John C. Baez's Blog, page 11

Well Temperaments (Part 6)

Andreas Werckmeister (1645–1706) was a musician and expert on the organ. Compared to Kirnberger, his life seems outwardly dull. He got his musical training from his uncles, and from the age of 19 to his death he worked as an organist in three German towns. That’s about all I know.

His fame comes from the tremendous impact of his his theoretical writings. Most importantly, in his 1687 book Musikalische Temperatur he described the first ‘well tempered’ tuning systems for keyboards, where ever...

Agent-Based Models (Part 6)

Today I’d like to start explaining an approach to stochastic time evolution for ‘state charts’, a common approach to agent based models. This is ultimately supposed to interact well with Kris Brown’s cool ideas on formulating state charts using category theory. But one step at a time!

I’ll start with a very simple framework, too simple for what we need. Later I will make it fancier—unless my work today turns out to be on the wrong track.

Today I’ll describe the motion of agents through a grap...

Well Temperaments (Part 5)

Okay, let’s study Kirnberger’s three well-tempered tuning systems! I introduced them last time, but now I’ve developed a new method for drawing tuning systems, which should help us understand them better.

As we’ve seen, tuning theory involves two numbers close to 1, called the Pythagorean comma (≈ 1.0136) and the syntonic comma (= 1.0125). While they’re not equal, they’re so close that practical musicians often don’t bother to distinguish them! They call both a comma.

So, my new drawing sty...

Agent-Based Models (Part 5)

Agent-based models are crucial in modern epidemiology. But currently, many of these models are large monolithic computer programs—opaque to everyone but their creators. That’s no way to do science!

Our team of category theorists, computer scientists, and public health experts has come up with a cool plan to create agent-based models out of small reusable modules which can be explained, tested, compared and shared. This will make it easier to compare different models and build new ones. As a...

Pied Butcherbird

 

As my friends are learning about my current obsession with tuning systems, they’re starting to ask interesting questions I don’t know the answers to.

For example, Michael Fourman asked me: if harmonies coming from simple fractions are so natural, do any bird or whale songs feature such harmonies?

It turns out an Australian bird called the pied butcherbird has long been a favorite of many composers! Jean-Michel Maujean figured out the frequency ratios that appear in the songs of this bird....

Well Temperaments (Part 4)

 

Now I want to start talking about some important well-tempered tuning systems invented by Johann Philipp Kirnberger. But first: who was this guy? As I tried to answer this question for myself I became sort of fascinated with his personality.

Kirnberger was a German music theorist who played an important role in formalizing baroque harmony and counterpoint. He was born in 1721.

As a child he studied the violin and harpsichord at home. He then moved to another town to study the organ, and...

Well Temperaments (Part 3)

Last time we saw the importance of some tiny musical intervals: irritating but inevitable glitches in our search for perfectly beautiful harmonies. Today I want to talk about a truly microscopic interval called the ‘atom of Kirnberger’.

It was discovered by Bach’s student Johann Kirnberger, and it has a frequency ratio absurdly close to 1:

2161 · 3-84 · 5-12 ≈ 1.0000088728601397

It arose naturally in Kirnberger’s attempt to find a tuning system close to equal temperament with only rational fre...

Summer Research at Topos

Are you a student wanting to get paid to work on category theory in Berkeley? Then you’ve got just one week left to apply! The application deadline for Research Associate positions at the Topos Institute is February 1st.

Details and instructions on how to apply are here:

● 2024 Summer RA Announcement

Alas, the Topos Institute can’t provide visas.

Here are some topics you could work on:

● Computational category theory using AlgebraicJulia (Julia skills recommended)
● Categorical statistics
● ...

Thaats

Recently I learned a little about ragas in Carnatic music, which is prevalent in the south of India. I discussed them here:

• Melakarta ragas.

Now let me give an equally superficial introduction to ragas in Hindustani music, which is prevalent in the north.

In Carnatic music we start with 72 different seven-note scales called ‘Melakarta ragas’, and then build many more ragas by leaving out notes and other tricks. Hindustani music is a bit similar: we start with 32 seven-note scales called ‘th...

Well Temperaments (Part 2)

Last time I ended with a question: why are certain numbers close to 1 so important in tuning systems? It helps to understand a bit about this before we plunge into the study of well temperaments. It turns out that in some sense western harmony evolved one prime at a time, so let’s look at the subject that way.

The prime 2

If all the frequency ratios in our tuning system were powers of 2:

2i

life would be very simple. Multiplying a frequency by 2 raises its pitch by an octave, so the only ...