From: pieterh@sci.kun.nl (Peter Herweijer)
Subject: Re: When is close close enough for just intonation
Keywords: Intonation, Frequency determination
Organization: University of Nijmegen, The Netherlands
Date: Thu, 27 Aug 1992 13:05:37 GMT

In <l9lrn8INNlcf@calvin.usc.edu> alves@calvin.usc.edu (William Alves) writes:
>In article <1992Aug25.134009.275@sci.kun.nl> pieterh@sci.kun.nl (Peter Herweijer) writes:
> >In article <fred-mckenzie-240892170014@128.159.169.79> fred-mckenzie@ksc.nasa.gov (Fred McKenzie) writes:
> >>In an earlier article, someone wrote:
> >>> >Before one even plunges into psychoacoustical waters, there is a purely
> >>> >acoustical issue.  Frequency as such makes mathematical sense strictly
> >>> >speaking only for periodic waves of infinite duration.  This is not
> >>> >just a matter of being pedantic.  For short sounds there is an
> >>> >indeterminacy principle (on the same mathematical basis as Heisenberg's
> >>> >in quantum mechanics) that says essentially that the shorter the sound
> >>> >the less definite the frequency.  

        [ ... ]

> >>The inference that it is not possible to determine pitch of a very short
> >>duration tone, may not be correct in the practical sense.

> >>A method that would give accurate results more rapidly, would be to measure
> >>time per cycle, rather than cycles per time.  Using a "clock" of one
> >>megahertz, for example, one could count clock pulses between cycles of an
> >>audio tone.

        [ example deleted of how a 440Hz tone can be measured to
          0.005 Hz accuracy in 0.1 second ]

> >This line of reasoning doesn't apply here because the method of measurement
> >presupposes the actual content of the waveform to be analysed: it should
> >be a simple periodic wave. 

>This is the problem precisely. How does a frequency counter know when
>the single waveform whose length it's measuring has ended? Zero crossings?
>There are plenty of very periodic waveforms that cross zero many times
>before repeating. A better solution might be pattern-matching, which is
>what most pitch-followers do, I understand. Such matching introduces its
>own frequency error margin and must allow for slight deviations from one
>wave to another. Even then, as was also pointed out, non-periodic waves
>or combinations of different frequencies will confuse such a counter.

>But then, for the purposes of the original discussion, we were not talking
>about building a frequency counter, but about how the ear perceives frequency.

> >because it cannot really analyse the sound. The human ear can, because it
> >works in a completely different way; in effect, it determines the frequency
> >content of the sound (although it is also sensitive to time-domain effects
> >like phase), effectively performing what is known in mathematics as a
> >"Fourier transform".

>I appreciate the "effectively" in there. It's more like thousands of
>little bandpass filters than the mathematical process of a Fourier
>transform - but I get your analogy.

I used that word on purpose; of course, the Fourier transform is more
like an idealization of what actually happens.

>>Now one of the properties of the Fourier transform is
>>the "indeterminacy principle" referred to. The frequency content of a 
>>wave of limited duration is not a sharp peak at the "real" frequency, but
>>a small frequency band. In the ear, all the cells beloning to that whole
>>band react to such a waveform, and no sharp frequency is established.

>Well, actually, the ear can discriminate frequency far more accurately
>than this model would indicate. Though a range of nerve endings are
>excited along the membrane, the brain perceives only the center fre-
>quency of the range. This is why the ear can discriminate frequency
>differences of less than a cent in the middle of its range when the
>two notes are played one after another. But when the two notes are
>played together, they seem to fuse into a single frequency which is
>the average of the two. Other clues tell us that it's not the same
>as either tone separately, especially beats. Hence the chorus effect -
>when an entire section is playing we perceive a single tone but with
>a different quality than a single instrument playing the same loudness.

I am aware that I only talked about what the *ear* could do without
paying attention to the post-processing that the *brain* performs. It
seems reasonable to suppose that, when a whole bunch of nerves
corresponding to a frequency band is excited, the brain makes a kind
of educated guess at "the" frequency - the middle of the band, probably.
This certainly matches the phenomena you mention. Two frequencies
close enough would form a single peak, but make their difference large
enough and slowly two peaks would become "visible". This would be the
point when the two tones come to be perceived separately. The actual
point depends on the "resolution" of the ear and the processing
capability of the brain, of course. The different sound quality of
a combination of inseparable notes should be explainabe by harmonic
content, small phase shifts, the shape of the excited band, etc.

Beats are something completely different, it's a simple mathematical
phenomenon that becomes important when two frequencies are close
together. It is quite independent of the "resolution" mentioned above,
but, as you suggest, surely plays an important role in the perception
of small frequency differences. Heck, I use it to tune my guitar.

>The question that I have is this: given this "uncertainty principle"
>of frequency discrimination, does the brain actually choose a precise
>pitch within the range of uncertainty which changes slightly as the
>duration increases, or do we hear a broad band of frequency which
>gradually narrows as the duration increases? I gave the example before

The only thing I can say is this. From what I know of the ear, this 
is more or less what happens: The ear "hears" a frequency band that
narrows as time increases (which is what would happen with a band-
filter model). I don't know what the brain does with that information,
though, although it certainly wouldn't surprise me if its pitch
accuracy would increase in time (up to a certain limit, of course).

>of a study that found that subjects could associate a frequency with
>a tone of *less than one wavelength duration*. Unless, as someone has
>since pointed out, that waveform was not sinusoidal and the partials
>helped the ear discriminate the frequency, I think this would argue
>for the former model above - that is, the brain "guesses" a frequency
>within the range of uncertainty.

That conclusion certainly seems the right one. Which doesn't mean
that the ear isn't subject to the limitations of the Fourier transform,
though. For example, given the model above, it should be possible to
verify that two notes with little difference are perceived as a single
pitch when heard for a short time, and as two different pitches when
heard over a longer time. (Hmmm... I could try that at home.)

But what about this: when one ear is fed white noise, and the other
exactly the same waveform but with 1mS delay, most persons "hear"
a tone of about 1 KHz. Apparently, either ear or brain also performs
time-domain analysis, possibly upsetting the "Fourier transform" picture.

>Bill Alves

Peter Herweijer
pieterh@sci.kun.nl
Newsgroups: rec.music.classical