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Pun aside, what is harmonic about the harmonic series or the harmonic average? I assume it has a direct connection to music, but I cannot see it.

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To add to Byron's link: in particular look at Henning's answer. –  Willie Wong Jul 26 '12 at 15:07
    
There is some discussion on why the harmonic mean is called harmonic on this page of Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics. –  Rahul Jul 26 '12 at 15:20
    
Also, I've edited the title to distinguish this from the question about harmonic functions. I tried to keep the pun I thought you were going for. –  Rahul Jul 26 '12 at 15:21
    
@RahulNarain I appreciate that! And the link was very helpful. –  Casey Jul 26 '12 at 15:24
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The harmonic progression describes the change in wavelength between notes. halving the wavelength doubles the frequency giving the octave, then taking a third of the original frequency gives the fifth (musically speaking) so C_0 has a wavelength of 2100cm, C_1 a wavelength of 1050cm, G_1 a wavelength of about 700cm (we use 704 because it makes the frequency a nicer number, but technically that's "off"), C_2 is then 525cm, etc.

Using this for any note you can find all the harmonics of the note based on their ratio to the lowest octave base note's wavelength (such as C_0 at 2100cm). It's these whole number ratios that give the sequence of numbers its name, and then the sum of them is the harmonic sereis.

http://www.phy.mtu.edu/~suits/notefreqs.html

here's a list of the notes and their frequency/wavelength so you can get an idea of what I'm talking about, but essentially that's the connection. Taking the wavelength of a base note X, then X*(1/n) gives the wavelength of a (musical) harmonic of the note. As a fun fact every power of 2 will return an octave of the note, no matter how many notes you allow to be within the octave (since you can do that, and still get all of the harmonics using the harmonic sequence.

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This is an excellent answer, but how does it pertain to the harmonic average? –  Casey Jul 26 '12 at 15:14
    
Note that pitch is usually defined by frequency, not by wavelength. Since the speed of sound in air varies slightly with temperature, so do the exact wavelengths of the standard pitches. Thus, orchestra wind instruments need to be retuned when the temperature changes, and must be acclimatized to the right temperature before a performance or risk losing tune during it. –  Henning Makholm Jul 27 '12 at 1:01
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Just to indicate the relation between a harmonic average and a harmonic progression: if the harmonic mean of some subset of terms of a harmonic progression happens to be another term of that sequence, then the position of the "mean" term is the (ordinary arithmetic) mean of the positions of that subset. More generally, if one imagines that terms of a harmonic progression can be interpolated at arbitrary (non-integral) positions, then the harmonnic mean can be computed by locating its operands in a harmonic progression, taking the average of their positions and looking up the term at that point in the progression.

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In short, $a, \frac{ab}2, b$ are in arithmetic progression; $a, \sqrt{ab}, b$ are in geometric progression; and $a, \frac{ab}{a+b}, b$ are in harmonic progression. –  Rahul Jul 26 '12 at 16:36
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