When things move, they tend to hit other things.

And then those things move, too.

When I pluck this string, it's shoving back and forth

against the air molecules around it

and they push against other air molecules

that they're not literally hitting so much as getting

too close for comfort until they get to the air

molecules in our ears, which push

against some stuff in our ear.

And then that sends signals to our brain to say,

Hey, I am getting pushed around here.

Let's experience this as sound.

This string is pretty special, because it

likes to vibrate in a certain way and at a certain speed.

When you're putting your little sister on a swing,

you have to get your timing right.

It takes her a certain amount of time to complete a swing

and it's the same every time, basically.

If you time your pushes to be the same length of time,

then even general pushes make your swing higher and higher.

That's amplification.

If you try to push more frequently,

you'll just end up pushing her when she's swinging backwards

and instead of going higher, you'll dampen the vibration.

It's the same thing with this string.

It wants to swing at a certain speed, frequency.

If I were to sing that same pitch,

the sound waves I'm singing will push against the string

at the right speed to amplify the vibrations so that that

string vibrates while the other strings don't.

It's called a sympathy vibration.

Here's how our ears work.

Firstly, we've got this ear drum that gets pushed around

by the sound waves.

And then that pushes against some ear bones

that push against the cochlea, which has fluid in it.

And now it's sending waves of fluid instead of waves of air.

But what follows is the same concept as the swing thing.

The fluid goes down this long tunnel,

which has a membrane called the basilar membrane.

Now, when we have a viola string, the tighter and stiffer

it is, the higher the pitch, which means a faster frequency.

The basilar membrane is stiffer at the beginning of the tunnel

and gradually gets looser so that it

vibrates at high frequencies at the beginning of the cochlea

and goes through the whole spectrum down to low notes

at the other end.

So when this fluid starts getting pushed around

at a certain frequency, such as middle C,

there's a certain part of the ear that vibrates in sympathy.

The part that's vibrating a lot is

going to push against another kind of fluid

in the other half of the cochlea.

And this fluid has hairs in it which get pushed around

by the fluid, and then they're like, Hey, I'm middle C

and I'm getting pushed around quite a bit!

Also in humans, at least, it's not a straight tube.

The cochlea is awesomely spiraled up.

OK, that's cool.

But here are some questions.

You can make the note C on any instrument.

And the ear will be like, Hey, a C.

But that C sounds very different depending

on whether I sing it or play it on viola.

Why?

And then there's some technicalities

in the mathematics of swing pushing.

It's not exactly true that pushing with the same frequency

that the swing is swinging is the only way

to get this swing to swing.

You could push on just every other swing.

And though the swing wouldn't go quite as high

as if you pushed every time, it would still swing pretty well.

In fact, instead of pushing every time or half the time,

you could push once every three swings or four, and so on.

There's a whole series of timings that work,

though the height of the swing, the amplitude, gets smaller.

So in the cochlea, when one frequency goes in,

shouldn't it be that part of it vibrates a lot,

but there's another part that likes to vibrate twice as fast,

and the waves push it every other time

and make it vibrate, too.

And then there's another part that

likes to vibrate three times as fast and four times.

And this whole series is all sending signals to the brain

that we somehow perceive it as a single note?

Would that makes sense?

Let's also say we played the frequency that's

twice as fast as this one at the same time.

It would vibrate places that the first note already

vibrated, though maybe more strongly.

This overlap, you'd think, would make

our brains perceive these two different frequencies as being

almost the same, even though they're very far away.

Keep that in mind while we go back to Pythagoras.

You probably know him from the whole Pythagorean theorem

thing, but he's also famous for doing this.

He took a string that played some note, let's

call it C. Then, since Pythagoras liked

simple proportions, he wanted to see

what note the string would play if you made it 1/2 the length.

So he played 1/2 the length and found

the note was an octave higher.

He thought that was pretty neat.

So then he tried the next simplest ratio

and played 1/3 of the string.

If the full length was C, then 1/3

the length would give the note G, an octave and a fifth above.

The next ratio to try was 1/4 of the string,

but we can already figure out what note that would be.

In 1/2 the string was C an octave up, then 1/2 of that

would be C another octave up.

And 1/2 of that would be another octave higher,

and so on and so forth.

And then 1/5 of the string would make the note E. But wait.

Let's play that again.

It's a C Major chord.

OK.

So what about 1/6?

We can figure that one out, too, using ratios we already know.

1/6 is the same as 1/2 of 1/3.

And 1/3 third was this G. So 1/6 is the G an octave up.

Check it out.

1/7 will be a new note, because 7 is prime.

And Pythagoras found that it was this B-flat.

Then 8 is 2 times 2 times 2.

So 1/8 gives us C three octaves up.

And 1/9 is 1/3 of 1/3.

So we go an octave and a fifth above this octave and a fifth.

And the notes get closer and closer

until we have all the notes in the chromatic scale.

And then they go into semi-tones, et cetera.

But let's make one thing clear.

This is not some magic relationship

between mathematical ratios and consonant intervals.

It's that these notes sound good to our ear

because our ears hear them together

in every vibration that reaches the cochlea.

Every single note has the major chord secretly contained

within it.

So that's why certain intervals sound consonant and others

dissonant and why tonality is like it is

and why cultures that developed music independently

of each other still created similar scales, chords,

and tonality.

This is called the overtone series, by the way.

And, because of physics, but I don't really

know why, a string 1/2 the length

vibrates twice as fast, which, hey,

makes this series the same as that series.

If this were A440, meaning that this

is a swing that likes to swing 440 times a second,

Here's A an octave up, twice the frequency 880.

And here's E at three times the original frequency, 1320.

The thing about this series, what

with making the string vibrate with different lengths

at different frequencies, is that the string is actually

vibrating in all of these different ways

even when you don't hold it down and producing

all of these frequencies.

You don't notice the higher ones, usually,

because the lowest pitch is loudest and subsumes them.

But say I were to put my finger right

in the middle of the string so that it can't vibrate there,

but didn't actually hold the string down there.

Then the string would be free to vibrate

in any way that doesn't move at that point,

while those other frequencies couldn't vibrate.

And if I were to touch it at the 1/3 point,

you'd expect all the overtones not divisible by 3

to get dampened.

And so we'd hear this and all of its overtones.

The cool part is that the string is pushing it

around the air at all these different frequencies.

And so the air is pushing around your ear

at all these different frequencies.

And then the basilar membrane is vibrating in sympathy

with all these frequencies.

And your ear puts it together and understands it

as one sound.

It says, Hey, we've got some big vibrations here and pretty

strong ones here, and some here and there and there.

And that pattern is what a viola makes.

It's the difference in the loudness of the overtones

that gives the same note a different timbre.

And simple sine wave with a single frequency

with no overtones makes an ooh sound, like a flute.

While reedy nasal sounding instruments

have more power in the higher overtones.

When we make different vowel sounds,

we're using our mouth to shape the overtones coming

from our vocal cords, dampening some while amplifying others.

To demonstrate, I recorded myself

saying ooh, ah, ay, at A440.

Now I'm going to put it through a low-pass filter, which

lets through the frequencies less than A441,

but dampens all the overtones.

Check it out.

[PLAYS BACK THROUGH FILTER]

OK.

Let's make ourselves an overtone series.

I'm going to have Audacity create a sine wave, A220.

Now I'll make another at twice the frequency, 440,

which is A an octave above.

Here it is alone.

[PLAYS BACK PITCH]

If we play the two at once, do you

think we'll hear the two separate pitches?

Or will our brain say, Hey, two pure frequencies

an octave apart?

The higher one must be an overtone of the lower one.

So we're really hearing one note.

Here it is.

[PLAYS BACK PITCH]

Let's add the next overtime.

3 times 220 gives us 660.

Here they are all at once.

[PLAYS BACK PITCHES]

It sounds like a different instrument

for the fundamental sine wave but the same pitch.

Let's add 880 and now 1000.

That sounds wrong.

All right.

880 plus 220 is 1100.

There, that's better.

We can keep going and now we have all these happy overtones.

Zooming in to see the individual sine waves,

I can highlight one little bump here

and see how the first overtone perfectly fits two bumps.

And the next has three, then four, and so on.

By the way, knowing that the speed of sound

is about 340 meters per second, and seeing

that this wave takes about 0.0009 seconds to play,

I can multiply those out to find that the distance between here

and here is about 0.3 meters, or one foot.

So now all these waves are shown at actual length.

So C-sharp, 1100 is about a foot long.

And each octave down is 1/2 the frequency or twice the length.

That means the lowest C on a piano, which

is five octaves lower than this C, has a sound wave

1 foot times 2 to the 5, or 32 feet long.

OK, now I can play with the timbre of the sound

by changing how loud the overtones are

relative to each other.