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SPEAKER 1: Let's dive into some music.

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Before we get working
on the actual problems

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you'll need to solve as part of this
problem set, let's take a moment

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and look at some background about music,
notes, and how we might represent them.

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What we're looking at here is
a portion of a piano keyboard.

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And you'll notice we have seven
different white notes and five

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black notes.

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And so as you may know, each
note is assigned a letter

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where we have C on the left
side, then D, E, F, G, A, and B,

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where the white keys on the piano are
each assigned to a different letter.

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A semi-tone, as we'll refer to
it, is just the interval from one

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note to the very next note, including
both the white and the black keys

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on the keyboard.

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So between C and D, that's
an interval of two semitones,

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because we go from C, first to
the black note in between C and D,

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and then to D. D and E are
also two semitones apart,

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because there's a black
note in between them.

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But E and F are one semi-tone
apart, because right next to E,

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then immediately following
it, is the note F.

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So semi-tones are just how we're going
to refer to a step of exactly one

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note on the piano keyboard.

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But how, then, are we going to represent
the black keys on the piano keyboard

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if the white keys are
all assigned to letters?

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Well, in order to do that,
we'll need to use accidentals.

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Accidentals adjust notes up or
down by exactly one semi-tone.

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So the sharp accidental,
represented here by the pound sign,

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means we're going to
move up one semi-tone.

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So these are how the black keys would
be represented in terms of sharps.

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Up one semi-tone from C is C sharp.

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And up one semi-tone from
D is D sharp, for example.

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Likewise, we can also represent
notes as flats, where a flat just

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means move it down one semi-tone.

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So here, we're looking at D flat,
which is one note down from D,

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and E flat, which is one
semi-tone down from E.

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And notice here that there are multiple
different ways to name the same note.

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That note in between C and D can be
represented as either C sharp or D

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flat.

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And they both mean the same thing.

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In fact, it's not just
the black keys that

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can be referred to as sharps and flats.

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Since F is one semitone above E, and
likewise E is one semi-tone below F,

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we can call F, E sharp,
and we can call E, F flat.

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And those are totally valid ways
of representing those notes.

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But as we know, a piano does
not just consist of these keys.

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It consists of far more.

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And what we say is that
the piano is divided

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into octaves, where an octave is just
going to be a sequence of these 12

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notes that gets repeated over and over
across from the low range of the piano

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to the high range of the piano.

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And so, generally, the
middle of the piano

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we'll call octave four
or the middle octave.

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And you'll see that we have a
C there, a D there, an E there,

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and all the other nodes that
are present in octave four.

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But that same pattern of notes
repeats again in octave five,

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to the right of octave four,
where all the notes sound similar,

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but are actually at a higher frequency.

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More on that later.

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We can actually refer to notes
not just by what letter they are

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and what accidental they have, but
also by what octave they're in.

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For example, we might call the C
that is located in octave four, C4.

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And we might call the D sharp that's
located in octave five as D sharp 5.

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And so what you're looking
at here is a representation

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of a bunch of notes on the piano
and how we would represent them

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in terms of what note they are,
what accidental they have, if any,

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and finally, what octave they're in.

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Finally, let's take a look
at how we would actually

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represent these notes if we
were to try to write them down

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in sheet music, a conventional way
for musicians to write down notes

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and how long they last,
and what they mean.

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This is an example of the treble clef
in sheet music, which is just part

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of the way that notes are represented.

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And each one of these lines
or spaces between a line

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is a space for an individual note.

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And so right below that bottom
line, if there were a note there,

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that would be the note D4.

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And after that, each time you go up
one line or space between a line,

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you just go up by one letter.

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So after D4, this note, which is
located on the bottom line, is E4.

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And the note immediately above that,
located in the space between those two

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lines, is F4.

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And next is G4.

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And remember, after G4,
we've run out of letters.

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And so we need to go to
A4 and loop back around.

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And then to B4.

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Next would be C4, but if you remember
from our picture from earlier,

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you'll notice that the octaves change
numbers once we get to C. Notice

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that we have B3, but the note
immediately after that is C4,

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because C is generally where we'll
call the beginning of the next octave.

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So if we take a look at
this sheet music again,

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we have D, E, F, G, A,
and B all in octave four.

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But the note immediately after
that is actually going to be C5.

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And now we're in the fifth octave.

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So following that is going to be D5.

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And following that is going to be E5.

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So every line and every
space between a line

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gets its own letter and octave
number in order to represent it.

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So this, for example, is A4.

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But how would we represent,
for example, A sharp 4, A sharp

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in the fourth octave?

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Well, if you remember
that we represented sharps

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as kind of just that
pound sign, we can just

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draw that symbol to the left of the
note, and now this note is A sharp 4.

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Likewise, if we wanted to
represent a note as flat,

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where flat just means
down one semi-tone,

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we could likewise to the left
of the note add the flat symbol.

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And now this note that we're
looking at here is A flat 4.

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And so using that representation,
we have the capacity

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to represent any note
using any accidentals,

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just by drawing the
notes on the correct line

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or space between the
line that corresponds

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to the right note in the right octave.

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Now, not all notes are
the same length, though.

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So how, in sheet music,
would we tell apart

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a note that is a short versus
a note that is a longer note?

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Well, there are a bunch of
different lengths of notes,

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and the notes look different
depending upon their length.

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So on your left, we're
looking at an eighth note.

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And an eighth note, we'll say,
lasts the duration of one eighth,

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and generally looks like a note with
a little curvy tail at the top of it.

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Next to that we have
a quarter note, which

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last twice as long as an eighth
note, for a duration of one fourth.

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And it looks just like an
eighth note but doesn't

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have that curvy tail at the top.

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Next to that, we have
a dotted quarter note,

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which is a quarter note with
a dot to the right of it.

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And that lasts three times
as long as an eighth note,

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or a duration of three
eighths, we might say.

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And finally, on the right hand
side, we have a half note,

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which looks like a quarter note except
the center of the note is not filled in

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and is hollow.

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And a half note lasts the
length of four eighth notes,

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or a duration of one half.

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There are more notes than just
these, but these are the ones

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that we're going to care about
for this particular problem set.

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And so using these different symbols
for notes that determine their length

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and placing them in the right place,
we have the ability to represent,

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in sheet music, a note, which note
it is, what accidental it has,

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what octave it is in, and
also how long the note lasts.

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And so one other thing
that you might find

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when it comes to the lengths of notes
is that sometimes when two eighth notes

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are next to each other.

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They'll be represented in
sheet music as just two

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notes that have a little bar
that connects the two of them.

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So if you ever take a look at that
and are confused by what that means,

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generally, that symbol on the
left just means two eighth notes.

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And that's all the background
we'll need in order

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to be able to dive
into the problems here.

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So with that in mind,
even if music notation

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might seem confusing or complicated at
first, I'm sure that in no time at all

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you'll be a natural.

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