WEBVTT
Kind: captions
Language: en
00:00:00.000 --> 00:00:03.580
you
00:00:03.580 --> 00:00:05.819
I have two seemingly unrelated
00:00:05.819 --> 00:00:08.740
challenges for you the first relates to
00:00:08.740 --> 00:00:10.570
music and the second gives a
00:00:10.570 --> 00:00:12.550
foundational result in measure theory
00:00:12.550 --> 00:00:14.500
which is the formal underpinning for how
00:00:14.500 --> 00:00:16.420
mathematicians define integration and
00:00:16.420 --> 00:00:18.970
probability the second challenge which
00:00:18.970 --> 00:00:20.439
I'll get to about halfway through the
00:00:20.439 --> 00:00:22.630
video has to do with covering numbers
00:00:22.630 --> 00:00:24.369
with open sets and is very
00:00:24.369 --> 00:00:26.560
counterintuitive or at least when I
00:00:26.560 --> 00:00:28.449
first saw it I was confused for a while
00:00:28.449 --> 00:00:30.939
for most I'd like to explain what's
00:00:30.939 --> 00:00:32.980
going on but I also plan to share a
00:00:32.980 --> 00:00:34.720
surprising connection that it has with
00:00:34.720 --> 00:00:37.510
music
00:00:37.510 --> 00:00:40.430
here's the first challenge I'm going to
00:00:40.430 --> 00:00:41.840
play a musical note with a given
00:00:41.840 --> 00:00:45.110
frequency let's say 220 Hertz then I'm
00:00:45.110 --> 00:00:47.150
going to choose some number between 1 &
00:00:47.150 --> 00:00:50.090
2 which we'll call our and play a second
00:00:50.090 --> 00:00:52.520
musical note whose frequency is R times
00:00:52.520 --> 00:00:55.910
the frequency of the first note 220 for
00:00:55.910 --> 00:00:59.180
some values of R like 1.5 the two notes
00:00:59.180 --> 00:01:01.400
will sound harmonious together but for
00:01:01.400 --> 00:01:03.530
others like the square root of 2 they
00:01:03.530 --> 00:01:06.440
sound cacophonous your task is to
00:01:06.440 --> 00:01:09.170
determine whether a given ratio R will
00:01:09.170 --> 00:01:10.759
give a pleasant sound or an unpleasant
00:01:10.759 --> 00:01:13.460
one just by analyzing the number and
00:01:13.460 --> 00:01:17.270
without listening to the notes one way
00:01:17.270 --> 00:01:19.009
to answer especially if your name is
00:01:19.009 --> 00:01:21.200
Pythagoras might be to say the two notes
00:01:21.200 --> 00:01:23.270
sound good together when the ratio is a
00:01:23.270 --> 00:01:25.580
rational number and bad when it's
00:01:25.580 --> 00:01:28.759
irrational for instance a ratio of three
00:01:28.759 --> 00:01:32.360
halves gives a musical v four thirds
00:01:32.360 --> 00:01:35.750
gives a musical fourth eight fifths
00:01:35.750 --> 00:01:39.740
gives a major sixth so on here's my best
00:01:39.740 --> 00:01:42.500
guess for why this is the case a musical
00:01:42.500 --> 00:01:43.940
note is made up of beats
00:01:43.940 --> 00:01:46.160
played in rapid succession for instance
00:01:46.160 --> 00:01:49.610
220 beats per second when the ratio of
00:01:49.610 --> 00:01:51.679
frequencies of two notes is rational
00:01:51.679 --> 00:01:53.660
there's a detectable pattern in those
00:01:53.660 --> 00:01:56.630
beats which when we slow it down we hear
00:01:56.630 --> 00:02:00.100
as a rhythm instead of as a harmony
00:02:00.100 --> 00:02:02.539
evidently when our brains pick up on
00:02:02.539 --> 00:02:04.369
this pattern the two notes sound nice
00:02:04.369 --> 00:02:06.399
together
00:02:06.399 --> 00:02:09.770
however most rational numbers actually
00:02:09.770 --> 00:02:13.900
sound pretty bad like 211 over 198 or
00:02:13.900 --> 00:02:18.200
1093 divided by 826 the issue of course
00:02:18.200 --> 00:02:19.880
is that these rational numbers are
00:02:19.880 --> 00:02:21.920
somehow more complicated than the other
00:02:21.920 --> 00:02:24.230
ones our ears don't pick up on the
00:02:24.230 --> 00:02:30.830
pattern of the beat one simple way to
00:02:30.830 --> 00:02:32.630
measure complexity of rational numbers
00:02:32.630 --> 00:02:34.670
is to consider the size of the
00:02:34.670 --> 00:02:36.680
denominator when it's written in reduced
00:02:36.680 --> 00:02:37.890
form
00:02:37.890 --> 00:02:40.959
so we might edit our original answer to
00:02:40.959 --> 00:02:42.459
only admit fractions with low
00:02:42.459 --> 00:02:47.290
denominators say less than 10
00:02:47.290 --> 00:02:49.360
even still this doesn't quite capture
00:02:49.360 --> 00:02:51.610
harmonious Ness since plenty of notes
00:02:51.610 --> 00:02:53.530
sound good together even when the ratio
00:02:53.530 --> 00:02:56.140
of their frequencies is irrational so
00:02:56.140 --> 00:02:58.060
long as it's close to a harmonious
00:02:58.060 --> 00:03:00.160
rational number
00:03:00.160 --> 00:03:02.740
and it's a good thing too because many
00:03:02.740 --> 00:03:05.380
instruments such as pianos are not tuned
00:03:05.380 --> 00:03:07.840
in terms of rational intervals but are
00:03:07.840 --> 00:03:10.500
tuned such that each half-step increase
00:03:10.500 --> 00:03:12.490
corresponds with multiplying the
00:03:12.490 --> 00:03:14.620
original frequency by the twelfth root
00:03:14.620 --> 00:03:17.860
of 2 which is irrational if you're
00:03:17.860 --> 00:03:20.200
curious about why this is done Henry
00:03:20.200 --> 00:03:21.580
admitted physics recently did a video
00:03:21.580 --> 00:03:25.570
that gives a very nice explanation this
00:03:25.570 --> 00:03:26.980
means that if you take a harmonious
00:03:26.980 --> 00:03:29.320
interval like a 5th the ratio of
00:03:29.320 --> 00:03:31.630
frequencies when played on a piano will
00:03:31.630 --> 00:03:33.850
not be a nice rational number like you
00:03:33.850 --> 00:03:36.490
expect in this case three-halves but
00:03:36.490 --> 00:03:38.800
will instead be some power of the 12th
00:03:38.800 --> 00:03:41.680
root of 2 in this case 2 to the 7 over
00:03:41.680 --> 00:03:44.980
12 which is irrational but very close to
00:03:44.980 --> 00:03:48.490
three-halves
00:03:48.490 --> 00:03:51.530
similarly a musical forth corresponds to
00:03:51.530 --> 00:03:53.900
two to the five twelfths which is very
00:03:53.900 --> 00:03:57.230
close to four thirds
00:03:57.230 --> 00:04:00.410
in fact the reason it works so well to
00:04:00.410 --> 00:04:02.390
have 12 notes in the chromatic scale is
00:04:02.390 --> 00:04:05.390
that powers of the 12th root of two have
00:04:05.390 --> 00:04:07.400
this strange tendency to be within a one
00:04:07.400 --> 00:04:09.470
percent margin of error of simple
00:04:09.470 --> 00:04:15.260
rational numbers so now you might say
00:04:15.260 --> 00:04:17.479
that a ratio R will produce a harmonious
00:04:17.479 --> 00:04:19.579
pair of notes if it is sufficiently
00:04:19.579 --> 00:04:21.440
close to a rational number with a
00:04:21.440 --> 00:04:24.470
sufficiently small denominator how close
00:04:24.470 --> 00:04:26.060
depends on how discerning your ear is
00:04:26.060 --> 00:04:28.729
and how smaller denominator depends on
00:04:28.729 --> 00:04:30.919
the intricacy of harmonic patterns your
00:04:30.919 --> 00:04:33.320
ear has been trained to pick up on after
00:04:33.320 --> 00:04:35.810
all maybe someone with a particularly
00:04:35.810 --> 00:04:38.030
acute musical sense would be able to
00:04:38.030 --> 00:04:39.889
hear and find pleasure in the pattern
00:04:39.889 --> 00:04:41.450
resulting from more complicated
00:04:41.450 --> 00:04:46.010
fractions like 23 over 21 or 35 over 43
00:04:46.010 --> 00:04:48.620
as well as numbers closely approximating
00:04:48.620 --> 00:04:52.910
those fractions this leads me to an
00:04:52.910 --> 00:04:55.340
interesting question suppose there is a
00:04:55.340 --> 00:04:57.860
musical savant who finds pleasure in all
00:04:57.860 --> 00:05:00.050
pairs of notes whose frequencies have a
00:05:00.050 --> 00:05:01.940
rational ratio even the
00:05:01.940 --> 00:05:04.070
super-complicated ratios that you and I
00:05:04.070 --> 00:05:08.120
would find cacophonous is it the case
00:05:08.120 --> 00:05:10.010
that she would find all ratios are
00:05:10.010 --> 00:05:12.110
between 1 & 2 harmonious even the
00:05:12.110 --> 00:05:15.080
irrational ones after all for any given
00:05:15.080 --> 00:05:16.970
real number you can always find a
00:05:16.970 --> 00:05:19.100
rational number arbitrarily close to it
00:05:19.100 --> 00:05:21.710
just like 3 halves is really close to 2
00:05:21.710 --> 00:05:26.990
to the 7 over 12 well this brings us to
00:05:26.990 --> 00:05:30.360
challenge number 2
00:05:30.360 --> 00:05:32.650
mathematicians like to ask riddles about
00:05:32.650 --> 00:05:34.509
covering various sets with open
00:05:34.509 --> 00:05:36.400
intervals and the answers to these
00:05:36.400 --> 00:05:38.139
riddles have a strange tendency to
00:05:38.139 --> 00:05:41.740
become famous lemmas and theorems by
00:05:41.740 --> 00:05:44.289
open interval I just mean the continuous
00:05:44.289 --> 00:05:46.270
stretch of real numbers strictly greater
00:05:46.270 --> 00:05:48.520
than some number a but strictly less
00:05:48.520 --> 00:05:50.680
than some other number B where B is of
00:05:50.680 --> 00:05:52.990
course greater than a my challenge to
00:05:52.990 --> 00:05:54.909
you involves covering all of the
00:05:54.909 --> 00:05:57.729
rational members between 0 & 1 with open
00:05:57.729 --> 00:06:00.400
intervals when I say cover all this
00:06:00.400 --> 00:06:02.740
means is that each particular rational
00:06:02.740 --> 00:06:04.839
number lies inside at least one of your
00:06:04.839 --> 00:06:09.039
intervals the most obvious way to do
00:06:09.039 --> 00:06:11.110
this is to just use the entire interval
00:06:11.110 --> 00:06:13.900
from 0 to 1 itself and call it done but
00:06:13.900 --> 00:06:15.939
the challenge here is that the sum of
00:06:15.939 --> 00:06:18.189
the lengths of your intervals must be
00:06:18.189 --> 00:06:24.789
strictly less than 1 to aid you in this
00:06:24.789 --> 00:06:27.279
seemingly impossible task you're allowed
00:06:27.279 --> 00:06:32.560
to use infinitely many intervals even
00:06:32.560 --> 00:06:34.389
still the test might feel impossible
00:06:34.389 --> 00:06:37.120
since the rational numbers are dense in
00:06:37.120 --> 00:06:39.580
the real numbers meaning any stretch no
00:06:39.580 --> 00:06:41.889
matter how small contains infinitely
00:06:41.889 --> 00:06:45.550
many rational numbers so how could you
00:06:45.550 --> 00:06:47.560
possibly cover all of the rational
00:06:47.560 --> 00:06:49.360
numbers without just covering the entire
00:06:49.360 --> 00:06:51.909
interval from 0 to 1 itself which would
00:06:51.909 --> 00:06:53.349
mean the total length of your open
00:06:53.349 --> 00:06:55.330
intervals has to be at least the length
00:06:55.330 --> 00:07:00.249
of the entire interval from 0 to 1 then
00:07:00.249 --> 00:07:02.409
again I wouldn't be asking if there
00:07:02.409 --> 00:07:05.430
wasn't a way to do it
00:07:05.430 --> 00:07:08.320
first we enumerate the rational numbers
00:07:08.320 --> 00:07:09.520
between zero and one
00:07:09.520 --> 00:07:11.440
meaning we organize them into an
00:07:11.440 --> 00:07:13.870
infinitely long list there are many ways
00:07:13.870 --> 00:07:16.090
to do this but one natural way that I'll
00:07:16.090 --> 00:07:18.010
choose is to start with one half
00:07:18.010 --> 00:07:20.140
followed by one-third and two-thirds
00:07:20.140 --> 00:07:23.770
then 1/4 and 3/4 we don't write down 2/4
00:07:23.770 --> 00:07:26.410
since it's already appeared as 1/2 then
00:07:26.410 --> 00:07:28.030
all reduced fractions with denominators
00:07:28.030 --> 00:07:30.430
I've I'll reduce fractions with
00:07:30.430 --> 00:07:32.590
denominators X continuing on and on in
00:07:32.590 --> 00:07:35.860
this fashion every fraction will appear
00:07:35.860 --> 00:07:38.110
exactly once in this list in its reduced
00:07:38.110 --> 00:07:40.180
form and it gives us a meaningful way to
00:07:40.180 --> 00:07:41.800
talk about the first ration remember
00:07:41.800 --> 00:07:43.990
that a second rational number the forty
00:07:43.990 --> 00:07:46.290
second rational number things like that
00:07:46.290 --> 00:07:49.060
next to ensure that each rational is
00:07:49.060 --> 00:07:50.950
covered we're going to assign one
00:07:50.950 --> 00:07:54.010
specific interval to each rational once
00:07:54.010 --> 00:07:55.419
we remove the intervals from the
00:07:55.419 --> 00:07:57.340
geometry of our setup and just think of
00:07:57.340 --> 00:07:59.320
them in a list each one responsible for
00:07:59.320 --> 00:08:01.540
one rational number it seems much
00:08:01.540 --> 00:08:03.100
clearer that the sum of their lengths
00:08:03.100 --> 00:08:04.960
can be less than one since each
00:08:04.960 --> 00:08:07.240
particular interval can be as small as
00:08:07.240 --> 00:08:09.310
you want and still cover its designated
00:08:09.310 --> 00:08:12.190
rational in fact the sum can be any
00:08:12.190 --> 00:08:15.250
positive number just choose an infinite
00:08:15.250 --> 00:08:17.169
sum with positive terms that converges
00:08:17.169 --> 00:08:20.680
to one like 1/2 plus 1/4 plus 1/8 on and
00:08:20.680 --> 00:08:25.000
on then choose any desired value of
00:08:25.000 --> 00:08:28.270
epsilon greater than zero like 0.5 and
00:08:28.270 --> 00:08:30.729
multiply all of the terms in the sum by
00:08:30.729 --> 00:08:32.919
Epsilon so that you have an infinite sum
00:08:32.919 --> 00:08:36.610
converging to Epsilon now scale the nth
00:08:36.610 --> 00:08:38.770
interval to have a length equal to the
00:08:38.770 --> 00:08:42.169
nth term in the sum
00:08:42.169 --> 00:08:45.000
notice this means your intervals start
00:08:45.000 --> 00:08:47.460
getting really small really fast so
00:08:47.460 --> 00:08:49.110
small that you can't really see most of
00:08:49.110 --> 00:08:51.300
them in this animation but it doesn't
00:08:51.300 --> 00:08:52.890
matter since each one is only
00:08:52.890 --> 00:08:54.900
responsible for covering one rationale
00:08:54.900 --> 00:08:59.040
I've said it already but I'll say it
00:08:59.040 --> 00:09:01.320
again because it's so amazing epsilon
00:09:01.320 --> 00:09:03.780
can be whatever positive number we want
00:09:03.780 --> 00:09:06.270
so not only can our sum be less than 1
00:09:06.270 --> 00:09:11.430
it can be arbitrarily small this is one
00:09:11.430 --> 00:09:13.320
of those results where even after seeing
00:09:13.320 --> 00:09:15.710
the proof it's still defies intuition
00:09:15.710 --> 00:09:18.690
the discord here is that the proof has
00:09:18.690 --> 00:09:20.490
us thinking analytically with the
00:09:20.490 --> 00:09:22.470
rational numbers in a list but our
00:09:22.470 --> 00:09:24.140
intuition has us thinking geometrically
00:09:24.140 --> 00:09:26.880
with all the rational numbers as a dense
00:09:26.880 --> 00:09:29.220
set on the interval where you can't skip
00:09:29.220 --> 00:09:31.440
over any continuous stretch because that
00:09:31.440 --> 00:09:34.880
would contain infinitely many rationals
00:09:34.880 --> 00:09:37.350
so let's get a visual understanding for
00:09:37.350 --> 00:09:40.950
what's going on beef side note here I
00:09:40.950 --> 00:09:42.540
had trouble deciding on how to
00:09:42.540 --> 00:09:45.240
illustrate small intervals since if I
00:09:45.240 --> 00:09:47.010
scale the parenthesis with the interval
00:09:47.010 --> 00:09:49.260
you won't be able to see them at all but
00:09:49.260 --> 00:09:50.910
if I just push the parenthesis together
00:09:50.910 --> 00:09:52.530
they cross over in a way that's
00:09:52.530 --> 00:09:55.350
potentially confusing nevertheless I
00:09:55.350 --> 00:09:57.330
decided to go with the ugly chromosomal
00:09:57.330 --> 00:10:00.090
cross so keep in mind the interval this
00:10:00.090 --> 00:10:02.280
represents is that tiny stretch between
00:10:02.280 --> 00:10:05.700
the Centers of each parenthesis okay
00:10:05.700 --> 00:10:08.940
back to the visual intuition consider
00:10:08.940 --> 00:10:10.830
when epsilon equals zero point three
00:10:10.830 --> 00:10:12.600
meaning if I choose a number between
00:10:12.600 --> 00:10:15.030
zero and one at random there's a 70%
00:10:15.030 --> 00:10:16.890
chance that it's outside those
00:10:16.890 --> 00:10:19.290
infinitely many intervals what does it
00:10:19.290 --> 00:10:23.010
look like to be outside the intervals
00:10:23.010 --> 00:10:25.709
the square root of 2 over 2 is among
00:10:25.709 --> 00:10:28.649
those 70% and I'm going to zoom in on it
00:10:28.649 --> 00:10:31.649
as I do so I'll draw the first 10
00:10:31.649 --> 00:10:33.630
intervals in our list within our scope
00:10:33.630 --> 00:10:36.570
of vision as we get closer and closer to
00:10:36.570 --> 00:10:38.760
the square root of 2 over 2 even though
00:10:38.760 --> 00:10:40.709
you will always find rationals within
00:10:40.709 --> 00:10:42.990
your field of view the intervals placed
00:10:42.990 --> 00:10:45.269
on top of those rationals get really
00:10:45.269 --> 00:10:48.329
small really fast one might say that for
00:10:48.329 --> 00:10:50.130
any sequence of rational numbers
00:10:50.130 --> 00:10:51.720
approaching the square root of 2 over 2
00:10:51.720 --> 00:10:54.420
the intervals containing the elements of
00:10:54.420 --> 00:10:56.850
that sequence shrink faster than the
00:10:56.850 --> 00:11:01.589
sequence converges notice intervals are
00:11:01.589 --> 00:11:03.630
really small if they show up late in the
00:11:03.630 --> 00:11:05.850
list and rationals show up late in the
00:11:05.850 --> 00:11:07.800
list when they have large denominators
00:11:07.800 --> 00:11:10.139
so the fact that the square root of 2
00:11:10.139 --> 00:11:13.110
over 2 is among the 70% not covered by
00:11:13.110 --> 00:11:15.930
our intervals is in a sense a way to
00:11:15.930 --> 00:11:18.540
formalize the otherwise vague idea that
00:11:18.540 --> 00:11:20.850
the only rational numbers close to it
00:11:20.850 --> 00:11:26.250
have a large denominator
00:11:26.250 --> 00:11:29.050
that is to say the square root of two
00:11:29.050 --> 00:11:36.760
over two is cacophonous in fact let's
00:11:36.760 --> 00:11:40.630
use a smaller epsilon say 0.01 and shift
00:11:40.630 --> 00:11:42.430
are set up to lie on top of the interval
00:11:42.430 --> 00:11:45.820
from 1 to 2 instead of from 0 to 1 then
00:11:45.820 --> 00:11:47.920
which numbers fall among that elite 1%
00:11:47.920 --> 00:11:52.570
covered by our tiny intervals almost all
00:11:52.570 --> 00:11:55.330
of them are harmonious for instance the
00:11:55.330 --> 00:11:57.460
harmonious irrational number 2 - the
00:11:57.460 --> 00:11:59.770
seven 12th is very close to three-halves
00:11:59.770 --> 00:12:01.840
which has a relatively fat interval
00:12:01.840 --> 00:12:03.910
sitting on top of it and the interval
00:12:03.910 --> 00:12:06.370
around 4/3 is smaller but still fat
00:12:06.370 --> 00:12:09.180
enough to cover 2 to the 5 twelfths
00:12:09.180 --> 00:12:12.190
which members of the 1% are cacophonous
00:12:12.190 --> 00:12:15.280
well the cacophonous rationals meaning
00:12:15.280 --> 00:12:16.920
those with high denominators and
00:12:16.920 --> 00:12:19.270
Irrational's that are very very very
00:12:19.270 --> 00:12:23.650
close to them however think of the
00:12:23.650 --> 00:12:25.600
savant who finds harmonic patterns in
00:12:25.600 --> 00:12:27.910
all rational numbers you could imagine
00:12:27.910 --> 00:12:30.250
that for her harmonious numbers are
00:12:30.250 --> 00:12:32.620
precisely those 1 percent covered by the
00:12:32.620 --> 00:12:34.570
intervals provided that her tolerance
00:12:34.570 --> 00:12:37.720
for error goes down exponentially for
00:12:37.720 --> 00:12:41.560
more complicated rationals in other
00:12:41.560 --> 00:12:43.930
words the seemingly paradoxical fact
00:12:43.930 --> 00:12:45.460
that you can have a collection of
00:12:45.460 --> 00:12:48.130
intervals densely populated range while
00:12:48.130 --> 00:12:50.310
only covering 1 percent of its values
00:12:50.310 --> 00:12:52.510
corresponds to the fact that harmonious
00:12:52.510 --> 00:12:54.850
numbers are rare even for the savant I'm
00:12:54.850 --> 00:12:57.040
not saying this makes the result more
00:12:57.040 --> 00:12:59.200
intuitive in fact I find it quite
00:12:59.200 --> 00:13:00.790
surprising that the savant I defined
00:13:00.790 --> 00:13:03.960
could find 99% of all ratios cacophonous
00:13:03.960 --> 00:13:06.220
but the fact that these two ideas are
00:13:06.220 --> 00:13:08.170
connected was simply too beautiful not
00:13:08.170 --> 00:13:10.560
to share