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

>From: "Robin Whittle" 
>To: csound@maths.ex.ac.uk, davids@pavell.com (David Schuyeteneer)
>Date: Sun, 28 Jun 1998 00:54:25 +1000
>Subject: Harmonics, physics, chords etc.
>Reply-To: rw@firstpr.com.au
>
>This afternoon, I sat down and wrote a treatise on my understanding 
>of why certain ratios of pitches sound good and others don't.  This 
>is all to do with physics of the sound and its harmonics - and 
>*nothing* to to with arbitrary human constructs such as scales. 
(snip)


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Thanks Paul for the commentry on "critical bandwidth", "no
perceptual rough beating" and "complex tones".  In my current
tired state, and without having read the source material in
"Science of Musical Sound" by John Pierce, I can't really comment 
without making lots of speculations which are probably off the 
track.

The business of listening to 800 Hz and 1000 Hz sine waves and 
hearing 200 Hz strikes me as likely to be very dependent on the 
level.  This is clear evidence of non-linear behaviour of the ear 
- which causes one signal to be in someway multiplied by the other.

At high signal levels, with dozens of harmonics ( AKA partials, 
overtones ) comeing into the ear, and each one intermodulating a 
bit with each of the others, it sure gets complex! 


- Robin


===============================================================

Robin Whittle     rw@firstpr.com.au  http://www.firstpr.com.au
                  Heidelberg Heights, Melbourne, Australia 

First Principles  Research and expression: music, Internet 
                  music marketing, telecommunications, human 
                  factors in technology adoption. Consumer 
                  advocacy in telecommunications, especially 
                  privacy. Consulting and technical writing. 

Real World        Electronics and software for music: eg.
Interfaces        the Devil Fish mods for the TB-303. 

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