Re: harmonic vs. inharmonic sounds

[Stuart Rosen <stuart@xxxxxxxxxxxxxxxxxxx>]

Following up Kevin, I too think it is essential to distinguish the 
subjective from the objective definitions. A subjective definition of 
'harmonicity' clearly requires a good model of pitch perception.

The objective definition is lots easier, as it is really all about 
periodicity.

Firstly, we should restrict the decision of harmonic vs inharmonic only 
to sounds that are continuous and consist of discrete spectral 
components only.

The fundamental frequency (f0) of a series of spectral components is the 
~greatest~ common factor of the component frequencies. In this case, the 
wave is periodic.

If no common factor exists, then the wave representing the series is 
inharmonic.

So, a sound wave with components of 300, 600 and 901 Hz has a 
fundamental frequency of 1 Hz, because it is periodic with a period of 1 
s. It's (subjective) pitch is a completely separate issue.

In fact, any spectrum you define with discrete spectral components 
written down with a fixed number of decimal places, will be periodic, if 
at a very low fundamental. Irrational numbers for the frequency of 
spectral components are very useful for giving examples of truly 
inharmonic sounds consisting only of discrete spectral components. So a 
sound with components at 100 Hz and sqrt(2)*100 Hz is truly inharmonic.

Unless you specify that the fundamental is the ~greatest~ common factor, 
there is no unique definition of the fundamental (or a single cycle), 
because any wave that is periodic over 1 ms, say, is also periodic over 
any multiple of that essential period. Without the 'greatest' 
constraint, a complex tone of fundamental of 100 Hz could be said to 
have a fundamental of 100 Hz or 50 Hz or 25 Hz or 12.5 Hz or .... which 
would be confusing!

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