Expert: Paul Klarreich - 12/23/2006

Question
Hello,

I have been trying to sort some things out on my own, and seem to have got a trifle confused, so here:

On the one hand, the harmonic progression is a sequence, for every three terms of which the middle one is the harmonic average of the other two; on the other, harmonic analysis is said to do with periodic functions and waves; is there a connection between the two, or is it just a coincidence of therms?

Thanks in advance,

Martin

Answer
Questioner:  Martin
Category:  Advanced Math
 
Subject:  Harmonic
Question:  Hello,

I have been trying to sort some things out on my own, and seem to have got a trifle confused, so here:

On the one hand, the harmonic progression is a sequence, for every three terms of which the middle one is the harmonic average of the other two; on the other, harmonic analysis is said to do with periodic functions and waves; is there a connection between the two, or is it just a coincidence of therms?

Thanks in advance,

Martin
.....................................................
Sorry, but I don't think there is any connection.  There might have been at one time, but the two are pretty well separate by now.

Harmonic analysis (not my area of expertise) is based on Fourier transforms and generalized functions.  And those are (loosely) based on periodic functions.

A harmonic sequence (my definition; simpler, I think, but yours is also correct) is a sequence whose terms are the reciprocals of an arithmetic sequence.  

Now if a violinist plays a low C, his instrument produces a number of 'harmonics' whose wavelengths are in a harmonic sequence. (Their frequencies are in an arithmetic sequence.)  So there is some basic connection there, but both subjects have diverged from that.