A rough rule-of-thumb for critical bandwidth which seems to work most of the
time is that it a bit like the Q of a filter. When two pitches are within a few
Hz, we get a beat. While this beat has a frequency < 5Hz, say, the effect is
similar to vibrato, and may be useful musically. As the frequencies separate
further, the beat frequency increases, until it reaches the point where it is at
the lower limit of our hearing - say 20Hz. Between these points we are hitting
the critical bandwidth - a beat at 15Hz is very disagreeable! Obviously, when
the generating tones are at a low pitch anyway, the difference of 15 or 20Hz
represents a significant interval. 

As the generating frequencies diverge still further, we eventually reach a new
region of cognitive interatcion - the resultant tone. Frequencies of 800 and
1000 will create a phantom pitch in our ears at 200Hz (it will not show up in an
FFT analysis, for example, though the effect on the visible waveform is clear).
These three pitches are in a harmonic relationship, so the net result is a
'harmonious' sound (in fact we can often get more than one resultant tone - in
this case, probably at 600Hz). Shift the upper pitch by a fraction - say, to
1005, and the resultant tone rises accordingly, and is now out of harmonic
relationship - the sound becomes inharmonious. The shift at the top is almost
inconsequential, but the effect on the resultant tone is vivid. Attempt to add a
new 'correct' bass tone at 200 Hz and we get a clash between that and the
resultant tone at 205Hz. On paper, we have three tones at 200, 800 and 1005 -
well separated - but the resultant tone messes it up very nicely.

As a flute player, I have to deal with this problem any time I play in an
orchestra. As I will often be playing the highest note in the wind section, any
deviation from good tuning will have a devastating effect on the harmoniousness
of the whole section. A group of expert players will be continually
'negotiating' tuning amonst themselves to reduce beats and inharmonic resultant
tones to a minimum, the ultimate determinant generally being the bass note - but
less so if that note is not the root of the chord. There has been little
research into exactly how this is managed, but I feel that knowledge of the
primary principles of harmony is crucial - we learn to hear what the harmonic
root note is, of any chord we are playing, whether at the top, the bass, or in
the middle, and negotiate tuning towards that note. Needless to say, equal
temperament has little to offer here - we create new mean-tone tunings
on-the-fly, as it were.

Just possibly, this could form the basis for an algorithm which at any point
will attempt to identify the root pitch (the Hindemith principles might work
well for this), and adjust secondary pitches accordingly. I will have to leave
that to the maths boffins, however!

Richard Dobson

Paul Winkler wrote:
> 
> Robin,
> 
> Interesting reading there. Comparing which overtones match up is an
> approach that is mentioned, but not shown in much detail, in that book I
> was talking about (Science of Musical Sound by John Pierce).
> 
> Pierce also mentions a phenomenon you've neglected, which I think might
> be significant as well: The potential presence of overtones which are
> both strong and within a critical bandwidth so that there is audible
> beating between them. What is the critical bandwidth? It's a ratio
> between sine tones above which there is no perceptual rough beating.
> It's been shown to vary with frequency: it's between 2 and 3 semitones
> from about 1,000 hz on up, but it can be much greater than 3 semitones
> at lower frequencies. In other words, above approx. 1,000 hz, two pure
> sine tones will have no discernible beating as long as they're at least
> 3 semitones apart. (This may explain your difficulty in tuning sine
> tones in octaves! Most people consciously or unconsciously listen for
> beating when trying to tune.)
> 
> For pure tones, maximum dissonance seems to occur in the neighborhood of
> 1/4 of the critical bandwidth. But for complex tones, these
> generalizations aren't valid. So how is the critical bandwidth and
> beating relevant to complex tones?  This could be shown in much the same
> way that you tabulated overtone correspondencies.  If you go looking for
> the overtones that DON'T correspond, you may notice something
> interesting. For instance: in complex tones one octave apart, most of
> the partials either coincide OR are separated by more than 3 semitones,
> so you won't hear much beating at all. Whereas if you look at two
> complex tones separated by 6 semitones (a diminished 5th or whatever you
> want to call it), you should see a much higher proportion of
> non-corresponding tones within a critical bandwidth. At least Pierce
> says so, but he doesn't show any examples, and I'm too lazy to do the
> math. :-]
> 
> Pierce also mentions a theory on where the critical bandwidth phenomenon
> comes from -- having to do with the physical workings of the ear. You
> might want to check it out.
> 
> Regards,
> 
> PW
>