Pythagorean Tuning

An important early tuning system is Pythagorean tuning, where we force all frequency ratios to involve only powers of 2 and 3. In music, 3/2 is the ‘fifth’: the most consonant of intervals except for the octave.

If we start with some frequency and go up and down by powers of 3/2, we create the ‘circle of fifths’ shown above. It’s almost a 12-pointed star, with one point for each note in the 12-tone equal-tempered scale.

Almost—but not quite! When we go up 12 fifths, we get a tone that’s alm...