Subtitles section Play video
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[flute plays Crash Course theme]
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That’s a familiar tune!
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How do instruments, like this guitar, create music?
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We’ve talked about the science of sound, and some of the properties of sound waves.
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But when we talk about sound waves in the context of music, there are all kinds of fascinating properties and weird rules to talk about.
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I’m talking about the music that comes from string, wind, and brass instruments.
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String instruments create sound when their strings vibrate in the air.
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And in order to understand how these instruments work, you have to realize that making music is not just an art.
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It’s ALSO a science.
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[Theme Music]
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Sound, you’ll recall, is a wave: a ‘longitudinal’ wave.
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This means that the medium that the wave travels through oscillates -- or moves back and forth
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-- in the same direction that the wave is moving.
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But string, wind, and brass instruments use a special kind of wave — they’re ‘standing waves.'
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A standing wave is a wave that looks like it isn't moving.
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Its ‘amplitude’ may change, but it isn't traveling anywhere.
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Standing waves are the result of two other things waves do, both of which we’ve talked about before: reflection and interference.
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Reflection is what happens when a wave reaches the end of a path, and then moves back along the same path.
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That’s what happens when you send a pulse down a fixed rope -- it reaches the end, and then comes right back.
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When we send a continuous wave down the rope, that’s when interference comes into play.
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The wave reaches the end of the rope and is reflected, but there are more peaks on the way.
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As the peaks pass each other, they interfere with one another, changing their sizes.
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Usually, you end up with crests and troughs that are different sizes and various distances apart.
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But at certain frequencies, the reflected waves interfere in such a way that you end up
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with a wave that seems to stay perfectly still, with only its amplitude changing.
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That’s a standing wave, and it can happen both in strings and in the air in pipes.
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And that's what makes music: Standing waves with different frequencies correspond to different musical notes.
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Now, in order to understand how standing waves operate, you should get to know their anatomy.
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The points of a standing wave that don’t oscillate are called nodes,
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and the points at the maximum height of the peaks are antinodes.
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And here’s something cool.
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If you look at a string on a stringed instrument, you can actually see where the nodes and antinodes are.
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The standing wave creates peaks along the string, and between those peaks, there are points that just stay still.
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So the peaks are the antinodes, and the points that don’t oscillate are nodes.
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And if one or both of the string’s ends are fixed, then each fixed end is a node too, because it’s stuck in place.
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Now, in a pipe, the standing waves are made of air molecules moving back and forth.
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But the areas where molecules oscillate the most
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(including those near any open ends of the pipe) form the peaks, and therefore the antinodes.
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And between those peaks, as well as at any closed ends of the pipe, are areas where molecules don’t move at all; those are the nodes.
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Generally, musicians make their music using the frequencies of these standing waves.
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But the nature of these waves depends a lot on what the ends of the string or pipe look like.
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Remember how a wave traveling down a rope gets reflected differently, depending on whether the end of the rope is fixed or loose?
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A fixed end will invert the wave -- turning crests into troughs, and vice versa --
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while a loose end will just reflect it without inverting it.
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The same thing holds true for air in a pipe: a closed end will invert the wave, while an open end won’t.
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So the properties of a standing wave will be a little different,
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depending on whether it’s made with a string with two fixed ends, or a pipe with two ends open,
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or a string or a pipe with one end fixed, and the other open.
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A string with two fixed ends -- like in a piano -- is probably the simplest way to understand standing waves.
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Because, we know that no matter what, the wave made by a fixed string will have at least two nodes -- one at each end.
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And in its most basic form, it would have just one antinode, in the middle.
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So the wave is basically a peak that moves from being a crest to a trough and vice versa
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like some kind of one-dimensional jump rope.
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This most basic kind of standing wave is known as the fundamental -- or the 1st harmonic.
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It’s the simplest possible standing wave you can have, with the fewest nodes and antinodes.
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There are other, more complex standing waves that you can have, too.
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These are known as overtones.
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Overtones build on the fundamental, incrementally: each overtone adds a node and an antinode.
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So each of these overtones is related to the fundamental wave -- and all of the overtones are related to each other.
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Together, the fundamental wave and the overtones make up what are known as harmonics.
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The fundamental is the 1st harmonic, and the overtones are higher-numbered harmonics.
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With each node-and-antinode pair that’s added to the standing wave,
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the number of the harmonic goes up: 2nd harmonic, 3rd harmonic, and so on.
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Now, physicists sometimes express harmonics in terms of wavelength.
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For example, for a string with two fixed ends, you’ll notice that the fundamental covers exactly half a wavelength.
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A full wavelength of the wave would span two peaks: a crest and a trough,
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but the fundamental spans exactly one peak, which is half the wavelength.
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So, for the fundamental of a string with two fixed ends, the length of the string is equal to half a wavelength.
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The second-simplest standing wave you can have on a string with two fixed ends has 3 nodes --
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one at each end, and one in the middle -- plus 2 antinodes in between the nodes.
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It’s called the 2nd harmonic, and the string holds exactly one wavelength.
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You can probably guess what the 3rd harmonic looks like: it has 4 nodes and 3 antinodes,
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and the string holds 1.5 -- or, 3/2 -- wavelengths.
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You may have started to notice a pattern: For a standing wave on a given length of string,
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the number of wavelengths that fit on the string is equal to the number of the harmonic, divided by 2.
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So, now we have an equation that relates the wavelength of a standing wave to the number of the harmonic and the length of the string.
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Once you get a handle on wavelength, you can figure out the aspect of the wave that musicians care about most -- the frequency.
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We’ve already established that a wave’s wavelength, times its frequency, is equal to its velocity,
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which will be the same for each harmonic, because a wave’s velocity only depends on the medium it’s traveling through.
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So a standing wave’s frequency will be equal to its velocity divided by its wavelength.
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For the fundamental with two fixed ends, we already know that the wavelength is twice the string's length.
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So the frequency of that fundamental standing wave -- known as the fundamental frequency
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-- is equal to the velocity, divided by twice the length of the string.
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We write it as f, with a subscript of 1.
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Now what about the frequency of the second harmonic -- the standing wave with 3 nodes and 2 antinodes?
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It will be equal to the velocity, divided by the length of the string.
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Which is twice the fundamental frequency.
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And the frequency of the third harmonic, with its 4 nodes and 3 antinodes,
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will be equal to three times the fundamental frequency.
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So, we’re starting to see another pattern here:
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The frequency of a standing wave with two fixed ends will just be equal to the number of the harmonic, times the fundamental frequency.
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In fact, that’s one way to define harmonics:
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The number of a harmonic is equal to the number you multiply by the fundamental frequency, to get the harmonic’s frequency.
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This math is what makes musical instruments work.
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When you press down a key on a piano, you make a hammer strike a string, creating standing waves in that string.
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Every string in the piano is tuned so that its fundamental frequency --
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which depends on the string’s mass, length, and tension -- corresponds to a given note.
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Middle C, for example, is 261.6 Hz.
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Guitars are also tuned so that the fundamental frequencies of their strings, correspond to set notes.
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And when you press down on the strings in certain places,
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you change the length of the active part of string so that its fundamental frequency corresponds to a different note.
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So, for a standing wave with two fixed ends, we can relate wavelength, frequency, velocity,
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the length of the string, and the number of the harmonic.
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And we can do the exact same thing for a standing wave with two loose ends -- in an open pipe, for example, like in a flute.
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A standing wave in a pipe with two open ends is kind of the opposite, of the wave with two fixed ends:
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Instead of having a node at each end, it has an antinode at each end.
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So the fundamental standing wave for a pipe with two open ends will have two antinodes,
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and one node in the middle of the wave.
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Then, the 2nd harmonic will have three antinodes and two nodes, and so on.
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But each harmonic still covers the same number of wavelengths.
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Remember how the fundamental wave for a string with two fixed ends covered half of a wavelength?
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The fundamental wave for a pipe with two open ends also covers half of a wavelength.
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That half is just in a different section of the wave.
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And just like a string with two fixed ends, the second harmonic for a pipe with two open ends also covers a full wavelength.
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It’s just that, in the case of the pipe, the wave starts and ends with a peak instead of a node.
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So the equations for wavelength and frequency for a standing wave with two open ends
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will be the same as they were for a standing wave with two fixed ends.
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So, we’ve covered guitars and pianos and flutes!
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But a pipe with one closed end and one open end works a little differently.
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These kinds of pipes are used in instruments like pan flutes, where you blow across the top of a closed pipe to make music.
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Here, standing waves need a separate set of equations, for a couple of reasons:
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First, the closed end of the pipe will be a node, because the air molecules aren’t oscillating there.
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And the open end will be an antinode, because that’s where there’s a peak in the oscillations.
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Which means that the simplest wave you can make in this pipe will stretch from one node, to one peak.
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But that’s only a span of a quarter of a wavelength in the pipe.
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Before, with both a string fixed at both ends, and an open pipe, the fundamental spanned half a wavelength.
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The fact that a pan-flute pipe only covers a quarter of a wavelength changes things.
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Because, remember: the frequency of each harmonic is equal to the number of the harmonic, times the fundamental frequency.
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But for a pipe that’s closed on one end, you can’t double the fundamental frequency,
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or quadruple it -- or multiply it by any even number.
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Because it would result in a wave that would need a node on both ends, or a peak on both ends.
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Which isn't possible.
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So, a pipe that’s closed on one end can’t have even-numbered harmonics.
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All of this helps explain why musical instruments sound different, even when they’re playing the same note.
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When you play a note, you’re creating the fundamental wave, plus some of the other harmonics -- the overtones.
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And for each instrument, different harmonics will have different amplitudes -- and therefore sound louder.
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But because of the physics of standing waves, instruments that have pipes with one closed end
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won't create the even-numbered harmonics at all.
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That’s why a C on the flute sounds so different from a C on, say, the bassoon!
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Today, you learned about standing waves, and how they’re made up of nodes and antinodes.
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We discussed harmonics, and how to find the frequency of a standing wave on a string with
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two fixed ends, a pipe with two open ends, and a pipe with one closed end.
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Finally, we explained why a pipe with one closed end can’t have even-numbered harmonics.
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Crash Course Physics is produced in association with PBS Digital Studios.
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You can head over to their channel and check out a playlist of the latest episodes from
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shows like First Person, PBS Game/Show, and The Good Stuff.
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This episode of Crash Course was filmed in the Doctor Cheryl C. Kinney Crash Course Studio
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with the help of these amazing people and our equally amazing graphics team is Thought Cafe.