I found it a lot easier to understand the harmonic and geometric averages when I learned about the "generalized f-mean". Many averages are arithmetic averages of a transformation of the value. "f" refers to the function which transforms your values. https://en.wikipedia.org/wiki/Quasi-arithmetic_mean

- The geometric average is the arithmetic average of the logarithm. It places emphasis on the ratio between numbers, rather than the absolute difference.

- The harmonic average is the arithmetic average of the multiplicative inverse. It averages values by a constant numerator rather than denominator. For example, the average fuel economy of multiple vehicles makes more sense per-distance, so miles/gallon should be rewritten as gallons/mile.

- The (RMS) root-mean-square is the arithmetic average of the square. Electrical power is proportional to the square of the amperage or voltage, so AC current and voltage uses the RMS average to make the power calculations correct.