(With thread drift apology)

Nick said
> Effectively in equal temperament G sharp and A flat are not exactly the same note

I hate to disagree with someone so knowledgeable, but it's the other way round. In equal temperament G sharp and A flat are exactly the same note. Just look at any ordinary piano (not a few specialised ones with more than 12 notes per octave).

In other temperaments such as mean tone (of which there are more than one variety) those notes can be different. Some early concertinas had separate buttons. Pythagorean tuning is great for a few notes but soon runs into difficulties. Many systems of temperament have been tried over the centuries, each with its pros and cons. Equal temperament is almost universal nowadays and gives almost perfect fifths but somewhat nasty thirds.