For my October 2022 diary, go here.

Diary — November 2022

John Baez

November 1, 2022

I've been away from my piano since September. I really miss playing it. So, I've been sublimating my desire to improvise on this instrument by finally learning a bunch of basic harmony theory, which I practice just by singing or whistling.

For example, I'm getting into modes. The following 7 modes are all obtained by taking the major scale and starting it at different points. But I find that's not the good way for me to understand the individual flavor of each one.

Much better for me is to think of each mode as the major scale (= Ionian mode) with some notes raised or lowered a half-step — since I already have an intuitive sense of what that will do to the sound. For example, anything with the third lowered a half-step (♭3) will have a minor feel. And Aeolian, which also has the 6th and 7th lowered (♭6 and ♭7), is nothing but my old friend the natural minor scale!

A more interesting mode is Dorian, which has just the 3rd and 7th notes lowered a half-step (3♭ and 7♭). Since this 6th is not lowered this is not as sad as minor. You can play happy tunes in minor, but it's easier to play really lugubrious tear-jerkers, which I find annoying. The major 6th of Dorian changes the sound to something more emotionally subtle. Listen to a bunch of examples here:

Some argue that the Dorian mode gets a peculiarly 'neutral' quality by being palindromic: the pattern of whole and half steps when you go up this mode is the same as when you go down:

w h w w w h w

This may seem crazily mathematical, but Leibniz said "Music is the pleasure the human mind experiences from counting without being aware that it is counting."

Indeed, there is a marvelous theory of how modes sound 'bright' or 'dark' depending on how many notes are sharped---that is, raised a half-tone---or flatted---that is, lowered a half-tone. I learned about it from Rob van Hal, here:

The more notes are flatted compared to the major scale, the 'darker' a mode sounds! The fewer are flatted, the 'brighter' it sounds. And one, Lydian, is even brighter than major (= Ionian), because it has no flats and one sharp!

So, let's list them from bright to dark. Here's a chart from Rob van Hal's video:

You can see lots of nice patterns here, like how the flats come in 'from top down' as the modes get darker: that is, starting at the 7th, then the 6th and then the 5th... but also, interspersed with these, the 3rd and then the 2nd.

But here's something even cooler, which I also learned from Rob van Hal, though he was surely not the first to discover it.

If we invert each mode   literally turn it upside down, by playing the pattern of whole and half steps from the top of the scale down instead of from bottom to top   the brighter modes become the darker modes, and vice versa!

Let's see it! Inverting the brightest, Lydian:

w w w h w w h

we get the darkest, Locrian:

h w w h w w w

Inverting the 2nd brightest, the happy Ionian (our familiar friend the major scale):

w w h w w w h

we get the 2nd darkest, Phrygian:

h w w w h w w

Inverting the third brightest, Mixolydian:

w w h w w h w

we get the third darkest, the sad Aeolian (our friend the natural minor):

w h w w h w w

And right in the middle is the palindromic Dorian:

w h w w w h w

What a beautiful pattern!

By the way, it's also cool how both the ultra-bright Lydian and the ultra-dark Locrian, and only these modes, have a note that's exactly half an octave above the 1. This is a very dissonant thing for a mode to have! In music jargon we say it like this: these modes have a note that's a tritone above the tonic.

In Lydian this note is the sharped 4th, which is a 'brighter than usual 4th'. In Locrian it's the flatted 5th, which is a 'darker than usual 5th'. But these are secretly the same note, or more technically 'enharmonic equivalents'. They differ just in the role they play   but that makes a big difference.

Why do both Lydian and Locrian contain the tritone above the tonic? It's not a coincidence: the tritone is mapped to itself by inversion of the octave, and inversion interchanges Lydian and Locrian!

This stuff is great, especially when I combine it with actually singing in different modes and listening to how they sound. Why am I learning it all just now, after decades of loving music? Because normally when I want to think about music I don't study theory   I go to the piano and start playing!

For music, more important than the mathematical patterns relating different modes is learning the 'personality' of individual modes and how to compose or improvise well in each mode.

Here are some introductions to that! Since I'm in awe of Rob van Hal I will favor his when possible. But there are many introductions to each mode on YouTube, and it's worth watching a lot, for different points of view.

Locrian is so unloved that I can't find a good video on how to compose in Locrian. Instead, there's a good one on how Björk created a top 20 hit that uses Locrian:

There's also a good one about Adam Neely and friends trying to compose in Locrian:

Jake Lizzio also creates a far-out tune in Locrian here:

November 8, 2022

Cohomology is often used to count holes in topological spaces, and — for example — classify line bundles over these spaces, based on the idea that a line bundle can 'twist' as it goes around a hole, as in the Möbius strip above. But modern number theory says that algebraic numbers like can be seen as functions on tiny spaces... which still have holes! So cohomology shows up here too.

The trick is to think of a 'number ring' like $$ \mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5} : \, a,b \in \mathbb{Z} \} $$ as the ring of functions on a kind of space, called a 'scheme'. Then ideals in your number ring act like line bundles on this space!

You can multiply a section of a line bundle by a function; similarly you can multiply an element of an ideal by an element of your number ring. That's how the analogy starts.

Carrying it much further, people developed 'class field theory', which applies cohomology to number theory. I've found Jürgen Neukirch's book Algebraic Number Theory to be one of the most fun ways to learn about this stuff. His love for the subject shines through.

Let me explain one tiny idea: how the ideal class group of a number ring is a cohomology group.

An ideal \(I\) of a number ring \(K\) gives a module of \(K\), just as a line bundle over a topological space gives a module for the ring of continuous functions on that space. In both cases we don't get just any old module, either: we get a locally free module of rank 1. This is an algebraic way of saying that 'locally', our module looks just like a copy of the ring it's a module of. But 'globally' it can twist around, like our pal the Möbius strip.

Now, some line bundles on a space are 'trivial'. These give modules of its ring of functions that don't twist around at all! These modules are not just 'locally free of rank 1' but actually free of rank 1. This is a fancy way to say they're isomorphic to the ring itself.

Similarly for ideals of a number ring \(K\). Some are actually free of rank 1: that is, isomorphic to \(K\) itself. These are the 'principal' ideals.

The analogy continuues: we can multiply line bundles, and also multiply ideals.

To multiply or 'tensor' two line bundles \(L\) and \(L'\) we take the tensor product of their fibers, which are 1d vector spaces, and get a new 1d vector space which we take as the fiber of a new line bundle \(L \otimes L'\).

To multiply two ideals \(I\) and \(I'\) of a number ring we simply multiply elements of \(I\) with elements of \(I'\) and take all possible sums, getting an ideal \(II'\).

When we multiply line bundles their twistiness gets added! This suggests that we should form an abelian group from line bundles, or some equivalence classes of line bundles. We can do it: just take isomorphism classes of line bundles and make it into a group using $$ [L] [L'] = [L \otimes L'] $$ We can do something similar with ideals of a number ring, but it's a bit subtler. We get a group called the 'ideal class group'.

The subtlety in the number theory case is that not every locally free module of rank 1 comes from an ideal: we need more general things called 'fractional ideals'. But if just talk about modules this subtlety disappears: we can just using locally free modules of rank 1.

So: in both topology and number theory, there's an important abelian group whose elements are the ways that a line bundle (or locally free module of rank 1) can be twisted!

And some good news: in both cases, this is a cohomology group!

If \(X\) is a topological space, its isomorphism classes of complex line bundles correspond to elements of the group \(H^1(X,\mathbb{C}^\ast)\). This is the first cohomology of \(X\) with coefficients in \(\mathbb{C}^\ast\), the group of invertible complex numbers!

If \(K\) is a number ring, its ideal class group is isomorphic to \(H^1(K,\mathbb{G}_m\). This is the first cohomology of \(K\) with coefficients in \(\mathbb{G}_m\).

What's \(\mathbb{G}_m\)? It stands for 'multiplicative group', and it's analogous to \(\mathbb{C}^\ast\). But here it actually stands for a sheaf of groups:

So: the ideal class group of a number field really is a first cohomology group \(H^1(K,\mathbb{G}_m)\).

And morally, not literally, it's first cohomology because it describes how something like a line bundle twists as we march around a 1-dimensional loop in some space — a scheme — and it has coefficients in \(\mathbb{G}_m\) because the 'twist' is an invertible number of some sort.

This is so cool. It means you can use your topological intuition in number theory!

For my December 2022 diary, go here.

© 2022 John Baez