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What is up with Noises? (The Science and Mathematics of Sound, Frequency, and Pitch): Accuracy not guaranteed. Get Audacity and play! http://audacity.sourceforge.net/ Correction: it is the "Basilar" membrane, which is what I say, but somehow between recording the script and actually drawing the stuff I got confused and thought I just pronounced my Vs poorly. Always sad to have such a simple and glaring error in something I put hundreds of hours of work into, but a "Vasilar" membrane can be the kind that a Vi draws to explain Viola Vibrations, I guess! Making up new words is just so prolightfully awstastic. Props to my Bro for excellent and creative swing pushing, and to my Mamma for filming it. Extra special thanks to my generous donators, without whom I would not have been able to create this video. Because of your support, I have the equipment, time, and take-out Thai food necessary for doing stuff like this.
相關課程
- (piano playing)
- 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
- although, they are not literally hitting so much as
- getting too close for comfort,
- until they get to be air molecules in our ear, which
- push against some stuff in our ear, and 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 pushing your little sister on a swing,
- you have to get your timing right.
- It take 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 gentle pushes make her 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
- 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 eardrum
- that gets pushed around by the sound waves.
- And then that pushed 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 gonna 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 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's 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 mathmatics
- of swing pushing. It's not exactly true that pushing
- with the same frequency as the swing is swinging
- is the only way to get the 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, but 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 there's another 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
- percieve it as a single note?
- Would that make 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 percieve
- these two different frequencies as being almost the same
- even though they are 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 at
- half the length.
- So he played half the length and found that the note was
- and octave higher. He thought that was pretty neat
- So then he tried the next simplest ratio and
- played a third of the string.
- If the full length was C, then a third the length
- would give the note G, an octave and a fifth above.
- The next ratio to try was one-fourth of the string
- But we can already figure out what note that would be.
- If half the string was C an octave up
- then half of that would be C another octave up.
- and half of that would be another octave higher and so on and so forth.
- And then one fifth of the string made the note E
- but wait, let's play that again...
- it's a C major chord
- Ok, so what about one-sixth?
- We can figure that one out too using ratios
- we already know.
- One-sixth is the same as half of one-third
- And one third was this G, so one-sixth
- is the G an octave up.
- Check it out! One-seventh will be a new note
- because 7 is prime and Pythagoras found that it was
- this B-flat.
- Then 8 is two x two x two
- so an eighth gives us C three octaves up
- And a 9 is a third of a third
- So we'd 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 on the chromatic scale
- and they go into semi-tones, etc.
- But let's make one thing clear.
- This is not some magical relationship between
- mathematical ratios and consonant intervals.
- It's that these notes sound good to our ears
- 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 other dissonant, and why tonality is like it is
- and why cultures that develop 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 half the length vibrates twice as fast,
- which, hey! makes this series the same as that series
- If this were A 440, 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 waves 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 in the middle of the string
- so that it can't vibrate there, but didn't actually hold the string down there
- then this string would be free to vibrate in any way
- that doesn't move at that point,
- while these other frequencies couldn't vibrate
- and if I were to touch it at the one-third point
- you'd expect all the overtones not divisible by three
- to get dampened, so we'd hear this and all its overtones
- The cool part is that the string is pushing around
- the air at all these different frequencies
- and so the air is pushing our ear at all these different frequencies
- and then the basilar membrane is vibrating in
- sympathy at all these different 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 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
- A simple sine wave with a single frequency with
- no overtones makes an "oooo" 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 chords
- dampening some, while amplifying others
- To demonstrate, I recorded myself saying
- "oooaaaeee" (singing different vowel sounds on the same pitch)
- at 440 Hz
- Now I'm going to put it through a low pass filter,
- which lets through the frequencies less than A 441
- but dampens all the overtones.
- Check it out!
- (sounds of vowel sounds again, but without higher overtones)
- Ok, lets make ourselves an overtone series.
- I'm going to have Audacity make a sine wave,
- A 220.
- Now I'll make another at twice the frequency, 440,
- which is A an octave above
- Here it is alone.
- If we play the two at once, do you think we'll hear
- the two separate pitches, or do you think our brain
- will 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.
- Let's add the next overtone.
- 3 time 220 gives us 660.
- Here they are all at once.
- It sounds like a different instrument from the
- fundamental sine wave, but the same pitch.
- Let's add 880.
- And now, 1000.
- Oh, that sounds wrong. Alright, 880 plus 220 is 1100
- 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 little 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#, 1100, is about a foot long,
- and each octave down is about half the frequency,
- or twice the length
- That means that the lowest C on a piano,
- which is five octaves lower than this C
- has a sound wave one foot times 2^5,
- or thirty-two feet long!
- Ok, now I can play with the timbre of the sound
- by playing with how loud the overtones are relative to each other
- What your ears are doing right now is pretty complicated
- All these sound waves get added up together into a single wave
- and if I export this file, we can see what it looks like.
- or I suppose you could graph it
- Anyway, your speakers or headphones have this little diaphram
- in them that pushes the air to make sound waves
- To make this shape, it pushed forward fast here,
- then does this little wiggly thing
- then another big push forward.
- A speaker, remember, is not pushing air from itself to your ear
- It bumps against the air, which bumps against more air
- and so on, until some air bumps into your eardrum,
- which moves in the same way as the diaphragm in the speaker did
- and that pushed the little bones
- that push the cochlea, which pushes the fluid
- which, depending on the stiffness of the basilar membrane
- at each point, is either going to push the basilar membrane
- in such a way that makes it vibrate a lot,
- and push the little hairs,
- or it pushes with the wrong timing,
- just like someone bad at playgrounds.
- This sound wave will push in a way that makes
- the A 220 part of your ear send off a signal
- which is pretty easy to see.
- Some frequencies get pushed the wrong direction sometime
- but the pushes in the right direction more than make up for it
- So now all these different frequencies that we added together
- and played are now separated out again.
- In the meantime, many other signals are being sent out
- from other noise, like the sound of my voice,
- and the sound of rain, and traffic,
- and noisy neighbors, and the air conditioner and so on
- But then our brain is like, "Yo! Look at these-I found a pattern!"
- "All these frequencies fit together into a series starting
- at this pitch, so I will think of them as one thing"
- "And it is a different thing when these frequencies,
- which fit the pattern of Vi's voice, and oh boy!
- that's a car horn!"
- Somehow this all works, and we're still pretty far
- from developing technology that can listen to lots of sound
- and separate it out into things nearly as well
- as our ears and brain can.
- Our brains are so good at finding these patterns
- that sometimes it find them when they're not there.
- Especially if it's subconsciously looking out for it
- and you're in a noisy situation
- In fact, if the pattern is mostly there,
- your brain will fill in the blanks and make you
- hear a tone that does not exist.
- Here I've got A 220 and its overtones
- Now I'm going to mute A 220.
- That frequency is not playing at all.
- But you hear the pitches A 220 below this A 440
- even though A 440 is the lowest frequency playing.
- Your brain is like, "Well, we've got all these overtones, so close enough"
- Let me mute the highest overtones one by one
- It changes the timbre but not the pitch
- until we leave only one left
- Somehow removing the higher note, you make the
- apparent pitch jump up.
- And, just for good measure....
- but you should try it yourself
- So there you have it.
- These notes, these notes given to us by simple ratios
- of strings by the laws of physics
- and how frequencies vibrate in sympathy to each other
- by the mathematics of how sine waves add up
- these notes are hidden in every spoken word
- tucked away in every song
- you can hear them in birdsong, bees buzzing,
- car horns, crickets, cries of infants...
- and most of the time you don't even realize they're there.
- There is a symphony contained in the screeching of a halting train,
- if only we are open to listen to it
- Your ears, perfected over hundreds of millions of years,
- capture these frequencies in such exquisite detail
- that it's a wonder that we can make sense of it all.
- But we do, breaking out the patterns that mathematics dictate
- finding order, finding beauty
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