The Icosidodecahedron

14 November, 2022

The icosidodecahedron can be built by truncating either a regular icosahedron or a regular dodecahedron. It has 30 vertices, one at the center of each edge of the icosahedron—or equivalently, one at the center of each edge of a dodecahedron. It is a beautiful, highly symmetrical shape. But it is just a shadow of a more symmetrical shape with twice as many vertices, which lives in a space with twice as many dimensions! Namely, it is a projection down to 3d space of a 6-dimensional polytope with 60 vertices.

Even better, it is also a slice of a more symmetrical 4d polytope with 120 vertices, which in turn is the projection down to 4d space of an even more symmetrical 8-dimensional polytope with 240 vertices: the so-called ‘E8 root polytope’. Note how the numbers keep doubling: 30, 60, 120 and 240.

To understand all this, start with the group of rotational symmetries of the icosahedron. This is a 60-element subgroup of the rotation group SO(3), so it has a double cover, called the binary icosahedral group, consisting of 120 unit quaternions. With a suitable choice of coordinates, we can take these to be

\displaystyle{ \pm 1 , \quad \frac{\pm 1 \pm i \pm j \pm k}{2}, \quad \frac{\pm i \pm \phi j \pm \Phi k}{2} }

together with everything obtained from these by even permutations of 1, i, j, and k, where

\displaystyle{ \phi = \frac{\sqrt{5} - 1}{2}, \quad \Phi = \frac{\sqrt{5} + 1}{2} }

are the ‘little’ and ‘big’ golden ratios, respectively. These 120 unit quaternions are the vertices of a convex polytope in 4 dimensions. In fact this is a regular polytope, called the 600-cell since it has 600 regular tetrahedra as faces.

If we slice the 600-cell with halfway between two of its opposite vertices, we get an icosidodecahedron. This is easiest to see by intersecting the 600-cell with the space of purely imaginary quaternions

\{ ai + bj + ck : \; a,b,c \in \mathbb{R} \}

Of the 600-cell’s vertices, those that lie in this 3-dimensional space are

\pm i, \pm j, \pm k

which form the corners of an octahedron, and

\displaystyle{ \frac{\pm i \pm \phi j \pm \Phi k}{2} ,  \quad  \frac{\pm j \pm \phi k \pm \Phi i}{2} , \quad  \frac{\pm k \pm \phi i \pm \Phi j}{2}   }

which form the corners of three ‘golden boxes’. A golden box is the 3d analogue of a golden rectangle: its three sides are in the proportions \phi, 1 and \Phi.

It is well-known that these points are the vertices of an icosidodecahedron. Here are the three golden boxes and octahedron inscribed in an icosidodecahedron, as drawn by Rahul Narain:

But we are not done with the binary icosahedral group—far from it!

Integer linear combinations of these 120 elements of the quaternions form a subring of the quaternions, which Conway and Sloane [CS] call the icosians. Since any icosian can be written as a + bi + cj + dk where the numbers a,b,c,d \in \mathbb{R} are of the form x + y \sqrt{5} with x,y rational, any icosian gives an 8-tuple of rational numbers. However, we do not get all 8-tuples of rationals this way, only those lying in a certain lattice in \mathbb{R}^8. And there is a way to think of this lattice as a rescaled copy of the famous E8 lattice! To do this, Conway and Sloane put a new norm on the icosians as follows. The usual quaternionic norm is

\|a + bi + cj + dk\|^2 = a^2 + b^2 + c^2 + d^2

But for an icosian this norm is always of the form x + \sqrt{5} y for some rationals x and y. Conway and Sloane define a new norm on the icosians by setting

|a + bi + cj + dk|^2 = x + y

With this new norm, Conway and Sloane show the icosians are isomorphic to a rescaled version of the E8 lattice in \mathbb{R}^8.

The 240 shortest nonzero vectors in this lattice are the vertices of an 8-dimensional convex polytope called the E8 root polytope:

However, if we remember that each of these 240 vectors came from a quaternion, we can also think of them as 240 quaternions. These turn out to be the vertices of two 600-cells in the quaternions! In the usual quaternionic norm, one of these 600-cells is larger than the other by a factor of \Phi.

In fact, there is an orthogonal projection from \mathbb{R}^8 down to \mathbb{R}^4 that maps the E8 root polytope to the 600-cell. So, in a very real sense, the 600-cell is the ‘shadow’ of a polytope with twice as many vertices, living in a space whose dimension is twice as large. And as a spinoff, this fact gives the same sort of relationship between the icosidodecahedron and a 6-dimensional polytope.

The key is to look at pure imaginary icosians: those of the form a i + b j + c k for real a,b,c. Since a,b and c are each of the form x + \sqrt{5}y with x and y rational, any pure imaginary icosian gives a 6-tuple of rational numbers. We do not get all 6-tuples of rationals this way, but only those lying in a certain lattice. We have

\|ai + bj + ck\|^2 = a^2 + b^2 + c^2

For a pure imaginary icosian this is always of the form x + \sqrt{5} y for some rationals x and y. So, we can define a new norm on the pure imaginary icosians by

|ai + bj + ck|^2 = x + y

With this new norm, the pure imaginary icosians are isomorphic to a rescaled version of a familiar lattice in \mathbb{R}^6, called the ‘D6 lattice’.

The 60 shortest nonzero vectors in the D6 lattice are called the roots of D6, and they are the vertices of a 6-dimensional convex polytope called the D6 root polytope. There is an orthogonal projection from \mathbb{R}^6 to \mathbb{R}^3 that maps this polytope to an icosidodecahedron. In fact 30 vertices of the D6 root polytope map to the vertices of this icosidodecahedron, while the other 30 map to vertices of a second, smaller icosidodecahedron.

Here is an image of the setup, created by Greg Egan:

Let’s see some details! The usual coordinatization of the D6 lattice in Euclidean \mathbb{R}^6 is

\mathrm{D}_6 = \left\{ (x_1, \dots, x_6) : \; x_i  \in \mathbb{Z}, \; \sum_i x_i \in 2\mathbb{Z} \right\} \subset \mathbb{R}^6

The roots of D6 are

(\pm 1, \pm 1, 0, 0, 0, 0)

and all vectors obtained by permuting the six coordinates. We shall see that these vectors are sent to the vertices of an icosidodecahedron by the linear map T \colon  \mathbb{R}^6 \to \mathbb{R}^3 given as a 3 × 6 matrix by

\left( \begin{array}{cccccc}  \Phi &  \Phi  & -1 & -1 & 0 &  0 \\  0 &  0  & \Phi &  -\Phi & -1 & 1 \\  -1 &  1 &  0 &  0 &  \Phi  & \Phi  \end{array} \right)

The rows of this matrix are orthogonal, all with the same norm, so after rescaling it by a constant factor we obtain an orthogonal projection. The columns of this matrix are six vertices of an icosahedron, chosen so that we never have a vertex and its opposite. For any pair of columns, they are either neighboring vertices of the icosahedron, or a vertex and the opposite of a neighboring vertex.

The map T thus sends any D6 root to either the sum or the difference of two neighboring icosahedron vertices. In this way we obtain all possible sums and differences of neighboring vertices of the icosahedron. It is easy to see that the sums of neighboring vertices give the vertices of an icosidodecahedron, since by definition the icosidodecahedron has vertices at the midpoints of the edges of a regular icosahedron. It is less obvious that the differences of neighboring vertices of the icosahedron give the vertices of a second, smaller icosidodecahedron. But thanks to the symmetry of the situation, we can check this by considering just one example. In fact the vectors defining the vertices of the larger icosidodecahedron turn out to be precisely \Phi times the vectors defining the vertices of the smaller one!

The beauties we have just seen are part of an even larger pattern relating all the non-crystallographic Coxeter groups to crystallographic Coxeter groups. For more, see the work of Fring and Korff [FK1,FK2], Boehm, Dechant and Twarock [BDT] and the many papers they refer to. Fring and Korff apply these ideas to integrable systems in physics, while the latter authors explore connections to affine Dynkin diagrams. For more relations between the icosahedron and E8, see [B2].

Acknowledgements

I thank Greg Egan for help with developing these ideas. The spinning icosidodecahedron was created by Cyp and was put on Wikicommons with a Creative Commons Attribution-Share Alike 3.0 Unported license. The 600-cell was made using Robert Webb’s Stella software and is on Wikicommons. The icosidodecahedron with three golden boxes and an octahedron inscribed in it was created by Rahul Narain on Mathstodon. The projection of the 240 E8 roots to the plane was created by Claudio Rocchini and put on Wikicommons with a Creative Commons Attribution 3.0 Unported license. The spinning pair of icosidodecahedra was created by Greg Egan and appears in an earlier blog article on this subject [B1]. The article here is an expanded version of that earlier article: the only thing I left out is the matrix describing a linear map S \colon \mathbb{R}^8 \to \mathbb{R}^4 that when suitably rescaled gives a projection mapping the E8 lattice in its usual coordinatization

\{ x \in \mathbb{R}^8: \, \textrm{all } x_i \in \mathbb{Z} \textrm{ or all } x_i \in \mathbb{Z} + \frac{1}{2} \textrm{ and } \sum_i x_i \in 2\mathbb{Z} \}

to the icosians, and thus mapping the 240 E8 roots to two 600-cells. For completeness, here is that matrix:

\left( \begin{array}{cccccccc}  \Phi+1 & \Phi -1 & 0  & 0 &  0 &  0 &   0  & 0 \\  0 & 0 & \Phi &  \Phi  & -1 & -1 & 0 & 0 \\  0 & 0  & 0 &  0  & \Phi &  -\Phi & -1 & 1   \\  0 & 0 & -1 &  1 &  0 &  0 &  \Phi  & \Phi  \end{array} \right)

The first image at the bottom of this post was also created by Greg Egan, on Mathstodon. The second shows an icosahedron and 3 golden rectangles morphing to an icosidodecahedron with 3 golden boxes, with an octahedron present at every stage. It was created by Vincent Pantaloni, on Geogrebra.

References

[B1] John Baez, Icosidodecahedron from D6, Visual Insight, January 1, 2015.

[B2] John Baez, From the icosahedron to E8, London Math. Soc. Newsletter 476 (2018), 18–23.

[BDT] Celine Boehm, Pierre-Philippe Dechant and Reidun Twarock, Affine extensions of non-crystallographic Coxeter groups induced by projection, J. Math. Phys. 54, 093508 (2013).

[CS] John H. C. Conway and Neil J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, Berlin, 2013.

[FK1] Andreas Fring and Christian Korff, Affine Toda field theories related to Coxeter groups of non-crystallographic type, Nucl. Phys. B729 (2005), 361–386.

[FK2] Andreas Fring and Christian Korff, Non-crystallographic reduction of generalized Calogero–Moser models, J. Phys. A39 (2006) 1115–1132.


The Circle of Fifths

12 November, 2022

The circle of fifths is a beautiful thing, fundamental to music theory.

Sound is vibrations in air. Start with some note on the piano. Then play another note that vibrates 3/2 times as fast. Do this 12 times. Since

(3/2)¹² ≈ 128 = 2⁷

when you’re done your note vibrates about 2⁷ times as fast as when you started!

Notes have letter names, and two notes whose frequencies differ by a power of 2 have the same letter name. So the notes you played form a 12-pointed star:

Each time you increase the frequency by a factor of 3/2 you move around the points of this star: from C to G to D to A, and so on. Each time you move about 7/12 of the way around the star, since

log(3/2) / log(2) ≈ 7/12

This is another way of stating the approximate equation I wrote before!

It’s great! It’s called the circle of fifths, for reasons that don’t need to concern us here.

But this pattern is just approximate! In reality

(3/2)¹² = 129.746…

not 128, and

log(3/2) / log(2) = 0.58496…

not 7/12 = 0.58333… So the circle of fifths does not precisely close:

The failure of it to precisely close is called the Pythagorean comma, and you can hear the problem here:

This video plays you notes that increase in frequency by a factor of 3/2 each time, and finally two notes that differ by the Pythagorean comma: they’re somewhat out of tune.

People have dealt with this in many, many ways. No solution makes everyone happy.

For example, the equal-tempered 12-tone scale now used on most pianos doesn’t have ‘perfect fifths’—that is, frequency ratios of 3/2. It has frequency ratios of

2^{7/12} \approx 1.4983

I have tried in this blog article to be understandable by people who don’t know standard music theory terminology—basic stuff like ‘octaves’ and ‘fifths’, or the letter names for notes. But the circle of fifths is very important for people who do know this terminology. It’s a very practical thing for musicians, for example if you want to remember how many sharps or flats there are in any key. Here’s a gentle introduction to it by Gracie Terzian:

Here she explains some things you can do with it:

Here’s another version of the circle of fifths made by “Just plain Bill”>—full of information used by actual musicians:

If you watch Terzian’s videos you’ll learn what all this stuff is about.


This Week’s Finds – Lecture 7

10 November, 2022

Today I’ll be talking about quaternions, octonions and E8. But a warning to everyone: today, November 10th, the seminar will be in a different building.

Today we’re in Lecture Theatre 1 of the Daniel Rutherford Building. This is a few minutes’ walk from the James Clerk Maxwell Building, just along from the Darwin Building.

Next week we will return to the usual place: Room 6206 of the James Clerk Maxwell Building, home of the Department of Mathematics of the University of Edinburgh.

As usual you can attend via Zoom:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: Yoneda36

And as usual, a video of today’s talk will appear here later.


Modes (Part 2)

6 November, 2022

When you first learn about the major scale it’s fairly straightforward, because they tell you about just one major scale. But the minor scale is more tricky, because they tell you about three—or actually four, two of which are the same!

The most fundamental of these is the natural minor scale. The C major scale goes

C D E F G A B C

The C natural minor scale goes

C D E♭ F G A♭ B♭ C

As you can see the 3rd, 6th and 7th notes of the scale are ‘flatted’: moved down a half-tone compared to the major scale. This gives the natural minor scale a darker, even ‘sadder’ quality compared to the major scale.

I prefer to work with note numbers instead of note names, not because I’m a mathematician so I love numbers, but because then we can simultaneously talk about different keys at once, not just the key of C. In this approach we call the notes of the major scale

1 2 3 4 5 6 7 8

and then the natural minor scale is

1 2 ♭3 4 5 ♭6 ♭7 8

Don’t ask me why the flats are written in front of the numbers now instead of after them—it’s just a convention.

Now, one thing about ‘common practice’ western harmony is the 7th tone plays a special role. It’s just a half-step below the 8, and we act like that dissonance makes it want very strongly to go up to the 8. The 8 is one octave above the 1, twice the frequency. Either the 1 or 8 instantly serves as a home base: we feel like a piece or passage is done, or momentarily at peace, when we play these notes. We say the 7 wants to ‘resolve’ to the 8, and we call it the ‘leading-tone’ for this reason: it suggests that we’ve almost reached the tonic, and makes us want to get there!

There’s much more we could say here, but it all combines to make people want a scale that’s like minor but contains the 7 instead of the ♭7. And since this scale is motivated by reasons of harmony theory, it’s called the harmonic minor scale. It goes like this:

1 2 ♭3 4 5 ♭6 7 8

However, now people singing this scale find it mildly awkward to jump up from ♭6 to the 7 because the distance between them is larger. In fact it’s 3 half-tones, larger than any step in the major or natural minor scale! One way to shrink this gap is to raise the ♭6 to a 6 as well. This gives the melodic minor scale:

1 2 ♭3 4 5 6 7 8

By now we’re almost back to the major scale! The only difference is the flatted 3. However, that’s still a lot: the ♭3 is considered the true hallmark of minorness. There are reasons for this, like the massive importance of the 1 3 5 chord, which serves to pound home the message “we’re back to 1, and this is the major scale, so we are very happy”. Playing 1 ♭3 5 says “we’re back to 1, but this is minor, so we are done but we are sad”.

However, singing up the scale is different from singing down the scale. When we sing up the melodic major scale we are very happy to sing the 7 right before the 8, because it’s the leading-tone: it tells us we’re almost home. But when we sing down we don’t so much mind plunging from the 8 down to ♭7, and then it’s not so far down to ♭6: these are both steps of a whole tone. If we do this we are singing in the natural minor scale. So what I called ‘melodic minor’ is also called melodic minor ascending, while natural minor is also called melodic minor descending.

Here I should admit that while this is an oft-told pedagogical story, the actual reality is more complex. Good composers or improvisers use whatever form of minor they want at any given moment! However, most western musicians have heard some version of the story I just told, and that does affect what they do.

To listen to these various forms of the minor scale, and hear them explained more eloquently than I just did, try this:

Grazie Terzian is the patient teacher of music theory I wish I’d had much earlier. You may feel a bit impatient listening to her carefully working through various scales, but that’s because she’s giving you enough time for the information to really sink into your brain!

Anyway: we’ve seen one form of major scale and three forms of minor, one of which has two names. All these scales differ solely in whether or not we flat the 3, 6 or 7. So, we can act like mathematicians and fit them into a cube where the operations of flatting the 3, 6 or 7 are drawn as arrows:

Here to save space I’ve written flatted notes with little superscripts like 3^\flat instead of ♭3: it makes no difference to the meaning.

This chart shows that flatting the 3 pushes our scale into minor territory, while flatting the 6 and then the 7th are ways to further intensify the darkness of the scale. But you’ll also see that we’re just using a few of the available options!

In part 1 I showed you another way to modify the major scale, namely by starting it at various different notes to get different ‘modes’. If we list them in order of the starting note—1, 2, 3, etc.—they look like this:

For example, Ionian is just major. But we saw that it is also very nice to list the modes from the ‘brightest’ to the ‘darkest’. Rob van Hal made a nice chart showing how this works:

Skipping over Lydian, which is a bit of an exception, we start with major—that is, Ionian—and then start flatting more and more notes. When we reach the Phrygian and Locrian we flat the 2 and then the 5, which are very drastic things to do. So these modes have a downright sinister quality. But before we reach these, we pass through various modes that fit into my cube!

Let’s look at them:

We’re now tracing out a different path from top to bottom. Ionian has no notes flatted. In Mixolydian we flat the 7. In Dorian we also flat the 3. Then in Aeolian we also flat the 6.

I mentioned that the ♭3 is considered the true hallmark of minorness. Thus, in the classification of modes, those with a flatted 3 are considered ‘minor’ while those without are considered ‘major’. So in our new path from the cube’s top to its bottom, we switch from major to minor modes when we pass from Mixolydian to Dorian.

Note that Ionian is just our old friend the major scale, and Aeolian is our friend the natural minor. We can combine the two cubes I’ve showed you, and see how they fit together:

Now we can get from the top to Dorian following two paths that pass only through scales or modes we’ve seen! Similarly we can get from melodic minor ascending to the bottom following two paths through scales or modes we’ve seen. In general, moving around this cube through the course of a piece provides a lot of interesting ways to subtly change the mood.

But two corners of our cube don’t have names yet! These are more exotic! But of course they exist, and are sometimes used in music. The mode

1 2 3 4 5 ♭6 7

is called harmonic major, and it’s used in the Beatles’ ‘Blackbird’. The mode

1 2 3 4 5 ♭6 ♭7

is called the melodic major scale, or also Mixolydian flat 6 or Aeolian dominant. It’s used in the theme song of the movie The Mask of Zorro, called ‘I Want to Spend My Lifetime Loving You’.

So, let’s add these two modes to our cube:

This is the whole enchilada: a ‘commuting cube’, meaning that regardless of which path we take from any point to any other point, we get the same mode in the end. We can also strip it of all the musical names and think of it in a purely mathematical way:

We could go further and study a 5-dimensional hypercube where we also consider the results of flatting the 2 and 5. That would let us include darker and scarier modes like Phrygian, Phrygian dominant and Locrian—but it would be tougher to draw!


Sundial Puzzle

4 November, 2022

I’ve quit explaining math on Twitter and moved my activities of that sort to Mathstodon. This is a branch of Mastodon, a federated social network that is run by its own users—not by an unpredictable self-centered billionaire.

It feels a lot like the internet of the late 80’s or early 90’s, with people pitching in to build things they themselves use, not serving as cogs in the giant machine of surveillance capitalism. I invite you to join us!

I plan to write things there, polish them up a bit and put them here. An example is my post about modes. Today I had fun solving a puzzle about sundials. My goal was to use the minimum amount of math.

Colin Beveridge posed this puzzle:

“Suppose I planted a metre-long straight stick vertically in the ground and traced the locus of the end of its shadow. What shape would it make? Happy to assume a locally flat Earth if it makes things easier.”

Equivalently: what curve is traced out by the shadow of the tip of a sundial during one day, if the shadow lands on flat ground?

The answer is: a hyperbola—or in one very special case a straight line!

To see this, work in Earth-centered coordinates and treat the Sun as a point S moving in a circle over the course of a day. Treat the ground as a plane P. Sunlight traces out a line L going from S to the sundial’s tip T and hitting this plane P at some point X.

As S goes around in a circle, what curve does X trace out?

That’s the math question I’m solving.

To solve it, we need an obvious math fact: as a point S goes around a circle, the line going through S and any point T traces out a cone.

And another less obvious but very famous fact: when we intersect a cone with a plane P we get a curve called a ‘conic section’, which can be a circle, ellipse, parabola, hyperbola, or a line.

So, the only question is which of these curves we can actually get!

As the Sun sets, the shadow of our sundial gets arbitrarily long—so we can only get a circle or ellipse if the Sun never sets.

We only get a parabola if the Sun sets in the exact same place on the horizon that it rises—since the two ‘ends’ of a parabola go off to infinity in the same direction.

All these cases are a bit unusual. In most circumstances the curve we get will be a hyperbola or a straight line.

We get a straight line only when the Sun rises at one point on the horizon, is straight overhead at noon, and sets at the opposite point of the horizon. This would happen every day if you lived at the equator and the Earth’s axis wasn’t tilted. But in reality this situation is rare.

So, the shadow traced out by a sundial’s tip is usually a hyperbola!

You can play around with these hyperbola-shaped shadows here:

• Intellegenti Pauca, Hyperbola shadows, Geogebra.

There’s a lot more one can say about this: for example, what happens with the change of seasons? But I wanted to keep this simple!

Click on this picture for some details about a nice sundial that shows off its hyperbolae:


Categories and Epidemiology

1 November, 2022

I gave a talk about my work using category theory to help design software for epidemic modeling:

Category theory and epidemiology, African Mathematics Seminar, Wednesday November 2, 2022, 3 pm Nairobi time or noon UTC. Organized by Layla Sorkatti and Jared Ongaro.

This talk was a lot less technical than previous ones I’ve given on this subject, which were aimed mainly at category theorists! You can see it here:

Abstract. Category theory provides a general framework for building models of dynamical systems. We explain this framework and illustrate it with the example of “stock and flow diagrams”. These diagrams are widely used for simulations in epidemiology. Although tools already exist for drawing these diagrams and solving the systems of differential equations they describe, we have created a new software package called StockFlow which uses ideas from category theory to overcome some limitations of existing software. We illustrate this with code in StockFlow that implements a simplified version of a COVID-19 model used in Canada. This is joint work with Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson.

Check out these papers for more:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson, Compositional modeling with stock and flow diagrams.

• Andrew Baas, James Fairbanks, Micah Halter, Sophie Libkind and Evan Patterson, An algebraic framework for structured epidemic modeling.

For some more mathematical talks on the same subject, go here.


Modes (Part 1)

1 November, 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 have a note that’s a 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!

The mathematics of modes

We clearly have an action of the 7-element cyclic group \mathbb{Z}/7 on the set of modes I’m talking about: they’re defined by taking the major scale and cyclically permuting its notes. But as we’ve seen, inversion gives an action of \mathbb{Z}/2 on the set of modes, with Dorian as its only fixed point.

Putting these two groups together, we get an action of the 14-element dihedral group \mathrm{D}_{14} on the modes. This is the semidirect product \mathbb{Z}/2 \ltimes \mathbb{Z}/7. More intuitively, it’s the symmetry group of the regular heptagon! The modes can be seen as the vertices of this heptagon.

We’ve also seen the modes have a linear ordering by ‘brightness’. However, this ordering is preserved by the symmetries I’ve described: only the identity transformation preserves this linear ordering.

All this should have been studied in neo-Riemannian music theory, but I don’t know if it has—so if you know references, please tell me! The \mathrm{D}_{14} group here is a baby version of the \mathrm{D}_{24} group often studied in neo-Riemannian theory. For more, see:

• Alissa S. Crans, Thomas M. Fiore and Ramon Satyendra, Musical actions of dihedral groups, American Mathematical Monthly 116 (2009), 479–495.

More on individual modes

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:

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

For more, read Modes (part 2).


This Week’s Finds – Lecture 6

28 October, 2022

 

Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. Here I use Dynkin diagrams to classify compact Lie groups—and especially compact semisimple Lie groups.

This is one of a series of lectures at the University of Edinburgh on topics drawn from my column This Week’s Finds:

http://math.ucr.edu/home/baez/twf/

Cover image by Tom Ruen, CC BY-SA 4.0


This Week’s Finds – Lecture 5

24 October, 2022

 

Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. Here I explain how Dynkin diagrams classify root lattices.

For more, read my paper “Coxeter and Dynkin diagrams” here:

http://math.ucr.edu/home/baez/twf/

To attend the talks on Zoom go here.


This Week’s Finds – Lecture 4

20 October, 2022

 

Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. Here I explain how Coxeter groups classify finite reflection groups: that is, finite groups of transformations of \mathbb{R}^n generated by reflections.

For more, read my paper “Coxeter and Dynkin diagrams” here:

http://math.ucr.edu/home/baez/twf/

To attend the talks on Zoom go here.