# Is there any meaningful history behind harmonic mean?

I am trying to understand the origin of harmonic mean and get an intuitively feel for why it was invented in the first place.

I have surfed the web but I keep seeing things like harmonic series / musical chords etc. Despite digging through, I can't really see any meaningful connection between harmonic series and harmonic mean.

That is, I understand that each term after the first in a harmonic serie is the harmonic mean of the neighboring terms but that doesn't really explain our current use of harmonic mean when solving problems.

It doesn't really give any gist as to why harmonic mean should even be thought of in the first place.

A mathematical relation linking the "harmonic series" with the "harmonic mean" is the following:

Let $$\{s_n\}$$ be the harmonic series

$$s_1=1,\; s_2 = \frac 12, \; s_3 = \frac 13,...$$

Then

The harmonic mean of $$s_{n-1}, \, s_{n+1}$$ is equal to $$s_n$$, $$H\left(s_{n-1}, \, s_{n+1}\right) = s_n$$.

PROOF $$H\left(s_{n-1}, \, s_{n+1}\right) = \frac {2}{\frac 1 {s_{n-1}} + \frac 1 {s_{n+1}}} = \frac{2}{n-1 + n+1} = \frac 1n = s_n$$

A characterization that we find in an ancient text, quoted in

https://en.wikipedia.org/wiki/Pythagorean_means#History

says

There are three means in music: one is arithmetic, second is the geometric, third is sub-contrary, which they call harmonic. The mean is arithmetic when three terms are in proportion such that the excess by which the first exceeds the second is that by which the second exceeds the third. (...) The mean is the geometric when they are such that as the first is to the second, so the second is to the third. (...) Subcontrary, which we call harmonic, is the mean when they are such that, by whatever part of itself the first term exceeds the second, by that part of the third the middle term exceeds the third.

So each satisfied a constant relation/was characterized by a distinct property, and the human mind needs no more to take an interest.

The definition above for the arithmetic mean is $$A:\;\; A -a = b-A \implies A(a,b) = \frac{a+b} 2.$$

The definition for the geometric mean is $$G:\;\; \frac a G = \frac G b \implies ab = G^2 \implies G(a,b) = (ab)^{1/2}.$$

The definition for the harmonic mean is more convoluted. It says for $$p$$ the "part", it is of the same value, namely it holds that $$H:\;\; \begin{cases} a-H = pa \\ H-b = pb \end{cases} \implies \begin{cases} p = \frac{a-H}{a} \\ p = \frac{H-b}{b}\end{cases}.$$

Equating right-hand-sides we get

$$\frac{a-H}{a} = \frac{H-b}{b} \implies H(a,b) = \frac {2ab}{b+a} = \frac {2}{\frac {b+a}{ab}} = \frac 2 {\frac 1a + \frac 1b}.$$

An apparently on-the-spot piece of literature, with references is

Komić, J. (2011). Harmonic Mean. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_645

• Thanks. Those where quite some resources you linked to. Commented Jan 29 at 18:34
• @Alexander You're welcome. Commented Jan 29 at 20:06