There is a long history of connection between the world of music and the world of mathematics.
A squared plus B squared equals C squared; that is of course the Pythagorean theorem from basic geometry, named for the Greek philosopher and religious teacher from 5th century BCE, Pythagoras. Pythagoras taught the belief that numbers were a guide to the interpretation of the universe. Mathematics could explain everything, including music.
Legend states that one day Pythagoras was walking past a smithy’s workshop, listening to the sound of the blacksmith’s hammers on the anvil. He turned his attention to the percussive sound that was produced and noted that some strikes sounded much higher than others. He was certain that there was a mathematical explanation for the different pitches he was hearing. So he entered the smithy’s shop and observed that they were using different sized hammers. Some of the hammers were large and others smaller, but they were ratios of each other: one being twice the size of another one, one being two-thirds the size of the last. Pythagoras declared these relationships as absolute intervals of music.
It’s a great story, but completely false. That’s not how these ratios actually work. But if we apply the story to lengths of string rather than hammers, we have something much more plausible.
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.
