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  • [PIANO ARPEGGIOS]

  • 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.