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[Music]
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in the last video I was talking about
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equal temperament which is the tuning
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system used in virtually all modern
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Western music and it was developed in
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the late 1500s before that there were
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two other families of tuning system one
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of which is called just intonation and
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I'm gonna be talking about that in a
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future video the other one is called
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Pythagorean tuning and that's the one
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I'm going to be dealing with today so to
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get a start to understanding what
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Pythagorean tuning is all about I want
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to take you first of all over to the
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keyboard so here's our keyboard and the
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first thing I should point out is that I
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can't play a Pythagorean scale on this
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keyboard because it's tuned to equal
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temperament but what it can do is show
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you a few things that will lead us on
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the path to understanding how
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Pythagorean tuning works so I want to
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focus on the scale of C major which
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starts on C and goes up on the white
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notes that's why I'm using C major don't
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have to use the black notes I can just
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go straight up on the white notes so c d
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e f g a b c brings me back to C so
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that's the familiar doremi far so a lot
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you know
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okay now that interval there the octave
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is what we call consonant consonant just
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means that it the notes fit together
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really well the term that's used is
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stable
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that's considered a stable interval
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because it doesn't want to go anywhere
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you're already where you want to be the
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other very stable and perfectly
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consonant interval is v which is the 5th
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note of the scale so which would be so
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so
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that one there so that is called a
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perfect countenance and the fifth is
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also called a perfect consonants now
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contrast that with the second so that's
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the one and two notes played together
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kind of jarring you know they don't sit
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well together you really want to go to
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another note like this for example which
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is the third also the seventh interval
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that's almost worse doesn't mean you
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can't use it in music it's just that
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when you play those two in isolation
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it's very very dissonant the term we use
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is dissonant so we have consonant and
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dissonant so you really want to go from
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that to that
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that's the stable sound so we've got two
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distances there the second and the
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seventh we've got two perfect
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consonances the octave and the fifth
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we've also got a few others we've got
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the third and that sounds nice together
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but that's called an imperfect
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consonants with any of the fourth which
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again that's fine but that's regarded as
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dissonant in some context and consonant
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in others and then we've got the sixth
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which also sounds fine that's regarded
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as an imperfect consonants as well
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and then of course we get eventually
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back to the octave again so some
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intervals are dissonant some are quite
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stable and are regarded as imperfect
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consonances and some are very stable in
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fact the two that are very stable are
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the octave and the fifth so what's so
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special about this interval the fifth is
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it just some quirkiness of our brains
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that make it seem like a pleasant
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interval to hear or is there something
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more fundamental well to understand that
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I'm gonna switch instruments now
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to a guitar because the beauty of the
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guitar is that we can actually see the
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things that are vibrating and namely the
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strings and when we look at the
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vibrating strings will understand a
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little bit more about the the physics of
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what's going on with these notes so
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here's our guitar with its six strings
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in a state of tension and if I play one
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of the strings it vibrates between two
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points one of the points is the bridge
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down here and the other one at the other
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end of the fretboard is the nut down
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here so the string is vibrating between
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those two points now what I'm going to
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do is press the string down at its
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halfway point and the halfway point is
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here and if I play it here this is what
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it sounds like there's the open string
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there I'm pressing it down here
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this point here is exactly halfway along
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the string so it's the same distance
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down to the bridge in this direction as
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it is from the fret that's this piece of
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metal here to the nut down here so the
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string is being pressed at its halfway
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point to get the octave now remember the
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octave is one of the perfect consonances
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of the scale the other one is the fifth
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and I've marked on the notes of the
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major scale here second third fourth
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fifth sixth seventh and then up to the
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octave so let me just play that whole
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scale for you so here's the open note
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Sol mi sol La Ti you'll see that I've
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marked on a couple of distances to the
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octave it's 32 point five centimeters in
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other words the distance from the nut to
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this fret here where the octave sounds
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is 32 point five centimeters the
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distance to the fifth
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is 21.5 centimeters now I'm going to use
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my trusty calculator to divide these two
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lengths so thirty two point five divided
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by twenty one point five equals one
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point five a little bit now I measured
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these distances with a tape measure so I
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can't guarantee you they're accurate to
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you know a millimeter they're probably a
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little bit out but that number there is
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very close to one and a half so here
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we've got a diagram that just summarizes
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what we've been seeing on the guitar
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that if you have a string and you stop
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it halfway along you're produced an
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octave so that's the simplest ratio is
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is want to giving you the octave which
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is one of the perfect consonances and if
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you stop the string two thirds of the
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way along you produce a fifth which is
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the next simplest ratio two to three and
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is the other perfect consonants and also
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the next one if we continue it if we
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stop the string three quarters of the
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way along so we have a ratio of three to
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four would produce a 1/4 now the Greeks
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of course didn't know all that we know
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about science and acoustics and
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frequencies and the way sound waves work
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but they did have stringed instruments
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and they could experiment with strings
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and they made this discovery that if you
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divide the string up in these simple
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ratios you get the most consonant sounds
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Pythagoras and his followers were
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obsessed by numbers and their
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relationship with the real world so when
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they made this discovery that the fifth
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that's the most consonant interval
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together with the octave correspond to a
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vibrating string stopped in this simple
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ratio of two to three they decided to
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base their whole system of tuning on the
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fifth and I want to quickly show you now
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how how this works so let's see if we
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can construct
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the scale of c-major using Pythagorean
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tuning we're going to start at C and
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we'll just choose a frequency and the
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frequency will uses is middle C which is
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260 one point six cycles per second or
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Hertz so that's gonna be our starting
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point now on the Pythagorean scale the
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first thing to do is to raise that by
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1/5 which means multiplying by 3 over 2
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so that gets us to this point here and
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the note G on the Pythagorean scale how
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do we produce our next note on the
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Pythagorean scale well we start from
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here and we raise it by another fifth
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that means multiplying by 3 over 2 again
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but now we've multiplied by 3 over 2
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times 3 over 2 that's 9 over 4 and 9
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over 4 is bigger than 2 so it pushes it
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into the next octave and we don't want
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that we want to be in this octave so how
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do we get the corresponding note in this
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octave from the one we've ended up in in
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the octave above well we have to divide
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by 2 because all the frequencies in this
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octave are half the frequencies in this
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one so overall we've multiplied by 3
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over 2 but then we've had to multiply by
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1/2 to get us back into this octave the
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corresponding note in this octave well
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if we've already multiplied by 3 over 2
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at this point and we then multiply it by
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another 3 over 2 but then by 1/2 overall
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from this point we've multiplied by 3
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over 4 which is the same as dropping by
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1/4 so going up by 1/5 and then dropping
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an octave is the same as dropping a
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fourth from your starting point so now
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overall we've gone 3 over 2 times 3 over
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2 times 1/2 so overall it's as if we
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started at C and multiplied by 9 over 8
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whichever way you want to think of it we
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end up at
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this point here now starting at this
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point we multiplied by 3 over 2 which
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gets us to here now that's fine because
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if we're still in the same octave so
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that's the next note on our Pythagorean
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scale to get the next one we multiply by
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another 3 over 2 but now we have the
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same problem because we end up in this
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octave so then we have to multiply by
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1/2 so again is the same as this process
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here effectively starting here we
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descend by 1/4 to get to the next note
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on our scale then we multiply by 3 over
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2 again
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to get the be on our Pythagorean scale
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and finally this is really the oddball
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one instead of going up and then back
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down we start at the C and we actually
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descend by 1/5 and then add an octave
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which is the same as starting from here
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and adding 1/4 or multiplying by 4 4
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over 3 so these are the ratios for
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producing all of the different notes on
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the Pythagorean scale starting from C
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and you'll notice that the frequencies
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that we end up with for example the D on
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the Pythagorean scale having started at
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middle C is close to that on the equal
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temperament scale but not quite the same
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and in fact you'll notice that all of
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the frequencies on the Pythagorean scale
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are a little bit different than the
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equal temperament scale that we use
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today not tremendously different we've
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got four 93.9 4 96.7 4 G we've got three
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nine three point zero three nine two
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point four so there's not a huge
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difference and if these scales were
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played separately you probably wouldn't
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really notice much of a difference but
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certainly if these two notes were played
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together then you would you would detect
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a difference between them okay well
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we've construct
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the Pythagorean scale of c-major based
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on repeated use of fifths you might want
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to replay the last part of this video
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and maybe go through the calculations
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yourself to make sure you've understood
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them what I haven't mentioned so far are
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the problems with Pythagorean tuning
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such things as the Pythagorean comma and
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the dreaded wolf interval I'll be
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talking about these in the next video
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thanks for watching and I'll see you
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next time
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[Music]
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you
