What is up with Noises? (The Science and Mathematics of Sound, Frequency, and Pitch)


[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.

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100 thoughts on “What is up with Noises? (The Science and Mathematics of Sound, Frequency, and Pitch)”

  1. Mini Kills44 says:


  2. James Rankin III says:

    Literally just blowed my mind

  3. Lee Strawbling says:

    I didn't understand any of this, but I felt like I learned something. (unlike school)

  4. 1UpsForLife says:

    4:39 nice little hint of prelude in c major there heh

  5. Andrea F Genovese says:

    thats the freakin astoria bridge at the end, i watched this video 5-6 times during my life and just realized that bridge is next to my home!! small world

  6. Heather Whiteley says:

    If one could gifts a youtube video, this video would be a perfect gift for at least two people I know. Funny how you can buy a short book, but short videos on youtube are free, which makes it awkward to give them as gifts.

  7. Cheila the Cat says:


  8. Hrisuk says:

    I really like you. I play viola and piano too, and I'm also into asking questions and finding answers. Found your channel a few days ago and I'm impressed by all your talents and habilities. I'm from South America but I learnt english when I was a kid. Really looking forward to watching more of your videos.

  9. Luka Puka says:

    I've been trying to look for a simple explanation to why fifths vibrate each other. This is amazing.

  10. Max J says:

    My speaker just talked about itself

  11. cetterus says:

    And then comes Pythagoras coma…

  12. TriLectricGaming says:

    Vi hart, I don't push my little sister on the swing.
    I don't have one.
    I have been offended.

  13. Micetticat says:

    A must see video to grasp the basics of harmonics and the links between math and music.

  14. Craig Keidel says:

    Hello from the future, @Vihart. I'm fairly certain Melodyne now uses the overtone series to identify pitches and instruments in their Direct Note Access tool – I don't think it existed yet in 2011. It's pretty neat, you should check it out.

  15. Vedanth Ashok says:

    we are tripping on acid on this now holy shit

  16. Christopher Ellis says:

    Hence we have Canto harmónico, or overtone singing , in which the strings cannot break.

  17. Sztefa001 says:

    A few times I caught myself on not paying attention to the meaning of what you were saying but just listening to this vid in awe.
    It's like watching one of those scientific programs about space/dimentions/quantum physics/etc on tv and at first you are actually trying to understand it but later on you are just like "whoa… I have no idea what they're talking about, but it sounds genious af" till the point when they switch to explaining sth else (and then it's the same story all over again).

  18. electriclightl says:

    wtf i just got the biggest mindblown nice video and nice explanation

  19. Janosch Gaia says:

    hey, i do some music visualizations,
    maybe you want to take a look on my channel 🙂
    thank you!

  20. Sam Blackhardt says:

    Bill Nye's daughter?

  21. Llama Llover says:

    I play the viola!

  22. 11 12 says:

    HEY! i also use audacity for audio

  23. Nicholas Powers says:

    11:19 I can explain why this happens if you care to talk about it 🙂

  24. Alison Carlisle says:

    1. AAAH PLEASE STOP DRAWING ON YOUR INSTRUMENTS!! I know it's dry erase but still!!
    2. Is this why I like the sound of an orchestra tuning/warming up so much??

  25. hehey nop says:

    can you talk about the undertone series

  26. Armani Buchanan says:

    Im here for Fine Arts –_

  27. Buttercupkat Productions says:

    I'm not using headphones 0-0 Anyways! Let's try and make the loudest sound you can! 🎹🎶 No cheating!
    📝☺️ 📃✏️🙄
    Or you will be thrown IN THE PIT OF FIRE. Nah jk But you will die someday. 🙄🔥

  28. ClassicalHeroes says:

    I love your music theory videos!!!

  29. Gillian Jones says:

    Everyone in math class thinks I'm crazy because I insist that triangles must have sounds because they have graphs and the graphs are waves and waves have sounds. But see. It's true

  30. Томас Андерсон says:

    5:41 /r/mapswithoutNZ

  31. Gordon Chan says:


  32. Vohasiiv says:

    Just 3 clicks from this video is another video on how to escape the matrix

  33. Ayryn says:

    I studied how the ear works some years ago and it blew my mind, seeing it in this video again and with all the other stuff blew my mind again, much more!!

  34. Reese Lauren says:

    when she writes on her violin 😖😱🤯

  35. Richard Carr says:

    What an excellent video. Bravo.

  36. Stephen Kramer says:

    I'm going to show all of your videos to my kids someday. Thanks, you're awesome.

  37. NOT mera says:

    Fistly, if ur a person that hears high frequency sounds (me and i can hear things like the older tvs (GEEZ they are LOUD)) would that mean that the membrabe in the cochlea is tighter, or just … something else?

  38. Dylan Watersnake says:

    1:46…agh you broke a string just for the views (#; ^ 😉

  39. Mel Sunday says:

    holy fucking shit

  40. Tucker Kelly says:

    So when a sound is too quiet is it now loud/powerful enough to vibrate all the air between the source of the sound and your ear?

  41. PepperShakerKetchupBottle says:

    1:47 every musician's nightmare.

  42. Sara Twiss says:

    Who else freaked when the string broke?

  43. Bas van Dijk says:

    I'm an auditory scientist and this is one of the best explanations of hearing and pitch that I've seen! Well done. Two comments though:
    1) Unless you are Spanish ( 🙂 ) , it's 'Basilar Membrane' , not 'Vasilar membrane'.
    2) The haircells are not on the bottom of the lower compartment, they are actually ON the basilar membrane. (in fact, to be completely correct there is a second membrane, called the tectorial membrane and the cochlea is split in 3 compartments rather than two, but this is a detail, the haircells should definitely be on the membrane, they actually 'feel' the movement of the membrane directly.

  44. Zeturic says:

    What about other fractions? Like 3/5 for example?

  45. Nicholas Powers says:

    Harmonic information over time. <– I like your rebuttal. I still want to be sure of it.

  46. Ruth Wale says:

    i forgot how to breathe

  47. Pseudonym ? says:



  48. Minichaud Berkimilov says:

    This video is the best content i've seen since go bwaaah.

  49. sad lesbian says:

    This is the COOLEST THING EVER. I never really thought about the airwaves hitting my ears and how fragile and specific the hairs in my ear are and its just blowing my mind for some reason to imagine that i can do something so complicated without even knowing!

  50. Rilian Lunsford says:

    so I used to talk about hearing "pieces" of sounds and no one knew what I was talking about, and i guess it was this.

  51. Jess Tollestrup says:


  52. Just1Mohr says:

    prolightfully awstastic


  53. Gunnykido7 says:


  54. Allen Hare says:

    Thanks, Vi. This helps a lot.

  55. Stirling Rutty says:


  56. Gabriela Luque says:

    She speaks way too fast, man

  57. Lavinia Bocaniala says:

    i still hear yanny, not laurel.

  58. Chayanika says:

    Well, Vi Hart's videos never get old. Today I'm re-watching this video after suuchh a long time, and it is even more better. Today is the first time I understood, every little niche and detail in the video (I think?), pausing the video several times, and musing over the stuff said. Got reminded of this video while I was studying ENT (Ear Nose Throat), as I am a medico. This has just made the subject seem so much more beautiful.

  59. Dragon Curve Enthusiast says:

    Math, Physics, Biology and Music. This video plucks all my strings!

  60. Matthew OR'egan says:

    Vandalise a viola

  61. Brady Forrest says:

    So I play the trumpet, and the whole 1 1/2 1/3 of the string etc, matches every note of the trumpet where you press no buttons, and now my mind has been blown.

  62. Milesgatman says:

    RIP Viola string. We thank you for giving your life for our education. Lest we forget.

  63. Connor King says:

    The Cochlea is sort of like a natural Fourier transform mechanism.

  64. jerckï72 says:

    saturation 🙁

  65. Ray Kent says:

    Great video. I knew all the stuff about harmonics but I didn't know about the variable stiffness of the membrane in the cochlea.

  66. sandile13809 says:

    did anyone else cringe when she broke that viola string? I died on the inside when she did that

  67. Bhaskara Rao says:

    Its a great work

  68. John Kotermanski says:

    For the love of god, marry me.

  69. chadjensenster says:

    The last note you played on Audacity sounded like Tuvan throat singing. I could totally hear the two tones.

  70. Max Ride II says:

    watching you write on your instrument physically pained me

  71. Maja Mo says:

    Did anyone else instantly flashback to the hearing test at the doctors when she started playing the A220 thing? – 7:57

  72. Sébastien MB says:

    I've enjoyed your videos very much. I hadn't seen one in maybe 6 years, and now I just saw this.

    You say you don't really know why « a string half the length oscillates at double the frequency ». The reason is simple : the pulse (perturbation) needs to travel half the distance, so since it travels at the same speed as before, if finishes a lap in half the time.

    I feel very proud that I could answer to something you mentioned not knowing. I wish you many good returns. Kind regards.

  73. Hrnek Bezucha says:

    So much maths and my head doesn't hurt! Not even a little bit! You're awesome

  74. Trevor Seines says:

    mutations in a thermodynamic closed system have never proven beneficial but over hundreds of millions of years, as you claim, would have led to complete degradation eons ago, not perfection. mathematics approves both the finite constraint and the infinite potentially.

  75. Amazing Little Kuin says:

    Don't press Read more


  76. Dany Y Pao says:

    this shit went waay too fast what the fuck is going onnnnnn

  77. SM4Guitar says:

    Damn, you're so smart!

  78. Ornithocowian King says:


  79. Alison TheRohirrimWillAnswer says:

    Yay, Bach!

  80. Seeing Dragons says:


  81. OMGFireDragon says:


  82. Boža Mori.Škof says:


  83. Jojogape says:

    This is my favorite video of yours. The production value, the sound design, the feel of an actual documentary… every detail counts.

  84. Feenix Fyre Art says:

    Bad at playgrounds. Oh my gosh. This is great quote from Vihart. Lol

  85. Kate Battersby Words And Music says:

    Wow! Love this. Thank you. Will come back repeatedly until I can keep up. 🙂

  86. Phonotical says:

    Admit it, you just wanted someone to push you on the swing for a while 😅

  87. Phonotical says:

    I would like to see this demonstration a different way, say having the pure tone coming from an individual speaker, this way the computer is not simply combining the output, same effect, different affect

  88. Aspiring Cloud Expert says:

    1:40 Correction: it is the "Basilar" membrane, not "Vasilar". See video description for more details.

  89. Threelly AI says:

    My CAT Decided What I ATE for 24 HOURS (And This Is What Happended…)

  90. NonTwinBrothers says:

    I remember watching this at a very young age, I had no idea about programs like audacity and other sound related stuff, but I was still really able to enjoy this video like I am today.

  91. Jeremiah Johnson says:

    Honestly fuck you for breaking that string it gave me anxiety for the whole rest of the night

  92. Julian Goulette says:

    5:36 The intervals of the harmonic series do generally get more dissonant as you go up, but not as fast as the diagram claims, what is to blame here is the fact that 12ED2 approximations instead of the real series. For example, the 11th harmonic is about 551¢ octave reduced, (1¢ = 1/100th of a 12ED2 semitone) while the closest thing 12ED2 has is 600¢, i.e. 2^(1/2), so using 12ED2 approximations does not give the harmonic scale justice.

  93. starzandearth says:

    When you did the C major cord thing, I literally said "No fucking way" and I was a densely packed library.

  94. sudhakar D says:

    wow . great production !!!

  95. RishiNandha says:

    yeah subconsciously looking out for Bach

  96. Paul Migneault says:

    If the stipulation is that you only check for halving then new notes don't just happen on prime numbers only, any odd numbered fraction will give you a new note. Work it out on a spreadsheet and you will see for yourself where (1/3, 1/5, 1/7, 1/9, 1/11, and so on) all give a new note. (See 5:13 where 9 is not a prime but is has a new note) The reason only odd fractions show a new note is that all even fractions are a half of a previous fraction therefore will only ever show a note that is an octave of the previous fraction, odd fractions are a new note position every time.

  97. Little Bunny says:

    casually draws on everything

  98. Gustavo Conti says:

    I love your content, every other video gives me the chills, you're so god at this. I've studied a lot of music theory and the physics of sound (waves in general), but your way of putting it was just so well written. Can't help but notice you haven't uploaded in some time, is it too early to say "please come back"? 😛

  99. Alexis E. says:

    Thank you so much, your video is a masterpiece

  100. Connor King says:

    It's sort of like a mechanical Fourier Transform.

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