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). DAn. On Wed, Jul 15, 2009 at 3:34 PM, James D. Miller 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 http://www.auditory.org/mhonarc/2008/msg00191.html http://www.auditory.org/mhonarc/2008/msg00189.html 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. DAn. On Wed, Jul 15, 2009 at 2:11 PM, James W. Beauchamp wrote: ``` ```Dear List, On the Wikipedia page http://en.wikipedia.org/wiki/Mel_frequency_scale 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 ``` ` ` ``` ------------------------------ Date: Wed, 15 Jul 2009 20:54:45 -0500 From: "James W. Beauchamp" Subject: Re: frequency to mel formula It would be good if someone could double check the O'Shaugnessy reference, as given by Dan earlier today: ``` ```O'Shaughnessy, D. (1978) Speech communication: Human and machine. Addison-Wesley, New York, page 150. ``` ``` I think the title is actually Speech Communications: Human and Machine. In the archived message http://www.auditory.org/mhonarc/2008/msg00189.html Dan gives the date of the book as 1987, so I'm not sure which is correct. At any rate, it is possible to buy a second edition of the book, which is copyrighted 2000. However, when perusing the Contents and the Index it looks like the page has changed. Pages for 'mel scale' in the Index are 128, 191, and 214. I hope the formula made it. Jim Original message: ``` ```From: Dan Ellis Date: Wed, 15 Jul 2009 15:55:25 -0400 To: AUDITORY@xxxxxxxxxxxxxxx Subject: Re: [AUDITORY] frequency to mel formula Comments: To: "James D. Miller" 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). DAn. ``` ``` ------------------------------ End of AUDITORY Digest - 14 Jul 2009 to 15 Jul 2009 (#2009-160) *************************************************************** ``` ```-- C. Douglas Creelman 416-690-9407 (phone & fax) 9 Fernwood Park Ave. 416-708-9407 (cell) Toronto, ON Canada creelman@xxxxxxxxxxxxxxxxx M4E 3E8```