Every voice is the same click. The only difference between any two of them is how often it happens. If you hear nothing on an iPad, check the little switch on the side.
Filled: built out of nothing but 2s, 3s and 5s. Struck out: needs a 7 or an 11 or a 13 or something worse. Dashed: passes the test but sits less than a semitone from 24, too close to be a step of its own. What survives is the scale.
You have already seen two things ticking at once turn into a chord. So here is the obvious next question, and it has a much better answer than it deserves: is there a rhythm that gives you a whole scale?
There is. Here it is.
24 27 30 32 36 40 45 48
Eight tickers. In the time the slowest one gets 24 clicks, the others get 27, 30, 32, 36, 40, 45 and 48. Speed the whole thing up and you have every note of a major scale — the eight notes people sing as do re mi fa sol la ti do. The last ticker is going exactly twice as fast as the first, which is the octave closing the loop.
Try the speed slider down at the bottom and listen honestly. Twenty-four against twenty-seven is not a rhythm any human being can follow. The pattern doesn’t come back around for twenty-four clicks, and you lost the thread around the fourth one.
Which is worth sitting with for a second. This scale has no slow version. It only becomes something you can understand once it is going too fast to count. Some relationships are only ever audible as pitch, and this is one of them.
Switch everything off except 24, 30 and 36. Those three numbers are 4, 5 and 6 with everything multiplied by six — so what you are hearing is four against five against six, which is a rhythm you could tap. It comes back around every four slow clicks.
Now try 32, 40 and 48. Divide by eight: 4, 5, 6 again. Identical rhythm, planted somewhere else.
And 27, 36, 45. Divide by nine: 3, 4, 5 — which is the same three notes as the last one, standing on a different one of themselves.
Between them those three little chords use every one of the eight numbers and nothing else. The major scale is one small chord, planted in three places. Nobody wrote a scale. Somebody built one thing and put it down three times.
Look at the strip of numbers above the reading, running from 24 to 48. Take each one apart into the smallest pieces it will break into. 30 is 2 × 3 × 5. 32 is five 2s. 45 is 3 × 3 × 5.
Now throw away every number that needs anything bigger than a 5 to build it. 28 needs a 7, so it goes. 33 needs an 11. 39 needs a 13. 37 is a prime all by itself and can’t be built out of anything.
Nine numbers survive: 24, 25, 27, 30, 32, 36, 40, 45, 48.
Eight of those nine are your scale. The one intruder is 25, and it fails for a completely different reason — it sits so close to 24 that it isn’t a separate step at all, less than a semitone away, crowding the note it came from. Throw out anything that close and what is left is exactly the major scale. Nothing missing, nothing spare.
So the scale is not a list handed down by somebody. It is what falls out of a sieve. Numbers made of 2s, 3s and 5s, no two of them crowding each other. That is the whole recipe, and you could run it on paper in about five minutes.
Tap any key on the strip and the bottom voice lands exactly on it. Now read the row of note names above the keyboard.
Three of the eight come out with a number stuck to them — they miss the nearest key by enough to be worth mentioning. And it is the same three every time, wherever you put the bottom note: the third one up, the sixth, and the seventh.
Look at what those three are built from. 5/4, 5/3, 15/8. Every one of them has a 5 in it. And the ones with no 5 — 9/8, 4/3, 3/2 — land within about two hundredths of a key, near enough that nobody would ever complain.
The piano is good at 3s and bad at 5s. The notes built out of threes it gets almost exactly right. The notes built out of fives it misses by something like a seventh of a step, every time, in every key, on every piano ever made. That isn’t a fault in the instrument. It is the bill for a keyboard where every key works the same as every other, and somebody decided a long time ago that it was worth paying.
Every set of tickers has a slowest thing they all agree on — the moment they all land together, which then repeats. The bench shows it as the floor underneath.
Turn on just 24, 30 and 36 and look at it. The floor sits comfortably below the chord, low but solidly a note, and the chord feels like it is standing on something.
Now turn on all eight. The floor drops through the bottom of hearing and turns into a flutter — too slow to be a note at all. There is nothing down there to stand on any more, which is exactly why all eight notes at once is a beautiful sort of fog rather than a chord.
A chord has a floor. A scale does not. That is most of the difference between them, and you can watch the number fall.
One last thing, and it may be the best of them.
Try the simplest possible rhythm instead: just counting. 8, 9, 10, 11, 12, 13, 14, 15, 16. Speed that up and you do get a scale — it is the scale a bugle plays, and the reason a bugle can play anything at all without valves.
But it is not this scale. It goes wrong in three places, and the three places are 11, 13 and 14. Every one of those needs a 7 or an 11 or a 13 to build it — the numbers the sieve threw away.
So plain counting gives you a scale. It just isn’t ours. The difference between the scale you grew up with and the one a bugle plays is entirely a matter of which primes somebody was willing to allow.