Even temparement, the twelth root of two thing, was introduced to allow different instruments of be played together, and to create a finite set of notes who keys. Under the previous system, you get to use exact simple ratios like 3:2 , which even temperament can only approximate, but repeatedly multiplying by 3:2 generates and infinite number of notes, not a cycle of twelve. Even temperament is always slightly off, but evenly so, hence the name. Guitars are naturally even tempered..a Pythagorean guitar would have frets zig zagging instead of parallel and evenly spaced.
Even temparement, the twelth root of two thing, was introduced to allow different instruments of be played together, and to create a finite set of notes who keys. Under the previous system, you get to use exact simple ratios like 3:2 , which even temperament can only approximate, but repeatedly multiplying by 3:2 generates and infinite number of notes, not a cycle of twelve. Even temperament is always slightly off, but evenly so, hence the name. Guitars are naturally even tempered..a Pythagorean guitar would have frets zig zagging instead of parallel and evenly spaced.