Above about twenty a second we stop saying clicks and start saying Hz. Hz just means per second — the number on screen never changes its meaning, only its name. If you hear nothing on an iPad, check the little switch on the side.
The note you hear is the ticking rate. Not the sound of the click itself.
Each click in here is a very short tone of its own, high up and gone in a few thousandths of a second — and you never hear it as a note. Move the click pitch slider while a note is playing: the character changes completely, and the note does not shift a hair.
Worth doing once, because it isn’t what you’d guess. The click decides what the note sounds like. The rate decides what the note is.
Top row is the slower voice, one dot per click. Bottom row is the faster one.
The two gold lines are the same instant — the moment both voices land together — drawn once at the start and again at the end, because that is where the pattern begins and where it begins again. Everything between them happens once and then repeats for as long as you leave it running.
While it is slow enough to follow, a thin line travels across so you can see where you are. Speed it up and the line goes away, because by then nobody could follow it anyway.
Start something ticking. Slowly — two ticks a second, about the speed of a dripping tap. Now start a second thing ticking a little faster, so that it gets six ticks in the time the first one gets five.
You can hear that. It’s a rhythm. If you pay attention you can find the moment where they both land at once, and you can count your way to the next one.
Now speed both of them up, keeping that six-against-five relationship exactly where it is.
Around eight ticks a second you lose the ability to count. Somewhere around twenty, something else happens, and it’s worth being ready for it, because it’s strange: the ticking stops being ticking. It turns into a note. A steady, held, musical note.
Nothing was added. Nobody put a note in there. There is no note in there. There’s a tick, happening a lot.
That’s the first thing, and most people never find it out: a note isn’t like a fast rhythm. It is one. Anything that repeats fast enough stops being events and becomes a pitch. Your ear has a speed limit, and on the other side of it, counting turns into hearing.
Look at what your two tickers are doing now. The slow one is going 200 times a second. The fast one, 240.
240 against 200 is 6 against 5. It’s the same six and five you started with, back when it was a rhythm you could tap along to. You changed the speed and nothing else. And what you’re hearing now is a chord.
There’s a word people use for “times per second” — hertz, usually written Hz. It isn’t a new idea. 200 Hz means 200 times a second. That’s all. The number didn’t change; the name for it did, because it got too fast to count and people wanted a different word for the fast version.
You might reasonably suspect that six and five got chosen by some committee. They didn’t, and this is the part everything else depends on.
Pluck a guitar string. The string is not vibrating at one speed. It’s vibrating at its speed, and at twice that speed, and three times, and four, and five, and six — all at the same time, getting quieter as they go up. Every one of them is happening in the same string at the same moment.
That stack of speeds is why a violin and a clarinet playing the same note don’t sound the same. They’re the same note with different amounts of each ingredient.
So six and five were never music theory. They’re two ingredients you’ll find inside a single plucked string, held up next to each other. The whole numbers were already there. Somebody just noticed.
Keep the stack in mind. Every note in this bench is dragging one along behind it, and in a moment that turns out to be the whole story.
Switch the bench to three against two — 200 and 300 — and it is about as easy on the ear as two notes get. Switch it to forty-five against thirty-two and it is horrible. It looks like the difference is the size of the numbers. It isn’t.
Two hundred a second doesn’t arrive alone. It brings 400, 600, 800, 1000. Three hundred brings 600, 900, 1200.
Look at 600. They both have it — exactly, not nearly. Several of their ingredients land on the same spot, and the ones that don’t coincide sit comfortably far apart from each other.
Now take 200 against 281, which is what forty-five against thirty-two comes to. Its stack is 562, 843, 1125. Nothing coincides with anything. And worse — this is the actual trouble — 562 lands right beside 600. Not on it. Beside it.
Two ingredients close but not equal refuse to blend. They wobble against each other. A slow wobble you hear as a wobble; a fast one you don’t hear as a wobble at all. Your ear gives up and reports it as grating.
So the scrape is not the numbers being big. It is ingredients grinding.
You can check this in about ten seconds without leaving the page. Pull the brightness slider all the way down — that strips the stacks away and leaves something close to two bare tones. Now switch between the smooth pairs and the rough ones. If the grinding is what makes the difference, most of the difference should go, because there’s almost nothing left to grind. Run brightness back up and it should come storming back.
Same two notes. Same fraction. Nothing changed but how many ingredients are in the room.
Both voices are ticking at their own rates. But the two of them together also make a pattern, and that pattern repeats at a rate of its own. With 200 and 300, the whole arrangement comes back around a hundred times a second.
A hundred times a second is a note.
You probably won’t hear it as a separate voice, and anyone who tells you it’s obviously there is overselling. It’s more of a lean. But look at everything in the air: 200, 300, 400, 600, 800, 900, 1200. That is very nearly the stack belonging to a note at 100, with a few members missing. Your ear has decent grounds for suspecting a 100 in the room, and it leans that way.
That lean is why two notes seven keys apart feel like they are standing on something.
For forty-five against thirty-two, the equivalent bottom would be around six a second. You already know what six a second is. It isn’t a note, it’s a flutter. Nothing to lean on — which is part of why that pair sounds like it is hanging in mid-air with nothing underneath it.
Here’s a thing about a keyboard that nobody mentions.
Go up four keys. Then go up three more. You’re at seven. That’s just adding: 4 + 3.
But in terms of what’s happening to the speed, four keys up is multiplying by 5/4, and three more is multiplying by 6/5. Multiply those together — 5/4 × 6/5 — and the fives cancel and you get 6/4, which is 3/2.
And 3/2 is exactly what sits seven keys up.
So: you added the positions and it multiplied the fractions. Every time. Walking up a keyboard is doing multiplication with your hand, and it works no matter which two you pick. Multiplying fractions is annoying. Counting to seven isn’t. The keyboard does the annoying part.
Take any of these fractions and break the top and bottom into the smallest pieces they’ll go into. 6/5 comes apart into 2 × 3, over 5.
The 2s aren’t very interesting — doubling a speed just moves you up an octave, which your ear treats as almost the same note. Throw them out. What’s left is the 3s and the 5s, and those are the parts that give an interval its flavour. Some intervals are made of 3s. Some are made of 5s. Six-against-five is made of one of each, which is why it sounds like two things at once, because it is.
That’s what the colours are doing. Not decoration. The 3s and the 5s.
One warning, though, and the last section is about it. The colours tell you how a note is built out of home. They do not tell you how it will sound. Those turn out to be two different questions.
Seven keys up on a real keyboard isn’t 3/2. It’s very slightly less.
You can watch this happen. Tap any key on the strip with three-against-two selected: the slower note reads dead on, and the faster one reads about +2¢. A cent is a hundredth of a key. So the true 3/2 sits two hundredths of a key above where the keyboard put its own.
Two hundredths of a key sounds like nothing. It isn’t nothing. Play a fifth on a real piano, somewhere in the middle, and listen underneath the notes rather than at them. There’s a slow pulse in there, roughly once a second, as though the sound is breathing. That is the two hundredths, made audible: the sixth ingredient of one note and the fourth of the other, close but not equal, wobbling against each other once a second. Piano tuners listen for exactly that pulse and count it out loud.
Somebody decided, a few hundred years ago, that it was better to be slightly wrong everywhere than perfectly right in one place and unbearable in the rest. That decision is inside every piano you have ever heard, and it has been beating once a second ever since.
Go up six keys — halfway up the octave, the middle of the middle. It’s the interval people used to call the devil’s interval, mostly because it refuses to sit still.
There are two buttons on this bench for that one spot. 7:5, which looks tidy and reasonable. And 45:32, which looks like something went wrong.
Play them one after the other.
They sound the same. Not similar — the same. They land within about a twelfth of a key of each other, which is under what anyone can hear. Two fractions that look nothing alike, arriving at one sound.
Which ruins a rule that was starting to look extremely solid. If small numbers caused sweetness, then 7:5 — small, tidy, perfectly respectable — ought to sound clearly better than 45:32. It doesn’t sound better. It doesn’t sound like anything different at all.
And this isn’t a fluke you stumbled on. Pick any spot on the keyboard and there is a tidy-looking fraction sitting close enough to it that nobody alive could tell the difference. The fractions aren’t lying to you. They just aren’t the thing that decides.
What decides is where the ingredients land. Real stacks, in real air, grinding or not grinding. Which is why you can take one pair of notes and make it mild or make it unbearable using a knob that doesn’t touch either note.
There’s a version of this you have almost certainly already heard. A flute playing those six-keys-apart notes sounds mild, a bit unsettled, nothing worse. A distorted electric guitar playing the identical two notes sounds like a threat. Distortion is a machine for adding ingredients — that is the entire job. Which is why rock guitarists build nearly everything out of fifths and leave most other intervals out: through that much distortion a fifth stays clean and almost everything else turns to mud.
Nobody worked that out on paper. They found it by ear, extremely loudly, some time around 1968. And it is the same fact you have sitting on a slider.
The simple fractions are still where the sweet intervals live. They are just not the reason.