Re: [Phys-l] Harmonics vs Overtones

[curtis osterhoudt <flutzpah@yahoo.com>]

   More muddying from me (very little of which has to do with the original discussion -- I apologize to the list)....

   The infinite series is just what I tend to think of when I hear "series". That's a peculiarity to me, and probably not worth anyone's time hashing out. 

   I'd be very careful (at least when talking to a more sophisticated audience)  in describing the sounds from a musical instrument as strictly being describable as sums of modes. For students, indeed, it's a great introduction to the idea of Fourier decomposition. Ideal instruments may work that way: "modes" are almost always necessarily linearly independent of each other (no coupling from one to another) -- thus your agreeable statement that these can be used as basis sets. 
   The problem is that in real instruments, nonlinearities are typically present (there is argument that they make *all* the difference), and so one hears "cross terms", "difference terms", and other bits which are certainly not present in the signal originally used to drive the instruments. The presence of these terms is, of course, dependent on the volume at which the instruments are played, and that's not usually covered by strictly modal decomposition. 

   As to your question about the possibility of an infinite series forming the sum which is heard:
      1) it's possible, so long as the sum is square-integrable (we have to conserve energy, after all);
      2) in principle, it doesn't matter, because our ears have upper limits on what they can hear anyway.

     Best regards, 
               Curtis Osterhoudt

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Down with categorical imperative!
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________________________________
From: Dan Crowe <Dan.Crowe@Loudoun.K12.VA.US>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Wednesday, April 1, 2009 1:45:16 PM
Subject: Re: [Phys-l] Harmonics vs Overtones

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.
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