Gunnar Fant's frequency map

[Arne Leijon <leijon@xxxxxx>]

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.

References
Fant, G. (1973). Acoustic description and classification of phonetic units, chapter 3, pages 32–83. 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. 47–65.
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):90–119.

Arne Leijon