Pythagorean comma

A Pythagorean comma is a microtonal musical interval, named after the ancient mathematician and philosopher Pythagoras. It is sometimes called a ditonic comma.

When ascending from an initial (low) pitch by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging twelve times, one eventually reaches a pitch approximately seven whole octaves above the starting pitch. If this pitch is then lowered precisely seven octaves, it will be discovered that the resulting pitch is (a very small amount over) 23.46 cents higher than the initial pitch. This microtonal interval is a Pythagorean comma:

$\left({\frac {3}{2}}\right)^{12}\!\!{\bigg /}\,2^{7}={\frac {3^{12}}{2^{19}}}={\frac {531441}{524288}}=1.0136432647705078125\!$

That is, twelve perfect fifths are not exactly equal to seven perfect octaves, and the Pythagorean comma is the amount of the discrepancy.

This interval has serious implications for the various tuning schemes of the chromatic scale, because in Western music, twelve perfect fifths and seven octaves are treated as the same interval. Equal temperament, today the most common tuning system used in the West, accomplished this by flattening each fifth by a twelfth of a Pythagorean comma (two cents), thus giving perfect octaves.

Chinese mathematicians had been aware of the Pythagorean comma as early as 122 BCE (its calculation is detailed in the Huainanzi), and in about 50 BCE, Ching Fang discovered that if the cycle of perfect fifths were continued beyond twelve all the way to fifty-three, the difference between this fifty-third pitch and the starting pitch would be much smaller than the Pythagorean comma, which was later named Mercator's comma.

Other intervals of similar sizes are the syntonic comma and the Holdrian comma.

External links

• Tonalsoft Encyclopaedia of Tuning
• "The Pythagorean Circle" from Brian Capleton's "Music, Mathematics, Philosophy, and Tuning: Harmonic Theory Pages"
• "Pythagorean Comma" — Phil Sloffer
• "The 'Pythagorean Comma'" — Jody Nagel