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 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 Herz then I'm going to choose some number between 1 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 harmonic together but for others like the 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 music musical fifth 4/3 gives a musical fourth 8 fths 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's 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 21 over 198 or 193 / 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 halfstep 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 minute physics 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 frequency when played on a piano will not be a nice rational number like you expect in this case three Hales but will instead be some power of the 12th root of two in this case 2 7/ 12th which is irrational but very close to three halves similarly a musical fourth corresponds to 2 to the 5 12ths 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 two have this strange tendency to be within a 1% 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 heart 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 Sant who finds pleasure in all pairs of notes whose frequency 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 1 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 Hales is really close to 2 7/2 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 lemas 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 of the rational numbers between 0 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 0 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're 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 0 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 0 to 1 then again I wouldn't be asking if there wasn't a way to do it first we enumerate the rational numbers between 0 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 1/2 followed by 1/3 and 2/3 then 1/4 and 3/4s we don't write down 24s since it's already appeared as 1/2 then all reduced fractions with denominator 5 all reduce 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 42 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 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 1/2 plus a 4 plus an e on and on then choose any desired value of Epsilon greater than 0 like 0. 5 and multiply all of the terms in the sum by Epsilon so that you 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 but I'll say it again cuz it's so amazing Epsilon 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 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 all the rational numbers as a dense set on the interval where 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 Epsilon equals 0. 3 meaning if I choose a number between 0 and 1 at random there's a 70% chance that it's outside those infinitely many intervals what does it look like to be outside the intervals the < TK of 2 over2 is among those 70% 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 2 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 < TK of 2/2 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 when they have large denominators so the fact that the < TK of 2 over2 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 a large denominator that is to say the < TK of 2 two is cacophanus 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 that Elite 1% covered by our tiny intervals almost all of them are harmonious for instance the harmonious irrational number 2 72th is very close to three Hales 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 to the 5 12ths which members of the 1% are cacophonous well the cacophonous rationals meaning those with high denominators and irrationals that are 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 the result more intuitive in fact I find it surprising that the Savant defined could find 99% of all ratios cacophonous but the fact that these two ideas are connected was simply too beautiful not to share