Without going back to Pythagoras, it's probably sufficient to say that Don's implication that A+A octave is a perfect interval is subjective to the context of the system of intonation, as Bruce has described.
There is a piece by Michael Harrison called "from different worlds" or something, that uses a piano cleverly engineered to play just intonation in all keys (by means of a very complex pedal board). It includes a Chorale section (Bach pastiche) that is quite astonishing, simply because of the intonation systems employed at each recapitulation. Listen to that, and it should tell you everything you need to know about pitch systems.
I'm currently working on a Scherzo for harpsichord in Kirnberger tuning. (starts getting wild about E),and I know people who play in Scholar's Lute etc etc. Most people seem to like Equal Temperament though. Blame Bach, blame the Accordion, I don't care. Sure it wiped out a lot of local pitch systems, but it did mean that more people were playing within the same pitch systems.
God, I'm boring. I still maintain that it doesn't matter whether E,B,E is a chord or not. Nobody says you have to play chords.
Best wishes, Jon