Coincidentally

frequency smearing was the name of one of the bands I played in in my 20s that's not true but it is more or less true that inside our ears we have a pair of pianos coiled up and upside down and sitting there behind each piano is a tiny genius mathematician without whose impressive work calculating ratios music would be meaningless noise. Thank

you. Yes, do

you have a

question? I do have a question if it's okay I am tasked by the CBC radio show ideas to find out the role of five and concepts of beauty and truth. Are you using the number five in the work you're doing here

this evening? Well thank you Tom for asking about the number five. In music theory the number five really pertains to the chord the five chord which is a very important chord. Oh how so?

Because

it's like a question mark. This

is Rebecca Hennessey with her band makeshift island I've interrupted their gig the tranzac club in Toronto for a dose of music

theory. It's like anything could happen on a five chord. Can

you demonstrate in some way? Well here check this

out I'm

going to play the piano for you. Okay okay.

So if we're in this key if this is the one key the the one chord the five chord that well this is the the one chord this would be the one the one scale maybe see major scale for those of you who care um when we get to hear what does that make you want to do anybody? Go to the next place. Go to it really makes you want to go there. So this is the question mark but we could go a lot of different places you can go. How about this one?

Yeah I mean I'm

hearing all the sounds but where's the five in all this? The five is how do we resolve what is the it's the the cadence? Well I think the thing Rebecca didn't answer is that the chord she's talking about is built on the fifth note of the scale. That's what the five chord is. I might need to hear it can you play a one chord and then and then take it to the five is that possible? Yeah

let's jam on the one chord everyone.

Okay now.

You can be flat, seat, seat, rest, seat okay.

One, one, okay.

One, one, one.

And then if you're going to the five chord

ready.

One, two, three, four.

One, two, three, four. Next one, one, two, three, four.

There you go. Take it to the five.

I'm playing at five.

One, two, three, four.

One,

two, three, four. Okay thank you. Thank you very much. Thank you for asking.

Anything could happen on the five things that happen on the five.

I'm now sitting at a standard upright acoustic piano. It has 88

keys. Here they all are. Try to count them as I

go.

Now unless you're extremely quick that was probably too fast for your account but in another part of your brain you may have been doing much more complicated math effortlessly at breakneck speed without meaning to. All of these keys are named using only seven letters. That's A. That's also called

A.

That's also

called A.

Why? We have 26 perfectly good letters of the alphabet and if we must use letters for some reason to describe these keys why not use them all. Even if you've had no musical training it's likely your brain already knows the answer or at least you can feel why. The frequency of each A is double the frequency of the last one. If my piano was in tune this A would be 440 hertz. Physically speaking waves of pressure, justling air particles, hitting your eardrum exactly 440 times per second and your brain somehow keeps count. That's remarkable. Even more remarkable here is 880 hertz. It's not universal that people pick up on the connection between these two sounds but most people do. It's your brain recognizing a ratio of 2 to 1. 880 is exactly twice 440. Impressive but multiplying by 2 is still fairly easy. What about a ratio of 3 to 2? 440 times 1.5. 660 hertz. That's called a perfect fifth. The first note in a musical scale and the fifth note of that scale. A, B, C, D, E. It's a multiplication relationship. The same multiplication relationship that 2 has with 3. Just in case you're not really sure you're tuned into this. That's the pair of frequencies with a more difficult relationship. It's square root of 2 to 1. Now square root of 2 is a famous irrational number. The first Greek mathematician to go on about it to his colleagues was reportedly murdered for causing trouble. And this sound, known as the devil's chord or the flat fifth,

is generally considered

to be jarring assuming that your brain is doing the math. Now that said, of course, you can acquire a taste for it. Okay, let's put

the devil back in the hole. We're now

getting to how the five chord means what it does in music. Both types of fifth note, the diabolical one and the perfect one, can claim to mark the halfway point between an A and the next A, or a B and the next B for that matter. The reason the perfect fifth is halfway is because that's where a ratio of 3 to 2 gets you. This E at 661 is halfway, mathematically, between 440 Hertz and 880. This is the sense of suspension at a halfway point that gives the five chord its personality. It's like a

question mark. Or if

a sum it's like the turnaround point in a journey that takes you out and then

back. Each

note in the five chord is 1.5 times the frequency of each note in the chord you started with. At that halfway point we can pause, stop for lunch maybe. We're not seriously uncomfortable for a while.

But the

tiny mathematician at the piano inside your ear can tell you we're not home. We're dealing here still with a fractional relationship and it feels good when we land safely back on that whole number, the ratio of 1 to 1, 2 to 1, 4 to 1, or 8 to 1. A whole number feels like home.

Hi, I'm Kate Kiragawa, historian of mathematics and I've

been writing a book with Timosie Revell, The

Secret Lives of Numbers.