Pythagoras: Triangles and Triads

Written By: Yingke Liang

Have you ever listened to western stringed music and enjoyed it? If you did, you owe it all to Pythagoras (yes, the triangle guy).

Pythagoras, using his knowledge of numbers and how to play the lyre, studied the ratios of string lengths and the resultant sounds produced at different ratios. He figured out that strumming a string stopped at exactly half the length of the original (a 1:1 ratio) produced a note that was an octave higher. Similarly, strumming a string stopped at a 1:2 ratio of the length produced a perfect fifth. Stopping the strings at 2:3 and 3:4 ratios also produced harmonious intervals. These ratios encompass the first four counting numbers which also happen to add up to 10 which Pythagoras found to be deeply profound and considered such ratios to be “Music of the Spheres”. Pythagorean tuning centers on the 2:3 ratio so tuning focuses on making the perfect fifths in tune. However, scales tuned in Pythagorean tuning will always have a “fifth” that sounds a quarter of a semitone flat and is named the “wolf interval”. It is a strange creature that could almost sound like a fifth but it doesn’t. The third, sixth, and seventh notes in an ascending major scale are also slightly sharp, but not jarring. Despite these slight hiccups, Pythagorean tuning is still employed by string orchestras today, because it is described as sounding more “natural” than other tuning schemes. The Pythagorean tuning was adapted during the medieval Ars Nova, a time when all formal music was religious so the “Music of the Spheres” wasn’t just a Pythagoras thing. (As a side-note, to circumvent this wolf interval the equal-tempered system was developed, which is the system keyboards are tuned in today. However, the trade-off is that the fifths just don’t sound as sweet and music doesn’t sound as natural as in Pythagorean tuning. I guess we can’t have it all).

Pythagoras wasn’t just lazing around developing this musical theory, however. Pythagoras lived during a tumultuous period of Greek history where people needed something to believe in, from the abstract to miscellaneous objects such as bronzed pets. Unsurprisingly, the cult of Pythagoras worshiped numbers, specifically the positive integers. Zero and the negative integers were considered in another realm and not an object of worship. Other than the rather banal object of worship, the Pythagoreans had normal cult-like rules, such as extremely strict vegetarianism, encouraging silence and the wearing of pure linen clothing, and a vendetta against beans because Pythagoras thought that each time a person farted they lost a portion of their soul. Despite the strange hijinks, the Pythagoreans were led by a talented Renaissance man and from this cult arose a system of musical harmonics that birthed western art music.


Medenhall, Margaret “The Music of the Spheres: Musical Theory and Alchemical Image” Mythological Studies Journal, vol. 4, 2013.

Hawkins, William. “Pythagoras, The Music of the Spheres, and the Wolf Interval.” Cleveland: Philosophical Club of Cleveland, 2012.

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