The strobe flashes once for every wobble of the bottom note. That note is therefore caught in the same place every time and never moves — it is the thing everything else is measured against. Nothing here is animated for effect: every drift you see is a real difference in frequency.
Take a loudspeaker and look at it from the side, so the cone is a single line pushing forward and back. Give it walls to move between — call them { and } — and the whole of what a note does is one bar sliding between them, very fast.
Too fast to see. So put a strobe on it: a light that flashes, briefly, at a rate you choose. Between flashes the room is dark and the bar might as well not exist. You only ever see it at the instants the light is on.
Set the strobe to flash once for every wobble of the note, and something odd happens. Every flash catches the bar at the same point of its journey. The bar stops. It is still moving — hundreds of times a second — and it looks like it has been nailed down.
Here is the part worth having, and it is why this bench draws a circle above every window.
A note going back and forth is not really a thing sliding along a line. It is easier to think of it as a point going round and round a circle at a steady speed — and the bar you see is just its shadow, cast sideways onto the wall. Round and round becomes back and forth.
That matters because the shadow throws information away. Near the edges the point is turning round and the shadow dawdles; through the middle it is moving straight across and the shadow races. Two quite different moments can put the shadow in the same place. The circle never lies like that.
So watch the circle, and read the bar as what a speaker would actually show you.
Now put a second note on, some fraction faster, and keep the strobe on the first.
The second note is not caught in the same place each time. It lands somewhere else, then somewhere else, and after a while it starts repeating — and the number of places it visits is exactly the bottom number of the fraction.
Three against two: two dots. Four against three: three dots. Six against five: five. The fraction stops being arithmetic and becomes something you count on a screen.
Better than that: the bench draws a line from each flash to the next, in the order they happened. For simple fractions you get a shape. A triangle, a square, a pentagon. And sometimes a star — because the dots are not always visited in order round the rim; seven against five skips two places each time, and skipping two places round five points is exactly how you draw a five-pointed star without lifting the pen.
So every interval has a shape. Not a picture of one — the shape is what the thing does.
Switch the tuning over to the way a keyboard does it and keep watching.
The shape stops closing. The dots never quite land where they landed before, so the figure creeps round the rim, slowly, and it will do that for as long as you leave it running. It never repeats. Not ever.
That is not a fault in the drawing. A keyboard’s intervals are deliberately not simple fractions — they are chosen so that every key works the same as every other, and the price is that nothing quite lines up. On a true fraction the shape closes and freezes. On a keyboard it crawls.
And here is the good part: the speed of the crawl is the size of the mistake. Start the sound and listen while you watch. There is a slow throb underneath the two notes, and the shape comes back round to where it started once per throb. Exactly once. The thing your eye is watching and the thing your ear is hearing are the same number.
One warning, and it is a real one.
Try seven against five, then forty-five against thirty-two. On this screen they could not look more different: five dots and a clean star, against a mess of thirty-two that barely holds still.
Now listen to them. They sound the same. They are the same, to closer than anyone can hear.
So this is a picture of the arithmetic, not a picture of the sound. It is honest about fractions and it will happily tell you two things are miles apart when your ears cannot tell them apart at all. Trust it about numbers. Do not trust it about music.
Which is worth knowing about pictures in general. A good one shows you something real. It does not follow that the thing it shows you is the thing you care about.