Pythagorean Fifth

The second case where harmonics are shared is when the second note is a factor of three higher in frequency than the bottom note, which one would also expect to be a very harmonious note. If we take that note down an octave, the frequency ratio of the higher note to the lower is now 3 to 2 ( ie 1.5 times the lower frequency) and these two notes again share a lot of harmonics. Every second harmonic of the top note is now the same as every third harmonic of the lower note. This ratio is the ratio called a perfect fifth, and is again considered a very harmonious combination of frequencies- not as harmonious as the the octave, but still harmonious. Again, looking at music theory throughout the ages, and looking at music from a wide diversity of cultures, two notes separated by this ratio are considered harmonious.

This fact of the relationship of frequencies (or actually in their case of the lengths of strings required to produce the notes) as whole number ratios for precisely the musical pitches considered to be harmonious was a discovery made by the Pythagoreans in ancient Greece (4th or 5th century BC) and was the key which showed them that the physical world was governed by mathematics.

This perfect fifth defined by the ratio of frequencies of 3 to 2 (ie 3/2) is called the Pythagorean fifth, and was considered throughout the middle ages to be the most harmonious interval outside the octave. Throughout history the fifth has formed the most consistent intervals in melodies and music.