I have two seemingly unrelated challenges for you. The first relates to music. And the second gives a foundational result in measure theory, which is the formal
underpinning for how mathematicians define integration and probability. The second challenge, which I’ll get to about halfway through the video, has to do
with covering numbers with open sets and is very counterintuitive. Or at least, when I first saw it, I was confused for a while.
Foremost, I’d like to explain what’s going on. But I also plan to share a surprising connection that it has with music. Here’s the first challenge. I’m going to play a musical note with a given frequency, let’s say 220 hertz. Then I’m going to choose some number between one and two, which we’ll call 𝑟, and
play a second musical note whose frequency is 𝑟 times the frequency of the first
note, 220. For some values of 𝑟, like 1.5, the two notes will sound harmonious together. But for others, like the square root of two, they sound cacophonous.
Your task is to determine whether a given ratio 𝑟 will give a pleasant sound or an
unpleasant one just by analyzing the number and without listening to the notes. One way to answer, especially if your name is Pythagoras, might be to say that two
notes sound good together when the ratio is a rational number and bad when it’s
irrational. For instance, a ratio of three-halves gives a musical fifth, four-thirds gives a
musical fourth, eight-fifths gives a major sixth, so on.
Here’s my best guess for why this is the case. A musical note is made up of beats played in rapid succession, for instance 220 beats
per second. When the ratio of frequencies of two notes is rational, there is a detectable pattern
in those beats. Which, when we slow it down, we hear as a rhythm instead of as a harmony. Evidently, when our brains pick up on this pattern, the two notes sound nice
together. However, most rational numbers actually sound pretty bad, like 211 over 198 or 1093
divided by 826. The issue, of course, is that these rational numbers are somehow more complicated
than the other ones. Our ears don’t pick up on the pattern of the beats.
One simple way to measure complexity of rational numbers is to consider the size of
the denominator when it’s written in reduced form. So we might edit our original answer to only admit fractions with low denominators,
say less than 10. Even still, this doesn’t quite capture harmoniousness. Since plenty of notes sound good together even when the ratio of their frequencies is
irrational, so long as it’s close to a harmonious rational number. And it’s a good thing too because many instruments, such as pianos, are not tuned in
terms of rational intervals. But are tuned such that each half-step increase corresponds with multiplying the
original frequency by the 12th root of two, which is irrational. If you’re curious about why this is done, Henry at MinutePhysics recently did a video
that gives a very nice explanation.
This means that if you take a harmonious interval, like a fifth, the ratio of
frequencies when played on a piano will not be a nice rational number like you
expect, in this case three-halves. But will instead be some power of the 12th root of two, in this case two to the seven
over 12, which is irrational but very close to three-halves. Similarly, a musical fourth corresponds to two to the five twelfths, which is very
close to four-thirds. In fact, the reason it works so well to have 12 notes in the chromatic scale is that
powers of the 12th root of two have the strange tendency to be within a one percent
margin of error of simple rational numbers.
So now you might say that a ratio 𝑟 will produce a harmonious pair of notes if it is
sufficiently close to a rational number with a sufficiently small denominator. How close depends on how discerning your ear is. And how small a denominator depends on the intricacy of harmonic patterns your ear
has been trained to pick up on. After all, maybe someone with a particularly acute musical sense would be able to
hear and find pleasure in the pattern resulting from more complicated fractions like
23 over 21 or 35 over 43. As well as numbers closely approximating those fractions.
This leads me to an interesting question. Suppose there is a musical savant who finds pleasure in all pairs of notes whose
frequencies have a rational ratio. Even the super complicated ratios that you and I would find cacophonous. Is it the case that she would find all ratios 𝑟 between one and two harmonious, even
the irrational ones? After all, for any given real number, you can always find a rational number
arbitrarily close to it, just like three-halves is really close to two to the seven
over 12. Well, this brings us to challenge number two.
Mathematicians like to ask riddles about covering various sets with open
intervals. And the answers to these riddles have a strange tendency to become famous lemmas of
theorems. By open interval, I just mean the continuous stretch of real numbers strictly greater
than some number 𝑎 but strictly less than some other number 𝑏, where 𝑏 is, of
course, greater than 𝑎. My challenge to you involves covering all of the rational numbers between zero and
one with open intervals. When I say cover, all this means is that each particular rational number lies inside
at least one of your intervals. The most obvious way to do this is to just use the entire interval from zero to one
itself and call it done. But the challenge here is that the sum of the lengths of your intervals must be
strictly less than one.
To aid you in this seemingly impossible task, you are allowed to use infinitely many
intervals. Even still, the task might feel impossible since the rational numbers are dense in
the real numbers. Meaning, any stretch, no matter how small, contains infinitely many rational
numbers. So how could you possibly cover all of the rational numbers without just covering the
entire interval from zero to one itself. Which would mean the total length of your open intervals has to be at least the
length of the entire interval from zero to one. Then again, I wouldn’t be asking if there wasn’t a way to do it.
First, we enumerate the rational numbers between zero and one, meaning we organize
them into an infinitely long list. There are many ways to do this. But one natural way that I’ll choose is to start with one-half, followed by one-third
and two-thirds, then one-fourth and three-fourths. We don’t write down two-fourths since it’s already appeared as one-half. Then all reduced fractions with denominator five, all reduced fractions with
denominator six, continuing on and on in this fashion. Every fraction will appear exactly once in this list, in its reduced form. And it gives us a meaningful way to talk about the first rational number, then a
second rational number, the 42nd rational number, things like that.
Next, to ensure that each rational is covered, we’re going to assign one specific
interval to each rational. Once we remove the intervals from the geometry of our setup and just think of them in
a list, each one responsible for one rational number, it seems much clearer that the
sum of their lengths can be less than one. Since each particular interval can be as small as you want and still cover its
designated rational. In fact, the sum can be any positive number. Just choose an infinite sum with positive terms that converges to one, like one-half
plus a fourth plus an eighth, on and on. Then choose any desired value of 𝜀 greater than zero, like 0.5. And multiply all of the terms in the sum by 𝜀 so that you have an infinite sum
converging to 𝜀.
Now scale the 𝑛th interval to have a length equal to the 𝑛th term in the sum. Notice, this means your intervals start getting really small really fast. So small that you can’t really see most of them in this animation. But it doesn’t matter, since each one is only responsible for covering one
rational. I’ve said it already, but I’ll say it again because it’s so amazing. 𝜀 can be whatever positive number we want. So not only can our sum be less than one, it can be arbitrarily small! This is one of those results where even after seeing the proof, it still defies
The discord here is that the proof has us thinking analytically, with the rational
numbers in a list. But our intuition has us thinking geometrically, with all the rational numbers as a
dense set on the interval. Well, you can’t skip over any continuous stretch because that would contain
infinitely many rationals. So let’s get a visual understanding for what’s going on.
Brief side note here. I had trouble deciding on how to illustrate small intervals. Since if I scale the parentheses with the interval, you won’t be able to see them at
all. But if I just push the parentheses together, they cross over in a way that’s
potentially confusing. Nevertheless, I decided to go with the ugly chromosomal cross. So keep in mind, the interval this represents is that tiny stretch between the
centers of each parenthesis. Okay, back to the visual intuition.
Consider when 𝜀 equals 0.3. Meaning, if I choose a number between zero and one at random, there is a 70 percent
chance that it’s outside those infinitely many intervals. What does it look like to be outside the intervals? The square root of two over two is among those 70 percent. And I’m going to zoom in on it. As I do so, I’ll draw the first 10 intervals in our list within our scope of
vision. As we get closer and closer to the square root of two over two. Even though you will always find rationals within your field of view, the intervals
placed on top of those rationals get really small really fast.
One might say that for any sequence of rational numbers approaching the square root
of two over two. The intervals containing the elements of that sequence shrink faster than the
sequence converges. Notice, intervals are really small if they show up late in the list. And rationals show up late in the list when they have large denominators. So the fact that the square root of two over two is among the 70 percent not covered
by our intervals is, in a sense, a way to formalize the otherwise vague idea that
the only rational numbers close to it have a large denominator. That is to say, the square root of two over two is cacophonous.
In fact, let’s use a smaller 𝜀, say 0.01, and shift our set-up to lie on top of the
interval from one to two instead of from zero to one. Then which numbers fall among that elite one percent covered by our tiny
intervals? Almost all of them are harmonious! For instance, the harmonious irrational number two to the seven twelfths is very
close to three-halves, which has a relatively fat interval sitting on top of it. And the interval around four-thirds is smaller but still fat enough to cover two to
the five twelfths.
Which members of the one percent are cacophonous? Well, the cacophonous rationals, meaning those with high denominators, and
irrationals that are very, very, very close to them. However, think of the savant who finds harmonic patterns in all rational numbers. You could imagine that for her, harmonious numbers are precisely those one percent
covered by the intervals. Provided that her tolerance for error goes down exponentially for more complicated
In other words, the seemingly paradoxical fact that you can have a collection of
intervals densely populate a range while only covering one percent of its values
corresponds to the fact that harmonious numbers are rare, even for the savant. I’m not saying this makes the result more intuitive. In fact, I find it quite surprising that the savant I defined could find 99 percent
of all ratios cacophonous. But the fact that these two ideas are connected was simply too beautiful not to