Re: [Phys-l] Harmonics vs Overtones

["Dan Crowe" <Dan.Crowe@Loudoun.K12.VA.US>]

A cutoff frequency implies that there are a finite number of modes, but a series does not have to be an infinite series.
 
The sound generated by a musical instrument can be modeled by a mathematical formula that is a sum of terms.  Each term in the sum represents a mode.  The modes themselves form a set, but the sound is represented by a sum.  Is this sum a finite series?  If not, why not?
 
I agree that there is a set of modes.  On the other hand, I think that it is useful to talk about representing sound as a modal series, which is a sum over the modes.  The modes are the basis set for the series.  I agree that the modal series is not, in general, a harmonic series.
 
Daniel Crowe
Loudoun County Public Schools
Academy of Science
dan.crowe@loudoun.k12.va.us 

> > > curtis osterhoudt <flutzpah@yahoo.com> 4/1/2009 1:53 PM >>>
  I may be muddying the waters, and John Denker may not have meant this at all, but he does have a point, referring to these "modes" as constituting a set, rather than a series.

    * When I see (or use) "series", I usually think, "Well, there's a relatively simple rule for generating each member of the series, be it arithmetic, geometric, or something more unusual." I also tend to think, "Ah, a series: there are an infinite number of elements." 
    * Each of the preceding thoughts about series can apply to "set", but they very definitely do not have to.

   In this context, "set" seems more appropriate to me, as the frequencies present in a given acoustic signal are often _nearly_, but not quite integer multiples of some sort of "fundamental". Additionally, one often has cutoff frequencies and so on which further complicate matters. 

   Of course, in teaching high schoolers (or even very bright undergraduates), a lot of this worry over nomenclature is wasted, and may be harmful pedagogically.