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