TITLE: What does "harmonic" mean in "harmonic mean"?
QUESTION [9 upvotes]: 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.

REPLY [2 votes]: 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.