Music And Measure Theory

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 counter-intuitive. 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 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 1 and 2, 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 this ratio r, like 1.5, the two notes will
sound harmonious together, but for others, like the square root of 2, they sound cacophonous.
Your task it 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 that two notes sound
good when the ratio is a rational number, and bad when it is irrational. For instance,
a ratio of 3/2 gives a musical fifth, 4/3 gives a musical fourth, of 8/5 gives a major
sixth, etc. 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, two notes sound nice together. However, most rational numbers actually sound
pretty bad, like 211/198, or 1093/826. The issue, of course, is that these rational number
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 the complexity of a rational number is to
consider the size of its denominator when it is 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 is 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 2, which is
irrational. If you’re curious about why this is done, Henry at minutephysics recently
did a video which 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 3/2, but will instead be some power of the 12th
root of 2, in this case 2^{7/12}, which is irrational, but very close to 3/2. Similarly,
a musical fourth corresponds to 2^{5/12}, which is very close to 4/3. In fact, the reason
it works so well to have 12 notes in the chromatic scale is that powers of the 12th root of 2
have a strange tendency to be within a 1% margin of error of simple rational numbers.
So now you might say 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 23/21 or 35/43, as well as numbers closely approximating these
fractions. This leads to an interesting question: Suppose
there is a musical savant, who find pleasure in all pairs of notes whose frequencies have
a rational ratio, even super complicated ratios that you and I would find cacophonous. Is
it the case that she would find all ratios r between 1 and 2 harmonious, even the irrational
ones? After all, for any given real number you can always find rational numbers arbitrarily
close it, just as 3/2 is close to 2^{7/12}. Well, this brings us to challenge number 2.
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 and 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 the rational numbers between 0 and 1 with
open intervals. When I say “cover”, all that means is that each particular rational
number lies in at least one of your intervals. The most obvious way to do this is to just
use the entire interval from 0 to 1 itself and call it done, but the challenge here is
that the sum of the lengths of your intervals must be strictly less than 1.
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 rational numbers without just covering the
entire interval from 0 to 1 itself, which would mean the total length of your open intervals
has to be at least the length of the entire interval from 0 to 1.
Then again, I wouldn’t be talking about this if there was not a way to do it.
First, we enumerate the rational numbers between 0 and 1, meaning we organize them into an
infinitely long list. There are many ways to do this, but one natural way I’ll choose
is start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don’t write down 2/4
since it has already appeared as ½, then all reduced fractions with denominator 5,
all reduced fractions with denominator 6, 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, the “second” rational
number, the 42nd rational number, things like that.
Next, to ensure that each rational is covered, we are 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 only one rational number, it seems much clearer
that the sum of their lengths can be less than 1, 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
1, like ½+¼+⅛+… on and on with powers of 2, then choose any desired value epsilon>0,
like 0.5, and multiply all terms by epsilon so that we have an infinite sum converging
to epsilon. 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, by I’ll say it again because it’s so amazing: epsilon can be
whatever positive number we want, so not only can our sum be less than 1, it can be arbitrarily
small! This is one of those results where even after
seeing the proof, it still defies intuition. 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 the rationals as a dense set on the interval, where you can’t skip
over any continuous stretch of numbers since each stretch contains infinitely many rationals.
So let’s get a visual understanding of what’s going on.
Brief side note here: I had trouble deciding on how to illustrate small open 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 it
potentially confusing. Nevertheless, I decided to go with the ugly chromosomal cross, so
keep in mind that the interval they represent is the tiny stretch between the centers of
each parenthesis. Okay, back to the visual intuition.
Consider when epsilon=0.3, meaning if I choose a number between 0 and 1 at random,
there is a 70% that it is outside all those infinitely many intervals. What does it look
like to be outside the intervals? Well, the square root of 2 over 2 is among those 70%,
and I’m going to zoom in it. As I do so I’ll draw the first 10 intervals in the
list within our scope of vision. As we get closer to the square root of 2 over 2, 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 2 over 2, the intervals covering the
elements of this sequence shrink faster than that sequence converges.
Notice, intervals are really small if they show up very 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 2 over 2 is among the 70% 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 large denominators.
That is to say, the square root of 2 over 2 is cacophonous.
In fact, let’s use a smaller epsilon, say 0.01, and shift our setup to lie on top of
the interval from 1 to 2 instead of from 0 to 1. Then which numbers fall among the elite
1% covered by our tiny intervals? Almost all of them are harmonious! For instance, the
harmonious irrational number 2^{7/12} is very close to 3/2, which has a relatively fat interval
sitting on top of it, and the interval around 4/3 is smaller, but still fat enough to cover
2^{5/12}. Which members of the 1% 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 1% covered by the intervals, provided
that her tolerance for error goes down exponentially for more complicated rationals.
In other words, the seemingly paradoxical fact that you can have a collection of intervals
densely populate a range while only covering 1% of its values corresponds to the fact that
harmonious numbers are rare, even for the savant. I’m not saying this makes it the
result more intuitive, in fact, I find it quite surprising that the savant I defined
could find 99% of all ratios cacophonous, but the fact that these two ideas are connected
was simply too beautiful not to share.

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