00:00:18.969 --> 00:00:20.650
It has such a wide range.

00:00:20.650 --> 00:00:24.190
It's the range of an orchestra.

00:00:34.190 --> 00:00:37.760
Any music I fall in love with,
I can have under my fingers.

00:00:37.760 --> 00:00:42.130
It's a beautiful instrument that is
sensitive, as the composer wants it to be,

00:00:42.130 --> 00:00:43.480
or as bombastic.

00:00:43.480 --> 00:00:46.039
Lots of clarity,
lots of warmth.

00:00:46.039 --> 00:00:49.360
There's so much
potential in this instrument.

00:00:49.840 --> 00:00:51.320
It's always astounding me.

00:00:51.320 --> 00:00:55.599
It is a machine that
is full of new sounds.

00:00:55.599 --> 00:01:02.119
An instrument that's easy to make music
with, even if you're not a virtuoso.

00:01:02.119 --> 00:01:05.079
You can play an entire
symphony on the piano.

00:01:05.079 --> 00:01:10.230
Since the piano is, in a sense, the
canon, it sets the law, which all other

00:01:10.230 --> 00:01:11.239
instruments follow.

00:01:11.239 --> 00:01:14.039
It was the instrument
of transposition.

00:01:14.039 --> 00:01:15.719
It was a way to
sing songs together.

00:01:15.719 --> 00:01:18.239
It was the television
of the 19th century.

00:01:19.119 --> 00:01:20.119
It's powerful.

00:01:20.119 --> 00:01:25.679
Like a member of your family who is
wonderful, but is more trouble than you

00:01:25.679 --> 00:01:26.760
would believe.

00:01:26.760 --> 00:01:28.679
The sands are always shifting.

00:01:28.679 --> 00:01:35.159
And different people's ears and the way
they listen and the way they hear will

00:01:35.159 --> 00:01:38.840
indicate a comfort zone for what
kind of lie they're looking for.

00:01:38.840 --> 00:01:43.320
That comes from
an imperfect sound.

00:01:43.320 --> 00:01:54.900
If we could have an instrument that
had only pure and perfect tones,

00:01:54.900 --> 00:01:56.590
we wouldn’t find
it very interesting.

00:01:56.590 --> 00:01:58.510
No color to it.

00:01:58.510 --> 00:02:00.230
No warmth.

00:02:00.230 --> 00:02:05.559
And yet, for centuries and centuries, it
was considered a crime against nature and

00:02:05.559 --> 00:02:06.670
against God.

00:02:06.670 --> 00:02:10.070
And so why was that the case?

00:02:10.710 --> 00:02:16.940
All these little lies are being told,
but in so doing, they falsified the

00:02:16.940 --> 00:02:21.750
perfection in order to create
equal imperfection everywhere.

00:02:21.750 --> 00:02:24.829
Acceptable imperfection.

00:03:17.820 --> 00:03:19.460
My name is David Clampett.

00:03:19.469 --> 00:03:23.710
I'm an associate professor in the
area of theory and composition.

00:03:23.949 --> 00:03:28.469
It's my principal interests lie
in mathematical music theory.

00:03:28.469 --> 00:03:35.780
Music theory is an ancient, really
ancient study, goes back to the ancient

00:03:35.780 --> 00:03:37.190
Greeks.

00:03:37.190 --> 00:03:41.889
And so we're
looking at patterns.

00:03:41.889 --> 00:03:49.750
But if you want to think about it just
very generally, music theory is studying

00:03:49.750 --> 00:03:50.829
patterns.

00:03:50.829 --> 00:03:56.800
Music theory is a system of analysis
with a vocabulary that allows us to

00:03:56.800 --> 00:04:02.869
explain what music is, its style, its
inner workings and its effects on people.

00:04:02.869 --> 00:04:07.829
Cognition and how people's minds work
when they listen to a piece of music.

00:04:08.829 --> 00:04:12.369
My name is Lincoln Hanks, and I'm a
composer and I teach music theory and

00:04:12.369 --> 00:04:13.550
composition.

00:04:13.550 --> 00:04:17.639
I know that's supposed to be a basic
question, and yet I think it's an

00:04:17.639 --> 00:04:22.590
incredibly interesting question that
doesn't really have a simple answer.

00:04:22.590 --> 00:04:27.849
Music theory has to do with my own
personal study of the relationship between

00:04:27.849 --> 00:04:29.750
mathematics and music.

00:04:29.750 --> 00:04:31.470
I'm Richard Plotkin.

00:04:31.470 --> 00:04:35.510
I am a professor
of music theory.

00:04:45.510 --> 00:04:48.599
This is the medieval concept of study,
you have the seven liberal arts, and

00:04:48.690 --> 00:05:02.150
four of them are the quadrivium,
three of them are the trivium.

00:05:08.150 --> 00:05:10.880
And music has always
existed on the math side.

00:05:10.970 --> 00:05:15.579
And one of the things that makes music
so interesting is that it can actually

00:05:15.579 --> 00:05:20.289
exist on both sides, the rhetorical
art forms and also on the math side.

00:05:20.289 --> 00:05:25.639
Music was not a purely
aesthetic enterprise.

00:05:25.639 --> 00:05:28.279
It was not just about
creating beautiful sounds.

00:05:28.279 --> 00:05:31.599
Music was also
about studying nature.

00:05:44.590 --> 00:05:49.920
Music was, in fact, studying
numbers through the study of sound.

00:05:49.929 --> 00:05:54.269
I am Eric Wang, professor of music
theory and early musical instrument

00:05:54.269 --> 00:05:55.329
specialist.

00:05:55.329 --> 00:05:58.769
Numbers are the
fabric of the universe.

00:05:58.769 --> 00:06:03.170
Numbers are the things that are
structuring everything in the world.

00:06:03.170 --> 00:06:06.849
So much of music can
be quantified as numbers.

00:06:07.409 --> 00:06:09.889
One frequency can be
a simple ratio of another.

00:06:09.889 --> 00:06:14.739
The time in which that frequency
sounds, the duration of notes, how we

00:06:14.739 --> 00:06:20.000
manipulate these parameters so that we
create expressive music that humans can

00:06:20.000 --> 00:06:20.650
relate to.

00:06:20.650 --> 00:06:26.239
And therefore, any musical instrument
is effectively a mathematical instrument

00:06:26.239 --> 00:06:27.010
as well.

00:06:34.010 --> 00:06:35.900
My name's Elaine Chu.

00:06:35.909 --> 00:06:37.590
I'm a professor of engineering.

00:06:38.550 --> 00:06:45.110
The work that I do has to do with using
mathematical models to demystify music.

00:06:45.110 --> 00:06:52.029
What does music consist of in
terms of its most basic pieces?

00:06:52.029 --> 00:06:59.880
And that, I would say, is
a sound, a note, a tone.

00:07:00.519 --> 00:07:03.199
How high or low a
particular pitch is.

00:07:03.199 --> 00:07:07.279
So this tone is
lower than this tone.

00:07:07.279 --> 00:07:13.730
And on top of that, I'd say the other
real two crafting elements of any music

00:07:13.730 --> 00:07:15.000
are dynamics.

00:07:15.000 --> 00:07:17.000
How loud or soft something is.

00:07:17.000 --> 00:07:24.110
And timbre, which is the particular sound
of a given instrument or person when

00:07:24.110 --> 00:07:27.000
they're singing
a particular note.

00:07:27.400 --> 00:07:33.019
That if I sing hello and you get a person
playing a French horn sounding that

00:07:33.019 --> 00:07:35.760
same note, I don't
sound like a French horn.

00:07:35.760 --> 00:07:37.239
I hope I don't sound
like a French horn.

00:07:37.239 --> 00:07:48.440
Chords are the simultaneous sounding
of multiple notes where those notes are

00:07:48.440 --> 00:07:52.750
intended to sound
together as a single unit.

00:07:56.780 --> 00:07:59.800
You're listening to two pitches at the
same time, usually three, to get the

00:07:59.920 --> 00:08:01.289
traditional chords, like the major or
minor diminished that we know in the

00:08:01.340 --> 00:08:04.039
traditional repertoire.

00:08:04.039 --> 00:08:06.440
My name is Tim Johnson.

00:08:06.440 --> 00:08:08.119
I'm a specialist
in music theory.

00:08:08.119 --> 00:08:12.880
Chords are sort of defined based on how
they stack into thirds traditionally.

00:08:13.440 --> 00:08:18.350
Once you stack a chord into thirds, the
lowest note determines what its root is

00:08:18.680 --> 00:08:23.119
So when we say a C chord, C would
be the lowest note of that chord.

00:08:23.119 --> 00:08:25.439
I am Michael Harrison.

00:08:25.439 --> 00:08:29.039
I am a composer utilizing
mathematically pure music.

00:08:29.039 --> 00:08:33.600
Music is moving airwaves.

00:08:34.359 --> 00:08:38.609
So when you strike a chord or a string
on a piano or play a note in a wind

00:08:38.609 --> 00:08:43.250
instrument or a percussion instrument,
what it does is it sets a series of

00:08:43.250 --> 00:08:44.319
waves into the air.

00:08:44.319 --> 00:08:52.239
And the relationships between these waves
are what create harmony or disharmony

00:08:52.239 --> 00:08:54.640
or what we call music.

00:08:55.359 --> 00:09:03.760
For example, if I play an interval like
an octave, it sounds very harmonious.

00:09:03.760 --> 00:09:09.029
And what is happening is the upper note,
the string is vibrating exactly twice

00:09:09.029 --> 00:09:11.239
as fast as the lower note.

00:09:12.029 --> 00:09:15.319
And because of this two to
one relationship in the waves

00:09:25.329 --> 00:09:26.800
it sounds very harmonious..

00:09:26.800 --> 00:09:34.809
Maybe for this purpose, it might be
useful for me to just demonstrate with the

00:09:34.809 --> 00:09:37.119
violin a couple of aspects.

00:09:37.119 --> 00:09:41.399
So the violin, we
have a vibrating string.

00:09:41.399 --> 00:09:45.399
This is the note D,
an open D string.

00:09:45.399 --> 00:09:50.800
So it's vibrating at a certain number
of cycles per second, let's say.

00:09:51.079 --> 00:09:56.720
But if you divide the string exactly in
half, as my pinky is doing here, we get

00:09:56.720 --> 00:09:59.680
the interval of an octave.

00:09:59.680 --> 00:10:02.199
So this is D, this is D also.

00:10:02.199 --> 00:10:08.590
And this is one of the fundamental
discoveries of the ancient Greeks.

00:10:09.989 --> 00:10:18.090
That there was this small number ratio,
the simplest of all sort of ratios in a

00:10:18.090 --> 00:10:21.470
sense, two to one, that was
characteristic of the octave.

00:10:21.470 --> 00:10:31.010
The next most important intervals, they
found that the three to one ratio was

00:10:31.010 --> 00:10:32.869
also very important.

00:10:33.229 --> 00:10:38.739
And if I divide the string exactly in
thirds, I've taken the open D string and

00:10:38.739 --> 00:10:42.750
then found a place
one third of the way up.

00:10:42.750 --> 00:10:46.270
And that gives me
the note up here, A.

00:10:46.270 --> 00:10:52.649
And presumably if this is some number
of vibrations per second, let's say it's

00:10:52.649 --> 00:10:56.750
200, it's not, but let's say it is, then
this would be three times that or 600.

00:10:58.470 --> 00:11:03.390
Other small ratios, four to
one gives us two octaves.

00:11:03.390 --> 00:11:11.140
You can see that all of music and the
kinds of accompaniment that we can put to

00:11:11.140 --> 00:11:16.510
melodies can be put together by stacking
pieces, which is a very abstract

00:11:16.510 --> 00:11:17.909
mathematical concept.

00:11:18.270 --> 00:11:23.569
And in the abstract world of music, we
use these building blocks to architect

00:11:23.569 --> 00:11:28.390
structures that develop over time to
create a piece of music that we pay

00:11:28.390 --> 00:11:29.670
attention to.

00:11:29.670 --> 00:11:36.350
So if you tune, for example, a piano
according to this basic principle, which

00:11:36.350 --> 00:11:38.270
we call just intonation.

00:11:43.190 --> 00:11:49.699
Just meaning pure or unaltered, then it
so happens by the physics of sound that

00:11:49.699 --> 00:11:55.149
not every interval will sound
harmonious with every other interval.

00:11:55.149 --> 00:12:01.729
Because when you do the math, there
end up being these gaps, which we call

00:12:01.729 --> 00:12:06.390
commas or sometimes in certain
contexts we call wolf tones.

00:12:07.949 --> 00:12:12.340
The classic example was actually
first figured out by the ancient Greek

00:12:12.340 --> 00:12:17.869
philosopher, mathematician Pythagoras
2500 years ago in ancient Greece.

00:12:17.869 --> 00:12:25.409
Way back in the 6th century BCE, there
was a character named Pythagoras that

00:12:25.409 --> 00:12:30.279
most of us learn about because in
school we learn this formula for right

00:12:30.279 --> 00:12:33.059
triangles, a squared plus b
squared equals c squared.

00:12:33.059 --> 00:12:40.079
But Pythagoras was a
much fuller character.

00:12:40.079 --> 00:12:44.239
He was a guru, wandered the streets
in white robes and a gold crown.

00:12:44.239 --> 00:12:47.770
He was said to have magical powers,
could stay the winds, could bring dead

00:12:47.770 --> 00:12:49.279
bodies back to life and so on.

00:12:49.279 --> 00:12:51.039
And so he had a cult following.

00:12:51.039 --> 00:12:55.039
We don't know a whole lot about the
Pythagoreans, mainly because they were

00:12:55.039 --> 00:12:56.239
sworn to secrecy.

00:12:56.800 --> 00:13:01.180
And in fact, if someone joined the
Pythagoreans, they were not allowed to speak

00:13:01.180 --> 00:13:02.559
at all for three years.

00:13:02.559 --> 00:13:09.239
What we do know, though, that they
were a kind of religious cult.

00:13:09.239 --> 00:13:17.520
They believed very strongly in a
harmony of the universe and of life.

00:13:17.520 --> 00:13:23.119
The group believed that the universe
was composed somehow of numbers.

00:13:23.559 --> 00:13:28.559
They thought that numbers were unchanging,
numbers did not seem to be part of

00:13:28.559 --> 00:13:30.600
the chaos of everyday life.

00:13:30.600 --> 00:13:34.119
Mathematical equations
always seem to hold.

00:13:34.119 --> 00:13:40.279
And so they believe somehow the gods
had created the universe out of numbers.

00:13:40.279 --> 00:13:45.750
I'm Michael Fowler, I teach physics, but
I'm also interested in the history of

00:13:45.750 --> 00:13:46.680
science.

00:13:46.680 --> 00:13:52.000
Pythagoras also developed the
mathematics of musical beauty.

00:14:15.920 --> 00:14:19.699
Pythagoras was walking by a blacksmith's
shop and he heard the smithies

00:14:19.890 --> 00:14:24.000
hammering anvils and he noticed that sometimes
the sounds of the anvils ringing were

00:14:24.220 --> 00:14:31.100
in beautiful harmony with each
other, and sometimes they weren't.

00:14:31.100 --> 00:14:34.020
And he wanted to
know why that happened.

00:14:34.899 --> 00:14:38.409
And he measured the weights of the
hammerheads and found that when they were in

00:14:38.409 --> 00:14:42.300
certain proportions to each
other, the sound was beautiful.

00:14:42.300 --> 00:14:45.340
And when they were not in those
proportions, the sound was ugly.

00:14:45.340 --> 00:14:51.489
He went home and he supposedly used
these proportions that he had discovered

00:14:51.489 --> 00:14:53.420
with string length.

00:14:53.699 --> 00:14:58.829
And some of these proportions were two
to one, that if one string is twice as

00:14:58.829 --> 00:15:04.470
long as another string, the shorter
string vibrates twice as fast as the lower

00:15:04.470 --> 00:15:07.380
string, and it creates
a beautiful harmony.

00:15:07.380 --> 00:15:11.180
That harmony is
known as the octave.

00:15:12.779 --> 00:15:17.489
Another proportion he found was three
to two, when the higher vibration is

00:15:17.489 --> 00:15:22.859
vibrating three times for every two
vibrations of the lower vibration, and that

00:15:22.859 --> 00:15:24.500
produces a fifth.

00:15:24.500 --> 00:15:28.640
And then he had one more formula that
produced a fourth, which is an upside

00:15:28.640 --> 00:15:29.539
down fifth.

00:15:29.539 --> 00:15:33.899
If you take the top note of a fifth,
put it on the bottom, you get a fourth.

00:15:33.899 --> 00:15:37.899
So Pythagoras had
his beautiful formulas.

00:15:37.899 --> 00:15:42.760
If you use these formulas, two to one,
three to two, four to three, you get

00:15:42.760 --> 00:15:43.579
harmony.

00:15:43.579 --> 00:15:51.690
What happens after Pythagoras establishes
these numbers is the church adopts

00:15:51.690 --> 00:15:52.909
them.

00:15:53.510 --> 00:15:57.550
By the second century, you have Clement
of Alexandria saying it was Christ who

00:15:57.550 --> 00:15:59.390
had tuned the
world in this way.

00:15:59.390 --> 00:16:04.420
And you have popes, St. Augustine, for
example, saying that churches should be

00:16:04.420 --> 00:16:08.260
built according to these proportions
that Pythagoras created because they are

00:16:08.260 --> 00:16:09.350
God's proportions.

00:16:09.350 --> 00:16:14.949
They become proportions that you can't
violate because now you have religious

00:16:14.949 --> 00:16:16.729
doctrine.

00:16:17.890 --> 00:16:21.770
So these become the
harmonies to use in music.

00:16:21.770 --> 00:16:29.100
However, there was a fault in this
formula that Pythagoras discovered.

00:16:29.100 --> 00:16:33.979
The fault was that if you stack octaves
up, two to one, two to one, two to one,

00:16:33.979 --> 00:16:39.309
two to one, two to one, and so on, and
you stack fifths, three to two, three to

00:16:39.309 --> 00:16:42.539
two, three to two, three to
two, you should at the very end,

00:16:42.539 --> 00:16:46.510
after you've gone through 12 different
pitches with the fifths, at the very end

00:16:46.510 --> 00:16:49.340
you should come
back to the same note.

00:16:49.340 --> 00:16:53.690
And so you should have the same note at
the top of the stack of octaves and at

00:16:53.690 --> 00:16:55.579
the top of the stack of fifths.

00:16:55.579 --> 00:16:58.739
And it turned out they were not the same
note, that they were out of tune with

00:16:58.739 --> 00:16:59.460
each other.

00:16:59.460 --> 00:17:04.600
So the frequency of the notes spiraled
upwards instead of connecting back with

00:17:04.600 --> 00:17:06.500
each other like a circle.

00:17:07.619 --> 00:17:12.800
This became known in ancient Greece as
a dark secret, terrible secret, that

00:17:12.800 --> 00:17:18.699
there was a curve in musical space and
that things didn't work perfectly the

00:17:18.699 --> 00:17:20.979
way they were supposed to.

00:17:35.970 --> 00:17:38.979
As time goes on, people start enjoying
the idea of having other beautiful

00:17:38.989 --> 00:17:42.050
harmonies as well.

00:17:42.889 --> 00:17:47.220
In the 13th century, you have especially
the English making use in choral music

00:17:47.220 --> 00:17:50.610
of an interval called the
third, and it's beautiful.

00:17:50.610 --> 00:17:55.609
And in fact, if we have something today
that we call a major third, it has this

00:17:55.609 --> 00:17:59.250
fifth, but also
has the third in it.

00:18:09.630 --> 00:18:13.619
And to me, that third, that sound of
the third, in a way, is the sound of the

00:18:13.619 --> 00:18:21.640
human heart, because you have the fifth,
which is sort of like the ceiling of

00:18:21.640 --> 00:18:23.619
heaven and the
floor of the earth.

00:18:23.619 --> 00:18:30.180
And then in between, you
have this warm, beautiful sound.

00:18:30.180 --> 00:18:36.460
The English loved this sound, and then
starting in the 13th century, they'd use

00:18:36.460 --> 00:18:40.809
it quite a lot in
their choral music.

00:18:40.809 --> 00:18:50.100
But just as octaves and fifths can't
co-exist together perfectly, thirds also

00:18:50.100 --> 00:18:51.029
can't.

00:18:51.029 --> 00:18:52.789
Thirds mess it up even more.

00:18:53.069 --> 00:18:58.840
And so this creates an even greater need
for temperament, for some way to make

00:18:58.840 --> 00:19:00.670
all of these sounds
blend together.

00:19:00.670 --> 00:19:05.350
When we came to putting pitches
together and we started creating these

00:19:05.350 --> 00:19:13.029
relationships of pitches, you
started getting very ugly sounds.

00:19:13.029 --> 00:19:15.670
Things sounded out of tune
and they were unpleasing.

00:19:16.670 --> 00:19:17.779
And when you're an actual musician having
to tune a harpsichord, or an organ,

00:19:17.779 --> 00:19:24.100
or a lute, or a harp, let's say you try
to actually tune that instrument all

00:19:24.100 --> 00:19:26.609
according to pure fifths,

00:19:26.609 --> 00:19:30.059
and you find out that a whole bunch
of intervals are out of tune, and it

00:19:30.059 --> 00:19:32.250
doesn’t really work well
for your piece of music,

00:19:32.250 --> 00:19:35.690
then you actually have to think, "Okay,
what do I have to do to adjust this?"

00:19:35.690 --> 00:19:39.050
And that's when theory and practice
can't coincide with each other.

00:19:39.050 --> 00:19:44.760
And so music theorists and composers
started developing ways of tempering that

00:19:44.760 --> 00:19:45.449
scale,

00:19:45.449 --> 00:19:49.050
kind of shifting pitches around a little
bit so that when you put two pitches

00:19:49.050 --> 00:19:50.649
together, they sound good.

00:19:50.649 --> 00:20:02.100
Temperament is an imperfect solution
to a very big problem in music.

00:20:02.100 --> 00:20:07.109
Temperament comes from the Latin temporare,
which is to moderate or to lessen

00:20:07.109 --> 00:20:10.899
something that is the
excessive quality of something.

00:20:10.899 --> 00:20:16.279
So in this case, in music, we're
talking about slightly detuning musical

00:20:16.279 --> 00:20:17.380
intervals.

00:20:17.380 --> 00:20:23.319
Tuning is the foundation of how
you're putting everything initially,l

00:20:23.319 --> 00:20:25.779
and temperament is
the tempering of intervals.

00:20:26.739 --> 00:20:31.359
So a major third, what they do, if
this is the lower note and this is the

00:20:31.359 --> 00:20:32.420
higher note,

00:20:32.420 --> 00:20:39.850
you temper this note by tuning it
in terms of tightening the string,

00:20:39.850 --> 00:20:44.260
or if you’re on a piano,
making this a little bit higher.

00:20:44.260 --> 00:20:49.059
So now the interval, instead of
being this wide, is this wide.

00:20:49.059 --> 00:20:54.250
And so this is the correct calculation
in Pythagorean terms, the tuning, the

00:20:54.250 --> 00:20:57.220
interval, the division of the string
by that super particular ratio.

00:20:57.220 --> 00:21:02.659
And this is the sound we actually
want. So tuned and tempered.

00:21:02.659 --> 00:21:07.779
So the sound we want is qualitative.
It depends on who's listening.

00:21:07.779 --> 00:21:12.720
It depends on what music has been
composed and what the people in that

00:21:12.720 --> 00:21:15.859
particular time
period find pleasing.

00:21:15.859 --> 00:21:21.779
And it means that in order to trick the
ear into thinking the piano's in tune,

00:21:21.779 --> 00:21:28.180
you have to introduce even more
falseness, even more untruthfulness to the

00:21:28.180 --> 00:21:32.899
tuning in order for it to sound
in tune to a specific individual.

00:21:32.899 --> 00:21:37.940
I am Mark Weiner, piano
tuner and technician.

00:21:37.940 --> 00:21:43.600
And specific individuals will have a
certain comfort zone for the lies you're

00:21:43.600 --> 00:21:48.420
telling in order to trick their ear
into thinking the piano's in tune.

00:21:48.420 --> 00:21:55.940
One of the things I tell people is that
tuning is one big lie made out of many,

00:21:55.940 --> 00:21:58.979
many multiple micro lies.

00:21:58.979 --> 00:22:04.190
And all those little lies come together
to create one big lie of being in tune

00:22:04.190 --> 00:22:05.699
and tricking you.

00:22:05.699 --> 00:22:11.630
But of course, nothing's in tune.
Absolutely nothing from the point of view of

00:22:11.630 --> 00:22:14.180
perfection or measurement.

00:22:14.180 --> 00:22:24.260
There's a sort of sadness inherent in
manipulating what, to me, is fundamental

00:22:24.260 --> 00:22:25.250
to the universe.

00:22:33.250 --> 00:22:37.500
The thing is that the Pythagorean
system is mathematically the simplest

00:22:37.500 --> 00:22:44.350
possible solution and it
works experimentally perfectly.

00:22:44.350 --> 00:22:50.720
So, for example, nature says that pi is,
well, 3.1415 something or other, right

00:22:50.720 --> 00:22:52.350
it is an irrational number.

00:22:52.350 --> 00:22:57.930
I could say, no, I've decided that pi
is 3, right? I could just impose that on

00:22:57.930 --> 00:22:58.510
the world.

00:22:58.510 --> 00:23:04.309
So which is the better scientific
approach? Temperament? That's just imposing

00:23:04.309 --> 00:23:07.229
something. Nature
isn't built that way.

00:23:07.229 --> 00:23:14.789
I'm Christopher Lehrich. I work a lot
on theory of religion. I used to play a

00:23:14.789 --> 00:23:18.829
lot of music and did
a lot of music theory.

00:23:18.829 --> 00:23:23.819
Don't think that temperament was anything
stable. There were different people

00:23:23.819 --> 00:23:26.909
tuning things in different
ways for quite a while.

00:23:26.909 --> 00:23:32.010
The sound we want is qualitative. It
depends on who's listening and what the

00:23:32.010 --> 00:23:36.109
people in that particular
time period find pleasing.

00:23:36.109 --> 00:23:42.260
What we want is also slightly defined
by theorists, a tuning theorist at the

00:23:42.260 --> 00:23:46.109
time, let's say Zarlino, who's
coming up with temperament.

00:23:46.109 --> 00:23:51.729
And he says, well, listen, these things
allow you to do more if you just mush

00:23:51.729 --> 00:23:56.319
this interval this way and it becomes
more useful to us if we mush this this

00:23:56.319 --> 00:23:56.829
way.

00:23:56.829 --> 00:24:02.050
What Zarlino does is to propose something
called the scenario, which bases the

00:24:02.050 --> 00:24:07.229
entire system now not on
4 as the high integer, but 6.

00:24:07.229 --> 00:24:14.510
So now we bring in the ratio of 4 to
5 or 3 to 5 or 5 to 6 and so forth.

00:24:14.510 --> 00:24:17.069
And this is supposed
to solve the problem.

00:24:17.069 --> 00:24:21.960
If we include 1, 2, 3, 4, 5, 6, we can
create all the ratios we need. And he

00:24:21.960 --> 00:24:24.750
had some good
justifications for it.

00:24:24.750 --> 00:24:28.649
6 is the first perfect number, being the
sum of all the numbers of which it is

00:24:28.649 --> 00:24:29.390
a multiple.

00:24:29.390 --> 00:24:33.409
There were six planets in the sky. Plato
and his philibus indicated that hymns

00:24:33.409 --> 00:24:36.270
should not celebrate
more than six generations.

00:24:36.270 --> 00:24:41.069
The world was created in six
days. Who could argue with this?

00:24:41.390 --> 00:24:46.670
This is the dominant solution to the
problem throughout roughly the middle of

00:24:46.670 --> 00:24:53.940
the 17th century, so
for about 100 years.

00:24:53.940 --> 00:24:58.739
One problem remained, which is the minus
6, which is in the proportion 8 to 5.

00:24:58.739 --> 00:25:02.609
But Zarlino couldn't bring stopping 7
and 8 in because there were all kinds of

00:25:02.609 --> 00:25:05.619
ugly proportions that you get
when you use those numbers.

00:25:06.579 --> 00:25:13.309
What you get after just intonation and
Zarlino's approach to temperament is

00:25:13.309 --> 00:25:17.100
mean tone temperament.

00:25:17.100 --> 00:25:22.059
Sort of this agreed-upon,
rough, standard.

00:25:22.059 --> 00:25:27.210
You temper the intervals towards the
middle of the instrument you're tuning and

00:25:27.210 --> 00:25:31.020
you let the other intervals
be wider towards the edges.

00:25:31.020 --> 00:25:34.880
So, in the area that you're playing the
most, many things sound good in many

00:25:34.880 --> 00:25:35.979
different keys.

00:25:35.979 --> 00:25:39.579
As you go towards the
edges, certain things won't.

00:25:39.579 --> 00:25:44.440
And good is a qualitative judgment based
on what those people liked to hear at

00:25:44.440 --> 00:25:45.260
the time.

00:25:45.260 --> 00:25:50.989
In the case of mean tone temperament,
what one has to do is take a fifth and

00:25:50.989 --> 00:25:52.300
slightly flatten it.

00:25:52.300 --> 00:25:59.420
So the keys that have the fewest sharps
and flats sound really wonderful.

00:26:08.420 --> 00:26:14.000
But if I keep going into the remote
keys, this is where the errors start to

00:26:14.220 --> 00:26:15.100
creep in.

00:26:37.100 --> 00:26:43.000
Johannes Kepler is chiefly famous
for his contributions to astronomy,

00:26:43.340 --> 00:26:47.659
but he probably spent as much effort
on music as he did on astronomy.

00:26:48.300 --> 00:26:52.140
His passion was, in fact,
to read the mind of God.

00:26:52.140 --> 00:26:57.979
In the sense, to understand why the
creator created the universe in the way he

00:26:57.979 --> 00:26:58.539
did.

00:26:58.539 --> 00:27:02.059
Kepler felt that harmony
was the key to the universe.

00:27:02.059 --> 00:27:04.059
So they connected
these two things.

00:27:04.059 --> 00:27:07.579
They saw harmony both in
the planets and also in music.

00:27:07.579 --> 00:27:12.430
Kepler decided that planets travel in
ellipses, and indeed we believe that is

00:27:12.430 --> 00:27:13.500
indeed true.

00:27:13.500 --> 00:27:16.380
They speed up when
they're closest to the sun,

00:27:16.380 --> 00:27:20.229
and they're at their slowest velocity
when they're the furthest point away from

00:27:20.229 --> 00:27:20.859
the sun.

00:27:20.859 --> 00:27:25.430
And he argues that Jupiter and Mars, when
they're together at the closest point

00:27:25.430 --> 00:27:26.699
away from the sun,

00:27:26.699 --> 00:27:29.819
will have a ratio of
an octave, two to one.

00:27:29.819 --> 00:27:35.020
Venus and the Earth, the ratio there
is an octave plus a major fifth.

00:27:35.020 --> 00:27:37.819
Venus and Mars, that
ratio is a major third.

00:27:37.819 --> 00:27:44.220
So he started naming the planets as
being part of this perfect scale.

00:27:44.220 --> 00:27:51.739
They were so intent on proving the
rightness of the Pythagorean Church-endorsed

00:27:51.739 --> 00:27:52.460
proportions

00:27:52.460 --> 00:27:56.140
that they allowed their scientific
thinking to be changed by it.

00:27:56.140 --> 00:28:06.170
It's not until you get a little
later, until you get Vincenzo Galilei,

00:28:06.170 --> 00:28:11.069
who's Galileo's father, that you get
somebody who actually claims that it's all

00:28:11.069 --> 00:28:12.250
subjective.

00:28:12.250 --> 00:28:17.210
God couldn't care less whether our
fifths were perfectly in tune or not.

00:28:17.210 --> 00:28:23.739
That's a radical statement, because how
dare you think outside the box in that

00:28:23.739 --> 00:28:24.090
way?

00:28:24.090 --> 00:28:29.930
In fact, there was a theologian in Italy,
Giordano Bruno, who was burned at the

00:28:29.930 --> 00:28:31.289
stake in 1600

00:28:31.289 --> 00:28:34.729
for saying there was a
possibility of infinite worlds.

00:28:36.010 --> 00:28:41.659
And in fact, Galilei was saying there's
a possibility of infinite musical

00:28:41.659 --> 00:28:42.329
worlds.

00:28:42.329 --> 00:28:53.220
Well, Vincenzo argued that in fact, what
sounds harmonious is to some extent a

00:28:53.220 --> 00:28:54.260
social construct.

00:28:54.260 --> 00:29:01.539
And indeed, Vincenzo Galilei pointed out
that in different cultures, the tunes,

00:29:01.539 --> 00:29:05.140
the popular tunes varied
in the keys that were used.

00:29:05.140 --> 00:29:10.899
And so sometimes things sounded more
euphonious to one group than another.

00:29:10.899 --> 00:29:17.159
So there were no longer these, like,
great religious icons in the sky, these

00:29:17.159 --> 00:29:17.779
numbers.

00:29:23.729 --> 00:29:29.409
In the late 17th century, you have Andreas
Werckmeister creating something called

00:29:29.729 --> 00:29:34.510
well-temperament. And this is a tuning
that allows you to play in any key you

00:29:35.010 --> 00:29:38.609
want, any of the 12 keys, but
they don't all sound the same.

00:29:38.609 --> 00:29:42.449
So some of the keys
sound a little harsh.

00:29:42.449 --> 00:29:48.210
Whereas in Meantone, some keys
work and other keys don't work.

00:29:48.210 --> 00:29:49.970
There's no two ways about it.

00:29:49.970 --> 00:29:53.170
In well-temperament, you can go
around the whole circle of keys

00:29:54.130 --> 00:29:58.210
and then some of them are more serene
and bright and beautiful, and some of

00:29:58.210 --> 00:29:59.729
them are harsher.

00:29:59.729 --> 00:30:03.489
My examples are going to be Bach's
Well-Tempered Clavier, book one.

00:30:03.489 --> 00:30:05.409
So the first one
goes like this.

00:30:10.399 --> 00:30:16.000
And on the well-temperament, you could
definitely hear the different qualities.

00:30:16.300 --> 00:30:27.500
It's in my way of thinking this was a
very short-lived idea, because you very

00:30:27.500 --> 00:30:28.090
quickly get

00:30:28.090 --> 00:30:32.850
people writing music that moves from key
to key to key to key, doesn't stay in

00:30:32.850 --> 00:30:33.690
one spot.

00:30:33.690 --> 00:30:40.520
And then if the sound changes constantly
from bright and light and airy to

00:30:40.520 --> 00:30:40.970
harsh

00:30:40.970 --> 00:30:44.250
in the middle of the music, it
doesn't always suit the music.

00:30:44.250 --> 00:30:45.850
It's not what the
composer wants.

00:30:45.850 --> 00:30:51.930
So this tuning is now failing us if we
want to take advances into music along

00:30:51.930 --> 00:30:53.130
Beethoven's lines.

00:30:53.130 --> 00:30:56.809
When we get to the Romantic era,
we're in really deep trouble.

00:30:56.809 --> 00:31:01.550
Here's a little bit of the famous Chopin
Prelude in A minor, in this Well-temperament

00:31:34.090 --> 00:31:36.050
Then the limitations
are really clear.

00:31:36.649 --> 00:31:39.050
What could
musicians do about it?

00:31:39.050 --> 00:31:47.380
Eventually, a proposal came along for
a different kind of tuning called equal

00:31:47.380 --> 00:31:47.940
temperament,

00:31:47.940 --> 00:31:50.899
the one that we have today.

00:31:50.899 --> 00:31:57.960
In this temperament, almost every
proportion that was considered important and

00:31:57.960 --> 00:32:00.020
sacred is destroyed.

00:32:00.020 --> 00:32:02.819
Everything is compromised.

00:32:03.859 --> 00:32:05.300
And two things happened.

00:32:05.300 --> 00:32:08.739
One was that this was
theologically reprehensible.

00:32:08.739 --> 00:32:15.699
The other, the music does become
robbed of certain beautiful qualities.

00:32:15.699 --> 00:32:25.779
What wound up happening is with
mathematics sort of coming of age,

00:32:25.779 --> 00:32:30.899
people were finally able to say, "Hey,
I don't want all these things that are

00:32:30.899 --> 00:32:32.819
off by just small amounts.

00:32:32.819 --> 00:32:36.340
I want to divide the
12 notes evenly."

00:32:36.340 --> 00:32:42.579
Therefore, each fifth is
made slightly out of tune,

00:32:42.579 --> 00:32:46.850
so that when you go around a series of
12 of them, you'll come back to the same

00:32:46.850 --> 00:32:47.350
place.

00:32:48.659 --> 00:32:52.899
So each fifth is detuned by 2%.

00:32:52.899 --> 00:32:57.859
Then, when you go around
a series of 12 of them,

00:32:57.859 --> 00:33:01.940
you'll have covered 2%
of a half-step for each fifth,

00:33:01.940 --> 00:33:06.899
times 12 fifths is 24% of a half-step,
or in music we call that 24 cents.

00:33:06.899 --> 00:33:13.890
And 24 cents is exactly the amount to
bring things back into sync with each

00:33:13.890 --> 00:33:14.819
other.

00:33:15.770 --> 00:33:22.609
Now, as a result, the fifths are all
slightly out of tune with themselves.

00:33:23.609 --> 00:33:27.029
And so the very fabric of our
equal-tempered system is compromised in its

00:33:27.340 --> 00:33:28.579
purity.

00:33:33.680 --> 00:33:38.640
Even great thinkers like Newton found
the solution of this equal temperament to

00:33:38.640 --> 00:33:40.319
be really unsatisfactory.

00:33:40.319 --> 00:33:44.899
Newton said that the musical tones that
you get in this case are like soiled

00:33:44.899 --> 00:33:45.920
and faded colors.

00:33:51.920 --> 00:33:58.279
So in this period from the late 17th
century to sort of the turn of the 17th,

00:33:58.289 --> 00:33:59.250
18th century,

00:34:00.210 --> 00:34:07.199
you would think, well, music
presumably is going to make all of its

00:34:07.199 --> 00:34:09.489
developments and move clearly,

00:34:09.489 --> 00:34:14.190
along with the sciences, because it's one of
the sciences, one of the quadrivium, right?

00:34:14.770 --> 00:34:19.010
We've got music and arithmetic
and astronomy and geometry.

00:34:19.010 --> 00:34:23.409
So music is one of the quadrivium, so
it seems like obviously that stuff's

00:34:23.409 --> 00:34:26.510
all being revolutionized by the science
end of things, it's going to go that

00:34:26.510 --> 00:34:26.850
way.

00:34:26.850 --> 00:34:31.809
But obviously at the same time, religion
is going to want to hold on to it.

00:34:31.809 --> 00:34:35.170
I mean, hymns and masses
and all that good stuff.

00:34:35.170 --> 00:34:39.899
So there's a sense in which music is
getting pulled in two quite different

00:34:39.899 --> 00:34:40.930
directions.

00:34:40.930 --> 00:34:45.140
And one of the things that's most
remarkable, and as far as I know, very ill-

00:34:45.140 --> 00:34:45.809
explained,

00:34:45.809 --> 00:34:50.750
is that in the 16th and 17th centuries,
it goes exactly the way you think it

00:34:50.750 --> 00:34:51.409
would.

00:34:52.050 --> 00:34:56.530
So it goes racing along into equal
temperament and nice scientific

00:34:56.530 --> 00:34:58.690
divisions and all that
good mathematical stuff.

00:34:58.690 --> 00:35:01.889
And once that's all been
completely accepted,

00:35:01.889 --> 00:35:06.690
music begins to collapse
out of the sciences utterly.

00:35:06.690 --> 00:35:11.329
Very hard to explain, very hard to account
for, and it's really not clear why.

00:35:11.329 --> 00:35:16.930
It seems to me that one thing that
begins to become apparent is that the way

00:35:16.930 --> 00:35:17.969
you evaluate

00:35:18.610 --> 00:35:23.489
which system you want to go with
is very much an aesthetic problem.

00:35:23.489 --> 00:35:30.739
So, for example, we say that Zarlino's
theory is messy from a scientific point

00:35:30.739 --> 00:35:31.329
of view.

00:35:31.329 --> 00:35:37.440
Well, because you have to have these,
like, platonic, abstract forms of foreign

00:35:37.440 --> 00:35:37.729
ness or

00:35:37.729 --> 00:35:41.329
whatever kind of in the middle of it
without any real justification, because

00:35:41.329 --> 00:35:42.369
the Bible says so.

00:35:42.369 --> 00:35:45.409
We don't accept that,
that's complicated.

00:35:45.969 --> 00:35:49.170
But there's another sense in which
there's a remarkably simple theory.

00:35:49.170 --> 00:35:52.850
You only need four numbers for
Pythagoras's theory, and you say, "Why?"

00:35:52.850 --> 00:35:53.809
Because it's four.

00:35:53.809 --> 00:36:00.130
And we have various different versions
of it, and the only way to choose is

00:36:00.130 --> 00:36:00.770
aesthetically,

00:36:00.770 --> 00:36:05.809
to say, "Well, this one just feels right
to me," or, "That one kind of chimes

00:36:05.809 --> 00:36:06.210
in well

00:36:06.210 --> 00:36:08.130
with this passage
I read in the Bible."

00:36:08.130 --> 00:36:14.930
This is not what the kind of scientific
mindset that's being developed here.

00:36:14.930 --> 00:36:16.210
That's not what they want.

00:36:16.210 --> 00:36:20.000
And so I think there's really something
of an attempt to sweep this under the

00:36:20.000 --> 00:36:20.449
rug.

00:36:20.449 --> 00:36:26.869
And then there's also a
very convenient excuse,

00:36:26.869 --> 00:36:31.819
which is that increasingly we're saying
aesthetic questions are not scientific

00:36:31.819 --> 00:36:32.550
questions.

00:36:32.550 --> 00:36:39.369
So increasingly we're saying that
music, at the crux of all of it,

00:36:39.369 --> 00:36:40.069
theoretically,

00:36:40.069 --> 00:36:43.110
seems to be coming down
to some aesthetic problems.

00:36:43.110 --> 00:36:44.789
We don't do aesthetics.

00:36:44.789 --> 00:36:48.309
For scientists, we leave
that to the art people.

00:36:48.309 --> 00:36:50.229
So it's not our
problem anymore.

00:36:50.229 --> 00:36:56.630
And so this sort of sweeps
the whole music problem away.

00:36:56.630 --> 00:37:00.880
And now that we've already got equal
temperament, what do we need science for

00:37:00.880 --> 00:37:01.670
anymore?

00:37:01.670 --> 00:37:04.710
My name is Miles Jackson.

00:37:04.710 --> 00:37:06.469
I am a history of
science professor.

00:37:07.829 --> 00:37:10.710
The interesting thing is that
when we get to the 19th century,

00:37:10.710 --> 00:37:16.639
we tend to move away from the physics of
the spheres, right, and perfect harmonies,

00:37:16.869 --> 00:37:20.550
And we enter the world of the industrial
revolution, of mass production.

00:37:20.550 --> 00:37:24.710
Certainly physicists are
still interested in music,

00:37:24.710 --> 00:37:27.430
but they're interested in
musical instrument manufacture.

00:37:27.430 --> 00:37:30.989
Although equal temperament, the theory,
the mathematics behind equal temperament,

00:37:31.270 --> 00:37:33.989
had been well known
since the 16th century,

00:37:33.989 --> 00:37:38.389
what a lot of people don't realize is
that actual tuning in equal temperament

00:37:38.389 --> 00:37:40.869
does not happen
until the 1830s.

00:37:40.869 --> 00:37:43.510
What really cemented
equal temperament

00:37:43.510 --> 00:37:49.670
was not the desire or need of composers
and musicians for this particular

00:37:49.670 --> 00:37:50.309
tuning,

00:37:50.309 --> 00:37:53.110
but it was the piano
manufacturers.

00:37:53.110 --> 00:37:58.149
What happened is in the
early to mid-19th century,

00:37:58.149 --> 00:38:02.869
John Broadwood and Sons was
a piano company based in London

00:38:02.869 --> 00:38:05.110
that was shipping
pianos all over the world.

00:38:05.110 --> 00:38:08.550
And in different parts of the world,
they would be tuned very differently.

00:38:08.550 --> 00:38:13.829
And they wanted their pianos to
sound the same wherever they were.

00:38:13.829 --> 00:38:20.420
So they started sending out their pianos
with a manual explaining how to tune

00:38:20.420 --> 00:38:20.710
it

00:38:20.710 --> 00:38:23.590
in what is now the standard
tuning, equal temperament.

00:38:23.590 --> 00:38:29.270
And this, more than anything,
codified the equal tempered system

00:38:29.270 --> 00:38:33.110
as the system that was used on
pianos and keyboard instruments.

00:38:33.110 --> 00:38:37.909
And since the piano is the
center of our educational system

00:38:37.909 --> 00:38:39.750
and our music and music theory,

00:38:39.750 --> 00:38:45.110
equal temperament ultimately dominated
the world of music and music theory

00:38:45.110 --> 00:38:50.550
to the point now that most musicians
don't even know that anything else exists.

00:39:21.850 --> 00:39:24.250
Music perception, of
course, is a huge topic.

00:39:24.250 --> 00:39:30.250
There are very definitely certain things
that one person will hear one way,

00:39:30.250 --> 00:39:31.369
another person hear another.

00:39:31.369 --> 00:39:35.449
That is the big question.
That's the whole question.

00:39:35.449 --> 00:39:37.369
What's happening?
What is happening?

00:39:37.369 --> 00:39:42.250
How are we interpreting these elements
that are in the outside world?

00:39:45.880 --> 00:39:51.000
Music is, for me, by definition,
what happens inside human beings.

00:39:51.000 --> 00:39:52.840
It's a subjective experience.

00:39:52.840 --> 00:39:59.239
What are the commonalities of the
way we interpret music we hear?

00:39:59.239 --> 00:40:01.559
Why does some
music evoke sadness?

00:40:01.559 --> 00:40:13.920
The human mind interprets music based
upon a set of cultural expectations,

00:40:13.920 --> 00:40:15.520
cultural norms.

00:40:15.520 --> 00:40:21.680
I work in what's called a
Western music tradition.

00:40:21.680 --> 00:40:27.199
So I'm working with music
that was developed in Europe,

00:40:27.199 --> 00:40:30.000
in the Western world,
from the medieval period,

00:40:30.000 --> 00:40:32.159
all the way through
the 20th century.

00:40:32.159 --> 00:40:36.000
And so when I listen
to a piece of music,

00:40:36.000 --> 00:40:40.319
I have that set of cultural
norms inside of me.

00:40:40.319 --> 00:40:44.639
So when I hear something, it's always
being referenced to those norms.

00:40:44.639 --> 00:40:50.119
Now, what if I was a man or a woman who
grew up in a village away from Western

00:40:50.119 --> 00:40:50.559
music

00:40:50.559 --> 00:40:55.680
and instead experienced music in a
ritualistic kind of fashion or a tribal

00:40:55.680 --> 00:40:56.960
context

00:40:56.960 --> 00:41:00.800
where music was performed for
a number of different reasons

00:41:00.800 --> 00:41:02.960
than they are in Western music?

00:41:02.960 --> 00:41:05.920
Different scales, different
rhythms and meters.

00:41:07.039 --> 00:41:10.399
If you took that person who had never
experienced Western music before

00:41:10.399 --> 00:41:13.920
and set them at a symphony
concert to listen to Prokofiev,

00:41:13.920 --> 00:41:17.840
they're not going to have the
same experience as I would.

00:41:17.840 --> 00:41:25.670
I'm James Delaney.

00:41:25.670 --> 00:41:27.989
I'm an associate
professor of philosophy.

00:41:27.989 --> 00:41:32.949
One thing that's interesting to think
about with human nature with respect to

00:41:32.949 --> 00:41:33.350
music

00:41:33.350 --> 00:41:35.350
is that why are we
attracted to music?

00:41:35.350 --> 00:41:36.630
Why do we like music?

00:41:36.630 --> 00:41:37.989
What is it that moves us?

00:41:37.989 --> 00:41:44.710
Are we moved by something very
technical, something mathematical,

00:41:44.710 --> 00:41:48.949
or are we moved at
some other emotional level

00:41:48.949 --> 00:41:53.430
that is separate from any kind of
technical or mathematical structure?

00:41:53.430 --> 00:41:54.550
I'm Marc Sabat.

00:41:54.550 --> 00:41:57.590
I'm a Canadian composer.

00:41:57.590 --> 00:42:03.590
Sounds, to begin with, gave us
information about things around us

00:42:04.229 --> 00:42:09.590
and that can be connected to,
for example, to fear response.

00:42:09.590 --> 00:42:10.710
You know, what's moving where?

00:42:10.710 --> 00:42:11.590
Where does it come from?

00:42:11.590 --> 00:42:16.949
But then it also, for humans, is
connected very much to communication,

00:42:16.949 --> 00:42:19.829
to singing to each other
and to speaking to each other,

00:42:19.829 --> 00:42:22.710
to comforting each other,
to teaching each other.

00:42:22.710 --> 00:42:28.149
And so I think all of those
connections tie the enjoyment of sound

00:42:28.149 --> 00:42:31.430
to feelings and to
the experience of life.

00:42:35.090 --> 00:42:41.170
What that is, is such an interesting web of all of these
things. That is what I think makes composion such a magical art.

00:42:41.170 --> 00:42:50.399
that it is as much mathematics, thinking, reflection,
communication, poetry, all of these things without a language.

00:42:53.279 --> 00:42:55.130
You know, there were a lot
of theories proposed about

00:42:58.130 --> 00:43:01.800
what makes some sounds beautiful
and some sounds not beautiful.

00:43:02.800 --> 00:43:05.000
And one of the theories is that when
you have these simple proportions

00:43:05.630 --> 00:43:08.140
that Pythagoras gave us,

00:43:09.140 --> 00:43:12.220
the waves of the
pitches that are vibrating,

00:43:13.030 --> 00:43:15.989
they don't knock up against
each other the way they do

00:43:15.989 --> 00:43:20.389
when you may have two
pitches where, you know,

00:43:20.389 --> 00:43:24.869
one is going like this and one is
going like this and they're, you know,

00:43:24.869 --> 00:43:29.909
the waves are knocking into each other
and that creates the dissonant sound.

00:43:29.909 --> 00:43:35.270
That doesn't explain
why, in equal temperament,

00:43:35.270 --> 00:43:38.469
where the numbers
are very strange,

00:43:38.469 --> 00:43:40.710
if you want to measure
the actual proportions,

00:43:40.710 --> 00:43:44.469
since they're not at all like
Pythagoras's, they've all been compromised.

00:43:44.469 --> 00:43:47.750
It doesn't explain
why they sound okay.

00:43:47.750 --> 00:43:49.829
They sound sort of okay,

00:43:49.829 --> 00:43:53.270
even though you get all of this knocking
of the waves against each other.

00:43:53.270 --> 00:43:58.069
And one explanation for
that, and it makes sense to me,

00:43:58.069 --> 00:44:01.829
is that the mind
hears a tempered fifth.

00:44:03.729 --> 00:44:06.930
And understands that
it's meant to be a real fifth.

00:44:06.930 --> 00:44:08.610
It's meant to
be a perfect fifth.

00:44:08.610 --> 00:44:13.409
And our minds just simply
interpret the sound correctly,

00:44:13.409 --> 00:44:15.329
and then we don't
get bothered by it.

00:44:15.329 --> 00:44:19.569
So this idea that we
have this filter in our brains

00:44:19.569 --> 00:44:21.809
that sort of interprets
what's going on,

00:44:21.809 --> 00:44:23.010
I think it has validity.

00:44:30.630 --> 00:44:37.270
In the end, if we recognize that all
of these scientific explanations,

00:44:37.270 --> 00:44:40.229
whether of celestial motion,

00:44:40.229 --> 00:44:43.510
but also of the scale
and consonances,

00:44:43.510 --> 00:44:45.430
if we recognize that all
these things, in the end,

00:44:45.430 --> 00:44:49.909
depend on certain, usually
hidden, aesthetic principles,

00:44:49.909 --> 00:44:53.750
we start ending up with
a very awkward bind,

00:44:53.750 --> 00:44:55.030
an awkward situation,

00:44:55.030 --> 00:44:59.989
in that it matches a
philosophical conception

00:44:59.989 --> 00:45:01.350
of how things ought to be.

00:45:01.350 --> 00:45:04.949
This is the kind of thing that
people don't like to talk about,

00:45:04.949 --> 00:45:06.309
in terms of how science works,

00:45:06.309 --> 00:45:08.389
is that an awful lot
of it, at some level,

00:45:08.389 --> 00:45:10.630
does come down to,
well, that sounds right.

00:45:10.630 --> 00:45:12.149
It really should be like that.

00:45:12.149 --> 00:45:19.190
With music, this
problem is central.

00:45:19.190 --> 00:45:22.630
Everybody knows
that from the beginning,

00:45:22.630 --> 00:45:24.229
I mean, all the way
back to Pythagoras,

00:45:24.229 --> 00:45:25.670
the whole question is,

00:45:25.750 --> 00:45:30.149
why do certain intervals
sound better than others?

00:45:30.149 --> 00:45:33.190
And all these
different explanations,

00:45:33.190 --> 00:45:35.510
whether it's Pythagoras
through Zarlino,

00:45:35.510 --> 00:45:36.949
these numerical explanations,

00:45:36.949 --> 00:45:40.469
there are various really odd
explanations along the way,

00:45:40.469 --> 00:45:43.750
all of these
explanations, in the end,

00:45:43.750 --> 00:45:47.909
have to rely on a problem,

00:45:47.909 --> 00:45:52.119
which is, how do we react?

00:45:52.119 --> 00:45:54.280
How does the soul work?

00:45:54.280 --> 00:45:56.920
Or the brain, or the
personality, or whatever it is.

00:45:56.920 --> 00:46:00.119
It seems as though our bodies,

00:46:00.119 --> 00:46:03.000
our minds, are
interpreting something

00:46:03.000 --> 00:46:05.319
in a very structured
and controlled way,

00:46:05.319 --> 00:46:07.880
that we can't
account for at all.

00:46:07.880 --> 00:46:08.920
It's just not there.

00:46:08.920 --> 00:46:11.719
So, at that point,
you have to say,

00:46:11.719 --> 00:46:15.880
well, is it something in the
way God constructed the soul?

00:46:15.880 --> 00:46:18.840
Is it something in the
nature of the universe

00:46:18.840 --> 00:46:20.280
as a system of numbers?

00:46:20.280 --> 00:46:23.960
Is it because we
have some kind of,

00:46:23.960 --> 00:46:28.840
you know, genetic
predisposition to like fifths?

00:46:28.840 --> 00:46:30.840
I mean, what is going on?

00:46:30.840 --> 00:46:32.680
And there's something
funny about the whole thing.

00:46:32.680 --> 00:46:37.159
Because there's something
in it that's entirely in nature,

00:46:37.159 --> 00:46:40.199
and none of these theories
can account for it at all.

00:46:41.319 --> 00:46:43.159
Except for
Pythagoras, actually.

00:46:43.159 --> 00:46:44.760
He says, because it's four.

00:46:44.760 --> 00:46:47.079
I like four.

00:46:47.079 --> 00:46:48.119
The universe’s number.

00:46:48.119 --> 00:46:51.960
I believe that there
is a perfect system,

00:46:51.960 --> 00:46:55.639
but it's one that
is not stationary.

00:46:55.639 --> 00:46:56.520
It's not fixed.

00:46:56.520 --> 00:46:59.880
And the instruments that
can work with that system

00:46:59.880 --> 00:47:02.920
are the instruments that
can actually, on the fly,

00:47:02.920 --> 00:47:05.719
change their pitches
in a given scale.

00:47:05.719 --> 00:47:08.199
And a great example
is the human voice.

00:47:09.000 --> 00:47:11.559
That really is the
only way, in my view,

00:47:11.559 --> 00:47:14.039
that you can have a
perfect tuning system.

00:47:14.039 --> 00:47:19.239
Well, I think that
perfection has to exist,

00:47:19.239 --> 00:47:21.960
if only as an
unachievable goal.

00:47:21.960 --> 00:47:25.719
I don't think there is anything
in this world that's perfect.

00:47:25.719 --> 00:47:29.159
You know, everything has
a positive and a negative.

00:47:29.159 --> 00:47:32.440
My name is Alex Francois.

00:47:32.440 --> 00:47:33.960
I'm a computer scientist.

00:47:33.960 --> 00:47:38.840
The concept of perfection,
I think, is very subjective.

00:47:38.840 --> 00:47:40.760
And it's not an invariant.

00:47:40.760 --> 00:47:42.760
It depends on time and space.

00:47:42.760 --> 00:47:44.840
What you think
is perfection today

00:47:44.840 --> 00:47:47.399
might not be perfection
anymore tomorrow.

00:47:47.399 --> 00:47:50.920
As music develops,

00:47:50.920 --> 00:47:56.039
it is increasingly important
to understand mathematics

00:47:56.039 --> 00:47:59.000
and to exploit it and to use it

00:47:59.000 --> 00:48:03.319
to explore new ways of
expressing ourselves in music.

00:48:03.319 --> 00:48:07.399
I do think that this
is a burgeoning field.

00:48:07.399 --> 00:48:11.479
This is what is totally open
to be explored right now

00:48:11.479 --> 00:48:12.600
in many different ways

00:48:12.600 --> 00:48:15.079
and between many
people, many artists.

00:48:15.079 --> 00:48:17.639
It's not something which
one person can come and say,

00:48:17.639 --> 00:48:20.520
"Look, I've found
the particular way."

00:48:20.520 --> 00:48:22.440
There is no such way like that.

00:48:22.440 --> 00:48:27.079
It is more that there is
no longer a single model.

00:48:27.079 --> 00:48:29.239
I'm Stephen Pope.

00:48:29.239 --> 00:48:31.319
I'm a composer and
a computer scientist.

00:48:31.319 --> 00:48:33.639
In the last century,

00:48:34.280 --> 00:48:36.840
we've seen every new idea

00:48:36.840 --> 00:48:39.319
that came out of mathematics
or computer science.

00:48:39.319 --> 00:48:43.000
Somebody has said, "How
can we use this for music?"

00:48:43.000 --> 00:48:44.520
So I think the
future looks bright

00:48:44.520 --> 00:48:47.639
because there are so
many new areas of math

00:48:47.639 --> 00:48:48.600
that are being described,

00:48:48.600 --> 00:48:50.840
whether it's higher
dimensional math,

00:48:50.840 --> 00:48:52.520
like the kind of
differential topology

00:48:52.520 --> 00:48:53.800
that's used in string theory,

00:48:53.800 --> 00:48:55.960
or new kinds of statistics.

00:48:55.960 --> 00:48:58.600
Some of them will
be extremely useful

00:48:58.600 --> 00:49:03.479
for content creation, music
analysis, tuning theory.

00:49:03.479 --> 00:49:04.680
I don't think we're all done.

00:49:04.680 --> 00:49:08.360
I think the rate of
change is increasing,

00:49:08.360 --> 00:49:11.719
so I think there will be certainly
new mathematical constructs

00:49:11.719 --> 00:49:13.719
that can be applied
either to composition,

00:49:13.719 --> 00:49:16.600
signal processing, or
anything in between.

00:49:16.600 --> 00:49:19.880
But this topic is bottomless.

00:49:19.880 --> 00:49:21.559
It just goes on forever.

00:49:21.559 --> 00:49:24.760
I wouldn't claim to
have the answers,

00:49:24.760 --> 00:49:26.680
but I think it's a
fascinating topic.

00:49:26.680 --> 00:49:32.199
And I think that the more
people actually investigate

00:49:32.199 --> 00:49:36.119
the history, the more
surprises we'll find.

00:49:36.119 --> 00:49:44.079
Today, the latest trends
in music and composition

00:49:44.079 --> 00:49:46.880
have to do with going
back to the original tunings,

00:49:46.880 --> 00:49:48.079
to the pure tunings.

00:49:48.079 --> 00:49:52.319
Those pure harmonies
emerge from the piano.

00:49:52.319 --> 00:49:53.600
And what happens sometimes

00:49:53.600 --> 00:49:57.039
is that the overtones of
those harmonies coalesce

00:49:57.039 --> 00:49:59.680
and create amazing effects.