2500 years ago Pythagoras discovered musical harmony in simple ratios of frequency. The story goes that he heard harmonious sounds from the blacksmiths' hammers; a hammer at half the weight sounded twice as high (Octave; 2/1), a pair at 2 to 3 sounded great (Perfect 5th; 2/3).

Pythagorean Temperament

Pythagoras made a system to divide the Octave through multiplying by 3/2 to get the Circle of Fifths and multiplying by 4/3 to get the Circle of Fifths in reverse order (Fourths).

Pythagorean Temperament worked nicely for music in those Medieval times. The Root, Fifth, and Octave were recognized, other Intervals, and changing key were of the devil!

The system has what you might call... interesting features:

• Derived from either Fifths or Fourths the 13th note, instead of an Octave, is 531441:524288, and is called the Pythagorean Comma.

• The Notes are the simpler ratio derived from either Fifths, or Fourths.

• Chords made with a Root and its Fifth sound more in tune compared to 12 Tone Equal Temperament!!! A Roots' Third however, is out to lunch!

Interesting Links:
Pythagorean Comma Indiana University
Pythagorean Comma
Pythagorean Scale Simon Fraser University
Squaring the Cirle - Pythagorus and Music of the Spheres Dartmouth University
Music of the Spheres Here attributed to Johannes Kepler