How Math Fixed Music – Rational Exponents Sound Good

Have you ever noticed how a guitar’s frets are increasingly closer together at the end of the fretboard, but farther apart at the top? Have you ever wondered why a grand piano is shaped kind of like a harp?

In this week’s episode of Wednesday’s Why we are going to switch it up a bit. Usually I take something in math that is misunderstood or just assumed to be true and unpack it in a way that you not only can understand why it’s true but also develop the ability to read math and unpack other things on your own. The purpose of Wednesday’s Why is to promote mathematical literacy.

But this week I’m going to show one way math is used I music, to make it consistent and thus, more pliable. But don’t worry, the math is only at the end and I’ll walk you through it, and there is not any homework or a pop quiz. And the music theory is as basic as can be, everybody probably already knows this stuff.

Throw a rock in a calm body of water and you see ripples. We can see them because they’re slow and water. Sound is similar to this, except in the air, we can’t see them. They’re also way too fast to see. Side note, an oscilloscope can be used to “see” sound. I’ll put a link to this in the description below. Way cool.

Anyway, the sounds we hear are like ripples and there are specific sounds we call notes or semi-tones. In Western Music we have twelve semi-tones. A, A#, B, C, C#, D, D#, E, F, F#, G, G#. (Instead of sharps #, we could use flats, but sharps are easier to type, so that’s what we’ll use.) Why they’re named these letters, I don’t know. If you know, leave a comment below, let me know!

And these notes are a big deal. They’re like a canvas for a painter. No canvas, no painting, right? The canvas for music is made up of these twelve notes (semi-tones). That’s it. (I’m making a very broad simplification here, but the spirit of the statement is true.)

Well, you might wonder, why are there so many keys on a piano?

That’s because the next note after G# is … A, again. But not the same A, an octave of A. The next D, for example, is an octave of the first D. It’s cyclic. A $–$ A# - B $–$ C $–$ C# - D … G# and back to A, again. Forever and ever and ever more, in both directions…

We hear the relationship between these tones. Musicians can use those relationships to make things sound happy, or sad, or angry. And if music is made outside of those twelve tones, it sounds like horrible, we would call it out of tune. (insert music appropriate for each and then untuned guitar)

The relationship between the notes and octaves is essential. It’s what makes music, well, music.

Before we can understand octaves, we must understand what notes are. Remember the deal with the ripples? Well, we can hear how fast a ripple in the air is. It’s similar to seeing how fast a ripple in the water is. How do we do that? Well, we compare how far apart the peaks and valleys are, right? (insert ripple clip) How fast those ripples, or waves, repeat is called their frequency.

It turns out, the tone A repeats 440 times a second! The next A, its octave, repeats 880 times per second. And if we went backwards, the A before our original A, repeats 220 times per second. The A before that, 110. Do you see what’s happening with octaves?

An octave is twice as “fast” as its base note. I’m sure there’s a technical term for this, but we’ll use base note here.

Music is nice and symmetrical, predictable and because of that we can combine different notes to establish moods. It’s very important then, that the relationship between A and C is the same, regardless of which A and C we are dealing with.

So the question is, how do we break apart the tones between one A and it’s octave?

Well, we need to break it up into twelve parts, so let’s just divide. Let’s start with the A that’s 220. It’s octave is 440. (The units are Hz, but we’re not going to worry about that for this demonstration). Let’s see, from 220 to 440 is a difference of 220. We need 12 parts out of that 220, so 220 divided by 12 is … gulp, it’s a fraction, but it’ll work out nice, you’ll see. To start at 220 and land on 440, with 12 equal steps, each step is an additional $18\frac{1}{3}$. Let’s see what this looks like. Don’t worry, I’ll do the math. (To make reading this table easier, I’ll use the repeating decimal notation instead of the fraction.)

 

 

Look at that, pretty cool, right?

Not really. There’s a huge problem. See, if we add eighteen and one-third to get our next octave, we end up at 660, not 880! In fact, we would have to add eighteen and one-third another 24 times before we ended up at 880.

Let’s take a look at another table and see what that would actually be like.

As you can see, this is a problem. The octave of C, when C is 275, should be 550. But with this method it ends up being 495. The note D# is 550. If you’re playing music and you need to hit a C note and you end up on D#, it will sound horribly wrong.

And not to mention, there are twelve semi-tones between a note and its octave, and octaves are defined as double the frequency. But between A, at 440, and its octave, there are twenty four “semi-tones.”

You see, by adding 18 and 1/3 to 220, twelve times, will allow us to arrive at 440, but nothing else works. And this isn’t just a weird misapplication of mathematics to music. This problem of normalizing music has been around since ancient times. The Greeks struggled with it. It was only recently that we figured it out.

So how do we come up with a way to space the notes so that this repeating pattern of notes and octaves works? How do we figure out how to break apart a set of 12 frequencies so that I can pick any frequency and find an octave and all of the semitones in-between have the same relationship to one another?

It turns out if you multiply a note’s frequency by 1.059460309436 it all works out perfectly. In the table below I will multiply A’s frequency by our ugly number of 1.0594603039436 and we will see that every note and its octave are doubled.

In the table below you can see three octaves of A as found by multiplying each frequency by our ugly number. Now, we should really distinguish one A for the next with a subscript because they’re not the same, they’re octaves of one another, but for the purposes of this exploration, what we have here should be clear enough.

 

Pick any note from the table below and find its octave. You’ll see the frequencies are indeed doubled. All of them, they all work perfect.

 

Great, right? But where does that ugly number come from? (insert pictures of math book here)

I promised to keep the math light, and I will. I’ll just mention that I spent hours and hours finding that number myself. It involved quadratics, binomial expansion, logarithms and all kinds of fun stuff. Math, when you don’t know the appropriate methods, is difficult!

Here’s how to understand that number, without all of the workings behind the scenes.

A note’s octave is twice the frequency, right? If we let f stand for the frequency of a note, say A, then 2f, with be the octave.

For the sake of clarity, let’s see that this works.

A’s frequency is 220. So f = 220.

2f = 2×220

2f = 440.

And if we wanted to find the next octave of A, we could do 4f.

A’s frequency is 220. So f = 220.

4f = 4×220

4f = 880

Pretty straight forward. Now, to introduce an exponent in a very non-threatening way. They’ll come into play in just a moment and incase you’ve not seen them since high school, a little refresher is good. You know that $2^2=2\times 2.$ So instead of writing 4, we could write $2^2.$

A’s frequency is 220. So f = 220.

$2^2$ f = 4×220

$2^2$ f = 880

Now, 2 is also $2^1.$ With me so far? Good. So if we wanted to find the third octave of A = 220, we could multiply 220 by $2\times 2\times 2,$ which is $2^3,$ which is of course just 8.

Do you see that $2\times 2\times 2=2^1\times 2^1\times 2^1$?

And this is $2^{1+1+1}=2^3.$

220×8 = 1,760

or

$220\times 2^3=1,760$

That is the third octave of A, either way.

Here’s the fact I’m trying to get you to see: With exponents, when the bases are the same (the big number at the bottom), you can add those exponents.

The problem is the steps between a note and its octave, the twelve semi-tones, right?

If we multiply a frequency by two we get the octave. But what do we multiply by to get the next semi-tone? The next semi-tone would be the first of twelve, semi-tones, of course.

Well, if we multiply by $2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$ we get the next semi-tone. Don’t go anywhere, just watch. We don’t even need to know what the exponent 1/12 is for this to work. If you understood that 2×2×2 = $2^3,$ then you got this! Watch…

Without using the frequencies, just the name of the note, let’s see how this works.

To go from A to A#, we multiply by $2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$. To go from A# to B we multiply by $2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$, again. From B to C, once again. Twelve times we can do this and we end up on A’s octave.

 

A

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

$\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

A#

B

C

C#

D

D#

E

F

F#

G

G#

A

 

Let’s take a quick look at that top column.

$\text{A}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times {\text{2}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

Do you remember about the whole adding exponents thing? We can add all of these 1/12^th here.

1/12 + 1/12 = 2/12, right?

1/12 + 1/12 + 1/12 = 3/12, right?

1/12 + 1/12 + 1/12 + 1/12 + 1/12 + 1/12 … twelve times is 12/12, right?

And is 12/12, one?

$2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}\times 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$

is the same as

$2^{\frac{1+1+1+1+1+1+1+1+1+1+1+1}{12}}$

and that is just

$2^{\frac{12}{12}}$.

Since 12/12 is one, and two the power of one is just two …

 

Do you see what we’ve done?

By taking a frequency, that ripple you can hear, and multiplying it by $2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$ we can find the next semi-tone. Do that twelve times and we get the twelve semi-tones and end up on the octave.

In case you haven’t guess, our ugly number, 1.059460309436, equals $2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}$.

Pretty cool, right?

So what about the guitar’s frets, again? They get closer together towards the bottom of a fretboard and are farther apart at the top? Do you notice how the frequencies get farther apart the lower the note and closer together the higher the note? That’s because notes aren’t spaced linearly. They’re not a constant ratio from one to the next. It’s a logarithmic relationship. You can see that on the fretboard, or in the shape of a piano’s string, or a harp.

I have no background in music theory. Everything I’ve claimed here is my own, all mistakes are mine. I am certain there are phrases and references in music that I’ve fumbled, but the idea is sound. My intent is only to promote interest in both music and math. They’re both beautiful things.

Thank you for reading.