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Re: Gunnar Fant's frequency map


Thank you so much for getting to the bottom of this with a trip to the National Library.

That's really interesting that the first use of the formula wasn't associated with the "mel" scale per se. You wouldn't get that impression from his 1959 paper (the one reprinted in the 1973 book), where he said:

"This formula, discussed in more detail earlier (FANT, 1949), is a better mel approximation than the Koenig scale which is exactly linear below 1,000 c/s and logarithmic above 1,000 c/s. The significance of the mel scale for incremental pitch judgments, masking, and intelligibility is discussed by KOENIG (1949), MUNSON and GARDNER (1950). Equal increments along the mel scale or one of its technical approximations above correspond closely to equal increments of auditory sensation."

Did Fant comment on the Koenig scale in his 1949 report? Or he didn't know about it yet?

Was the Koenig scale pitched as a "mel" scale? Does anyone have a copy of the Koenig paper? W. Koenig (1949). "A new frequency scale for acoustic measurements". Bell Telephone Laboratory Record 27: 299-301.


At 10:31 AM +0000 3/18/11, Arne Leijon wrote:
Some time ago Richard Lyon asked about the origin of
Prof. Gunnar Fant's "frequency-to-mel" scale

x(f)= k log( 1+f/f_0 ), with f_0=1000 Hz,

most easily accessible in Ch 3 of Fant (1973), page 48.

Yesterday, I spent a couple of hours studying the original
lab report of Fant (1949) where he introduced and motivated this scale.

In this report, G Fant did not use the term "mel". It is quite clear that
he saw his scale as a cochlear map function, describing
the cochlear characteristic place of acoustic frequency components.

He discussed the scale mailny as a way to display speech power density spectra
in terms of power per length unit along the basilar membrane.

He motivated his choice of x(f) function with a table of correction values in dB,
that would be needed to transform a power density spectrum measured with
constant bandwidths in Hz, into power per uniform steps along the x-scale.

In this table he presented these correction values as

L(f) = 10 log_10 ( BW(f) / BW( f_0) ), with f_0=1000 Hz,

using auditory bandwidth estimates BW(f) from four sources:
A: Difference Limens for frequency, ref Stevens and Davis (1938).
B: Critical Bandwidths, ref Fletcher (1929)
C: Bandwidths with equal intellibigility contributions, ref Beranek (1947)
D: Bandwidths with equal intelligibility contributions, ref French & Steinberg (1947)
in comparison with the corresponding values derived from his proposed mapping.

Fant (1949) interpreted the sources A-D just as different methods to estimate
the same thing, namely, the cochlear map. He did not discuss any use of his proposed scale x(f) as a "Numerical Scale of Pitch" (in mels), as suggested by Stevens & Davis (1938) in the pages that
Fant did not refer to.

Fant also noted as an advantage of the scale, that the vowel formant bandwidths are roughly constant
if measured in units of the proposed scale.

Fant, G. (1973). Acoustic description and classification of phonetic units, chapter 3, pages 32­83. MIT Press, Cambridge, MA. Fant, C. G. M. (1949). Analys av de svenska konsonantljuden. Technical Report protokoll H/P 1064, LM Ericsson.
citing among others:
Stevens, S. and Davis, H. (1938). Hearing. New York, pp. 94-99 and pp. 127-130.
Fletcher, H. (1940). Auditory patterns. Rev Mod Phys, 12, pp. 47­65.
Beranek, L. L. (1947). The design of speech communication systems. Proceedings of the I.R.E., (Sept). French, N. and Steinberg, J. (1947). Factors governing the intelligibility of speech sounds. J Acoust Soc Amer, 19(1):90­119.

Arne Leijon