How Math Fixed Music – Rational Exponents Sound Good
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
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
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
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,
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
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 . 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?
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
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
Pick any note from the table below and find its
octave. You’ll see the frequencies are
indeed doubled. All of them, they all
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.
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.
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 So instead of writing 4, we could write
frequency is 220. So f = 220.
Now, 2 is also With me so far? Good.
So if we wanted to find the third octave of A = 220, we could multiply
220 by which is which is of course just 8.
Do you see that ?
And this is
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 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 = 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 . To go from A# to B we multiply by ,
again. From B to C, once again. Twelve times we can do this and we end up on
Let’s take a quick look at that top column.
Do you remember about the whole adding exponents
thing? We can add all of these 1/12th
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?
the same as
that is just
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 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 .
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.
For a great book on the relationship between math and music, and how it came to be, try Harmonograph.