Not meaning to confuse things further, but pianos have a special problem which requires "stretch tuning" to make them sound in tune with themselves.

The higher notes are tuned slightly sharp and the lower notes slightly flat. Of course this means that the pitch ratio in an octave is no longer exactly two (the use of the word "two" rather than 2.00 is intentional for mathematical reasons - "two" is exact, 2.0000000 etc, is never exact).

One of the best references I've found on this is on a guitar site (the problem also occurs with guitars, but to a lesser extent).

http://www.doolinguitars.com/intonation/intonation5.html

Since there are quite a few guitarists on this site, they may find this site interesting.

Glenn