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What is up with noises? (The science and mathematics of sound, frequency, and pitch)

Video transcript
[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. 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. 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 diaphragm in them that pushes the air to make sound waves. To make this shape, it pushes forward fast here, then does this wiggly thing, and then another big push forwards. The speak, remember, is not pushing air from itself to your ears. It bumps against the air, which bumps against more air, and so on, until some air bumps into your ear drum, which moves in the same way that the diaphragm in the speaker did. And that pushes 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 A220 part of your ear send off a signal, which is pretty easy to see. Some frequencies get pushed the wrong direction sometimes, 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. And 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 air conditioner and so on. But then our brain is like, Yo, look at these! I found a pattern! And 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 than these frequencies, which fit the patterns 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 anywhere near as well as our ears and brains can. Our brains are so good at finding these patterns that sometimes it finds 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 A220 and his overtones. [PLAYS PITCH] Now I'm going to mute A220. That frequency is not playing at all. But you hear the pitches A220 below this A400, even though A440 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 by removing a higher note, you make the apparent pitch jump up. And just for good measure. [PLAYS SEQUENCE OF PITCHES] 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 with each other. By the mathematics of how sine waves add up. These notes are hidden in every spoken word, tucked away in every song. We 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 listening 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. Picking out the patterns that mathematics dictates. Finding order. Finding beauty.