The Pythagorean Comma
Pythagoras discovered that by stopping a string at 2/3 of its length it was possible to create a 5th. In theory by repetitively calculating a fifth one can calculate the twelve tones of the scale (the circle of fifths). By dividing the results by 2 for each octave above the base octave it possible to calculate the results for twelve tones in one octave.
Unfortunately the circle of fifths is not a circle, it is a spiral which ends up 23 cents sharp once the tonic is reached. To use this demonstration input your desired starting note on the bottom left corner and click on Calculate The blue lines indicate values raised by a 5th (3/2 ratio) . The error is displayed both in cents and hertz (cents will always stay the same). Compare the starting value at the bottom left of the "home octave with the value for VIII at the top right of the home octavee. VIII should be double I. It never is. This is the Pythagorean comma.