The Malvern Windchime
Mystery of the Pythagorean Comma
The concise way to describe the Pythagorean comma is that you cannot tune a circle of perfect 5ths and end up where you started.
When ascending from a low pitch by a cycle of tuned perfect fifths (ratio 3:2), passing each alternately twelve times, eventually one reaches a pitch roughly seven whole octaves above the starting pitch.
If this pitch is then lowered precisely seven octaves, the resulting pitch is 23.46 cents (a very small amount) higher than the initial pitch. Called a Pythagorean comma, this microtonal interval named after Pythagoras, the ancient mathematician and philosopher, is sometimes known as a ditonic comma.
Therefore, twelve perfect fifths are not exactly equal to seven perfect octaves, and the Pythagorean comma is the amount of the discrepancy.
Chinese mathematicians knew of the Pythagorean comma as early as 122 BC. Circa 50 BC, Ching Fang discovered that continuing the cycle of perfect fifths beyond 12 all the way to 53, the difference between this 53rd pitch and the starting pitch would be much smaller than the Pythagorean comma, later named Mercator's comma.
Download a full White Paper on the Mystery of the Pythagorean Comma here
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