you
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
for most 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 1 &
2 which we'll call our 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 2 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 the 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 v 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'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 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 beat 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
harmonious Ness 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 twelfth root
of 2 which is irrational if you're
curious about why this is done Henry
admitted physics recently did a video
that gives a very nice explanation this
means that if you take a harmonious
interval like a 5th 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 2 in this case 2 to the 7 over
12 which is irrational but very close to
three-halves
similarly a musical forth 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
this 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 smaller 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 are
between 1 & 2 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 3 halves is really close to 2
to the 7 over 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 of the
rational members between 0 & 1 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 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're allowed
to use infinitely many intervals even
still the test 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 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 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 1/4 and 3/4 we don't write down 2/4
since it's already appeared as 1/2 then
all reduced fractions with denominators
I've I'll reduce fractions with
denominators X 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 ration remember
that a second rational number the forty
second 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 1/2 plus 1/4 plus 1/8 on and
on then choose any desired value of
epsilon greater than zero 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 rationale
I've said it already but 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's 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 beef side note here I
had trouble deciding on how to
illustrate small intervals since if I
scale the parenthesis with the interval
you won't be able to see them at all but
if I just push the parenthesis 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 zero point three
meaning if I choose a number between
zero and one 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 square root of 2 over 2 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 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 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 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 a large denominator
that is to say the square root of two
over two is cacophonous in fact let's
use a smaller epsilon say 0.01 and shift
are set up 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 - the
seven 12th is very close to three-halves
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 twelfths
which members of the 1% are cacophonous
well the cacophonous rationals meaning
those with high denominators and
Irrational's 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 percent 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 populated range while
only covering 1 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% of all ratios cacophonous
but the fact that these two ideas are
connected was simply too beautiful not
to share