Bust of Pyathagoras
Allan Gluck, CC BY 4.0 , via Wikimedia Commons

The most incomprehensible thing about the universe is that it is comprehensible.

Pythagoras was a Greek philosopher and mathematician who lived in the Greek region of Ionia (now part of Turkey) from 570 - 495 BC. His achievements encompass many mathematical demonstrations and proofs that remain relevant today (notably, the Pythagorean Theorem) but also include many discoveries relevant to music.

According to legend, one day, Pythagoras was passing by a blacksmith’s shop when he heard harmonious, consonant sounds coming from inside. He entered and discovered the sounds were made by hammers, ringing after striking an anvil. After speaking with the blacksmith, he found the hammers weighed 12 pounds, 9 pounds, 8 pounds, and 6 pounds.

The 6-pound and 9-pound hammers resonated with a harmony that matched that of the 8-pound hammer with the 12-pound hammer. Pythagoras realized this must be because both pairs related to each other with a ratio of 2:3.

The 6-pound and 12-pound hammers, furthermore, with a ratio of 1:2, resonated together with such consonance they seemed to be producing the same pitch.

Pythagoras realized that musical harmonies relate to each other through a harmonic series, a pattern of whole integer ratios. An example can be seen by starting on the note A, which has a frequency of 110 Hz, and continually increasing the frequency by this same amount.

The mathematical ratios Pythagoras had discovered relate to musical harmonies. Frequencies related by a ratio of 1:2 are an octave apart. A 2:3 ratio produces a perfect fifth. Two pitches related at 3:4 sound a perfect fourth apart. The ratio of 4:5 creates a major third and a ratio of 5:6 a minor third. The difference between a perfect fourth and a perfect fifth has a ratio of 8:9, called a whole tone or a whole step. The difference between two whole steps and a perfect fourth has a ratio of 243:256 and is called a semitone or half-step.

Using just the first two ratios, Pythagoras attempted to create the perfect musical scale. From a fundamental starting pitch, a scale can be formed by raising notes by a 2:3 ratio. The frequencies of the resulting pitches can then be adjusted at a 1:2 ratio to bring them all within a more limited range. (This example uses frequencies derived from a standard A-440.)

The resulting scale, placing the notes in ascending order, contains frequencies related to each other by only two ratios, the 8:9 whole tone and the 243:256 semitone.

Continuing the 2:3 series up for twelve notes will eventually reach the seventh octave of the starting pitch. (Frequencies in this example are rounded to the nearest whole number.)

However, a discrepancy arises between the 2:3 series and a simple doubling of octaves at a 1:2 ratio.

This discrepancy is called a Pythagorean comma. It would take two thousand years for musicians to develop a system that avoids this, and then only by compromising the integrity of pure musical harmony. For most of musical history, Pythagorean tuning was the standard.