John Baez
Mathematics Colloquium of the Claremont Colleges, Claremont McKenna College, January 30, 2026
The Mathematics of Tuning Systems
Leibniz said "Music is the pleasure the human mind experiences from counting without being aware that it is counting." The first step is choosing a tuning system — the frequency ratios between pitches in a scale. Different kinds of music sound best in different tuning systems! In music from the Middle Ages until today, new musical styles have gone hand in hand with mathematical innovations in tuning systems. Here I will focus on a few of the most important and beautiful Western systems, from Pythagorean tuning to today's reigning champion: equal temperament. Can you hear the difference? What will come next?
If you look at a piano keyboard you'll see groups of 2 black notes alternating with groups of 3. So the pattern repeats after 5 black notes, but if you count you'll see there are also 7 white notes in this repetitive pattern. So: the pattern repeats each 12 notes.
Some people who never play the piano claim it would be easier if had all white keys, or simply white alternating with black. But in fact the pattern makes it easier to keep track of where you are — and it's not arbitrary, it's musically significant.
For one thing white notes give a 7-note scale all their own. Most very simple songs use only this scale! The black notes also form a useful scale. And the white and black notes together form a 12-tone scale.
Starting at any note and going up 12 notes, we reach a note whose frequency is almost exactly double the one we started with. Other spacings correspond to other frequency ratios.
I don't want to overwhelm you with numbers. So I'm only showing you a few of the simplest and most important ratios. These are really worth remembering.
We give the notes letter names. This goes back at least to Boethius, the guy famous for writing The Consolations of Philosophy before he was tortured and killed at the order of Theodoric the Great. (Yeah, "Great".) Boethius was a counselor to Theodoric, but he really would have done better to stay out of politics — he was quite good at math and music theory.
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