(Or why the Fundamental Theorem of Arithmetic means we shouldn’t get so stressed when big projects go tits-up. A journey via the physics of music, the ingenious Bach, and a whole lot of faking.)
Take a string. Tie it tightly between two fixed points. If you have a guitar to hand, even better, but it’s not essential. In fact, you should be able to grasp all of what follows without any props at all, providing you’re prepared to take on trust some of the descriptions.
Please now forget everything you know, assume, or half-remember being taught about music theory.
Pluck the string and listen to the sound. That note – the clear, strong one you can hear? This is our starting point. Our fundamental. Hold your finger lightly against the string, half-way along. Pluck again, in the centre of one of the halves. Hear that new note?
If the string’s tight enough, it should be higher than the open-string note, but in a way, its twin. Sounding very similar in character – in perfect harmony in fact – just higher. (“Twice” as high in a sense. The string is vibrating exactly twice as quickly as it did when it was twice the length.)
If you laid the sound waves of the open-string note side-by-side with those of the higher note, every other ‘peak’ of the waves would be in exactly the same place. That makes for great harmony.
(All going swimmingly so far; let’s mix it up by finding another note).
This time hold your finger lightly against the string, but a third of the way along (doesn’t matter from which end). Pluck the shorter part of the string with your other hand. Now this is quite different. You get a note that’s even higher than the ‘half-way’ note. Play the ‘half-way’ note again. Then the “third-the-way-along” again.
Listen to the difference. If they were played together at the same time (you’d obviously need two identical strings) they’d also sound in pleasant harmony. The higher note’s sound-waves are packed together a little more tightly, but laying the two side-by-side, the peaks would again coincide regularly, this time in a 3-to-2 ratio, rather than 2-to-1 as before. Let’s introduce some labels here to cut down on the hyphens… Call the ‘half-way’ note the ‘octave’.
(Try and wipe from your mind the oct- prefix as being anything to do with the number eight. All that can come later, as I hope will become clear). Call the ‘third-the-way’ note the ‘dominant’. What we’ve done so far, without using any musical theory to speak of, is build a relationship that mathematically links two notes together. Two notes that are in perfect harmony.
This simple description will be the building block of what follows. Which should get quite a bit more weird soon – bear with me. (By the way, the demonstration with the tied string and so on couldn’t have been done using notes you’d find on a piano. Not an ordinary piano anyway. Because a piano has to ‘fake’ its notes, just a tiny bit. All will become clear, I hope.)