John C. Baez's Blog, page 15

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...

Lectures on Applied Category Theory

Want to learn applied category theory? You can now read my lectures here:

• Lectures on applied category theory

There are a lot of them, but each one is bite-sized and basically covers just one idea. They’re self-contained, but you can also read them along with Fong and Spivak’s free book to get two outlooks on the same material:

• Brendan Fong and David Spivak, Seven Sketches in Compositionality: An Invitation to Applied Category Theory.

Huge thanks go to Simon Burton for making my lectures ...

The Moduli Space of Acute Triangles

Recently Quanta magazine came out with an article explaining modular forms:

• Jordana Cepelewicz, Behold modular forms, the ‘fifth fundamental operation’ of math, Quanta, 23 September 2023.

It does a heroically good job. One big thing it doesn’t do is explain these funny looking ‘fundamental domains’ in the upper half-plane:

That is: where does this picture come from and why is it important?

By sheer coincidence, I just wrote a little article explaining the concept of ‘moduli space’ through an...

Life’s Struggle to Survive

I’m giving a public talk, the second of my Leverhulme Lectures at the International Centre for Mathematical Sciences:

• Life’s struggle to survive. Tuesday September 26, 6 pm UK time. Room G.03 on the ground floor of the Bayes Centre, 47 Potterrow, Edinburgh.

Abstract. When pondering our future amid global warming, it is worth remembering how we got here. Even after it got started, the success of life on Earth was not a foregone conclusion! In this talk I recount some thrilling, chilling...

The Triassic-Jurassic Extinction

214 million years ago an asteroid hit what is now Canada. Now the crater is a ring-shaped reservoir 70 kilometers in diameter: Manicouagan Reservoir.

Did this impact cause a mass extinction? The asteroid was 5 kilometers across, while the one that killed the dinosaurs much later was 10 kilometers across. But this is still huge!

For a while people thought this impact may have caused the Triassic-Jurassic mass extinction event. But now that the crater has been carefully dated, they don’t think...

In hydraulis by Antoine Busnois

You may remember this post of mine:

• Renaissance polyphony: the Franco-Flemish school.

This school of music flourished for two whole centuries, roughly from 1400 to 1600. Though I haven’t been posting about it lately, I continue to enjoy it.

Antoine Busnois (1430–1492) is one of the most famous composers in the second generation of the Franco-Flemish school. He’s almost up there with Johannes Ockeghem. And I just ran into a piece by him called In hydraulis. I really like it! But why did h...

Hoàng Xuân Sính’s Birthday

It’s Hoàng Xuân Sính’s 90th birthday today! Here she is in front of Grothendieck in 1967:

He taught algebraic geometry in the countryside in Vietnam while Hanoi was being bombed, and she took notes. After he left she did her thesis with him by correspondence! When the war ended, she went to Paris to defend her thesis and get her Ph.D. Then she returned to Hanoi and started the first private university in Vietnam.

A while ago Hà Huy Khoái, director of the Mathematics Institute at the Vietn...

The Ring of Brodgar

I showed you my pictures of Skara Brae, but another great archaeological site I saw on my trip to Orkney was the stone circle called the Ring of Brodgar. This is located nearby:

But unlike Skara Brae it’s not on the west shore, on the Norwegian Sea. Instead, it’s on a small isthmus between the Loch of Stenness and the Loch of Harray—a natural point for travelers to pass:

photo by Jim Richardson

Most of the megalithic structures on the British Isles do not contain stone circles. The three m...

Seminar on “This Week’s Finds”

Summer is coming to a close! It’s almost time to continue my seminars on topics from This Week’s Finds in Mathematical Physics. As before, I’ll be doing these on Thursdays at 3:00 pm UK time in Room 6206 of the James Clerk Maxwell Building, home of the School of Mathematics at the University of Edinburgh. The first will be on September 22nd, and the last on November 30th. I’ll skip October 19th… and any days there are strikes.

We’re planning to

1) make the talks hybrid on Zoom so that you ca...

The Syntonic Comma

Tuning systems in music is the subject where you get mad at irrational numbers. Nothing works perfectly—and it’s not your fault: it’s math’s fault. You’re left pushing around lumps in the carpet.

An ‘octave’ is the chord where the high note vibrates 2 times as fast as the low note. In a ‘perfect fifth’ it vibrates 3/2 times as fast. In a ‘perfect fourth’ it vibrates 4/3 as fast. In a ‘major third’ it vibrates 5/4 as fast. Our ears love these simple fractions.

But if you go up 4 perfect f...