Pythagoras is credited with discovering that musical harmony is based on simple integer ratios, a finding he made by experimenting with a vibrating string. He found that a string halved in length produced a note an octave higher (a 2:1 ratio), a string two-thirds the length produced a perfect fifth (a 3:2 ratio), and a three-quarters length string created a perfect fourth (a 4:3 ratio). This established the foundation for western tuning systems and the concept of consonant intervals.Â
Pythagoras’s musical discoveries
•The Octave:
A string that is exactly half the length of another produces a note that is an octave higher. The ratio of their frequencies is 2:1.
• The Perfect Fifth:
A string with a length ratio of 3:2 produces a perfect fifth, which is a highly consonant interval.
• The Perfect Fourth:
A string with a length ratio of 4:3 creates a perfect fourth.
• The tetractys:
Pythagoras and his followers believed the numbers 1, 2, 3, and 4 held cosmic significance, and their sum, 10, was seen as a perfect number. The musical ratios they discovered are based on these fundamental numbers.
Pythagoras is attributed with discovering that a string exactly half the length of another will play a pitch that is exactly an octave higher when struck or plucked. Split a string into thirds and you raise the pitch an octave and a fifth. Spilt it into fourths and you go even higher – you get the idea. This concept is known as the overtone series or harmonic series and it is a feature of physics, affecting waves and frequencies in ways we can see and hear and ways we can’t.
Pythagoras believed that the planets themselves, all heavenly bodies, rang out notes of vibration based on their orbit and distance to each other. We humans simply lack the ability to hear this music of the spheres.
These mathematical ratios helped to define every system of intonation throughout history. In other words, we tune our modern day instruments using the mathematics that Pythagoras discovered almost 2,500 years ago.
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