From: alves@calvin.usc.edu (William Alves)
Newsgroups: rec.music.classical
Subject: Re: When is close close enough for just intonation
Date: 25 Aug 1992 19:39:36 -0700
Organization: University of Southern California, Los Angeles, CA
NNTP-Posting-Host: calvin.usc.edu
Keywords: Intonation, Frequency determination

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.

>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.

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
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.

Bill Alves
Newsgroups: rec.music.classical