I was in New York recently, and there’s no way I could have
visited without taking a trip to the Museum of Mathematics, or MoMath. It was a
wonderful museum, and I was very impressed with MoMath’s particular brand of
“low floor, high ceiling” math – the kind of math that I love most. A lot of
the exhibits on display were plays off of concepts I’d seen or done before, but
the central sculpture on floor -1 was completely new, beautiful, and
fascinating. It’s called the Harmony of the Spheres, and all I knew is that it
had something to do with the mathematics of music. It wasn’t working properly
that day, but it was supposed to play notes and light up depending on which
sphere you touched.
I was immediately drawn the sculpture, and asked a passing
volunteer what it meant. They didn’t know exactly what each node and connection
represented, and then it became a puzzle to me and I stopped asking questions, pulled
out a notebook, and sat down and stared at it.
The picture doesn’t do it justice – 3D sculptures are pretty
hard to flatten into a picture without losing a lot. That didn’t stop me from
trying it in a different way, and it came out 90% aesthetic, 10% helpful. The vertices
(where two line segments meet) are the large circular nodes in the picture, and
the line segments themselves are the connections between the nodes. The lengths
of the line segments doesn’t mean anything – this was actually the shortest I
could get them and still keep the lines straight.
At the very least, it should give you a vague understanding
of about how many nodes there are and how they’re oriented.
I color-coded the circles to be the same as the colors they
are in the picture in this set of representations, which should help a lot.
This first node, at the center of it all, is the rightmost sphere in the
picture, the red one. It’s tiny, compared to the picture, but that’s actually
the biggest I could get it without covering up the details on the rest of the
drawing.
Then, there’s two pink nodes, connected to each other and
the red node.
Then, there’s three purple nodes, connected in a line (because the ones on the ends aren't connected to each other). The ones
on the ends are connected to the pink nodes on that edge, and the one in the
middle is connected to both.
Next is four blue nodes, connected in a line, with a similar
“reverse pyramid” set of connections to the purple nodes.
You might be noticing now that I arranged them in a square,
and the purple ones in an equilateral triangle, and so the next set must be:
Five cyan nodes, arranged in a pentagon.
And then six green nodes, arranged in a hexagon. At this
point, if we ignore all the other nodes, this is a pretty simple construction.
It looks like this.
But now (instead of a heptagon) another hexagon of yellow nodes
come in, and it complicates things. They are not only linked in a line, like
the previous set was – the ends are linked together.
I couldn’t draw it as a regular hexagon – that would mean
the rest of it would have to be curved, but don’t read into that. This 2D picture
is pretty, but consequences like this irregular hexagon originate from the squishing
of 3D to 2D, not from any interesting or intrinsic properties.
Speaking of messy squishing, look at what the simplified
version looks like now.
But all that’s left is a triangle of three orange nodes,
And one final red node.
And here’s the simplified version, in case it’s easier to
read. It gets pretty messy at the end.
So, we’ve got this weird set of nodes and connections, there’s
some sort of color coding involved, and no one knows what it’s supposed to mean.
It’s supposed to be tied to music somehow, but there’s nothing fundamental
about music that points to this strange structure. What’s even the point of all
this?
After about an hour of scribbling and walking repeatedly
around the sculpture to get different angles, I figured it out. This
sculpture is a complete mapping of three-part harmony! Warning, there’s a
teensy bit of music theory up ahead, but I tried to keep it very simple.
Let me explain how it works. Imagine you hit a piano with three
fingers, each pressing down a key. There’s a couple of possibilities of what that
would sound like, right? They could be playing a major chord, a very discordant
set of notes, or even all the same note.
With that in mind, here’s the kicker: each node on this
sculpture is a possible arrangement of your three fingers.
But wait a minute, I hear your fictional voice say, there’s
a ton of keys on the piano, and many more combinations of three! There’s so few
nodes on that picture! And you’re right: for this sculpture, we’re only
concerned about the intervals, not the notes themselves. If your fingers fall
on three consecutive keys on the right side of the piano, and then three consecutive
keys on the left side, those are the same for our purposes. It’s the same with
relation to itself, you just picked it up and moved it with relation to the
piano, and that doesn’t matter. Changes in octave also don’t matter. A piano
has the same twelve notes repeated over and over – the pitches are different,
but we consider a high C and a low C and a middle C all the same.
The obvious next question is “what do two connected nodes
mean?” followed shortly by “what do the colors mean?” This sculpture is ingeniously
designed to answer just those questions. The connections connect two “adjacent”
nodes in harmony – it means that one finger changes its note by a half-step.
If you’re mashing down three notes, and one finger decides to move to the next
key either above or below it, you just traveled from one node to another by
using one of the connections.
Well, which nodes are which? Let’s start with the very first
node I introduced. The red node, at the rightmost side of the photo, the bottom of the diagram. This node is unison – all three fingers are on the same key. Let’s
call it E, for convenience.
If you want to go to one of the pink nodes, it doesn’t
matter which one of your fingers you move. It does matter, however, whether you
choose to go up or down with it. Let’s say that the right pink node is going down,
and the left pink node is going up. That means the right pink node is “two fingers on one key, one finger on the key just below it” and the left one
is “two fingers on one key, one finger on the key just above it”. So far so
good.
Now, let’s try to get to one of the purple nodes. If you
have two fingers on E and one on F (putting you on the left pink node), and
you move one of your E fingers up to F, you actually move over to the other
pink node, because now you have two on F and one on E. No, to move to a purple
node, you have to get farther away. You can move the F up to F#, which gets you
to the left purple node, or one of your E fingers down to D#, which gets you
to the center purple node, with one finger on each of three consecutive keys. Those
are the only places you can go from this node, outside of moving the F back
down to E and returning to the red node. The right purple node can only be
reached through the right pink node, as you can see on the drawing and the
sculpture!
I’m not going to go through all of the nodes, but you should
be starting to see how clever this sculpture is! The color-coding is a measure
of how far away from unison and each other the notes are getting. Pink means that
one step was taken away from unison, purple means that two steps were taken,
etc.
And eventually, because there are only twelve notes, any
note that’s getting higher or lower comes back to unison! Look at this progression,
of keeping two fingers on the same note, and increasing the note on the other
one continuously – it comes back around!
The last, furthest away node, also colored red in the
sculpture, is the farthest any three notes can get from each other before they
start coming back. It’s the intervals for an augmented chord – a full four
semitones between each of the three notes! That node is connected to two orange
nodes – those are what we know as major and minor chords!
And that is the story of the beautiful sculpture at the
Museum of Mathematics and how much fun I had figuring out how it worked!
