Pythagoras’ Ideas About “Perfect” Musical Harmony Are Not Quite Right After All

Pythagoras is famed for the theorem that probably gave you a headache in high school, clever cup designs, and his… erm… unusual beliefs about legumes. But the “father of numbers” had his fingers in all sorts of pies, and one other area where he made great strides was in the world of music. All these centuries later, however, new research is suggesting that Pythagoras’ ideas about musical harmony may not be as universal as once thought.

“Our findings challenge the traditional idea that harmony can only be one way, that chords have to reflect these mathematical relationships. We show that there are many more kinds of harmony out there, and that there are good reasons why other cultures developed them,” explained study co-author Dr Peter Harrison, Director of Cambridge University’s Centre for Music and Science, in a statement.

A lot of Western music theory relies on the idea of “consonance”, of creating combinations of notes that sound pleasant together. Pythagoras identified the link between the ratio of frequencies of musical notes and consonance.

For example, if we play a two-note chord where one of the notes has exactly half or double the frequency of the other, we in the Western world understand this as an octave. The notes sound concordant with each other – nothing is clashing, and the sound is perceived as pleasant.

Different ratios produce other intervals that are considered consonant, such as the perfect fifth (3:2 ratio). But whilst a lot of the music we know has been built on these principles, in reality it seems humans prefer things to be a little rougher round the edges.

Through online behavioral experiments with over 4,000 people from the US and South Korea, the researchers gathered data on how people perceive the pleasantness of different chords.

“We prefer slight amounts of deviation. We like a little imperfection because this gives life to the sounds, and that is attractive to us,” Harrison said.

As well as this, a lot of these traditional ideas about harmony simply don’t apply when we look at instruments that are less familiar to Western musicians. The team focused particularly on the bonang, an instrument comprising a collection of small gongs that forms part of the traditional Indonesian percussion ensemble gamelan.  

A musician playing the bonang in a gamelan performance in Yogyakarta, Indonesia.
Image credit: aditya_frzhm/

“When we use instruments like the bonang, Pythagoras’s special numbers go out the window and we encounter entirely new patterns of consonance and dissonance. The shape of some percussion instruments means that when you hit them, and they resonate, their frequency components don’t respect those traditional mathematical relationships,” Harrison explained.

“That’s when we find interesting things happening.”

While the bonang’s harmonic patterns map perfectly onto the musical scale used in its native Indonesia, the chords it can play cannot be recreated on a Western piano, for instance, because it simply isn’t tuned that way.

But even people who’ve never heard gamelan music before can still appreciate its tonal consonance, as the team discovered, opening up a world of exciting possibilities for composers and musicians to try out new combinations of instruments and sounds.  

“Quite a lot of pop music now tries to marry Western harmony with local melodies from the Middle East, India, and other parts of the world,” Harrison explained.

“Musicians and producers might be able to make that marriage work better if they took account of our findings and considered changing the ‘timbre’, the tone quality, by using specially chosen real or synthesized instruments. Then they really might get the best of both worlds: harmony and local scale systems.”

The study is published in the journal Nature Communications.

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