Pythagorean Tuning

Pythagorean tuning is the oldest tuning system in European music heritage. According to legend, "Pythagoras discovered concords in the simple ratios among the divisions of a sounding string." Because the ancient Greeks believed numbers were the answer to all things, including the physical and the spiritual, so music was aligned with numbers.  The ancient Greeks recognized three primary interval distances: the fourth, fifth and octave. Their corresponding ratios are:

                           3/4, 2/3, 2/1


If a vibrating string or air column sounds a C at its full length, it will sound an F with three quarters of its length, G with two thirds and C an octave higher with half of its length. Because intervals with these ratios do not cause beats, they are called pure intervals. Pythagorean scales were built upon pure fourths and fifths. The ratios mentioned above are related to the string length, but modern acoustics prefers to use the ratios of frequency. To convert the length into frequency ratios, just invert them. Thus, in relation to the fundamental tone, the frequency ratios of the fourth, fifth and octave would be:

4/3, 3/2, and 2