Re: Frequency to Mel Formula

[Donald D Greenwood <ddg@xxxxxxxxxxxxx>]

Dick,

Your comments and questions are well taken.  Stevens discarded the  
"half-pitch" mel scale of 1937 completely, as I commented.  Siegel  
1964 or 65) did the only published effort I know about to replicate  
the 1937 experiment of Stevens, as mentioned by Richard Warren and as  
described also in Footnote 4 of my 1997 paper, where I present  
arguments against half-pitch judgments that you and others have.   
Stevens replaced the 1937 scale with the 1940 mel scale, that was  
based on experiments intended to divide frequency intervals into four  
(4 not 2) equal perceptual intervals.  That scale appears to have been  
methodologically biased, as I commented.  That scale is also the one  
that was approximated by Fant and used thereafter, as Pierre says, by  
the "speech science and technology community". I agree with Pierre  
that they have been ill-advised to use it, but no doubling or halving  
was involved in arriving at the 1940 scale.

In 1940 Stevens, as you have noted, did include a separate experiment  
(reported in the same paper as the equisection data) asking subjects  
to make what he called half-pitch judgments, but Stevens provided 40  
Hz signals to the subjects to provide them with "zero" pitch  
approximations.  That seems to have converted those subject judgments  
into bisection experiments, i.e. 40 Hz on one end, standard tone on  
the other, with subject setting the "middle" tone.  In any case, when  
Stevens provided the "zero point" you call for, the resulting pseudo  
half-pitch data fit in with his 4 part equisection results, but (as he  
says) they were not used to make the 1940 scale, which used 4 part  
equisection data only.

The use of narrow noise bands signals to try equisection experiments  
sounds like an experiment that could be tried, whether the outcome  
would be useful or not.

Donald Greenwood
On 29 Jul, 2009, at 8:15 PM, Richard F. Lyon wrote:

> Diana,
>
> Certainly the circular or helical aspect of pitch is crucial, in  
> many aspects of  pitch perception.  But there's also this one- 
> dimensional scale that can be valid in some contexts.  I hadn't said  
> or known anything about this "half-pitch" concept, which would  
> certainly bring in the whole octave equivalence complication.  But  
> is that what was used for the mel-scale tests and such?  I didn't  
> think so.  Rather, the idea was to subdivide intervals into  
> perceptually equal intervals ("equisection").  Of course, if the  
> intervals are like 2 octaves or such, or the subject is musically  
> savvy, that's going to bias the judgements based on the pitch  
> circularity.  But if the signals are something like narrow noise  
> bands, maybe it would be possible to do the task while avoiding  
> those cues of "consonance" and such?
>
> The "half pitch" idea presumes a well-defined, or well-perceived at  
> least, zero point, as well as a nonlinear mapping to try to get at.  
> Plus it puts the likely result right where the octave is, at least  
> for low frequencies.  Did anyone actually use that approach?   
> Richard Warren and Snorre Farner say several studies did so; I'm  
> surprised; it seems like a bad idea.  Wouldn't you almost always get  
> a result of half pitch equal to half frequency?  Is that the  
> explanation for why the linear-to-log breakpoint ended up so high?   
> Or did they really do equisection of intervals defined by two  
> nonzero tone frequencies?
>
> Stevens says they did both, but the curve he plots show only the  
> equisection results:
> http://books.google.com/books?id=r5JOHlXX8bgC&pg=PA166&dq=pitch+curve+equisection&lr=&as_brr=3&ei=VudwStWOPIrykATalqz4Dg
>
> Dick