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Pythagorean comma

When you ascend by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging 12 times, you eventually reach a note around seven octaves above the note you started on, which, when lowered to the same octave as your starting point, is 23.46 cents higher than the initial note. This interval, 531441:524288 or approximately 1.0136:1, is called a Pythagorean comma.

This interval has serious implications for the various tuning schemes of the chromatic scale, because in western music, 12 perfect fifths and seven octaves are treated as the same interval. Equal temperament, today the most common tuning system used in the west, gets around this problem by flattening each fifth by a twelfth of a pythagorean comma, thus giving perfect octaves.

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